Stress–strain relation of entangled polymer networks (original) (raw)

Stress Relaxation of Entangled Polymer Networks

1999

The non-linear stress-strain relation for crosslinked polymer networks is studied using molecular dynamics simulations. Previously we demonstrated the importance of trapped entanglements in determining the elastic and relaxational properties of networks. Here we present new results for the stress versus strain for both dry and swollen networks. Models which limit the fluctuations of the network strands like the tube model are shown to describe the stress for both elongation and compression. For swollen networks, the total modulus is found to decrease like (V 0 /V) 2/3 and goes to the phantom model result only for short strand networks.

Elasticity of polymer networks

We develop and solve a new molecular model for nonlinear elasticity of entangled polymer networks. This model combines and generalizes several succeseful ideas introduced over the years in the field of the rubber elasticity. The topological constraints imposed by the neighboring network chains on a given network are represented by the confining potential that changes upon network deformation. This topological potential restricts fluctuations of the network chain to the nonaffinely deformed confining tube. Network chains are allowed to fluctuate and redistribute their length along the contour of their confining tubes. The dependence of the stress σ on the elongation coefficient λ for the uniaxially deformed network is usially represented in the form of the Mooney stress, f*(1/λ) ) σ/(λ -1/λ 2 ). We find a simple expression for the Mooney stress, f*(1/λ) ) Gc + Ge/(0.74λ + 0.61λ -1/2 -0.35), where Gc and Ge are phantom and entangled network moduli. This allows one to analyze the experimental data in the form of the universal plot and to obtain the two moduli G c and Ge related to the densities of cross-links and entanglements of the individual networks. The predictions of our new model are in good agreement with experimental data for uniaxially deformed polybutadiene, poly(dimethylsiloxane), and natural rubber networks, as well as with recent computer simulations.

Contribution of Entanglements to Polymer Network Elasticity

Macromolecules, 2017

Trapped entanglements, cross-linker functionality, and elastically effective chains are the sources of elasticity of polymer networks and gels. However, despite more than 80 years of theoretical and experimental research in this field, still little is known about their relative contribution to network elasticity. In this work, we use double quantum nuclear magnetic resonance (DQ NMR) experiments to characterize the elasticity of model polymer networks prepared with cross-linkers of mixed functionality and control of structural defects. An order parameter that condensates the elastic response within the theoretical framework of the entangled phantom theory for rubber elasticity was identified. Standard lore dictates that low molecular weight precursors for the elastically active chains leads to a negligible contribution of trapped entanglements. Here we show that the contribution of trapped entanglements may equal the contribution coming from elastically active material and that it is independent of network topology.

Elasticity of Highly Entangled Polymer Networks and Gels: Review of Models and Theory of Nonaffine Deformations

Polymer Science, Series C, 2023

The main models of phantom and topologically entangled polymer networks are surveyed. A theory of anisotropic and nonaffine deformation of both swollen and deswollen (with partial solvent removal) strongly entangled polymer networks in athermal and θ-solvents has been developed. It is shown that under weak anisotropic deformations of the deswollen network, the entanglement tube consists of fractal loopy globules. In a θ-solvent, slight deformations of the network lead to a decrease in the overlap of loopy globules without changing their sizes. Deformations of swollen networks, as well as strong deformations of deswollen networks, are described in terms of the slip-tube model. An effective Hamiltonian has been derived that determines the entropy of fractal loopy globules. Based on the Hamiltonian, it is shown that topological constraints can be described using the polymer-quantum diffusion analogy. The connection between topological and quantum entanglements is demonstrated.

Dynamic mechanical response of polymer networks

2000

The dynamic-mechanical response of flexible polymer networks is studied in the framework of tube model, in the limit of small affine deformations, using the approach based on Rayleighian dissipation function. The dynamic complex modulus G* is calculated from the analysis of a network strand relaxation to the new equilibrium conformation around the distorted primitive path. Chain equilibration is achieved via a sliding motion of polymer segments along the tube, eliminating the inhomogeneity of the polymer density caused by the deformation. The characteristic relaxation time of this motion separates the low-frequency limit of the complex modulus from the high-frequency one, where the main role is played by chain entanglements, analogous to the rubber plateau in melts. The dependence of storage and loss moduli, G' and G'', on crosslink and entanglement densities gives an interpolation between polymer melts and crosslinked networks. We discuss the experimental implications of the rather short relaxation time and the slow square-root variation of the moduli and the loss factor tan at higher frequencies.

