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David E Hanson - One of the best experts on this subject based on the ideXlab platform.
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a new paradigm for the molecular basis of Rubber Elasticity
Contemporary Physics, 2015Co-Authors: David E Hanson, John L BarberAbstract:The molecular basis for Rubber Elasticity is arguably the oldest and one of the most important questions in the field of polymer physics. The theoretical investigation of Rubber Elasticity began in earnest almost a century ago with the development of analytic thermodynamic models, based on simple, highly-symmetric configurations of so-called Gaussian chains, i.e. polymer chains that obey Markov statistics. Numerous theories have been proposed over the past 90 years based on the ansatz that the elastic force for individual network chains arises from the entropy change associated with the distribution of end-to-end distances of a free polymer chain. There are serious conceptual objections to this assumption and others, such as the assumption that all network nodes undergo a simple volume-preserving linear motion and that all of the network chains have the same length. Recently, a new paradigm for Elasticity in Rubber networks has been proposed that is based on mechanisms that originate at the molecular leve...
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the entropy of the rotational conformations of poly isoprene molecules and its relationship to Rubber Elasticity and temperature increase for moderate tensile or compressive strains
Journal of Chemical Physics, 2013Co-Authors: David E Hanson, John L Barber, Gopinath SubramanianAbstract:Molecular networks comprised of crosslinked cis-1,4 polyisoprene, often referred to as “natural Rubber,” are one of the most common systems for the study of Rubber Elasticity. Under moderate tensile or compressive strain, network chains begin to assume straighter paths, as local molecular kinks are removed. Isoprene units along the chain backbone are mechanically forced from their equilibrium distributions of 18 possible rotational states into a smaller subset of states, restricted to more linear conformations with the greatest end-to-end distances. There are two consequences to this change: both the configurational entropy and average internal energy decrease. We find that the change in entropy, and resulting change in free energy, gives rise to an elastic force. We derive an expression for a chain extension force constant that we have incorporated in an explicit, three-dimensional meso-scale network simulation code. Using this force model, our simulations predict a macroscopic stress-strain relationship that closely matches published experimental values. We also predict a slight increase in temperature resulting from the change in average internal energy in the affected isoprene units that is consistent with experiments.
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the molecular kink paradigm for Rubber Elasticity numerical simulations of explicit polyisoprene networks at low to moderate tensile strains
Journal of Chemical Physics, 2011Co-Authors: David E HansonAbstract:Based on recent molecular dynamics and ab initio simulations of small isoprene molecules, we propose a new ansatz for Rubber Elasticity. We envision a network chain as a series of independent molecular kinks, each comprised of a small number of backbone units, and the strain as being imposed along the contour of the chain. We treat chain extension in three distinct force regimes: (Ia) near zero strain, where we assume that the chain is extended within a well defined tube, with all of the kinks participating simultaneously as entropic elastic springs, (II) when the chain becomes sensibly straight, giving rise to a purely enthalpic stretching force (until bond rupture occurs) and, (Ib) a linear entropic regime, between regimes Ia and II, in which a force limit is imposed by tube deformation. In this intermediate regime, the molecular kinks are assumed to be gradually straightened until the chain becomes a series of straight segments between entanglements. We assume that there exists a tube deformation tensi...
