Theory Of Rubber Elasticity Research Articles

Comparison of recent rubber-elasticity theories with biaxial stress–strain data: the slip-link theory of Edwards and Vilgis

The Edwards–Vilgis (EV) slip-link theory (1986) derives the elastic free energy of a rubber-like network model containing stable and sliding network junctions (crosslinks and slip-links) and predicts both low-strain softening and high-strain hardening. The four-parameter stress–strain relations calculated by the theory for geometrically different deformation modes up to high strains were tested experimentally using published biaxial stress–strain data on simple covalently crosslinked networks. For networks with low degrees of strain softening and low extensibilities, the experimental dependencies could be described rather well but, generally, a simultaneous satisfactory fit to uniaxial, pure shear and equibiaxial data was not obtained. Systematic experiment–theory deviations exceeding 10% were observed and some of the parameters had a tendency to assume values lying outside the reasonably expected range. The prediction of a pronounced maximum in the strain dependence of stress supported by slip-links seems to be a reason for the discrepancy. Also, modeling of the high-strain singularity in entropy is done in the EV theory using a rather simple approximation. As a result, the finite extensibility contribution to the stress of a slip-link-free network model becomes improbably high and significantly exceeds that following, at a given modulus and locking stretch, from the rigorously derived Langevin-statistics-based eight-chain-network elasticity theory of Arruda and Boyce.

Network development during epoxy curing: experimental ultrasonic data and theoretical predictions

Ultrasonic waves propagation acts as a dynamic mechanical deformation on a material. When, during ultrasonic wave propagation, chemical or physical changes occurs, the evolution of an elastic modulus can be monitored. Therefore, this technique can be considered a powerful tool for non-destructive cure monitoring with a potential for “in situ” applications. In this work, the isothermal cure of a model epoxy resin cured with an amine is studied using propagation of longitudinal ultrasonic wave. The epoxy to amine ratio is optimized in order to reach full conversion of the amine groups during curing. The relative changes in the ultrasonic velocity and attenuation, measured by the transmission technique, have been applied to the calculation of the longitudinal modulus. The ultrasonic modulus has been compared with the degree of reaction measured using Differential Scanning Calorimetry (DSC). Furthermore a correlation between the ultrasonic modulus and the crosslinking density is presented combining DSC data with the stoichiometry of reactants according with the statistical theory of Miller and Macosko. The plot of the ultrasonic modulus as a function of the crosslinking density suggested that the theory of rubber elasticity can not be applied to the ultrasonic bulk longitudinal modulus as a consequence of the small deformation involved in the propagation of the ultrasonic waves.

An elementary molecular-statistical basis for the Mooney and Rivlin–Saunders theories of rubber elasticity

By relaxing the assumption that the end-to-end vectors of molecules transform as macroscopic material line elements, we arrive at a generalization of the molecular-statistical theory of rubber elasticity. This generalization includes as special cases continuum-mechanical theories proposed by Mooney and by Rivlin and Saunders as improvements upon the classical neo-Hookean theory.

Phase diagrams of monomer and uv-cured difunctional-acrylate-nematic-liquid-crystal systems.

This paper deals with the thermal properties of systems made of the difunctional monomer 1,6-hexanedioldiacrylate (HDDA) and the low-molecular-weight liquid crystal E7. Experimental phase diagrams of uv-cured and uncured solutions of HDDA/E7 systems are established with a polarized optical microscope and a differential scanning calorimeter and the data analyzed within a theoretical formalism that combines the Flory-Huggins model of isotropic mixing and the Maier-Saupe model of nematic order. Ultraviolet-curing samples with a difunctional monomer such as HDDA leads to a crosslinked polymer network and consequently an elastic contribution to the free energy is introduced according to the Flory-Rehner theory of rubber elasticity. The amount of liquid crystal segregated is evaluated to assess the efficiency of the phase separation mechanism.

Analysis of Network Structure of UV-Cured Acrylates by1H NMR Relaxation,13C NMR Spectroscopy, and Dynamic Mechanical Experiments

The network structure of UV-cured acrylates was analyzed by dynamic mechanical experiments, 1H T2 NMR relaxation, and 13C NMR spectroscopy. Photocured mixtures of difunctional (poly(ethylene glycol) diacrylate) and monofunctional (2-ethylhexyl acrylate) were studied. The mean cross-link density and fraction of dangling chains were varied in these networks by changing the content of monofunctional acrylate. Examining networks by 13C NMR spectroscopy, we observed that the conversion is high, and the contribution or the effect of side reactions that generate extra cross-links is negligible. The spatial distribution of cross-links is very heterogeneous in the cured samples; i.e., polyacrylate ziplike network junctions are interconnected by polyether chains. An increase in the content of monofunctional monomer causes a significant decrease in the cross-link density because the monofunctional monomer acts as a chain extender. NMR T2 relaxation method shows good correlation with mechanical tests with respect to molar mass between cross-links, Mc. Obtained values of Mc are significantly smaller compared to molar mass of diacrylate. As anticipated, it is apparent that classical rubber elasticity theories are not applicable for characterization of this type of networks containing ziplike network junctions.