Rubber elasticity of cross-linked networks with trapped entanglements and dangling chains

Macromolecules, 1989

This is a theoretical study on the effects of the network structure and deformation on the stress response of permanent elastomeric networks containing trapped entanglements and chain defects. In these systems, elastically active chains with both ends cross-linked coexist with segments that are tethered at one end only. It is assumed that under strain, internal strands preserve their deformation with time while dangling segments restore their initial length by retraction and continue their conformational renewal through chain fluctuations. The dynamics of the tethered branches cause disengagement of untrapped entanglements on neighboring segments, thus undermining the structural solidity of the elastomer. The calculated stress under constant deformation depends on the strain value, the molecular weight between cross-links and between entanglements, the relative participation of the tethered chains in the network, and the degree of the relaxation completion. The theory accommodates the observed deviations from neo-Hookean behavior, and the predicted effects of the network architecture are consistent with the experimental evidence.

Nonaffine deformation and elasticity of polymer networks

We demonstrate that the origin of the nonlinear elasticity of polymer networks rests in their nonaffine deformations. We introduce the affine length Raff, which separates the solid-like elastic deformations on larger scales from liquid-like nonaffine deformations on smaller scales. This affine length grows with elongation λ as Raff ∼ λ 3/2 and decreases upon compression as Raff ∼ λ 1/2 . The behavior of networks on scales up to Raff is that of stretched or compressed individual chains (we call them affine strands). The affine strands are stretched in the elongation direction and confined and folded in the effective tubes in the compression direction. The fluctuations of affine strands determine the diameters of the confining tubes a, which change nonaffinely with the network deformation a ∼ λ 1/2 . Our model gives a unified picture of deformations of both phantom and entangled networks and leads to a stressstrain relation that is in excellent agreement with experiments.

Theory of Flexible Polymer Networks: Elasticity and Heterogeneities

Polymers, 2020

A review of the main elasticity models of flexible polymer networks is presented. Classical models of phantom networks suggest that the networks have a tree-like structure. The conformations of their strands are described by the model of a combined chain, which consists of the network strand and two virtual chains attached to its ends. The distribution of lengths of virtual chains in real polydisperse networks is calculated using the results of the presented replica model of polymer networks. This model describes actual networks having strongly overlapping and interconnected loops of finite sizes. The conformations of their strands are characterized by the generalized combined chain model. The model of a sliding tube is represented, which describes the general anisotropic deformations of an entangled network in the melt. I propose a generalization of this model to describe the crossover between the entangled and phantom regimes of a swollen network. The obtained dependence of the Mo...

Extensional stress growth and stress relaxation in entangled polymer solutions

Journal of Rheology, 2003

We report an evaluation of the double constraint release model with chain stretch ͑DCR-CS͒ suggested by Ianniruberto and Marrucci ͓J. Rheol. 45, 1305-1318 ͑2001͔͒, in predicting the transient stress growth and stress relaxation behavior of two well-characterized entangled polymer solutions undergoing homogeneous uniaxial extensional flow. The experiments are conducted using a filament stretching rheometer. The DCR-CS model belongs to a family of simplified single-segment models that incorporate constraint release, double reptation, and segmental stretching into the basic reptation mechanism proposed in the original Doi-Edwards theory and seeks to extend the predictive capacity of the theory to more complex flow fields. We show that the single-mode DCR-CS differential model performs well in predicting the transient extensional stress growth and steady-state extensional viscosity over a range of stretch rates. The model also predicts the observed stress relaxation following cessation of stretching satisfactorily. We further show that the model predicts shear thickening even in steady shear flow.

Transiently Trapped Entanglements in Model Polymer Networks

The relaxational dynamics of trapped entanglements in model silicone polymer networks is studied through the residual dipolar couplings (RDC) obtained by double quantum nuclear magnetic resonance (DQ NMR). These experiments were performed on model polymer networks containing linear pendant chains. The model networks where synthesized by end-linking a mixture of R,ω-divinyl poly (dimethylsiloxane) (B 2 ) and ω-vinyl poly(dimethylsiloxane) (B 1 ) with trifunctional (A 3 ) or tetra-functional (A 4 ) cross-linkers. At the time scale of the NMR experiments only a small fraction of the linear pendant chains B 1 loses the memory of its early configuration. Then, the unrelaxed topological constrains involving pendant material render a nonzero average dipolar coupling that contributes to the solid-like behavior of the NMR response. Irrespective of the functionality of the cross-linkers, upon the presence in the network of pendant chains induced by the insertion of the B 1 monofunctional poly(dimethylsiloxane) an important reduction in the RDC is observed as a consequence of the transiently trapped entanglements. It was also verified that, according to the viscoelastic response, the networks prepared with A 4 cross-linkers show systematically higher values of the residual dipolar couplings than trifunctional cross-linked networks.