Rubber and rubber elasticity

Ohne Zusammenfassung

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Rubber and rubber elasticity

     

Microscopic theory of rubber elasticity

A microscopic integral equation theory of elasticity in polymer liquids and networks is developed which addresses the nonclassical problem of the consequences of interchain repulsive interactions and packing correlations on mechanical response. The theory predicts strain induced softening, and a nonclassical intermolecular contribution to the linear modulus. The latter is of the same magnitude as the classical single chain entropy contribution at low polymer concentrations, but becomes much more important in the melt state, and dominant as the isotropic-nematic liquid crystal phase transition is approached. Comparison of the calculated stress-strain curve and induced nematic order parameter with computer simulations show good agreement. A nearly quadratic dependence of the linear elastic modulus on segmental concentration is found, as well as a novel fractional power law dependence on degree of polymerization. Quantitative comparison of the theory with experiments on polydimethylsiloxane networks are presented and good agreement is found. However, a nonzero modulus in the long chain limit is not predicted since quenched chemical crosslinks and trapped entanglements are not explicitly taken into account. The theory is generalizable to treat the structure, thermodynamics and mechanical response of nematic elastomers.     

The role of pressure in rubber elasticity

We describe a series of molecular dynamics computations that reveal an intimate connection at the atomic scale between difference stress (which resists stretches) and pressure (which resists volume changes) in an idealized elastomer, in contrast to the classical theory of rubber elasticity. Our simulations idealize the elastomer as a "pearl necklace," in which the covalent bonds are stiff linear springs, while nonbonded atoms interact through a Lennard-Jones potential with energy epsilon(LJ) and radius sigma(LJ). We calculate the difference stress t(11)-(t(22)+t(33))/2 and mean stress (t(11)+t(22)+t(33))/3 induced by a constant volume extension in the x(1) direction, as a function of temperature T and reduced density rho(*)=Nsigma(IJ) (3)/nu. Here, N is the number of atoms in the simulation cell and nu is the cell volume. Results show that for rho(*)<1, the difference stress is purely entropic and is in good agreement with the classical affine network model of rubber elasticity, which neglects nonbonded interactions. However, data presented by van Krevelen [Properties of Polymers, 3rd ed. (Elsevier, Amsterdam, 1990), p. 79] indicate that rubber at standard conditions corresponds to rho(*)=1.2. For rho(*)>1, the system is entropic for kT/epsilon(LJ)>2, but at lower temperatures the difference stress contains an additional energy component, which increases as rho(*) increases and temperature decreases. Finally, the model exhibits a glass transition for rho(*)=1.2 and kT/epsilon(LJ) approximately 2. The atomic-scale processes responsible for generating stress are explored in detail. Simulations demonstrate that the repulsive portion of the Lennard-Jones potential provides a contribution sigma(nbr)>0 to the difference stress, the attractive portion provides sigma(nba) approximately 0, while the covalent bonds provide sigma(b)<0. In contrast, their respective contributions to the mean stress satisfy Pi(nbr)<0, Pi(nba)>0, and Pi(b)<0. Analytical calculations, together with simulations, demonstrate that mean and difference stresses are related by sigma(nbr)=-APi(nbr)P(2)(theta(b)), sigma(b)=BPi(b)P(2)(theta(b)), where P(2)(theta(b)) is a measure of the anisotropy of the orientation of the covalent bonds, and A and B are coefficients that depend weakly on rho(*) and temperature. For high values of rho(*), we find that [sigma(nbr)]>[sigma(b)], and in this regime our model predicts behavior that is in good agreement with experimental data of D.L. Quested et al. [J. Appl. Phys. 52, 5977 (1981)] for the influence of pressure on the difference stress induced by stretching solithane.     

Two tube models of rubber elasticity

Polymer entanglements lead to complicated topological constraints and interactions between neighboring chains in a dense solution or melt. Entanglements can be treated in a mean field approach, within the famous reptation model, since they effectively confine each individual chain in a tube-like geometry. In polymer networks, due to crosslinks preventing the global reptation and constraint release, entanglements acquire a different topological meaning and have a much stronger effect on the resulting mechanical response. In this article we discuss two different models of rubber elasticity, both utilizing the reptation ideas. First, we apply the classical ideas of reptation statistics to calculate the effective rubber-elastic free energy of an entangled rubbery network. In the second approach, we examine the classical Rouse dynamics of chains with quenched constraints at their ends by crosslinks, and along the primitive path by entanglements. We then proceed to average a microscopic stress tensor for the network system and present it in a manageable form in the equilibrium t → ∞ limit. Particular attention is paid to the treatment of compressibility and hydrostatic pressure in a sample with open boundaries. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2679–2697, 2006     

A hydrogel with rubber elasticity transition

Reaction between dimethyldivinylsilane and 3,6-diazaoctane in the presence of 3-lithio-3,6-diazaoctane yields a new telechelic oligomer, poly(silamine), which consists of alternating 3,3-dimethyl-3-silapentane and N,N′-diethylethylenediamine units in the main chain. Poly(silamine) shows unique phase transition properties in response to the degree of protonation of amino groups in the polymer. Poly(silamine) also shows a strong interaction with several anions. Due to the interaction between poly(silamine) and anions along with the protonation of amino groups in the poly(silamine), the rubber elasticity of poly(silamine) is drastically changed. A discrete volume change can be observed when the environmental pH of the poly(silamine) gels is varied. This can be explained not only by a change in ionic osmotic pressure but also by a change in the rubber elasticity of the networks in the gel.     

