Stress Relaxation Dynamics Research Articles

Molecular Model for Linear Viscoelastic Properties of Entangled Polymer Networks.

A molecular Kuhn-scale model is presented for the stress relaxation dynamics of entangled polymer networks. The governing equation of the model is given by the general form of the linearized Langevin equation. Based on the fluctuation-dissipation theorem, the stress relaxation modulus is derived using the normal mode representation. The entanglements are introduced as additional entropic springs connecting internal beads of the network strands. The validity of the model is assessed by comparing predicted stress relaxation modulus and viscoelastic storage and loss moduli with the estimates from molecular dynamics (MD) simulations, using the same computer models. A finite element procedure is proposed and used to assemble the network connectivity matrix, and its numerically solved eigenvalues are used to predict the linear stress relaxation dynamics. Both perfect (fully polymerized stoichiometric) and imperfect networks with different soluble and dangling structures and loops are studied using mapped Kuhn-scale network models with up to several dozen thousand Kuhn segments. It is shown that for the overlapping ranges of times and frequencies, the model predictions and MD estimates agree well.

Recyclable dual-curing thiol-isocyanate-epoxy vitrimers with sequential relaxation profiles

Two series of poly(thiourethane)s-poly(thiol-epoxy) hybrids were prepared from dual-curing, stoichiometric thiol:isocyanate:epoxy mixtures. 1-methylimidazole (MI) was used as initiator for the thiol-isocyanate and thiol-epoxy curing reactions. A salt of tetraphenylborate derived from highly basic amine 1,5,7 triazabicyclo [4,4,0]dec-5-ene (TBD) was used as catalyst for the activation of bond exchange reactions. The curing kinetics and the glass transition temperature (Tg) of the samples were studied by differential scanning calorimetry (DSC). Thermal stability was characterized by thermogravimetric analysis (TGA). Stress relaxation dynamics related to the bond exchange reactions were studied by dynamic mechanical analysis (DMA). Recyclability of the fully cured samples according to the DMA results was carried out by hot-pressing at elevated temperatures. The results show that the curing process takes place in a controlled and sequential way: the thiol-isocyanate reaction takes place first, at moderate temperatures, followed by the thiol-epoxy reaction, which is completed at higher temperatures. The analysis of stress relaxation evidences a complex two-step relaxation behavior depending on the contribution of isocyanate and epoxy groups to the mixture. Isocyanate-rich materials show a simple relaxation process corresponding to the trans-thiocarbamoylation exchange reactions. Epoxy-rich materials show, in contrast, a two-step relaxation processes evidencing that trans-thiocarbamoylation alone may not be sufficient to relax the stress completely, due to an apparently permanent network structure. However, complete stress relaxation is possible for epoxy-rich materials through additional bond exchange reactions such as transesterification or dynamic thiol-Michael, but at a slower rate. Depending on the composition and the temperature, full recycling of the material can be achieved at moderate temperatures in a timescale of minutes to hours.

Unexpected Slow Relaxation Dynamics in Pure Ring Polymers Arise from Intermolecular Interactions.

Ring polymers have fascinated scientists for decades, but experimental progress has been challenging due to the presence of linear chain contaminants that fundamentally alter dynamics. In this work, we report the unexpected slow stress relaxation behavior of concentrated ring polymers that arises due to ring-ring interactions and ring packing structure. Topologically pure, high molecular weight ring polymers are prepared without linear chain contaminants using cyclic poly(phthalaldehyde) (cPPA), a metastable polymer chemistry that rapidly depolymerizes from free ends at ambient temperatures. Linear viscoelastic measurements of highly concentrated cPPA show slow, non-power-law stress relaxation dynamics despite the lack of linear chain contaminants. Experiments are complemented by molecular dynamics (MD) simulations of unprecedentedly high molecular weight rings, which clearly show non-power-law stress relaxation in good agreement with experiments. MD simulations reveal substantial ring-ring interpenetrations upon increasing ring molecular weight or local backbone stiffness, despite the global collapsed nature of single ring conformation. A recently proposed microscopic theory for unconcatenated rings provides a qualitative physical mechanism associated with the emergence of strong inter-ring caging which slows down center-of-mass diffusion and long wavelength intramolecular relaxation modes originating from ring-ring interpenetrations, governed by the onset variable N/ND, where the crossover degree of polymerization ND is qualitatively predicted by theory. Our work overcomes challenges in achieving ring polymer purity and by characterizing dynamics for high molecular weight ring polymers. Overall, these results provide a new understanding of ring polymer physics.

