Tube Model Of Rubber Elasticity Research Articles

30 Years of Macromolecular Theory and Simulations

30 Years of <i>Macromolecular Theory and Simulations</i>

A generalized tube model of rubber elasticity.

In the present paper, a new type of micro-mechanically motivated chain network model for rubber-like materials is proposed. The model captures topological constraints of polymer network chains, in particular, entanglements. The model demonstrates how the local molecular packing constraints modify under deformation and shows the impact of these changes on the macroscopic elasticity of the material. To this end, we combine concepts of a confining tube and a slip-link (reptation) model. In these models, entanglements of polymer chains play an important role. The nature of entanglements is discussed, and relationships governing entanglements are formulated in terms of molecular physics. In the context of nonlinear elasticity, we apply a non-affine concept which captures the liquid-like behavior of polymer networks at smaller scales in a more realistic way. Model predictions show good agreement with experimental results from uniaxial and biaxial tension tests.

Micromechanics of Stress-Softening and Hysteresis of Filler Reinforced Elastomers with Applications to Thermo-Oxidative Aging.

A micromechanical concept of filler-induced stress-softening and hysteresis is established that describes the complex quasi-static deformation behavior of filler reinforced rubbers upon repeated stretching with increasing amplitude. It is based on a non-affine tube model of rubber elasticity and a distinct deformation and fracture mechanics of filler clusters in the stress field of the rubber matrix. For the description of the clusters we refer to a three-dimensional generalization of the Kantor–Webman model of flexible chain aggregates with distinct bending–twisting and tension deformation of bonds. The bending–twisting deformation dominates the elasticity of filler clusters in elastomers while the tension deformation is assumed to be mainly responsible for fracture. The cluster mechanics is described in detail in the theoretical section, whereby two different fracture criteria of filler–filler bonds are considered, denoted “monodisperse” and “hierarchical” bond fracture mechanism. Both concepts are compared in the experimental section, where stress–strain cycles of a series of ethylene–propylene–diene rubber (EPDM) composites with various thermo-oxidative aging histories are evaluated. It is found that the “hierarchical” bond fracture mechanism delivers better fits and more stable fitting parameters, though the evolution of fitting parameters with aging time is similar for both models. From the adaptations it is concluded that the crosslinking density remains almost constant, indicating that the sulfur bridges in EPDM networks are mono-sulfidic, and hence, quite stable—even at 130 °C aging temperature. The hardening of the composites with increasing aging time is mainly attributed to the relaxation of filler–filler bonds, which results in an increased stiffness and strength of the bonds. Finally, a frame-independent simplified version of the stress-softening model is proposed that allows for an easy implementation into numerical codes for fast FEM simulations

NEW INSIGHTS INTO RUBBER NETWORK STRUCTURE BY A COMBINATION OF EXPERIMENTAL TECHNIQUES

ABSTRACTRobust quantitative cross-link density characterization becomes necessary for the complete understanding of the structure and optimization of final properties of rubber compounds for industrial applications. A combination of different experimental techniques have been used to establish the quantitative consistency on the correlations between the results obtained by the individual methods within a reliable unique (physically based) platform reclined on the concept of rubber elasticity that considers the impact of entanglements in technical rubbers. The contribution of cross-links and elastically active entanglements to mechanical properties has been quantified by the analysis of uniaxial stress–strain measurements by means of the extended tube model of rubber elasticity. In a complementary manner, rubber network structure has also been investigated by state-of-the-art multiple-quantum low-field NMR experiments and classical T1 and T2 relaxation measurements. In addition, equilibrium swelling data were analyzed by the classical phantom and Flory–Rehner limits as well as by applying the theoretical approach proposed by Helmis, Heinrich, and Straube that takes into account topological constraints during swelling. Correlations among these complementary techniques have been reported, and the interpretation of the obtained differences is addressed. The baseline study focuses on unfilled NR, setting the basis for the investigation of unfilled SBR matrices and filled rubbers.

Stress softening of multigraft copolymers

The hysteresis behaviour of multigraft (MG) copolymers, with a polyisoprene backbone and polystyrene (PS) side chains, was investigated by applying a modified softening model proposed by Elías-Zúñiga, which uses an approach of Ogden and Roxburgh. The model was combined with the non-affine tube model of rubber elasticity of Kaliske and Heinrich. Four parameters are obtained: chemical and physical cross-link moduli ( G c, G e), the number of statistical segments between two successive entanglements ( n e/ T e) and a softening parameter ( b). The model was proven to be valid by a comparison with other methods evaluating hysteresis behaviour. The characterization of the multigraft copolymers revealed a branch point and molecular architecture dependence of the softening parameter. b was low for tetrafunctional MG copolymers with cylindrical microdomains, and it was further reduced for a spherical morphology and for more complex molecular architectures. The magnitude of b also depends on the PS arm molecular weight for hexa- and tetrafunctional multigraft copolymers.

