Elastic properties of swollen real polymer networks

Summary In this paper a new set of stress-strain relations (uniaxial, equi-biaxial, unequibiaxial extensions and pure shear) for swollen networks were presented. Which are derived from the molecular theory of rubber elasticity with constraints of junctions and trapped entanglements and with crosslinks, trapped entanglements and carbon black-polymer interactions. They successed in relating the elastic equation of state to the volume fraction of polymer in swollen networks by three molecular parameters C₁, C₂ and C₃. The relation of stress-strain for uniaxial extension was verified by experiments. It is shown that this relation can successfully predict the dependence of V2 on the C₁, C₂ and C₃, and the contribution of modulus for swollen networks from the trapped entanglement which it shows that role of entanglements can only approach to a limited value, never to zero.     

Two-component hydrogels

Summary The elastic equations of state for 2-component networks are derived from a joint Gaussian distribution function composed of Gaussians for each component. The free energy of elasticity is then applied to the case of uni-directional deformations for compatible systems. The stress-strain equations are shown to be equivalent to those for a single component network except in the case where the macroscopic strain is sufficient to cause the ratio of average chain lengths of the components to change with the strain. The stress-strain equations are developed for both dry and swollen networks.The authors dedicate this paper to Prof. Dr. Hans Batzer on the occasion of his 60th birthday     

Experimental and theoretical investigations on petroleum-based elastomers with two cross-linking systems of different lengths viewed as bimodal networks

A review with 36 references discussing the chemistry and the structure-property relationship of elastomers cured with two cross-linking systems of different chain lengths such as sulfur and the polymerization products of p-benzoquinone and viewed as bimodal networks. These exceptional networks have shown remarkable improvements in the overall mechanical properties which are anticipated to be due to the non-Gaussian effects known for bimodal networks and evident by the anomalous upturn in the modulus values in Mooney-Rivlin stress-strain data representations. Proton and ¹³C NMR as well as energy minimization calculations were used to study the chemical structures and single chain contributions of polyquinones. Nuclei bending of these oligomers have shown to be greatly influenced by the restricted torsional behavior due to the presence of the hydrogen bonds between the benzenoid nuclei. Intrinsic atomic-level forces for the networks were evaluated using molecular dynamics techniques and showed that while the forces acting on the junction points of the cross-linking segments and the elastomeric chains had no apparent change as a consequence of the networks' bimodal formation, forces acting on the short chains of the bimodal networks are of much higher values as compared to those of unimodal networks. The presence of the relatively long polyquinone chains in the bimodal networks has caused the short sulfur chains to stretch to its maximum extensibility and no longer can increase its end-to-end distance separation by simple rotations about its skeletal bonds. Limited chain extensibility of the short chains resulting from the deformation of the bond angles and bond lengths has lead to higher potential energies. Studies on the swollen bimodal networks have validated the above conclusions since swelling of the networks will prevent the elastomeric chains from undergoing possible strain-induced crystallization during the stress-strain experiments and any abnormalities in the mechanical behavior of these networks must be therefore the result of the limited extensibility of the short chains of the networks.     

Elasticity of urethane networks

Various polymer networks have been prepared based on poly(propylene oxide) which has been linked either by urethane groups or ester groups. These networks have been examined by stress-strain measurements to find the degree of deviation from ideal rubber elasticity (as specified by the statistical theory) and by neutron scattering to find the mobility of the poly(propylene oxide) chain. The stress-strain results showed the urethane-linked networks to be identical to the ester-linked networks as long as there were no short chain polyols in the formulation. Networks containing short chain polyols have a greater deviation from ideal rubber elasticity. The neutron scattering results showed the ester-linked poly(propylene oxide) to have the same mobility as an uncrosslinked poly(propylene oxide), whereas urethane-linked networks have a lower mobility, similar to an analogous urethane-linked polymer. A small effect of crosslinking was observed. The results are explained in terms of some phase separation occurring in the polyurethanes only when short chain polyols are present. In the simple urethane-linked networks the poly(propylene oxide) chain mobility is reduced by hydrogen bonding to the urethane groups, but this has no effect on the equilibrium stress-strain properties.     

Polyisobutylene model elastomers prepared using demonstrably complete end-linking reactions

Summary Linear polyisobutylene (PIB) molecules with either hydroxyl or isopropenyl groups at the chain ends were prepared using cationic polymerizations with bifunctional initiator-chain transfer agents. Extensive spectroscopic analyses confirmed the essentially perfect difunctionality of the two types of polymers. The former polymers were end-linked using an aromatic triisocyanate, and the latter by means of a tetrafunctional silane. The resulting trifunctional and tetrafunctional model PIB networks were found to have absolutely negligible sol fractions, which demonstrates that the end-linking reactions used to prepare them were essentially complete. The networks were studied with regard to their equilibrium stress-strain isotherms in uniaxial extension at 25 °C. The results thus obtained are in satisfactory agreement with theory and yield no evidence whatever for significant elastic contributions from inter-chain entanglements.     

