Kinetic Theory Of Rubber Elasticity Research Articles

Recent Studies on Degradation Mechanism of Rubber Vulcanizates by Chemorheology

Abstract A critical discussion is presented of the influence of chemical degradation mechanisms on the mechanical properties, especially stress relaxation, of vulcanized rubber. The response of the vulcanizate is interpreted in terms of the kinetic theory of rubber elasticity. Different chemical mechanisms are explored. The influence of diluent is interpreted in terms of the kinetic theory of rubber elasticity. Different chemical mechanisms are explored. The influence of diluent is considered. Comparisons are made to experiments on polyisoprenes and polyurethanes.

Biorenewable ROMP-based thermosetting copolymers from functionalized castor oil derivative with various cross-linking agents

A new norbornenyl-functionalized castor oil alcohol (NCA) was synthesized and ring-opening metathesis copolymerized separately with two norbornene-based cross-linking agents: dicyclopentadiene (DCPD) and a bifunctional norbornene crosslinker (CL). Isothermal differential scanning calorimetry (DSC) was used to examine the cure behavior of NCA/DCPD and NCA/CL resins, through which a reasonable cure schedule was determined. The glass transition temperature (Tg) and storage modulus (E′), characterized by dynamic mechanical analysis (DMA), increased significantly in both copolymer systems with the addition of cross-linking agents. Cross-link density of the two systems was evaluated using a modified empirical equation from the kinetic theory of rubber elasticity. Differences in tensile stress-strain behavior and thermal stability between polymerized NCA/DCPD and NCA/CL were correlated to the structural rigidity and cross-linking density resulting from the cross-linking agents.

A micromechanically motivated diffusion-based transient network model and its incorporation into finite rubber viscoelasticity

This paper presents the development of a physical-based constitutive model for the representation of viscous effects in rubber-like materials. The proposed model originates from micromechanically motivated diffusion processes of the highly mobile polymer chains described within the formalism of Brownian motion. Following the basic assumption of accounting for the elastic and the viscous effects in rubber viscoelasticity by their representation through a separate elastic ground network and several viscous subnetworks, respectively, the kinetic theory of rubber elasticity is followed and extended to represent also the viscous contribution in this work. It is assumed that the stretch probability of certain chain segments within an individual viscous subnetwork evolves based on the movement of the chain endpoints described by the Smoluchowski equation extended in this work from non-interacting point particles in a viscous surrounding to flexible polymer chains. An equivalent tensorial representation for this evolution is chosen which allows for the closed form solution of the macroscopic free energy and the macroscopic viscous overstress based on a homogenization over the probability space of the introduced micro-objects. The resulting model of the viscous subnetwork is subsequently combined with the non-affine micro-sphere model and applied in homogeneous and non-homogeneous tests. Finally, the model capacity is outlined based on a comparison with in the literature available experimental data sets.

Network structure of gelatin gel at high pressure determined by rheological measurements

The present paper reports on rheological investigations of gelatin gels cured under high-pressure conditions. Measurement of the storage modulus showed that gels developed under elevated pressure revealed higher values of G′ in comparison to gels cured at ambient pressure. Based on the kinetic theory of rubber elasticity, it is presumed that the higher values of the storage modulus originate from the increased number of triple helix junction zones per molecular chain. Additionally, considering the triple helix nucleation process, it is hypothesised that the average length of the triple helix junction zones is lower, at least in the initial period of gel network building, if the gel is setting at high pressure.

Finite Element Analysis of Transversely Isotropic Viscoelastic Material for Thermoforming Process

In this work, a dynamic finite element method is used in the modeling and numerical simulation of the transversely viscoelastic behavior of a thin, isotropic, and incompressible thermoplastic membrane. The transversely isotropic viscoelastic functional based on the kinetic theory of rubber elasticity is presented. The model is expressed in terms of four material parameters and equations relating these parameters. The thermoforming of the sheet is performed under the action of perfect gas flows. The Lagrangian formulation, together with the assumption of the membrane theory, is used in the finite element implementation. The numerical validation is performed by comparing the theoretical solution for the uniaxial and equibiaxial Hencky deformation with numerical results. Moreover, the influence of fiber direction in the transversely viscoelastic material on the thickness and on the stress distribution in the thermoforming sheet are analyzed.