Hydrophobically modified complex coacervates for designing aqueous pressure-sensitive adhesives.

The rheology of complex coacervates can be elegantly tuned via the design and control of specific non-covalent hydrophobic interactions between the complexed polymer chains. The well-controlled balance between elasticity and energy dissipation makes complex coacervates perfect candidates for pressure-sensitive adhesives (PSAs). In this work, the polyanion poly(3-sulfopropyl methacrylate) (PSPMA) and the polycation quaternized poly(4-vinylpyridine) (QP4VP) were used to prepare complex coacervates in water. Progressive increase of hydrophobicity is introduced to the polyanion via partial deprotection of the protected precursor. Hence, the polymer chains in the complex coacervates can interact via both electrostatic (controlled by the amount of salt) and hydrophobic (controlled by the deprotection degree) interactions. It was observed that: (i) a rheological time-salt-hydrophobicity superposition principle is applicable, and can be used as a predictive tool for rheology, (ii) the slowdown of the stress relaxation dynamics, due to the increase of hydrophobic stickers (lower deprotection degree), can be captured by the sticky-Rouse model, and (iii) the systematic variation of hydrophobic stickers, amount of salt, and molecular weight of the polymers, enables the identification of optimizing parameters to design aqueous PSA systems. The presented results offer new pathways to control the rheology of complex coacervates and their applicability as PSAs.

Constitutive models for well-entangled living polymers beyond the fast-breaking limit

In well-entangled living polymers, there is a complex relationship between reversible polymerization reactions and stress relaxation dynamics. This relationship is already well-understood in the “fast-breaking” limit, where polymers tend to break apart much faster than they can relax interior tube segments by reptation. For well-entangled living polymers that are not necessarily fast-breaking, we introduce a new suite of computationally efficient partial differential equation models for linear and nonlinear rheology. For linear rheology calculations, we retain full-chain depictions of standard stress relaxation processes (reptation, double reptation, contour length fluctuations, etc.) and replace the reaction terms with a simple “shuffling” approximation. Besides predicting bulk rheology, these shuffling models also yield new insights into the rheological contribution from chains at different sectors of the molecular weight distribution. Generalizing to nonlinear rheology models, additional approximations must be made with respect to reptation and constraint release in order to facilitate applications in computational fluid dynamics. To evaluate... a pair of constitutive models with complementary strengths and weaknesses: LRP-f (living Rolie-Poly, fitted) and STARM-E (simplified tube approximation for rapid-breaking micelles, extended). Nonlinear rheology calculations are provided for both models over a range of flow conditions in both fast-breaking and semi-slow breaking systems. In spite of their differing assumptions and approximations, we find that both models are capable of producing similar results. From this, we conclude that the predictions of the LRP-f and STARM-E models reflect their shared physical basis, and hence either model can be used with reasonable confidence for describing nonlinear rheology in systems of well-entangled living polymers across the fast/slow breaking spectrum.

Relaxation dynamics of Pd–Ni–P metallic glass: decoupling of anelastic and viscous processes

The stress relaxation dynamics of metallic glass Pd40Ni40P20 was studied in both supercooled liquid and glassy states. Time-temperature superposition was found in the metastable liquid, implying an invariant shape of the distribution of times involved in the relaxation. Once in the glass state, the distribution of relaxation times broadens as temperature and fictive temperature decrease, eventually leading to a decoupling of the relaxation in two processes. While the slow one keeps a viscous behavior, the fast one shows an anelastic nature and a time scale similar to that of the collective atomic motion measured by x-ray photon correlation spectroscopy (XPCS). These results suggest that the atomic dynamics of metallic glasses, as determined by XPCS at low temperatures in the glass state, can be related to the rearrangements of particles responsible of the macroscopically reversible anelastic behavior.

