Tube relaxation: A quantitative molecular model for the viscoelastic plateau of entangled polymeric media

The relaxation of an entangled polymeric medium in the viscoelastic plateau is investigated theoretically by using the slip-link representation of topological constraints. In addition to the chain retraction process introduced by Daoudi and investigated theoretically by Doi, we show that two processes contribute significantly to the relaxation: The first, “equilibration across slip-links,” is a longitudinal reequilibration between parts of the chain which have been differently extended or compressed, depending on their initial orientation relatively to the strain tensor. The second, “tube relaxation,” is a mean-field representation of the loss of topological constraints on one chain due to the retraction of the others. Closed analytical expressions for the stress accounting for these three processes are derived and compared with previous theories: the relaxation should be much more progressive than previously predicted, and the terminal time for retraction is reduced significantly by tube relaxation.     

Recent developments in polymer dynamics investigations of architecturally complex systems

The tube model for linear and branched architectures is nowadays able to predict in high precision the linear viscoelastic relaxation time spectrum. For linear chains, the involved time scales fit to the commonly accessible dynamic scattering techniques. This makes it possible to microscopically investigate the correlation between structures and relaxation processes. In branched systems, however, the hierarchical nature of relaxations limits direct investigation via these microscopic methods as the dynamic processes are prolongated to much longer relaxation times that are no more accessible to usual dynamic scattering methods. A way to overcome this difficulty is offered by the use of static small angle neutron scattering. Here, the combination of annealing and quenching steps after a step deformation provides unique information of the structure at particular times along the relaxation spectrum. This, however, necessitates the availability of architecturally clean and specifically deuterium labelled model polymers due to the sensitivity of the scattering method. Therefore, we outline in this contribution first the current status on the synthesis and analysis of such compounds with relation to neutron scattering. Secondly, we present exemplary neutron scattering results from in situ stress relaxation studies inside the neutron beam on linear and H-shaped branched polymers which were molecularly designed to highlight specific relaxation processes. We discuss the relevance of the tube model parameters in linear and non-linear studies.     

Hydration behavior of heart muscle studies by nuclear magnetic relaxation: Deuterium exchange of water protons in the myocardium

Proton transverse relaxation, deuterium and oxygen-17 NMR measurements on functional animal heart muscle were employed to study the distribution and exchange of water protons in the heart. Our nonlinear regression analysis of such data showed the presence of three proton transverse relaxation components that are likely to correspond, respectively, to two major types of water compartments in the heart muscle and the heart muscle matrix. A deuterium exchange study was undertaken to obtain additional information concerning the chemical exchange of water protons/deuterium within these two water compartments, and the effects of proton intermolecular dipolar interactions on the transverse relaxation of water protons. Our results are likely to influence the analysis and interpretation of MRI data for myocardium since it provides details of the microscopic water distribution in the myocardium which is important to the heart function.     

Theory of nonlinear elasticity, stress-induced relaxation, and dynamic yielding in dense fluids of hard nonspherical colloids

We generalize the microscopic na?ve mode coupling and nonlinear Langevin equation theories of the coupled translation-rotation dynamics of dense suspensions of uniaxial colloids to treat the effect of applied stress on shear elasticity, cooperative cage escape, structural relaxation, and dynamic and static yielding. The key concept is a stress-dependent dynamic free energy surface that quantifies the center-of-mass force and torque on a moving colloid. The consequences of variable particle aspect ratio and volume fraction, and the role of plastic versus double glasses, are established in the context of dense, glass-forming suspensions of hard-core dicolloids. For low aspect ratios, the theory provides a microscopic basis for the recently observed phenomenon of double yielding as a consequence of stress-driven sequential unlocking of caging constraints via reduction of the distinct entropic barriers associated with the rotational and translational degrees of freedom. The existence, and breadth in volume fraction, of the double yielding phenomena is predicted to generally depend on both the degree of particle anisotropy and experimental probing frequency, and as a consequence typically occurs only over a window of (high) volume fractions where there is strong decoupling of rotational and translational activated relaxation. At high enough concentrations, a return to single yielding is predicted. For large aspect ratio dicolloids, rotation and translation are always strongly coupled in the activated barrier hopping event, and hence for all stresses only a single yielding process is predicted.     