Time dependent large principal deformation of polymers

A constitutive model which describes principal deformations of polymeric materials under small and large deformations is presented. It uses the kinetic theory of rubber elasticity as a starting point for the large deformation behaviour and generalizes this theory to include the effect of broad-spectral rate dependence. The constitutive model reduces to (phenomenological) small strain linear viscoelasticity under vanishing strains, while at long times the response corresponds to that of a rubber-elastic material. The behaviour of the model under constant strain rate is examined in both the loading and unloading mode. As a result of appealing to physical-molecular concepts it provides naturally for simulation of a rate dependent yield phenomenon through intrinsic relaxation behaviour without the specific introduction of a yield parameter or yield function. The effect of generalizing the rubber elasticity model to include compressibility is discussed. It is found that the strain energy functional cannot be generalized arbitrarily through an additive term containing the dilatational component. The investigation renders a material description, which, while possibly lacking in detail of species-specific behaviour, reproduces major features of nonlinear and large deformation viscoelastic characteristics. These general features make the model suitable for investigating the influence of rate dependent yielding and large deformation viscoelasticity in the solution of boundary value problems

In Memoriam: Herman F. Mark. May 3, 1895-April 6, 1992

Abstract Professor Herman F. Mark died on April 6, 1992, one month before his 97th birthday. For almost 70 years Herman Mark was in the forefront of polymer chemistry. He was an outstanding scientist, a superb teacher, a fascinating lecturer, an excellent organizer, and a remarkable human being. In the early days of his scientific career he determined the crystal structures of cellulose and a number of other natural and synthetic polymers. He also played a leading role in the develop- ment of the kinetic theory of rubber elasticity, and he was responsible for the simple determination of polymer molecular weights based on viscosity measurements, as represented by the Mark-Houwink equation. What Mark did not actively investi- gate himself, he helped disseminate throughout the rapidly developing field of poly- mer science.

Crosslinking‐property relationships in PMR polyimide composites. Part I

AbstractThis study examines for the first time how matrix crosslinking affects the composite physical and mechanical properties of a graphite fiber reinforced PMR polyimide composite during long‐term isothermal aging. Unidirectional composite specimens of Celion 6000/PMR‐P1 were isothermally exposed at 288°C in air for various time periods up to 5000 h. The matrix crosslink densities were estimated from the kinetic theory of rubber elasticity and shifts in the glass transition temperatures (Tgs). The Tg, coefficient of thermal expansion, density, weight loss, moisture absorption, and elevated temperature flexural and interlaminar shear properties were also determined. Several linear relationships were found between the matrix crosslink density and composite physical and mechanical properties. The Tg, initial weight loss and density, and elevated temperature interlaminar shear strength increase with an increase in crosslink density. Conversely, the initial moisture absorption and coefficient of thermal expansion decrease with increasing crosslink density. As expected, the elevated temperature flexural strength and modulus show no direct correlations with crosslink density. Further, after achieving the highest matrix crosslink density, several of the composite properties begin to decrease rapidly. These findings suggest that time‐temperature dependent nature of attaining the maximum matrix crosslinking is closely linked to the onset of the composite property degradation. Though much more work is needed, a fundamental understanding of the relationships between matrix crosslinking and composite physical and mechanical property can provide a scientific basis for the prediction of the extent of composite service life not only for PMR polyimides but also for other thermosetting matrix resins, such as epoxies and bismaleimides.

Determination of the entanglement molecular weight of thermoplastic elastomers by tensile retraction measurements

AbstractThe molecular weight between crosslinks was measured as a function of the maximum extension to which samples of solution cast films of ABA triblocks were tested. These measurements were done by applying the kinetic theory of rubber elasticity to the retraction portion of a tensile hysteresis curve. The molecular weight between chain restrictions, as determined by extrapolation to 0% elongation, was found to be identical to the entanglement molecular weight normally associated with the polybutadiene or polyisoprene that was used for the center block. The disentanglement of the small amount of AB diblock present could also be followed by fitting the data collected to what could be pictured kinetically as a simultaneous first‐and second‐order loss of crosslink density.