Effects of paclitaxel on the viscoelastic properties of mouse sensory nerves

Paclitaxel is an effective and widely used chemotherapeutic, but also causes debilitating peripheral sensory neuropathy. Due to its influence on microtubule stability, we and others have hypothesized that paclitaxel alters neuromechanical properties. A prior study suggested that paclitaxel increases the tensile moduli of rat sensory nerves. However, the effects of paclitaxel on tissue level viscoelasticity have not been tested. In this study, sural branches of C57BL/6J mouse sciatic nerves were bilaterally excised. One nerve was treated with Ringer’s solution containing paclitaxel, and the contralateral nerve with Ringer’s alone. Nerves were then subject to a passive loading protocol in which peak stress, relaxed stress, and stress-relaxation dynamics were monitored at increasing strain. Elastic and tangent tensile moduli were calculated from both peak and relaxed stress-strain curves as well as failure stress were significantly elevated in paclitaxel-treated nerves compared to controls. Double-exponential fits (with τm and τn indicating fast and slow time constants, respectively) were successfully applied to model stress-relaxation. Though no significant differences in the τm and τn were found between groups, paclitaxel treatment significantly increased the variability of τm, suggesting heterogeneous effects on nerve biomechanical properties. Our data establish that paclitaxel effects at the cellular level influence tensile viscoelastic properties of nerves at the tissue level. These results have implications for understanding biomechanical influences on the progression and physical rehabilitation of paclitaxel-induced neuropathy.

A full-chain tube-based constitutive model for living linear polymers

We present a new strategy for introducing population balances into full-chain constitutive models of living polymers with linear chain architectures. We provide equations to describe a range of stress relaxation processes covering both unentangled systems (Rouse-like motion) and well entangled systems (reptation, contour length fluctuations, chain retraction, and constraint release). Special attention is given to the solutions that emerge when the 'breaking time' of the chain becomes fast compared to various stress relaxation processes. In these 'fast breaking' limits, we reproduce previously known results (with some corrections) and also present new results for nonlinear stress relaxation dynamics. Our analysis culminates with a fully developed constitutive model for the fast breaking regime in which stress relaxation is dominated by contour length fluctuations. Linear and nonlinear rheology predictions of the model are presented and discussed.

Microscopic dynamics of stress relaxation in a nanocolloidal soft glass

We report x-ray photon correlation spectroscopy (XPCS) experiments with in situ rheometry performed on a soft glass composed of a concentrated suspension of charged silica nanoparticles subjected to step strains that induce yielding and flow. The XPCS measurements characterize the particle-scale and mesoscale motions within the glass that underlie the highly protracted decay of the macroscopic stress following the step strains. These dynamics are anisotropic, with slow, convective particle motion along the direction of the preceding shear that persists for surprisingly large times and that is accompanied by intermittent motion in the perpendicular (vorticity) direction. A close correspondence between the convective dynamics and stress relaxation is demonstrated by power-law scaling between the characteristic velocity of the collective particle motion and the rate of stress decay.

Particle diffusion in extracellular hydrogels.

Hyaluronic acid is an abundant polyelectrolyte in the human body that forms extracellular hydrogels in connective tissues. It is essential for regulating tissue biomechanics and cell-cell communication, yet hyaluronan overexpression is associated with pathological situations such as cancer and multiple sclerosis. Due to its enormous molecular weight (in the range of millions of Daltons), accumulation of hyaluronan hinders transport of macromolecules including nutrients and growth factors through tissues and also hampers drug delivery. However, the exact contribution of hyaluronan to tissue penetrability is poorly understood due to the complex structure and molecular composition of tissues. Here we reconstitute biomimetic hyaluronan gels and systematically investigate the effects of gel composition and crosslinking on the diffusion of microscopic tracer particles. We combine ensemble-averaged measurements via differential dynamic microscopy with single-particle tracking. We show that the particle diffusivity depends on the particle size relative to the network pore size and also on the stress relaxation dynamics of the network. We furthermore show that addition of collagen, the other major biopolymer in tissues, causes the emergence of caged particle dynamics. Our findings are useful for understanding macromolecular transport in tissues and for designing biomimetic extracellular matrix hydrogels for drug delivery and tissue regeneration.

Optical Tweezers Microrheology Maps the Dynamics of Strain-Induced Local Inhomogeneities in Entangled Polymers.