Butadiene rubbers: topological constraints and microscopic deformation by mechanical and small angle neutron scattering investigation

Summary Results of stress-strain measurements are reported for butadiene rubbers of varying crosslink density. The fluctuation of the effective tube diameter of the polymer networks was investigated under uniaxial elongation by mechanical measurements as well as by small angle neutron scattering (SANS) which probe the local orientation on a segmental scale. The effect of topological constraints on the microscopic deformation of the butadiene network chains is well described within a tube approach. For the first time, experiments at large deformations and for polydisperse sample are presented. Excellent agreement between the statistical mechanical model and the experimental results is obtained. Received: 20 November 2001 /Revised version: 5 March 2002/ Accepted: 5 March 2002     

Monte Carlo simulation of two interpenetrating polymer networks: Structure, swelling, and mechanical properties

The swelling and mechanical properties of various interpenetrating polymer networks (IPNs) were studied. Six networks made from permutations of a moderately crosslinked polyelectrolyte network (ref), a moderately crosslinked neutral polymer network (net1), and a highly crosslinked polyelectrolyte network (net2) were first swollen in water and structural properties such as end-to-end chain lengths and radial distribution functions were compared with the component networks' equilibrium properties. The swelling of composite IPNs was discussed in terms of a balance between the osmotic pressure due to mobile counterions and the restoring force of the network chains, which act in parallel to counteract the osmotic swelling. For the ref-net2 system, the strong stretching of net2 chains increases the network restoring force and the further swelling due to the counterions is suppressed. The swollen networks were then uniaxially stretched, and equilibrium stress-strain plots were obtained up to high extension ratios. The equilibrium volume decreased upon uniaxial extension, and the elastic moduli of IPNs of the A-A type were slightly greater than that of their respective single networks.     

Characterization of ethylene–propylene rubber and ethylene–propylene–diene rubber networks

The stress–strain (S/S) and the swelling equilibrium behavior in a series of ethylene propylene rubber (EPR) and ethylene propylene diene monomer (EPDM) networks were investigated and the results were employed to evaluate the effects of varying the cure conditions on the crosslinking efficiency in these networks. The S/S curve of completely swollen vulcanizates is in agreement with the predictions of rubber elasticity theory, while that of dry or partially swollen vulcanizates is fully described by the Mooney-Rivlin equation. ? values determined in benzene were found to vary linearly with v_r (v_r = equilibrium volume fraction of rubber in swollen sample). Crosslinking efficiency, moles of crosslinks produced per moles of crosslinking agent used, ranges from 3.7 in peroxide-cured EPDM (55% wt ethylene and 2.6% unsaturation) to 0.15 in similarly cured EPR (43% ethylene). Efficiency in the latter system improves to 0.6 by addition of a coagent (sulfur) to the cure formula. Crosslinking efficiency in EPDM (55% ethylene) was found to increase in the order: peroxide- > resin- > sulfur-cured. In the EPDM sulfur vulcanizates, changing the terpolymer in the cure formula resulted in significant changes in the crosslinking efficiency.     

Molecular modelling of the stress-strain behaviour of elastomers in pure shear

It is demonstrated that experimental stress-strain data for several types of polymer network under pure shear display an approximately universal behaviour. The Monte-Carlo network-modelling method of Stepto, Taylor and Cail is extended to treat stress-strain behaviour in pure shear and its predictions are in good agreement with the experimental data. The predictions of Gaussian network theory are shown to be seriously in error.     

Polyisobutylene model elastomers prepared using demonstrably complete end-linking reactions

Summary Cationic polymerizations with a trifunctional initiator-chain transfer agent were used to prepare three-arm polyisobutylene [C(CH₃)₂CH₂] (PIB) molecules with hydroxyl groups at all three chain ends. Extensive spectroscopic analyses confirmed the essentially perfect trifunctionality of the polymers, which were then end-linked using an aromatic diisocyanate to give trifunctional model networks. The PIB elastomers were found to have negligible sol fractions, which demonstrates that the end-linking reactions used to prepare them were essentially complete. They were studied, swollen, with regard to their equilibrium stressstrain isotherms in uniaxial extension at 25°C. As was found to be the case for trifunctional and tetrafunctional PIB networks prepared from the linear chains, the results were in satisfactory agreement with theory and yielded no evidence that inter-chain entanglements contribute to the modulus at elastic equilibriums.     