Swelling equilibria of ionized poly(methacrylic acid) gels in the absence of salt

The kinetic theory of rubber elasticity for polyelectrolytes has been modified by including an effect of the electrostatic persistence length. The relations for swelling equilibria as a function of the degree of neutralization in polyelectrolyte gels are derived. We find that the volume fraction of the polymer in the swollen gel does not decrease monotonically with increasing α, but instead exhibits a minimum near α ∼ 0.1. Beyond this value of α the extent of swelling decreases with increasing degree of neutralization. The theory has been verified by measurements of the swelling behaviour of weakly crosslinked polymethacrylic acid gels as a function of varying degree of neutralization.

A two material constant, nonlinear elastic stress constitutive equation including the effect of compressibility

A new constitutive equation for compressible, nonlinear elasticity is derived. The stress-deformation relation takes the final form σ ij = 3k 2J (J 2 3 - 1) δ ij + μ J (B ij - j 2 3 δ ij) where B ij is the left Cauchy-Green deformation tensor. J is the Jacobean of the deformation, and k and μ are the two governing mechanical properties having the same interpretation as in linear, isotropic elasticity theory. In this nonlinear theory, k and μ are separately determinable from large deformation conditions of pure dilatation and simple shear, respectively. Under isochoric conditions, the distortional part of the theory takes the form of the kinetic theory of rubber elasticity. Under conditions of small volume change, the theory coincides with the molecular theory of Flory derived for elastomers, but given explicit from only in the one dimensional case.

Recent Studies on Degradation Mechanism of Rubber Vulcanizates by Chemorheology

Abstract A critical discussion is presented of the influence of chemical degradation mechanisms on the mechanical properties, especially stress relaxation, of vulcanized rubber. The response of the vulcanizate is interpreted in terms of the kinetic theory of rubber elasticity. Different chemical mechanisms are explored. The influence of diluent is interpreted in terms of the kinetic theory of rubber elasticity. Different chemical mechanisms are explored. The influence of diluent is considered. Comparisons are made to experiments on polyisoprenes and polyurethanes.

The contribution to rubber elasticity of topological entanglements

A microscopic explanation of the departures from the statistical kinetic theory of rubber elasticity is given using a topological description of entanglements. This is accomplished in terms of linking numbers which can be topologically defined between pairs of entangled polymer molecules. A statistical mechanical treatment of the linking numbers is presented and an expression derived for the elastic modulus of an entangled coil. Our results show the expected deformation softening in both uniaxial extension and compression. We identify the dependence of this behaviour on the density, the entangling ability of the molecule and on the details of fabrication of the network. We also show that, with the framework of the concepts developed in the paper, the deformation softening behaviour implies that the system is under-entangled with respect to an intrinsic degree of entanglement characteristic of the system. This intrinsic degree of entanglement is defined in the paper and for states of entanglement exceeding this quantity a deformation hardening is predicted.

Physical and Chemical Properties of Rubber-Like Elastic Fibres From the Octopus Aorta

ABSTRACT We investigated the physical and chemical properties of highly extensible elastic fibres from the octopus aorta. These fibres are composed of an insoluble rubber-like protein which we call the octopus arterial elastomer. The amino acid composition of this protein is different from that of other known protein rubbers, being relatively low in glycine and high in acidic and basic residues. Up to extensions of 50%, mechanical data from native elastic fibres fit a theoretical curve for an ideal Gaussian rubber with elastic modulus G = 4·65 × 105 Nm−2, and this is unchanged by prolonged exposure to formic acid. Thermoelastic tests on this protein indicate that the elastic force arises primarily from changes in conformational entropy, as predicted by the kinetic theory of rubber elasticity. Analysis of the non-Gaussian behaviour of the elastic fibres at extensions greater than 50% suggests that the molecular chains in this octopus protein are somewhat less flexible than those in resilin or elastin. Some speculations on the molecular design of these protein rubbers are made.