Optical tweezers microrheology (OTM) offers a powerful approach to probe the nonlinear response of complex soft matter systems, such as networks of entangled polymers, over wide-ranging spatiotemporal scales. OTM can also uniquely characterize the microstructural dynamics that lead to the intriguing nonlinear rheological properties that these systems exhibit. However, the strain in OTM measurements, applied by optically forcing a microprobe through the material, induces network inhomogeneities in and around the strain path, and the resultant flow field complicates the measured response of the system. Through a robust set of custom-designed OTM protocols, coupled with modeling and analytical calculations, we characterize the time-varying inhomogeneity fields induced by OTM measurements. We show that homogenization following strain does not interfere with the intrinsic stress relaxation dynamics of the system, rather it manifests as an independent component in the stress decay, even in highly nonlinear regimes such as with the microrheological large-amplitude-oscillatory-shear (MLAOS) protocols we introduce. Our specific results show that Rouse-like elastic retraction, rather than disentanglement and disengagement, dominates the nonlinear stress relaxation of entangled polymers at micro- and mesoscales. Thus, our study opens up possibilities of performing precision nonlinear microrheological measurements, such as MLAOS, on a wide range of complex macromolecular systems.

Fabrication of thermo- and pH-sensitive cellulose nanofibrils-reinforced hydrogel with biomass nanoparticles

Cellulose nanofibrils (CNFs) have been widely used as the reinforcing agent. However, there is still a challenge to control the mechanical properties of CNFs-reinforced hydrogels by changing the environmental variables. Here, we demonstrated a thermo- and multi-pH-responsive CNFs-reinforced hydrogel containing lignin nanoparticles as the functional dynamic cross-linkers in the polymeric network. The elastic and viscous modulus of the hydrogel can be tailored: (i) the modulus increases upon increasing pH from 4 to 7, (ii) with pH further increasing from 7 to 9, the modulus decreases. The as prepared hydrogel also responded to the environment temperature, with the gelation at 26.4 °C. The significant differences in CNFs-reinforced hydrogel mechanics at various environment conditions are a direct result of the distinct stress-relaxation dynamics dictated by the dynamic hydrogen bonding and swelling/shrinking of lignin nanoparticles. This work also provides a new approach to design stimuli-responsive hydrogels.

Tailoring relaxation dynamics and mechanical memory of crumpled materials by friction and ductility.

Crumpled sheets show slow mechanical relaxation and long lasting memory of previous mechanical states. By using uniaxial compression tests, the role of friction and ductility on the stress relaxation dynamics of crumpled systems is investigated. We find a material dependent relaxation constant that can be tuned by changing ductility and adhesive properties of the sheet. After a two-step compression protocol, nonmonotonic aging is reported for polymeric, elastomeric and metal sheets, with relaxation dynamics that are dependent on the material's properties. These findings can contribute to tailoring and programming of crumpled materials to get desirable mechanical properties.

Coarse-Grained Simulations of Three-Armed Star Polymer Melts and Comparison with Linear Chains.

Melts of three-armed star polymers have been simulated using a coarse-grained model parameterized by atomistic simulations of polyethylene. The bonds between the highly coarse-grained, and hence soft, polymer beads are explicitly prevented from crossing by the TWENTANGLEMENT algorithm. The three melts of symmetric stars, differing in the lengths of the arms, are compared against five melts of linear polymers with comparable dimensions to study the impact of branched architecture on self-diffusion and bulk rheological properties. Differently from the power-law relation between the viscosity and molecular mass of linear chains, the star polymers in our simulations follow an exponential mass-viscosity relation and show qualitative agreement with the storage and loss moduli for stars with far longer arms from experiments. The stress relaxation dynamics of the stars are also compared with theoretical analysis in terms of Rouse modes.

Correlation between stress relaxation dynamics and thermochemistry for covalent adaptive networks polymers

A simple scaling relationship between normalized relaxation time and reaction kinetics is established for CANs polymers.

Transition from stress-driven to thermally activated stress relaxation in metallic glasses

The short-range ordered but long-range disordered structure of metallic glasses yields strong structural and dynamic heterogeneities. Stress relaxation is a technique to trace the evolution of stress in response to a fixed strain, which reflects the dynamic features phenomenologically described by the Kohlrausch-Williams-Watts (KWW) equation. The KWW equation describes a broad distribution of relaxation times with a small number of empirical parameters, but it does not arise from a particular physically motivated mechanistic picture. Here we report an anomalous two-stage stress relaxation behavior in a Cu46Zr46Al8 metallic glass over a wide temperature range and generalize the findings in other compositions. Thermodynamic analysis identifies two categories of processes: a fast stress-driven event with large activation volume and a slow thermally activated event with small activation volume, which synthetically dominates the stress relaxation dynamics. Discrete analyses rationalize the transition mechanism induced by stress and explain the anomalous variation of the KWW characteristic time with temperature. Atomistic simulations reveal that the stress-driven event involves virtually instantaneous short-range atomic rearrangement, while the thermally activated event is the percolation of the fast event accommodated by the long-range atomic diffusion. The insights may clarify the underlying physical mechanisms behind the phenomenological description and shed light on correlating the hierarchical dynamics and structural heterogeneity of amorphous solids.