Letter to The Editor: Nomenclature in the theory of rubber elasticity

Letter to The Editor: Nomenclature in the theory of rubber elasticity

Poly(ester-siloxane)urethane network structure from tensile properties

New poly(ester-siloxane)urethane structures, which exhibit the alternation of the flexible and hard segments in the polymer chain, using the ‘prepolymer’ polyaddition procedure were obtained. These networks were analyzed on the basis of molecular theory of rubber elasticity from stress–strain data. The cumulated effect of the physical and chemical crosslinkings induced by a tetrafunctional compound was pointed out by the average critical molecular weight of the linear segments existing among the junctions of copolymer networks. The correlation between the structure of soft and hard segments and tensile properties, density of chemical crosslinks and the hardness of these polymers was established.

Free volume and transport properties of heterogeneous poly(ethylene- co-octene)s

n-Hexane desorption measurements were performed on heterogeneous poly(ethylene- co-octene)s with hexyl branch contents between 0.8 and 3.9 mol% and crystallinities between 30 and 60%. Crystal core contents obtained by Raman spectroscopy were lower than the density-based crystallinity, particularly for the samples with the highest degree of branching. A modified Cohen–Turnbull–Fujita free volume model adequately described the diffusivity data. The free volume of the penetrable phases was strongly dependent on their total volume fraction, suggesting the presence of an interfacial penetrable component with low fractional free volume. The dependence of the fractional free volume of the penetrable phases on the phase composition suggests that mass transport takes place from the liquid-like component to the interfacial component and that the penetrant molecules are trapped at the interfacial sites. The decrease in geometrical impedance factor with increasing crystallinity may be explained by the presence of extraordinarily wide dominant crystal lamellae in the heterogeneous low crystallinity samples. The saturation concentration of n-hexane in a wide range of polyethylenes (including the heterogeneous poly(ethylene- co-octene)s, linear polyethylenes and poly(ethylene- co-butene)s reported earlier; crystallinity range: 30–94%) showed a complex non-linear dependence on crystallinity that was qualitative in accordance with rubber elasticity theory considering also molecular cilia.

Rheology and structure development during gelation of low-methoxyl pectin gels: the effect of sucrose

A non-isothermal kinetic model described G′ values during gelation induced by cooling dispersions of low-methoxyl (LM) pectin+Ca 2+, and 10 and 30% (wb) sucrose at pH 4 and {2(Ca 2+)/(COO −)}=0.58. Two different gelation regimes, corresponding to two different temperature ranges, were observed. During the aging process, increasing the pectin concentration resulted in a significant increase in G′ increasing the sucrose content also increased G′. Using the plateau values of G′ and the rubber elasticity theory, the number average molecular weight of chains between cross-links, M c, was estimated. At the same pectin concentration, M c of 0% sucrose gels was greater than those of 10 and 30% sucrose gels. These results reflect the important role of the hydroxyl groups of sucrose in gel formation and that sucrose may stabilize the structure of junction zones.

The stress-strain curve of Luflexen in tension

This work investigated the stress-strain properties of a low density polyethylene based on novel metallocene catalysts, marketed by BASF under the trade name of Luflexen. It was found that the polymer exhibited rubber-like properties similar to those associated with the styrene-butadiene block copolymers. The stress-strain properties were modelled using the HT model proposed by Haward and Thackray. The model employs a dashpot to represent a viscosity in parallel with a spring defined by the theory of rubber elasticity for which non-Gaussian chain statistics were initially selected. It was shown that at extension ratios λ > 2 the model generates curves in good agreement with experimental results. A significant feature of the treatment was the ability of the model to provide an estimate of the limiting extension ratio λmax as represented, in the model, by the parameter n1/2.