Protein elasticity based on conformations of sequential polypeptides: The biological elastic fiber

Fibrous elastin of the biological elastic fiber is a cross-linked condensed state in which there is roughly one-half polypeptide and one-half water. The precursor protein tropoelastin, a chemical fragmentation product α-elastin, and a sequential polypeptide (l·Val1-l·Pro2-Gly3-l·Val4-Gly5) n , which is a prominent primary structural feature of tropoelastin, are each soluble in all proportions in water at 20°C. On heating to physiological temperatures, each undergoes aggregation and forms a dense viscoelastic phase, which as the fiber itself, is about 60% water. This reversible heat-elicited condensed phase is called the coacervate. Circular dichroism studies show coacervation to be a process of increasing intramolecular order. Electron microscopy (light, scanning, and transmission) shows coacervation to be a process of increasing order intermolecularly. Thus a rise in temperature between 20 and 40°C results in an increase in order of the polypeptide. Coacervation is an inverse temperature transition, and the condensed state is anisotropic at the molecular level. Thermoelasticity studies in water on bovine ligamentum nuchae fibrous elastin and on γ-irradiation cross-linked polypentapeptide coacervates show increases in elastomeric force,f, over the same 20–40°C temperature range in which the inverse temperature transition gives rise to the coacervate, and the constancy off/T with temperature, once the transition is effectively completed, suggests a high-entropy component to the elastomeric force. Thus the data argue for an anisotropic-entropic elastomer. Detailed conformational studies on the polypentapeptide result in the development of a β-spiral conformation in which there are regularly recurring β-turns in loose helical array (a structure that forms on raising the temperature) and in which there are recurring dynamic suspended segments that are the focal point of large, low-energy oscillatory motions called librations. The structure gives rise to a librational entropy mechanism of elasticity wherein the amplitudes of the rocking motions become damped on stretching. This perspective is substantiated by dielectric relaxation studies on the coacervate state and by characterization of synthetic analogs of the polypentapeptide. Dielectric relaxation studies on a concentrated state of about 60% water show the development of a regular structure over the same temperature range as for the development of the coacervate state, and the development of the regular structure with increasing temperature is seen to parallel the development of elastomeric force with increasing temperature. Increasing elastomeric force coincides with increasing regularity of structure! Synthetic analogs of the polypentapeptide, designed to interfere with the librational processes of the suspended segment, impair elastic function, and an analog that makes the β-turn more rigid results in increased elastic modulus. This development of a librational entropy mechanism for protein elasticity is a departure from the kinetic theory of rubber elasticity, the random network perspective that has dominated the traditional view of biological elasticity for the past several decades. The new perspective opens the way to insightful consideration of new elastomeric biomaterials with numerous biomedical applications.

Rheological Monitoring of the Formation of Polyacrylamide/Cr+3 Gels

Abstract The formation of polyacrylamide/chromium-ion gels has been followed rheologically. The time dependence of the storage modulus has been used to analyze the kinetics of the gelation process. The kinetic theory of rubber elasticity has been used to determine the crosslink density in the gel from the measured value of the storage modulus. The effects of changing polymer, chromium ion, and reducing agent concentrations have been studied.

The diffusion of a fluid through a highly elastic spherical membrane

A MATHEMATICAL theory of elastic materials valid for membrane like bodies has been presented in the treatise on non-linear elasticity by Green and Adkins [l]. This theory has been used extensively in solving specific boundary value problems. The same theory has also been employed to solve problems involving membranes of viscoelastic materials [2]. This theory is essentially two dimensional in that the governing equations, which hold on the middle surface in the limit of vanishing thickness, are assumed to hold even when the ratio of the thickness to a typical dimension is small but finite. This implies that the variation of the stretch ratios and related quantities are negligible along the thickness and are hence taken to be constant. There are several problems of practical interest of which the problem of filtration is one where a membrane theory for diffusion would be appropriate. It is our aim to develop such a membrane theory which would be applicable in the case of interacting continua. The membrane theory as developed by Green and Adkins [l] for elastic solids forms the basis for our work. It is, however, found that modifications have to be made to the theory if it is to be compatible with the theory of interacting continua. The problem of the diffusion of an ideal fluid through a spherical shell of non-linearly elastic material has been previously studied [3,4]. It was found that the stretch ratios were not constant in the radial direction. The gradient of the stretch ratio in the radial direction though small, was not negligible (about 10%). These gradients in the stretch ratios would be present even if the thickness of the shell were to become very small, i.e. in the membrane approximation. This new feature is a consequence of the presence of diffusive body forces which depend on the gradients of the densities and stretch tensors. Hence, in this work, we develop a membrane theory which depends on the values at the deformed middle surface of physical quantities and their gradients in the thickness direction. In order to avoid the question of flow within the membrane middle surface due to gradients in this surface, we confine attention to the axially symmetric problem of the diffusion of an ideal fluid through a spherical membrane of a nonlinear elastic material. After a brief review of the notations and basic equations relevant to a mixture of interacting continua in Section 2, we introduce a specific constitutive relation which is useful in describing the behavior of rubber-like nonlinearly elastic solids in Section 3. The forms of the constitutive relations employed are obtained by incorporating expressions suggested in the kinetic theory of rubber elasticity (cf. Treloar [5]) for the specific internal energy function. The phenomenon of swelling is introduced in Section 4. The problem inherent to specifying the traction boundary condition for interacting continua is overcome by assuming that the swollen state of the mixture is a saturated state. This permits the use of a relation between the surface tractions and the amount of stretching in the saturated state and helps resolve the problem of specifying the boundary condition. The kinematical quantities and the associated geometric relations pertinent to the deformed state are developed in Section 5. The equilibrium equations for the two constituents and the membrane approximations are developed in Section 6. The