Use of monomer fraction data in the parametrization of association theories

Association theories such as the CPA (cubic-plus-association), NRHB (non-random hydrogen bonding) equations of state and the various variants of SAFT (statistical associating fluid theory) have been extensively applied to phase equilibrium calculations. Such models can also be used for estimating the monomer fraction of hydrogen bonding compounds and their mixtures. Monomer fraction data are obtained from spectroscopic measurements and they are available for a few compounds such as pure water and alcohols as well as for some alcohol–alkane and similar mixtures. These data are useful for an understanding of the capabilities and limitations of association models. The purpose of this work is two-fold: (i) to compare the performance of three models, CPA, NRHB and sPC-SAFT, in predicting the monomer fraction of water, alcohols and mixtures of alcohol-inert compounds and (ii) to investigate whether “improved” model parameters can be obtained if monomer fraction data are included in the parameter estimation together with vapor pressures and liquid densities. The expression “improved” implies parameters which can represent several pure compound properties as well as monomer fraction data for pure compounds and mixtures. The accuracy of experimental monomer fraction data is discussed, as well as the role of monomer fraction data in clarifying which association scheme should be used in these equations of state. The results reveal that the investigated association models (CPA, sPC-SAFT and NRHB) can predict, at least qualitatively correct, monomer fractions of associating compounds and mixtures. Only, small differences are observed between the models. In addition, it has been shown that, using a suitable association scheme, a single set of parameters can describe satisfactorily vapor pressures, liquid densities and monomer fractions of water and alcohols. The 4C scheme is the best choice for water, while for methanol there is small difference between the 2B and 3B association schemes.     

Experimental analysis of the molecular theory of rubber elasticity

Stress-strain and birefringence-strain experiments, on polybutadiene and poly(methyl-3,3,3, trifluoropropyl siloxane) elastomeric networks of various degrees of crosslinking have been carried out in order to evaluate the recent molecular theory of rubber elasticity of Erman and Flory, which is based on the idea of constraints imposed by entangled network chains on crosslink fluctuations. Previously published results on polyoxypylene and poly(dimethylsiloxane) networks have also been analyzed. In general, the theory describes the elastic behavior of all these systems satisfactorily. Stress-strain and birefringence data can be interpreted using the same set of parameters. The most important theoretical parameter κ, which is a measure of the severity of constraints, varies with the elastic modulus as predicted by theory. However, the relationship is not a universal one, as was originally suggested.     

Contribution of entrapped entanglements to equilibrium elasticity of rubber networks

Synopsis Theoretical analysis of the deformation of a one-entanglement network as compared with one-junction network is given. The model which is a modification of the Flory-Rehner tetrahedron shows that the elastic behavior of an entanglement entrapped between permanent crosslinks is generally different than the behavior of a localized crosslink. Elastic effectivity of entanglements is lower than that of localized crosslinks and the more so the higher is the applied deformation. Some consequences for the form of the non-linear constitutive equation of rubber elasticity are indicated.With 4 figuresDedicated toHerman F. Mark on his 80th birthday     

Analytic stress-strain relationship for isotropic network model of rubber elasticity

James and Guth [3] have used the inverse Langevin function-based expression of tension in a polymer chain to build the so-called 3-chain model of uncompressible rubber elasticity. It is an analytical model, but it is not isotropic. If one considers an isotropic superposition of an infinite number of chains in all the directions, one obtains an isotropic model, but it is no more tractable analytically. Following Cohen [12], we propose to replace the inverse Langevin function by its Padé approximant, the two functions being very close. The isotropic model is then analytically integrable, and yields large strain strain-stress relationship.     

Effect of junction constraints on rubber elasticity in multiaxial states of stress

The strains obtained in an amorphous polymer network subject to multiaxial states of stress are calculated according to a recent molecular theory which takes account of constraints on junctions in real networks. Experimental measurements for biaxial states of stress, for pure shear superposed on tension, and for combined torsion and simple extension, are seen to compare favorably with the predictions of the theory.     