Analysis of Stress-Strain Data for Dry and Swollen Rubbers by a New Tube Model of Rubber Elasticity Based on Finite Chain Extensibility

Abstract New stress-strain equations have been extensively tested for both unswollen and swollen rubbers, which were very recently derived from a new non-Gaussian tube model of rubber elasticity that incorporates the effect of finite chain extensibility. Particularly, as core parameters, they involve the junction-fluctuation suppression parameter (h) of Dossin and Grassley and the tube deformation index (γ) of Gaylord. The best-fits of the equations to experimental literature data for unswollen and swollen networks have been done. The results are very good with sufficiently small departures of theory from experiment and the two fitting parameters can consistently well explain the molecular situations of the networks.

Measurements of freezing‐point depression to evaluate rubber network structure. Crosslinking of natural rubber with dicumyl peroxide

AbstractAn experimental approach based on the freezing‐point depression of a solvent in a swollen gel has been developed to characterize the structure of rubber networks. This property depends on the conditions required for the formation of crystalline nuclei, which are limited by the elastomer network restrictions. Information about the functionality, spatial distribution, and number of crosslinks can be obtained by the use of this easy and ready experimental method. Application of the tube model of rubber elasticity in the uniaxial stress–strain experiments of natural rubber samples vulcanized with dicumyl peroxide yields the characteristic parameters of the rubber networks, which are in concordance with the network structures predicted by the freezing point method. Finally influence of vulcanization conditions in network structure and its relationship with the mechanical properties was evaluated. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 544–556, 2007

Experimental and Self-Consistent-Field Theoretical Study of Styrene Block Copolymer Self-Adhesive Materials

This work addresses thermodynamic and mechanical properties of styrene block copolymer (SBC) self-adhesive materials. Mixtures of polystyrene-block-polyisoprene-block-polystyrene (SIS triblock) copolymers with resin and their blends with polystyrene-block-polyisoprene (SI diblock) copolymers were investigated experimentally and theoretically. The experiments involved small-angle X-ray scattering, atomic force microscopy, and tensile measurements, while the theoretical analysis invoked a mesoscopic representation of SBC materials combined with a continuum, 3D real-space self-consistent-field (SCF) approach. The calculations predict that the SBC systems should become microphase separated, with PS-rich spherical domains forming a bcc lattice in SIS triblock/SI diblock blends and in SIS triblock/resin systems with low resin content (20 wt %), while at high (60 wt %) resin content a disordered structure of PS-rich domains is expected. Extensive comparisons of SCF predictions concerning the type and the geometric characteristics of the morphologies obtained, the composition profiles, and the effect of diblock and resin content on the long-range space ordering of the PS-rich domains are performed against the experimental data. SCF theory predicts that approximately 79% of SIS molecules are connecting different PS-rich domains (i.e., are forming bridges) in SIS/SI blends, almost independently of the SI diblock content. The SCF results are combined with a slip tube model of rubber elasticity to predict the elastic behavior of SBC materials. In particular, the bridging properties are used to estimate the PS-rich domain cross-linking contribution, Gc, to the total shear modulus. The entanglement contribution, Ge, is evaluated from experimental and theoretical results reported in the literature concerning the properties of entanglements. The predicted stress−strain curves are in good qualitative agreement with the tensile experimental data for small values of strain.

Hyperelasticity and stress softening of filler reinforced polymer networks

AbstractStarting from an analysis of filler networking in bulk polymers, a constitutive micro‐mechanical model of stress softening and hysteresis of filler reinforced polymer networks is developed. It refers to a non‐affine tube model of rubber elasticity, including hydrodynamic amplification of the rubber matrix by a fraction of hard, rigid filler clusters with filler‐filler bonds in the unbroken, virgin state. The filler‐induced hysteresis is described by an anisotropic free energy density, considering the cyclic breakdown and re‐aggregation of the residual fraction of soft filler clusters with already broken, damaged filler‐filler bonds. Experimental investigations of the quasi‐static stress‐strain behaviour of silica and carbon black filled rubbers up to large strain agree well with adaptations found by the developed model. The microscopic material parameters obtained appear reasonable, providing information on the mean size and distribution width of filler clusters, the tensile strength of filler‐filler bonds and the polymer network chain density.