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the molecular kink paradigm for Rubber Elasticity numerical simulations of explicit polyisoprene networks at low to moderate tensile strains
Journal of Chemical Physics, 2011Co-Authors: David E HansonAbstract:Based on recent molecular dynamics and ab initio simulations of small isoprene molecules, we propose a new ansatz for Rubber Elasticity. We envision a network chain as a series of independent molecular kinks, each comprised of a small number of backbone units, and the strain as being imposed along the contour of the chain. We treat chain extension in three distinct force regimes: (Ia) near zero strain, where we assume that the chain is extended within a well defined tube, with all of the kinks participating simultaneously as entropic elastic springs, (II) when the chain becomes sensibly straight, giving rise to a purely enthalpic stretching force (until bond rupture occurs) and, (Ib) a linear entropic regime, between regimes Ia and II, in which a force limit is imposed by tube deformation. In this intermediate regime, the molecular kinks are assumed to be gradually straightened until the chain becomes a series of straight segments between entanglements. We assume that there exists a tube deformation tension limit that is inversely proportional to the chain path tortuosity. Here we report the results of numerical simulations of explicit three-dimensional, periodic, polyisoprene networks, using these extension-only force models. At low strain, crosslink nodes are moved affinely, up to an arbitrary node force limit. Above this limit, non-affine motion of the nodes is allowed to relax unbalanced chain forces. Our simulation results are in good agreement with tensile stress vs. strain experiments.
Mary C Boyce - One of the best experts on this subject based on the ideXlab platform.
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deformation of elastomeric networks relation between molecular level deformation and classical statistical mechanics models of Rubber Elasticity
Macromolecules, 2001Co-Authors: J S Bergstrom, Mary C BoyceAbstract:In this work a specialized molecular simulation code has been used to provide details of the underlying micromechanisms governing the observed macroscopic behavior of elastomeric materials. In the simulations the polymer microstructure was modeled as a collection of unified atoms interacting by two-body potentials of bonded and non-bonded type. Representative Volume Elements (RVEs) containing a network of 200 molecular chains of 100 bond lengths are constructed. The evolution of the RVEs with uniaxial deformation was studied with molecular dynamics techniques. The simulations enable observation of structural features with deformation including bond lengths and angles as well as chain lengths and angles. The simulations also enable calculation of the macroscopic stress-strain behavior and its decomposition into bonded and non-bonded contributions. The distribution in initial end-to-end chain lengths is consistent with Gaussian statistics treatments of Rubber Elasticity. It is shown that application of an axial strain of +/ 0.7 (a logarithmic strain measure is used) only causes a change in the average bond angle of /+ 5 degrees indicating the freedom of bonds to sample space at these low-moderate deformations; the same strain causes the average chain angle to change by /+ 20 degrees. Randomly selected individual chains are monitored during deformation; their individual chain lengths and angles are found to evolve in an essentially ane manner consistent with Gaussian statistics treatments of Rubber Elasticity. The average chain length and angle are found to evolve in a manner consistent with the eight-chain network model of Elasticity. Energy quantities are found to remain constant during deformation consistent with the nature of Rubber Elasticity begin entropic in origin. The stress-strain response is found to have important bonded and non-bonded contributions. The bonded contributions arise from the rotations of the bonds toward the maximum principal stretch axis(es) in tensile(compressive) loading.
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deformation of elastomeric networks relation between molecular level deformation and classical statistical mechanics models of Rubber Elasticity
Macromolecules, 2001Co-Authors: J S Bergstrom, Mary C BoyceAbstract:In this work, molecular simulations are conducted to provide details of the underlying micromechanisms governing the observed macroscopic behavior of elastomeric materials. The polymer microstructure is modeled as a collection of unified atoms interacting by two-body potentials of bonded and nonbonded type. Representative volume elements (RVEs) containing a network of 200 molecular chains of 100 bond lengths are constructed. The evolution of the RVEs with uniaxial deformation was studied using a molecular dynamics technique. The simulations enable observation of structural features with deformation including bond lengths and angles as well as chain lengths and angles. The simulations also enable calculation of the macroscopic stress−strain behavior and its decomposition into bonded and nonbonded contributions. The distribution in initial end-to-end chain lengths is consistent with Gaussian statistics treatments of Rubber Elasticity. It is shown that application of an axial strain of ±0.7 (a logarithmic st...