Hierarchical Modeling of Entangled Polymers

Summary : We demonstrate that it is possible to link multi-chain molecular dynamics simulations with the tube model using a single chain slip-links model as a bridge. This hierarchical approach allows significant speed up of simulations, permitting us to span the time scales relevant for a comparison with the tube theory. Fitting the mean-square displacement of individual monomers in molecular dynamics simulations with the slip-spring model, we show that it is possible to predict the stress relaxation. Then, we analyze the stress relaxation from slip-spring simulations in the framework of the tube theory. In the absence of constraint release, we establish that the relaxation modulus can be decomposed as the sum of contributions from fast and longitudinal Rouse modes, and tube survival. Finally, we discuss some open questions regarding possible future directions that could be profitable in rendering the tube model quantitative, even for mildly entangled polymers.     

Significance of cross correlations in the stress relaxation of polymer melts

According to linear response theory, all relaxation functions in the linear regime can be obtained using time correlation functions calculated under equilibrium. In this paper, we demonstrate that the cross correlations make a significant contribution to the partial stress relaxation functions in polymer melts. We present two illustrations in the context of polymer rheology using (1) Brownian dynamics simulations of a single chain model for entangled polymers, the slip-spring model, and (2) molecular dynamics simulations of a multichain model. Using the single chain model, we analyze the contribution of the confining potential to the stress relaxation and the plateau modulus. Although the idea is illustrated with a particular model, it applies to any single chain model that uses a potential to confine the motion of the chains. This leads us to question some of the assumptions behind the tube theory, especially the meaning of the entanglement molecular weight obtained from the plateau modulus. To shed some light on this issue, we study the contribution of the nonbonded excluded-volume interactions to the stress relaxation using the multichain model. The proportionality of the bonded/nonbonded contributions to the total stress relaxation (after a density dependent "colloidal" relaxation time) provides some insight into the success of the tube theory in spite of using questionable assumptions. The proportionality indicates that the shape of the relaxation spectrum can indeed be reproduced using the tube theory and the problem is reduced to that of finding the correct prefactor.     

Nonlinear stress relaxation of an entangled linear polystyrene in single step‐strain flow: A quantitative theoretical investigation

The nonlinear stress relaxation of a nearly monodisperse, moderately entangled polystyrene solution (i.e., roughly seven entanglements per chain at equilibrium) in single step‐strain flow is investigated quantitatively by a detailed comparison of an existing set of experimental data with a simulation based on the tube model. The proposed simulation enables the effects of primary nonlinear relaxation mechanisms other than chain retraction to be identified more clearly and investigated individually. Two peculiar nonlinear relaxation behaviors are observed in this experiment. One is concerned with an apparent enhancement in the stress relaxation at short times, and the other is responsible for a seeming slowdown of the stress relaxation at long times. These findings are discussed within the tube model, in view of the effects of convective constraint release, partial strand extension, and nonaffine deformation. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1281–1293, 2003     

Prediction of viscoelastic properties with coarse‐grained molecular dynamics and experimental validation for a benchmark polyurea system

To explore the relationship between microscopic structure and viscoelastic properties of polyurea, a coarse‐grained (CG) model is developed by a structure matching method and validated against experiments conducted on a controlled, benchmark material. Using the Green‐Kubo method, the relaxation function is computed from the autocorrelation of the stress tensor, sampled over equilibrium MD simulations, and mapped to a real time scale established by matching self‐diffusion rates of atomistic and CG models. Master curves computed from the predicted stress relaxation function are then compared with dynamic mechanical analysis experiments mapped to a wide frequency range by time–temperature superposition, as well as measurements of ultrasonic shear wave propagation. Computational simulations from monodisperse and polydisperse configurations, representative of the benchmark polyurea, show excellent agreement with the experimental measurements over a multidecade range of loading frequency. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2016 , 54, 797–810     

Local Transient Jamming in Stress Relaxation of Bulk Amorphous Polymers

Bulk amorphous polymers become stretched and parallel-aligned under loading stress, and their intermolecular cooperation slows down the subsequent stress relaxation process. By means of dynamic Monte Carlo simulations, we employed the linear viscoelastic Maxwell model for stress relaxation of single polymers and investigated their intermolecular cooperation in the stress relaxation process of stretched and parallel-aligned bulk amorphous polymers. We carried out thermal fluctuation analysis on t...     