A Nonlinear Theory of Viscoelasticity for Application to Elastomers

A simple nonlinear theory of viscoelasticity has been developed for application to elastomers. The theory is the viscoelastic generalization of the kinetic theory of rubber elasticity, and it is used to model time and rate-dependent effects. The method of derivation reveals that the theory is applicable to stress-imposed rather than strain-imposed conditions. Thus the creep test provides the logical means for deducing the material properties of the theory, while the relaxation-test technique is not applicable. When the long time asymptotic state is at the same level of deformation for both tests, the sources and degree of nonlinearity in relaxation tests are shown to be more severe than those involved in creep tests. A simple means is deduced for obtaining the creep function from the nonlinear creep response of the elastomer.

Optical properties of single elastin fibres indicate random protein conformation.

In describing the properties of fibrous proteins it has been common practice to attribute the macroscopic mechanical properties to the organization at the molecular level. Hence, the high tensile stiffness of collagen and silk has been viewed as an inevitable consequence of their crystalline structure. For elastin, however, there has been considerable controversy and confusion in assigning a conformation to this rubber-like protein. The mechanical and thermodynamic properties of elastin are consistent with the kinetic theory of rubber elasticity, and this theory is based on an isotropic network of kinetically free, random-coil molecules. In contrast, numerous electron microscope studies of negatively stained elastins obtained by autoclave and alkali purification, as well as coacervates of the various soluble elastins, reveal a highly ordered (anisotropic) structure, consisting of 3- to 5-nm filaments that apparently run parallel to the long axis of elastin fibres. These filaments have been accepted as evidence for an anisotropic molecular structure in elastin that is dramatically different from the random network of the kinetic theory. We have now used polarized light microscopy to distinguish these two types of structure. We find that both purified and unpurified, water-swollen, single elastin fibres are optically isotropic, in agreement with the predictions of the kinetic theory of rubber elasticity.

Preparation and properties of poly-( N-butylmethacrylamide) networks

The gel-effect influences the copolymerization of N-butylmethacrylamide with methylene- bis-acrylamide similarly to other vinyl-divinyl copolymerizations. The fraction of crosslinks wasted in cycles is very high at the gel point, reaching 0.84–0.95; it decreases with increasing content of the crosslinking agent and with decreasing content of dimethylsulphoxide (DMSO) in the polymerization mixture. The effect of residual time processes on the mechanical characteristics in the rubberlike region decreases with increasing concentration of elastically active network chains, ν e . The equilibrium deformational behaviour obeys the kinetic theory of rubber elasticity. With increasing dilution of the system, the efficiency of the cross-linking reaction decreases and so does the effect of trapped entanglements. A comparison of the gel points and the mechanical characteristics for poly( N-butylmethacrylamide) and poly( N-ethylmethacrylamide) networks swollen in DMSO indicates that the substituent at the amide nitrogen atom has a dilution effect depending on its size. The interaction parameter of the system poly( N-butylmethacrylamide)-DMSO increases linearly with the volume fraction of the polymer in the swollen gel, ν 2; χ = 0.498 + 0.362 ν 2 for ν 2 ϵ (0.1, 0.33) and 25°.