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constitutive models of Rubber Elasticity a review
Rubber Chemistry and Technology, 2000Co-Authors: Mary C Boyce, Ellen M ArrudaAbstract:Abstract A review of constitutive models for the finite deformation response of Rubbery materials is given. Several recent and classic statistical mechanics and continuum mechanics models of incompressible Rubber Elasticity are discussed and compared to experimental data. A hybrid of the Flory—Erman model for low stretch deformation and the Arruda—Boyce model for large stretch deformation is shown to give an accurate, predictive description of Treloar's classical data over the entire stretch range for all deformation states. The modeling of compressibility is also addressed.
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direct comparison of the gent and the arruda boyce constitutive models of Rubber Elasticity
Rubber Chemistry and Technology, 1996Co-Authors: Mary C BoyceAbstract:Abstract The Arruda and Boyce eight-chain network constitutive model for Rubber elastic materials is compared to the new Gent constitutive model for Rubber Elasticity. The salient features of each of the two models are compared. The ability of both models to predict three dimensional large strain deformation is demonstrated showing the near equivalence of these two model constructions as well as their abilities to predict complex three-dimensional deformation with only two material constants.
Mikhail Itskov - One of the best experts on this subject based on the ideXlab platform.
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analytical network averaging of the tube model Rubber Elasticity
Journal of The Mechanics and Physics of Solids, 2016Co-Authors: Vu Ngoc Khiem, Mikhail ItskovAbstract:Abstract In this paper, a micromechanical model for Rubber Elasticity is proposed on the basis of analytical network-averaging of the tube model and by applying a closed-form of the Rayleigh exact distribution function for non-Gaussian chains. This closed-form is derived by considering the polymer chain as a coarse-grained model on the basis of the quantum mechanical solution for finitely extensible dumbbells ( Ilg et al., 2000 ). The proposed model includes very few physically motivated material constants and demonstrates good agreement with experimental data on biaxial tension as well as simple shear tests.
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a Rubber Elasticity and softening model based on chain length statistics
International Journal of Solids and Structures, 2016Co-Authors: Mikhail Itskov, A G KnyazevaAbstract:Abstract The classical statistical theory of polymerization predicts a random distribution of polymer chain lengths. This distribution has long ago been known in the polymerization theory but, to the best of our knowledge, has not so far been utilized in mechanics of polymers. In the present paper, we incorporate this chain length statistics into full network Rubber models which are based on continuous directional distributions of polymer chains. The free energy of the full network results as an integral of single chain energies over the unit sphere. In the case of an initially isotropic spatial arrangement of chains and ideally elastic behavior an analytical solution in terms of micro-structural parameters of the network is obtained. Introducing a softening criterion formulated in terms of the minimal number of chain segments available in the distribution we can describe not only elastic behavior but also inelastic phenomena especially pronounced in filled Rubbers. These are, for example, the Mullins effect, permanent set and strain induced anisotropy. In this case, numerical integration over the unit sphere is applied. Predictions of the model demonstrate good agreement with experimental data with respect to the above mentioned phenomena.
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a simple and accurate approximation of the inverse langevin function
Rheologica Acta, 2015Co-Authors: Ehsan Darabi, Mikhail ItskovAbstract:The inverse Langevin function cannot be represented in an explicit form and requires an approximation by a series, a non-rational or a rational function as for example by a Pade approximation. In the current paper, an analytical method based on the Pade technique and the multiple point interpolation is presented for the inverse Langevin function. Thus, a new simple and accurate approximation of the inverse Langevin function is obtained. It might be advantageous, for example, for non-Gaussian statistical theory of Rubber Elasticity where the inverse Langevin function plays an important role.
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a full network Rubber Elasticity model based on analytical integration
Mathematics and Mechanics of Solids, 2010Co-Authors: Mikhail Itskov, Alexander E Ehret, Roozbeh DargazanyAbstract:Full-network Rubber Elasticity models generally require numerical integration over the unit sphere. In the present paper, a procedure for analytical integration of power series in terms of stretch square is proposed instead. This procedure is applied both to the inverse Langevin function and its rounded Pade approximation. The integrated power series demonstrates fast convergence to the analytical solution so far as it is available or to the numerical one based on a high resolution integration scheme. Good agreement with experimental data on silicone Rubber is obtained as well. The integration procedure is also implemented to average the stretch on the basis of a q-root operator. This operator is usually applied in order to introduce a non-affine relation between micro and macro stretches into a network model.