Theory of aging in structural glasses

The random first-order transition theory of the dynamics of supercooled liquids is extended to treat aging phenomena in nonequilibrium structural glasses. A reformulation of the idea of "entropic droplets" in terms of libraries of local energy landscapes is introduced which treats in a uniform way the supercooled liquid (reproducing earlier results) and glassy regimes. The resulting microscopic theory of aging makes contact with the Nayaranaswamy-Moynihan-Tool nonlinear relaxation formalism and the Hodge-Scherer extrapolation of the Adam-Gibbs formula, but deviations from both approaches are predicted and shown to be consistent with experiment. The nonlinearity of glassy relaxation is shown to quantitatively correlate with liquid fragility. The residual non-Arrhenius temperature dependence of relaxation observed in quenched glasses is explained. The broadening of relaxation spectra in the nonequilibrium glass with decreasing temperature is quantitatively predicted. The theory leads to the prediction of spatially fluctuating fictive temperatures in the long-aged glassy state, which have non-Gaussian statistics. This can give rise to "ultraslow" relaxations in systems after deep quenches.     

Nonlinear Viscoelasticity Raised at Low Temperatures by Intermolecular Cooperation of Bulk Amorphous Polymers

We performed dynamic Monte Carlo simulations of stress relaxation in parallel-aligned and uniaxially stretched bulk amorphous polymers at low temperatures.We observed an extra-slowing down in the early stage of stress relaxation,which causes nonlinear viscoelasticity as deviated from Debye relaxation and Arrhenius-fluid behaviors observed previously at high temperatures.Meanwhile,fluctuation analysis of stress relaxation revealed a substantial increase in the stretch fractions of polymers at the transient periods of high-temperature Debye relaxation.Structural analysis of free volume further revealed the scenario that,at low temperatures,the modulus of polymer entropy elasticity decreases with temperature and eventually loses its competition to the imposed modulus (Deborah number becomes larger than one),and hence upon stress relaxation under constant strains,monomers are firstly accumulated nearby two stretching ends of polymers,resulting in tentative global jamming like physical cross-linking there,and thus retarding the coming transient state of stress relaxation.We concluded that intermolecular cooperation raises physical crosslinking for nonlinear viscoelasticity of polymer stress relaxation as well as the rubbery states unique to bulk amorphous polymers.The new microscopic mechanism of the fluid-rubbery transition of polymers may bring insights into the intermolecular cooperation mechanism of glass transition of small molecules,if the fluid-rubbery transition is regarded as an extrapolation of glass transition from low to high molecular weights.     

Glassy dynamics and mechanical response in dense fluids of soft repulsive spheres. II. Shear modulus, relaxation-elasticity connections, and rheology

We apply the quiescent and mechanically driven versions of nonlinear Langevin equation theory to study how particle softness influences the shear modulus, the connection between shear elasticity and activated relaxation, and nonlinear rheology of the repulsive Hertzian contact model of dense soft sphere fluids. Below the soft jamming threshold, the shear modulus follows a power law dependence on volume fraction over a narrow interval with an apparent exponent that grows with particle stiffness. To a first approximation, the elastic modulus and transient localization length are controlled by a single coupling constant determined by local fluid structure. In contrast to the behavior of hard spheres, an approximately linear relation between the shear modulus and activation barrier is predicted. This connection has recently been observed for microgel suspensions and provides a microscopic realization of the elastic shoving model. Yielding, shear and stress thinning of the alpha relaxation time and viscosity, and flow curves are also studied. Yield strains are relatively weakly dependent on volume fraction and particle stiffness. Shear thinning commences at values of the effective Peclet number far less than unity, a signature of stress-assisted activated relaxation when barriers are high. Apparent power law reduction of the viscosity with shear rate is predicted with a thinning exponent less than unity. In the vicinity of the soft jamming threshold, a power law flow curve occurs over an intermediate reduced shear rate range with an apparent exponent that decreases as fluid volume fraction and/or repulsion strength increase.     

Structural relaxation behavior of strain hardened shape memory polymer fibers for self‐healing applications

In this study, shape‐memory polyurethane (SMPU) fibers were strain hardened by cold‐drawing programming (CDP) process. The programmed fibers are experimentally studied on the physical and thermomechanical properties. Structural relaxation, which determines shape memory capability of the SMP fibers, is quantified by conformational entropy change. Based on the entanglement tube theory and reptation theory, the entropic force is derived as a “bridge” to link the stress relaxation and structural relaxation, and thus structural relaxation can be evaluated by stress relaxation. It was found that the CDP SMPU fibers would still have good crackclosing capability after 13 years of hibernation in polymer matrix composite. © 2013 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2013 , 51, 966–977     

The influence of transverse differential swelling stresses on the kinetics of sorption of penetrants by polymer membranes

The influence of transverse differential swelling stresses on the kinetics of sorption of a penetrant in a polymer membrane exhibiting linear viscoelasticity is described by a model developed from the much simpler one of Crank. Sorption and transverse swelling kinetic curves are computed numerically. The character of absorption and desorption curves is examined systematically mainly as a function of (i) the magnitude of the stresses set up and of the stress-dependence of the diffusion coefficient, (ii) the relative rates of stress relaxation and of diffusion, and (iii) the degree of plasticization or “softening” of the polymer by the penetrant. It is shown that important general features of experimental sorption kinetic curves can be reproduced satisfactorily under well defined conditions. Attention is also given to transverse swelling kinetic curves. Their correlation with the corresponding sorption curves is examined briefly but systematically and discussed with reference to experimental data.     