Structure-mechanical property correlations of model siloxane elastomers with controlled network topology

We review our recent studies towards the molecular understanding of mechanical properties-structure relationships of elastomers using model polydimethylsiloxane (PDMS) networks with controlled topology. The model elastomers with controlled lengths of the network strands and known amounts of cross-links and dangling chains are obtained by end-linking the functionally terminated precursor PDMS with known molecular weights using multi-functional cross-linkers. Several modern entanglement theories of rubber elasticity are assessed in an unambiguous manner on the basis of the nonlinear stress-strain behavior of the model elastomers under general biaxial strains. The roles of cross-links and entanglements in the large-scale structure of the swollen state are revealed from small angle X-ray scattering spectra. A remarkably stretchable elastomer with the ultimate strain over 3000% is obtained by optimizing the network topology for high extensibility, i.e., by reducing the amounts of trapped entanglements and the end-to-end distance of the network strands. The model elastomers with unattached chains exhibit a pronounced viscoelastic relaxation originating from the relaxation by reptative motion of the guest chains. The relaxation spectra provide a definite basis to discuss the dynamics of guest linear chains trapped in fixed polymer networks. The temperature- and frequency-insensitive damping elastomers are made by introducing intentionally many dangling chains into the networks.     

Molecular modelling of the elastic behaviour of poly(ethylene terephthalate) network chains

The Monte-Carlo (MC) method developed to model the elastomeric stress-strain behaviour of polyethylene (PE) and poly(dimethyl siloxane) (PDMS) networks and the stress-optical behaviour of PE networks is now applied to the stress-strain behaviour of poly(ethylene terephthalate) (PET) networks. In keeping with the previous results for PE and PDMS networks, increases in the proportions of fully extended chains with macroscopic deformation are found to give rise to steady decreases in the rates of Helmholtz energy changes, causing reductions in moduli at moderate macroscopic deformations. There is no need to invoke a transition from affine to phantom chain behaviour as deformation increases.By using rotational-isomeric-state (RIS) models of the network chains and the MC method, stress-strain behaviour can be related to chemical structure. In this respect, the greater conformational flexibility of the PET chain leads to lower network moduli and smaller deviations from Gaussian network behaviour than for PE networks. In addition, the stiff, aromatic section of the PET repeat unit structure is seen to endow particular characteristics on the end-to-end distribution functions of PET chains. These characteristics are taken fully into account in evaluating the elastomeric properties of the PET networks. Subsequent publications will apply the present results to interpreting the measured stress-strain and the stress-optical properties of entangled PET melts.     

Assessment of the sliding link model of chain entanglement in polymer networks

The stress-strain relations derived from the sliding link model of chain entanglement proposed by Ball et al. (Polymer 1981, 22, 1010) are compared with experimental results. There is good agreement with Ogden's empirical treatment of several results obtained previously from general pure strain tests on rubber vulcanizates. New uniaxial data for random polyisoprene networks at various stages from the gel point indicate that a significant fraction of trapped entanglements does not behave as the sliding links of the model but gives the same contribution to the stress as the chemical cross-links.     

Experimental studies and molecular modelling of the stress-optical and stress-strain behaviour of poly(ethylene terephthalate). Part III: Measurement and quantitative modelling of birefringence-strain, stress-strain and stress-optical properties

Experimental measurements and Monte-Carlo (MC) modelling of the birefringence-strain, stress-strain and stress-optical behaviour of poly(ethylene terephthalate) (PET) are used, together with the analysis of orientation-strain and conformation-strain behaviour reported in Paper I, to give a detailed, quantitative interpretation and characterisation of its deformation-related properties. The difference between the stress-strain and stress-optical behaviour of PET that had been reported previously is confirmed. Except for the stress, the measured values of all the properties studied are in agreement with those calculated using the MC modelling, which suggests that not all of the junctions or the chains in the entangled PET network are elastically active.The results given by Kuhn and Grün theory are compared with those given by the MC modelling. The expected shortcomings of Kuhn and Grün theory are found. However, distinct from the behaviour reported previously for polyethene, the theory can be used to evaluate, semi-empirically, the stress-optical coefficient of PET.     

Swelling and mechanical properties of cellulose hydrogels. II. The relation between the degree of swelling and the creep compliance

The mechanical properties of swollen cellulose hydrogels have been studied. The degree of swelling of the gels was varied between 0.75 and 6.3 g water/g dry gel (g/g) by partial drying followed by reswelling in water. Creep rate was measured in uniaxial compression in the time interval 15–900 s for gels in equilibrium with water. Isochronous relations between stress and reversible strain were found to be linear, and creep compliance was calculated from the slopes. Both the creep compliance and the creep rate increase with an increased degree of swelling. General observations, such as the high strain limit of linearity in the stress–strain curves and the magnitude of the creep compliance, indicate similarities between swollen cellulose gels and rubbery networks. It is therefore assumed that the statistical theories for swollen networks can describe the amorphous matrix of the gels. In order to obtain creep compliance values representative of the amorphous matrix, the experimental values were corrected for the presence of crystalline regions. It is also suggested that non-load-bearing microvoids are present at high swelling levels. According to calculations based on the theory, the network chains of the amorphous regions in a gel swollen to 2.4 g/g contain about 11 monomer units and the Flory-Huggins interaction parameter χ equals 0.2.     