J S Bergstrom - One of the best experts on this subject based on the ideXlab platform.
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deformation of elastomeric networks relation between molecular level deformation and classical statistical mechanics models of Rubber Elasticity
Macromolecules, 2001Co-Authors: J S Bergstrom, Mary C BoyceAbstract:In this work a specialized molecular simulation code has been used to provide details of the underlying micromechanisms governing the observed macroscopic behavior of elastomeric materials. In the simulations the polymer microstructure was modeled as a collection of unified atoms interacting by two-body potentials of bonded and non-bonded type. Representative Volume Elements (RVEs) containing a network of 200 molecular chains of 100 bond lengths are constructed. The evolution of the RVEs with uniaxial deformation was studied with molecular dynamics techniques. The simulations enable observation of structural features with deformation including bond lengths and angles as well as chain lengths and angles. The simulations also enable calculation of the macroscopic stress-strain behavior and its decomposition into bonded and non-bonded contributions. The distribution in initial end-to-end chain lengths is consistent with Gaussian statistics treatments of Rubber Elasticity. It is shown that application of an axial strain of +/ 0.7 (a logarithmic strain measure is used) only causes a change in the average bond angle of /+ 5 degrees indicating the freedom of bonds to sample space at these low-moderate deformations; the same strain causes the average chain angle to change by /+ 20 degrees. Randomly selected individual chains are monitored during deformation; their individual chain lengths and angles are found to evolve in an essentially ane manner consistent with Gaussian statistics treatments of Rubber Elasticity. The average chain length and angle are found to evolve in a manner consistent with the eight-chain network model of Elasticity. Energy quantities are found to remain constant during deformation consistent with the nature of Rubber Elasticity begin entropic in origin. The stress-strain response is found to have important bonded and non-bonded contributions. The bonded contributions arise from the rotations of the bonds toward the maximum principal stretch axis(es) in tensile(compressive) loading.
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deformation of elastomeric networks relation between molecular level deformation and classical statistical mechanics models of Rubber Elasticity
Macromolecules, 2001Co-Authors: J S Bergstrom, Mary C BoyceAbstract:In this work, molecular simulations are conducted to provide details of the underlying micromechanisms governing the observed macroscopic behavior of elastomeric materials. The polymer microstructure is modeled as a collection of unified atoms interacting by two-body potentials of bonded and nonbonded type. Representative volume elements (RVEs) containing a network of 200 molecular chains of 100 bond lengths are constructed. The evolution of the RVEs with uniaxial deformation was studied using a molecular dynamics technique. The simulations enable observation of structural features with deformation including bond lengths and angles as well as chain lengths and angles. The simulations also enable calculation of the macroscopic stress−strain behavior and its decomposition into bonded and nonbonded contributions. The distribution in initial end-to-end chain lengths is consistent with Gaussian statistics treatments of Rubber Elasticity. It is shown that application of an axial strain of ±0.7 (a logarithmic st...
Takamasa Sakai - One of the best experts on this subject based on the ideXlab platform.