Phase transition and elasticity of protein-based hydrogels

Reported are specific materials characterizations of three protein-based polymers comprised of repeating pentapeptide sequences, namely (GVGVP)251, (GVGIP)260 and (GVGVP GVGVP GEGVP GVGVP GVGVP GVGVP)n](GVGVP) where G = glycine, V = valine, P = proline, I = isoleucine, and E = glutamic acid, which had been previously prepared and γ-irradiation cross-linked into elastic matrices. These polymers exhibit a hydrophobic folding and assembling transition on raising the temperature above a critical temperature, designated by Tt. Their equilibrium swelling ratio, uniaxial tensile and dynamic shear behavior were studied. The effect of the transition on swelling and mechanical properties was demonstrated. Equilibrium swelling ratio below the transition temperature decreased with the increase of γ-irradiation dose. Above the transition temperature, hysteresis and frequency dependence were found in tensile loading-unloading tests and dynamic shear measurements, respectively. The rubber elasticity theory of random chain networks was applied to the data only below the transition temperature, where they may properly be considered random network hydrogels, to estimate molecular weight between cross-links and the Flory-Huggins interaction parameter.

高伸長下における純ゴムの非エントロピー弾性

This research investigates the temperature dependence of pure rubber in a stretched state. The motivation for this investigation is a desire to validate the theory on the reinforcement mechanism of rubber by carbon black which we proposed previously in 1982. The structure model of this reinforcement system is a super-network of stretched molecular chains. The reinforcement effect is represented by a separated term, called the γ term, in a strain energy density function (W function) and it has been observed that this γ term has no temperature dependence. The classical rubber elasticity theory, however, predicts that the molecular chains display entropy elasticity even in the nongaussian region, when in a highly extended state. This implies that the γ term must be temperature dependent. In order to clarify this problem, we have determined a W function for isoprene rubber that is highly stretched from the measured results by the strip-biaxial extension method. Then we have the conclusion that the W function in the highly extended state is not temperature dependent, because the temperature dependence of ∂W/∂I1 becomes very small for the extension ratio greater than 6.5(I1=43).

Dependence of solid particle erosion on the cross-link density in an epoxy resin modified by hygrothermally decomposed polyurethane

Solid particle erosion of epoxy resin (EP) modified by various amounts of hygrothermally decomposed polyester-urethane (HD-PUR) was studied at oblique (30, 45, and 60°) and normal impact (90°). The cross-linked structure of the modified EP was characterised by the mean molecular weight between cross-links ( M c). It was established that the initial brittle erosion of the densely cross-linked EPs ( M c≤200 g/mol) changed to a rubbery one by increasing HD-PUR modification due to which a less cross-linked network was formed ( M c>1000 g/mol). This change was reflected by the impact angle dependence: compositions of low M c showed the minimum resistance at normal (90°), whereas those of high M c at 30° impact angle. The resistance to solid particle erosion changed linearly with increasing fracture energy ( G c) of the resins. G c, on the other hand, followed the prediction of the rubber elasticity theory and increased linearly with M c 1/2.

Use of hygrothermal decomposed polyester–urethane waste for the impact modification of epoxy resins

Hygrothermally decomposed polyurethane (HD-PUR) of a polyester type was used as an impact modifier in tri- and tetrafunctional epoxy (EP) resins. Between 5 and 80 wt % of the PUR modifier was added to the EP prior to its crosslinking with a diamine compound (diaminodiphenyl sulfone, DDS). The mean molecular weight between crosslinks (Mc ) was determined from the rubbery plateau modulus of the dynamic mechanical thermal analysis (DMTA) spectra. The fracture toughness (Kc) and energy (Gc) of the modified resins were determined on static-loaded compact tension (CT) specimens at ambient temperature. The change in the Kc and Gc as a function of Mc followed the prediction of the rubber elasticity theory. The efficiency of the HD-PUR modifier was compared with that of a carboxyl-terminated liquid nitrile rubber (CTBN). Attempts were also made to improve the functionality of the modifier by hygrothermal decomposition of PUR in the presence of glycine and ε-caprolactam, respectively. DMTA and fractographic results showed that HD-PUR functions as an active diluent and a phase-separating additive at the same time. As HD-PUR can be regarded as an amine-functionalized rubber, it was used as the hardener (by replacing DDS) in some EP formulations. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 78: 1139–1151, 2000

Characterization of hydrogels formed from acrylate modified poly(vinyl alcohol) macromers

Poly(vinyl alcohol) was modified with pendent acrylate groups to form a macromer that was crosslinked via photopolymerization. Polymerization behavior was studied for several initial macromer concentrations using DSC and Near-IR spectroscopy. Under mild photoinitiating conditions (e.g. 0.05wt% initiator and less than 20mW/cm2 of 365nm light), the hydrogels polymerized to 100% conversion in less than 5min. To characterize the network structure, the hydrogels formed from the acrylated poly(vinyl alcohol) macromer were compared to gels that were chemically crosslinked with glutaraldehyde and gels that were physically crosslinked by semi-crystalline regions introduced through freeze–thaw cycles. The equilibrium swelling ratio and compressive modulus were characterized for all of the resulting PVA hydrogels, and related to the network structure (i.e. Mc) through a modified Flory–Rehner equation and rubber elasticity theory.