Some simulations on filler reinforcement in elastomers

This review illustrates how elastomer reinforcement can be modeled using Monte Carlo simulations on rotational isomeric state chains to characterize their spatial configurations in the vicinity of filler particles. The results are distributions of the chain end-to-end distances as perturbed by this excluded-volume effect, and the results obtained are in agreement with experimental results gotten by neutron scattering. The use of these distributions in standard molecular theories of rubberlike elasticity then produces stress-strain isotherms suitable for comparison with those in elongation experiments. Such simulations have now been carried out for elastomeric matrices reinforced by spherical filler particles (either on a cubic lattice or randomly dispersed), or by prolate or oblate particles on cubic lattices (either with their axes oriented or randomized). The simulated mechanical properties are consistent with experimental results available at the present time, and should provide encouragement and guidance for additional simulations and experiments.     

Long time stress relaxation of amorphous networks under uniaxial tension: The Dynamic Constrained Junction Model

The constrained junction model that represents the stress-strain relations of amorphous networks in equilibrium is modified to analyze stress relaxation. Deviation of stress from equilibrium when a network is stretched suddenly is represented by a time dependent constraint contribution that is of the same form as that of the equilibrium theory. The time dependent motions of the junctions are assumed to obey the Langevin equation. The only new term in the model is a time dependent κ parameter that vanishes at long times. Results of the model are compared with uniaxial stress relaxation experiments on polyisoprene networks with different degrees of cross-linking. Experiments show that the time dependent κ parameter obeys a stretched exponential form, with β = 0.4 and τ = 40 s, both of which are the same for all extensions and cross-link densities studied. The front factor κ₀ depends on cross-link density in the same way as in the equilibrium case. Comparison with stress relaxation experiments shows satisfactory agreement at a wide range of extensions and for different degrees of cross-linking. The relatively low value of the stretched exponent parameter, β = 0.4, is interpreted in terms of a molecular picture where entanglements contribute to relaxation at a wide spectrum of time scales.     

Stress-strain relationships in uniaxial extension of high cis-1,4-polyisoprene networks with different molecular weight precursors

Equilibrium stress-strain relationships in uniaxial extension for high cis-1,4-polyisoprene (Shell IR 307) networks were obtained by extrapolation of relaxation measurements to infinite time through a BKZ constitutive equation. Three series of networks were investigated, each series being characterized by its polymer precursor molecular weight. Influence of crosslinking density was studied through varying amounts of dicumyl peroxide as crosslinking agent. These results were used to test Flory and Erman's recent molecular elasticity theory of imperfect networks with constraints on junctions. It was shown that this later theory treating entanglements as restrictions on junction fluctuations could be reasonably used to characterize network topology. A universal value of 0.50 for the interpenetration parameter I is confirmed and an interpretation of parameter ζ in terms of network inhomogeneity is tentatively given.     

Langevin-elasticity-theory-based description of the tensile properties of double network rubbers

A previously proposed and successfully tested constitutive equation denoted by the ABGIL code (a combination of the Arruda-Boyce equation based on the Langevin elasticity theory and a constraint term based on tube theories; strain-induced increase in the finite extensibility parameter is assumed) has been found to provide a good basis for quantitative interpretation of the stress-strain data recently obtained by Mott and Roland on double networks of natural rubber, prepared by introducing additional crosslinks (second network) into a first network stretched to various extents. Experimental information on properties of the first and second networks has been used to obtain their ABGIL parameters and to calculate, under the common assumption of additivity of contributions, the stress-strain properties and residual stretch of the resulting double networks. The predictive ability of the ABGIL equation has been found to be very good. Effects of the finite extensibility of network chains appear to be significant in double networks while the possible role of orientational crystallization cannot be excluded.     

Elastic properties of random-linked cis-PB networks: A characterization and gel point study

cis-Polybutadiene networks with microstructure 98% cis, cross-linked in the bulk state by gamma irradiation, were characterized in terms of their mechanical properties. Data from stress-strain analyses, swelling degrees, stress relaxation, and random-linking theory are in accordance and the estimated gel points practically coincide. Trapped entanglements lead to a 30% increase in the network modulus, from which also the plateau modulus, 0.71 ± 0.09 MPa, was determined. The tensile data were obtained from Mooney-Rivlin representations without junction constrained corrections. Preliminary stress-relaxation experiments indicated a slow relaxation at room temperature. It can be concluded that the assumption of additive contributions from chemical and entanglement cross-link densities in the Langley theory works well in the range of doses investigated here. © 1993 John Wiley & Sons, Inc.