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experimental observation of two features unexpected from the classical theories of Rubber Elasticity
Physical Review Letters, 2017Co-Authors: Kenta Fujii, Ungil Chung, Mitsuhiro Shibayama, Kengo Nishi, Takamasa SakaiAbstract:Although the elastic modulus of a Gaussian chain network is thought to be successfully described by classical theories of Rubber Elasticity, such as the affine and phantom models, verification experiments are largely lacking owing to difficulties in precisely controlling of the network structure. We prepared well-defined model polymer networks experimentally, and measured the elastic modulus G for a broad range of polymer concentrations and connectivity probabilities, p. In our experiment, we observed two features that were distinct from those predicted by classical theories. First, we observed the critical behavior G∼|p-p_{c}|^{1.95} near the sol-gel transition. This scaling law is different from the prediction of classical theories, but can be explained by analogy between the electric conductivity of resistor networks and the Elasticity of polymer networks. Here, p_{c} is the sol-gel transition point. Furthermore, we found that the experimental G-p relations in the region above C^{*} did not follow the affine or phantom theories. Instead, all the G/G_{0}-p curves fell onto a single master curve when G was normalized by the elastic modulus at p=1, G_{0}. We show that the effective medium approximation for Gaussian chain networks explains this master curve.
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Rubber Elasticity for percolation network consisting of Gaussian chains.
The Journal of chemical physics, 2015Co-Authors: Kengo Nishi, Takamasa Sakai, Hiroshi Noguchi, Mitsuhiro ShibayamaAbstract:A theory describing the elastic modulus for percolation networks of Gaussian chains on general lattices such as square and cubic lattices is proposed and its validity is examined with simulation and mechanical experiments on well-defined polymer networks. The theory was developed by generalizing the effective medium approximation (EMA) for Hookian spring network to Gaussian chain networks. From EMA theory, we found that the ratio of the elastic modulus at p, G to that at p = 1, G0, must be equal to G/G0 = (p − 2/f)/(1 − 2/f) if the position of sites can be determined so as to meet the force balance, where p is the degree of cross-linking reaction. However, the EMA prediction cannot be applicable near its percolation threshold because EMA is a mean field theory. Thus, we combine real-space renormalization and EMA and propose a theory called real-space renormalized EMA, i.e., REMA. The elastic modulus predicted by REMA is in excellent agreement with the results of simulations and experiments of near-ideal diamond lattice gels.
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Rubber Elasticity for incomplete polymer networks.
The Journal of chemical physics, 2012Co-Authors: Kengo Nishi, Yukiteru Katsumoto, Kenta Fujii, Ungil Chung, Takamasa Sakai, Hiroshi Noguchi, Masashi Chijiishi, Toshio Nakao, Mitsuhiro ShibayamaAbstract:We investigated the relationship between the elastic modulus, G and the reaction probability, p for polymer networks. First, we pointed out that the elastic modulus is expressed by G = {(fp/2 − 1) + O((p − 1)2)} NkBT/V (percolated network law), which does not depend on the local topology of the network structure or the existence of the loops. Here, N is the number of lattice point, V is the system volume, f is the functionality of the cross-link, kB is the Boltzmann constant, and T is the absolute temperature. We also conducted simulations for polymer networks with triangular and diamond lattices, and mechanical testing experiments on tetra-poly(ethylene glycol) (PEG) gel with systematically tuning the reaction probability. Here, the tetra-PEG gel was confirmed to be a potential candidate for ideal polymer networks consisting of unimodal strands free from defects and entanglements. From the results of simulations and experiments, it was revealed, for the first time, that the elastic modulus obeys this law...
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examination of the theories of Rubber Elasticity using an ideal polymer network
Macromolecules, 2011Co-Authors: Yuki Akagi, Takuya Katashima, Yukiteru Katsumoto, Kenta Fujii, Takuro Matsunaga, Ungil Chung, Mitsuhiro Shibayama, Takamasa SakaiAbstract:We evaluate the homogeneity of Tetra-PEG gel and examine the models predicting Rubber Elasticity. Infrared spectroscopy revealed the near absence of dangling chains. Concentration dependence of the number of elastically effective chains revealed the near absence of elastically ineffective loops and the validity of the phantom network model. These data suggest that Tetra-PEG gel is close to an ideal polymer network and that the phantom network model is the true model describing the contribution of the chemical cross-link to the Rubber Elasticity of a swollen polymer network.