Cross-linking assessment after accelerated ageing of ethylene propylene diene monomer rubber

The ageing of filled and cross-linked ethylene propylene diene elastomer (EPDM) has been studied under accelerated UV irradiation (λ ≥ 290 nm) at 60 °C, thermal ageing at 100 °C and in nitric acid vapours for different time intervals. Hardness measurements were performed. DSC-thermoporosimetry was used to estimate the mesh size distribution and cross-linking densities for each ageing. The development of functional groups was monitored by ATR spectroscopy. An increase in oxidation with exposure time after the different types of ageing was observed. The thermal stability of EPDM was assessed by TGA and evolved volatile gases were identified using FTIR spectroscopy.     

Effect of initial monomer concentration on the equilibrium swelling and elasticity of hydrogels

The linear swelling ratio α and the effective network chain length N of a series of poly(N,N-dimethylacrylamide) (PDMAAm) hydrogels were investigated as a function of the gel preparation concentration . PDMAAm hydrogels were prepared at a fixed cross-linker ratio but at various initial monomer concentrations. It was found that α is not a monotonic function of . As is increased, α first decreases up to about and remains constant in a narrow range of , but then it increases continuously. The -dependence of α is due to the variation of the network chain length N depending on the gel preparation concentration. In the range of below 0.1, N follows the scaling relationship , while at higher concentrations, N varies only slightly with . The increase of α with N obeys the relation , as predicted by the Flory-Rehner theory.     

Thermal behavior of core‐shell rubber/styrene monomer gels

The thermal behavior of core‐shell rubber (CSR)/styrene monomer mixtures was studied using subambient differential scanning calorimetry (DSC). Interaction between the styrene and CSR material was observed as a depression of the freezing and melting points of the styrene monomer and as a shift in the glass transition temperature of the rubbery phase in the CSR materials. The depression of the freezing point of the styrene in the CSR/styrene mixtures was related to the size of the critical nuclei required for crystallization. The heat of crystallization was found to decrease linearly with decreasing styrene content, but calculations showed that not all of the styrene present in the mixtures crystallized upon cooling, confirming that there was an interaction between the CSR material and the styrene monomer. At temperatures below the glass transition temperature of the system, the mixtures contain a pure styrene crystalline phase and an amorphous CSR/styrene phase. The styrene was found to act as a plasticizer, reducing the glass transition temperature of the rubbery core material. The variation of the glass transition temperature of the CSR/styrene systems was determined with respect to the composition of the amorphous phase. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 3136–3150, 2000     

The influence of flame retardants on the properties of sealing ethylene-propylene-diene monomer rubber

This paper presents the results of investigating the structure, the combustibility, and the basic physical and mechanical properties of sealing ethylene-propylene-diene monomer rubber specimens filled with flame retardants such as aluminum and magnesium hydroxides of different dispersity. It is shown that an increase in the flame-retardant particle size causes a change in the structure and the properties of the rubbers and increases their combustibility.     

Oxidative stress relaxation and the measurement of sol fraction in rubber vulcanizates

A simple method to determine the chain scission mechanism of the oxidative degradation of rubber vulcanizates is proposed. The method involves the measurement of oxidative stress decay and the change in sol fraction, which allow us to distinguish whether scission occurs randomly along the main chain, near crosslinks or of crosslinks. The applicability of this method was well established using natural rubber vulcanizates as the reference samples. The chain scission of cis-1,4-polyisoprene vulcanizaties was proved to take place randomly along the main chain irrespective of their crosslink structure. On the other hand, the chain scission of dicumyl peroxide cured cis-1,4-polybutadiene takes place selectively near crosslinks. It is suggested that the unusual behavior of cis-1,4-polybutadiene vulcanizates is due to the characteristic structure of the crosslinks.     

Silicone networks with junctions of high functionality and the theory of rubber elasticity

Results of Krik, Bidstrup, Merrill, and Meyers on polydimethylsiloxane networks of high functionality ? yield values of the reduced force [f*] in the high-extension limit that are directly proportional to (v_s/V) (1 ? 2/?) where v_s/V is the number of effective chains in volume V. The contention that trapped entanglements contribute significantly to the modulus is refuted by adherence of the results to this proportionality down to the lowest degrees of interlinking. Features of the relationship of stress to strain that appear to be in conflict with current theory are attributable to crowding of chains about the junctions of high functionality and of large linear dimension in the networks investigated by these authors.     

An alternative approach to rubber elasticity theory based on a lattice model

An alternative rubber elasticity theory for bulk elastomers has been developed. It is based on the model of a cubic lattice which is filled completely with long flexible chains. The dependence of the configurational entropy on the orientation factor of Hermans has been derived for a uniaxial homogeneous orientation of the system. With the assumption of a constant stress-optical coefficient, the dependence of the true stress Γ on the deformation λ = L/L₀, with L the length of the sample, can be described approximately as τ ∝︁ in λ both for uniaxial extension and compression, which agrees quite well with experimental results on bulk elastomers.