Mechanical and Optical Behavior of Double Network Rubbers

Stress, strain, and optical birefringence were measured for a series of peroxide-cured natural rubbers, having both isotropic and double network structures. The residual stretch (permanent set) for the latter ranged from 2.0 to 4.5, with elastic moduli that were an increasing function of this residual strain, as found in previous works. A small birefringence, ca. 10-5, was observed for the unstressed double networks, and its magnitude increased with increasing residual strain. The sign of this birefringence corresponded to extension of the (undeformed) double networks. Under stress, the birefringence followed the stress optical law. The double network properties were interpreted using the constrained-chain model of rubber elasticity, with the assumption of independent, additive contributions from the two component networks. The calculated results differed from the experimental findings, in particular underestimating the residual strain. This failure is a consequence of the overprediction of the stresses during compression, a limitation common to molecular theories of rubber elasticity. The modeling of the double networks does account qualitatively for the sign of their unstressed birefringence, which is due to the stress- optical coefficient being larger in tension than in compression. This particular deviation from the stress- optical law is known from both theory and experiment.

Use of hygrothermal decomposed polyester-urethane waste for the impact modification of epoxy resins

Hygrothermally decomposed polyurethane (HD-PUR) of a polyester type was used as an impact modifier in tri- and tetrafunctional epoxy (EP) resins. Between 5 and 80 wt % of the PUR modifier was added to the EP prior to its crosslinking with a diamine compound (diaminodiphenyl sulfone, DDS). The mean molecular weight between crosslinks (Mc) was determined from the rubbery plateau modulus of the dynamic mechanical thermal analysis (DMTA) spectra. The fracture toughness (Kc) and energy (Gc) of the modified resins were determined on static-loaded compact tension (CT) specimens at ambient temperature. The change in the Kc and Gc as a function of Mc followed the prediction of the rubber elasticity theory. The efficiency of the HD-PUR modifier was compared with that of a carboxyl-terminated liquid nitrile rubber (CTBN). Attempts were also made to improve the functionality of the modifier by hygrothermal decomposition of PUR in the presence of glycine and «-caprolactam, respectively. DMTA and fractographic results showed that HD-PUR functions as an active diluent and a phase-separating additive at the same time. As HD-PUR can be regarded as an amine-functionalized rubber, it was used as the hardener (by replacing DDS) in some EP formulations. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 78: 1139 -1151, 2000

Shear Modulation Force Microscopy Studies on Cross-linked Elastomer Thin Films

ABSTRACTThe atomic force microscope in the shear force modulation microscopy (SMFM) mode has been used to characterize the surface modulus and cross-link density of elastomer thin films, where standard rheological methods cannot be applied. Brominated poly(isobutylene-co-4-methylstyrene) (BIMS) is a synthetic terpolymer which can be stoichiometrically cross-linked by N, N'-dicinnamylidene-1,6-hexanediamine. The results on several types of BIMS elastomers with different bromide content are reported. The cross-linking reaction at the surface is found to be significantly faster than that of bulk. The estimated shear moduli of the thin films were found to be proportional to the cross-link density, as expected from the rubber elasticity theory for bulk materials.

The elastic property of polyvinyl alcohol gel with boric acid as a crosslinking agent

The elastomers of polyvinyl alcohol gel were made from the polyvinyl alcohol polymer, with boric acid added as a crosslinking agent, in the mixed solvent of dimethyl sulfoxide and water. From the experimental results, the viscosity of polyvinyl alcohol solution is found to increase not only with an increment of boric acid content, but also with the temperature in the range of 70°C ∼ 100°C, although the viscosity is decreased in the range of 30°C ∼ 70°C. Moreover, the molecular mass between junctions of polyvinyl alcohol gel is calculated from the rubber elastic theory and found to be decreased with the increment of boric acid content. We also evaluated the values of Young's modulus of polyvinyl alcohol gel, E, E*, and the elastic parameters C1 and C2 of the Mooney-Rivlin equation, according to Hook's law and theory of rubber elasticity. Based on these, the polyvinyl alcohol gel behaves as a good rubberlike elastic property. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 74: 3046–3052, 1999

Mechanical behaviour of crosslinked poly(ester-siloxane) urethanes

The mechanical behaviour of crosslinked poly(ester-siloxane) urethanes is studied based on molecular theories of rubber elasticity. The correlation between mechanical parameters and concentration of urethane groups is discussed, the optimum concentration of these groups for better mechanical properties being established. Several polyurethane networks are compared from a flexibility point of view.