Combined use of NMR relaxation measurements and hydrodynamic calculations to study protein association. Evidence for tetramers of low molecular weight protein tyrosine phosphatase in solution

We describe a novel method for determining weak association constants of oligomeric protein complexes formed transiently under equilibrium conditions. This type of equilibrium process is recognized as being biologically important, but generally hard to study. Heteronuclear spin relaxation rates measured at multiple protein concentrations are analyzed using relaxation rates predicted from hydrodynamic calculations, yielding equilibrium constants and structural characterization of the protein complexes. The method was used to study the oligomerization equilibrium of bovine low molecular weight protein tyrosine phosphatase. X-ray structures of monomeric and dimeric forms of the protein have been reported previously. Using longitudinal and transverse (15)N relaxation rates measured at four different protein concentrations, we detected the monomer, dimer, and a previously unknown tetramer and measured the dissociation constants of the equilibria involving these species. A comparison of experimental and predicted relaxation rates for individual backbone amide (15)N spins enabled delineation of the tetramerization interface. The results suggest a novel concept for substrate modulation of enzymatic activity based on a "supramolecular proenzyme". The fast and reversible switching of the "supramolecular proenzyme" would have obvious advantages for the regulation of enzymes involved in cell signaling pathways.     

Mass and momentum transfer in polymers. III. Effects of liquid penetrants on tensile stress relaxation of amorphous polymers

Sorption, diffusion, swelling, and tensile stress relaxation measurements were made at room temperature (23°C) for the systems poly(n-butyl methacrylate) (PBMA) with liquid methanol and ethanol, and poly(methyl acrylate) (PMA) with liquid water. Stress relaxation curves for the fully swollen polymers could be superimposed approximately with those for the dry polymers by appropriate shifting along the long axes. For PMA–water the measured curve for stress relaxation with concurrent sorption could be predicted accurately by using a moving boundary theory with data measurements of stress relaxation of the unswollen and swollen polymer combined with sorption data. The modified moving boundary theory is generalized to include the effects of dimension changes through swelling and the larger effects of plasticization associated with sorption of liquids. This improved theory accurately predicts measured curves of stress relaxation with concurrent sorption for the PBMA–alcohol systems from individual stress relaxation, sorption, diffusion and swelling data. The general approach should be applicable to other amorphous polymer–liquid swelling agent systems. The anisotropic nature of swelling of polymer films and its effect on calculated diffusion coefficients are discussed briefly.     

Macroscopic–microscopic mechanical relaxation behavior of hybrid organic–inorganic materials

The mechanical properties of transparent hybrid organic–inorganic nanocomposites made from siloxane and zirconium oxopolymers are investigated at two different length scales. The complex interface that associates the two phases is made of covalent Zr O Si bonds and hydrogen bonding. The rubbery properties studied by creep and recovery present specific behaviors in comparison with model elastomers. This is a result of the complex crosslinking units. The stress relaxation phenomenon has been studied at the molecular scale by ²H quadrupolar NMR. During stress relaxation, the anisotropy of the molecular motion decreases slowly. This study demonstrates the straightforward relationship existing between the macroscopic and microscopic relaxation phenomena. © 2001 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 39: 645–650, 2001     

Constrained Rouse model of rubber viscoelasticity

In this work we use a new approach to investigate the equilibrium and linear dynamic-mechanical response of a polymer network. The classical Rouse model is extended to incorporate quenched constraints on its end-boundary conditions; a microscopic stress tensor for the network system is then derived in the affine deformation limit. To test the model we calculate the macroscopic stress in equilibrium, corresponding to the long-time limit of relaxation. Particular attention is paid to the treatment of compressibility and hydrostatic pressure in a sample with open boundaries. Although quite different in general, for small strains the model compares well with the classic equilibrium rubber-elasticity models. The dynamic shear modulus is obtained for a network relaxing after an instantaneous step strain by keeping track of relaxation of consecutive Rouse modes of constrained network strands. The results naturally cover the whole time range--from the dynamic glassy state down to the equilibrium incompressible rubber plateau.