Influence of Preparation Conditions on Network Parameters of Sulfur-Cured Natural Rubber

The effect of initial chain length before cross-linking and sulfur/accelerator amount during curing of natural rubber samples on network parameters (effective chain density, gel fraction, amount of chain ends, entanglement density, trapping factor, and relaxation and correlation times) is investigated by means of proton NMR relaxation, equilibrium swelling, and stress−strain analysis. Remarkable differences are observed in the two sample series depending on the variable initial molar mass. The stress−strain data are evaluated with respect to a non-Gaussian tube model of rubber elasticity that considers the finite extensibility of network chains by referring to the path integral approach of Edwards and Vilgis (Rep. Prog. Phys. 1988, 51, 243; Polymer 1986, 27, 483). According to several experimental indications, we assume a nonaffine tube deformation law as first derived by Heinrich et al. (Adv. Polym. Sci. 1988, 85, 33). The NMR relaxation data are analyzed by considering three types of chains (gel, sol and chain ends). The best fit is obtained by assuming an anisotropic motion of the inter-cross-link chains and chain ends. The swelling data are analyzed by assuming phantom like chains. Within the framework of experimental errors, the network parameters evaluated from the three experimental techniques show fair agreement for both sample series.

On the Length Scale Dependence of Microscopic Strain by SANS

We present a SANS study on the length scale dependence of chain deformation patterns in dense cross-linked elastomers. Three different polyisoprene networks of long primary block-copolymer chains of the HDH-type were analyzed. The total length of the primary chains is identical, while the length of the deuterated middle block was varied in order to cover several length scale regimes of interest. The scattering data are analyzed in the frame of the tube model of rubber elasticity in combination with the random phase approximation (RPA), which is used in order to account for the interchain correlations. We show experimentally that for the longest labeled path the rubber elastic response is dominated by both cross-link and entanglement contributions, whereas it is of clear phantomlike chain behavior if the labeled middle block of the chain is shorter than the distance between two successive entanglements, i.e., the tube diameter. We further propose schemes to quantify nonaffine deformations on various length scales and chain scissioning processes during the radical cross-linking process.

A generalized tube model of rubber elasticity and stress softening of filler reinforced elastomer systems

An advanced micro-mechanical model of hyperelasticity and stress softening of reinforced rubbers is presented that combines a non-Gaussian tube model of rubber elasticity with a damage model of stress-induced filler cluster breakdown. The path integral formulation of rubber elasticity is briefly reviewed. Within this framework the consideration of tube-like, topological constraints (packing effects) as well as finite chain extensibility of rubber networks is described. The results are compared to the classical Mooney-Rivlin and inverse Langevin approaches of rubber elasticity. The effect of the filler is taken into account via hydrodynamic reinforcement of the rubber matrix by rigid, self-similar filler clusters, which leads to a quantitative description of stress softening by means of a strain or pre-strain dependent hydrodynamic amplification factor, respectively. Thereby, the pronounced stress softening or high hysteresis of reinforced rubber is referred to an irreversible breakdown of filler clusters during the first deformation cycle. It is shown that the developed concept is in fair agreement with experimental data of unfilled NR-samples in uni-, equibiaxial and pure shear stretching mode. The pronounced stress softening of carbon black filled E-SBR- and EPDM-samples is well described on a quantitative level by an exponential filler cluster decay law.

A generalized tube model of rubber elasticity and stress softening of filler reinforced elastomer systems

An advanced micro-mechanical model of hyperelasticity and stress softening of reinforced rubbers is presented that combines a non-Gaussian tube model of rubber elasticity with a damage model of stress-induced filler cluster breakdown. The path integral formulation of rubber elasticity is briefly reviewed. Within this framework the consideration of tube-like, topological constraints (packing effects) as well as finite chain extensibility of rubber networks is described. The results are compared to the classical Mooney-Rivlin and inverse Langevin approaches of rubber elasticity. The effect of the filler is taken into account via hydrodynamic reinforcement of the rubber matrix by rigid, self-similar filler clusters, which leads to a quantitative description of stress softening by means of a strain or pre-strain dependent hydrodynamic amplification factor, respectively. Thereby, the pronounced stress softening or high hysteresis of reinforced rubber is referred to an irreversible breakdown of filler clusters during the first deformation cycle. It is shown that the developed concept is in fair agreement with experimental data of unfilled NR-samples in uni-, equibiaxial and pure shear stretching mode. The pronounced stress softening of carbon black filled E-SBR- and EPDM-samples is well described on a quantitative level by an exponential filler cluster decay law.

The equilibrium and non-equilibrium elastic behavior of cross-linked polymer networks

The tube model of rubber elasticity (GAYLORD 1979, MARRUCCI 1981, GAYLORD 1982) is developed in detail and is shown to be able to describe both the equilibrium stress-strain, shear modulus and swelling behavior and the non-equilibrium elastic behavior of cross-linked polymer networks.

The constant volume tube model of rubber elasticity

A tube model is presented which provides a surprisingly good description of the simple extension and compression behavior of a cross-linked rubber, while remaining conceptually and mathematically simple.