Finite cohesion due to chain entanglement in polymer melts.

Three different types of experiments, quiescent stress relaxation, delayed rate-switching during stress relaxation, and elastic recovery after step strain, are carried out in this work to elucidate the existence of a finite cohesion barrier against free chain retraction in entangled polymers. Our experiments show that there is little hastened stress relaxation from step-wise shear up to γ = 0.7 and step-wise extension up to the stretching ratio λ = 1.5 at any time before or after the Rouse time. In contrast, a noticeable stress drop stemming from the built-in barrier-free chain retraction is predicted using the GLaMM model. In other words, the experiment reveals a threshold magnitude of step-wise deformation below which the stress relaxation follows identical dynamics whereas the GLaMM or Doi-Edwards model indicates a monotonic acceleration of the stress relaxation dynamics as a function of the magnitude of the step-wise deformation. Furthermore, a sudden application of startup extension during different stages of stress relaxation after a step-wise extension, i.e. the delayed rate-switching experiment, shows that the geometric condensation of entanglement strands in the cross-sectional area survives beyond the reptation time τd that is over 100 times the Rouse time τR. Our results point to the existence of a cohesion barrier that can prevent free chain retraction upon moderate deformation in well-entangled polymer melts.

Frequency analysis of stress relaxation dynamics in model asphalts.

Asphalt is an amorphous or semi-crystalline material whose mechanical performance relies on viscoelastic responses to applied strain or stress. Chemical composition and its effect on the viscoelastic properties of model asphalts have been investigated here by computing complex modulus from molecular dynamics simulation results for two different model asphalts whose compositions each resemble the Strategic Highway Research Program AAA-1 asphalt in different ways. For a model system that contains smaller molecules, simulation results for storage and loss modulus at 443 K reach both the low and high frequency scaling limits of the Maxwell model. Results for a model system composed of larger molecules (molecular weights 300-900 g/mol) with longer branches show a quantitatively higher complex modulus that decreases significantly as temperature increases over 400-533 K. Simulation results for its loss modulus approach the low frequency scaling limit of the Maxwell model at only the highest temperature simulated. A Black plot or van Gurp-Palman plot of complex modulus vs. phase angle for the system of larger molecules suggests some overlap among results at different temperatures for less high frequencies, with an interdependence consistent with the empirical Christensen-Anderson-Marasteanu model. Both model asphalts are thermorheologically complex at very high frequencies, where they show a loss peak that appears to be independent of temperature and density.

Understanding Constraint Release in Star/Linear Polymer Blends

In this paper, we exploit the stochastic slip-spring model to quantitatively predict the stress relaxation dynamics of star/linear blends with well-separated longest relaxation times and we analyze the results to assess the validity limits of the two main models describing the corresponding relaxation mechanisms within the framework of the tube picture (Doi’s tube dilation and Viovy’s constraint release by Rouse motions of the tube). Our main objective is to understand and model the stress relaxation function of the star component in the blend. To this end, we divide its relaxation function into three zones, each of them corresponding to a different dominating relaxation mechanism. After the initial fast Rouse motions, relaxation of the star is dominated at intermediate times by the “skinny” tube (made by all topological constraints) followed by exploration of the “fat” tube (made by long-lived obstacles only). At longer times, the tube dilation picture provides the right shape for the relaxation of the stars. However, the effect of short linear chains results in time-shift factors that have never been described before. On the basis of the analysis of the different friction coefficients involved in the relaxation of the star chains, we propose an equation predicting these time-shift factors. This allows us to develop an analytical equation combining all relaxation zones, which is verified by comparison with simulation results.

Slow dynamics of stress and strain relaxation in randomly crumpled elasto-plastic sheets

Stress and strain relaxation in randomly folded paper sheets under axial compression is studied both experimentally and theoretically. Equations providing the best fit to the experimental data are found. Our findings suggest that, in an axially compressed ball folded from an elastic or elasto-plastic material, the relaxation dynamics is ruled by activated processes of an energy foci rearrangement in the crumpling network. The dynamics of relaxation is discussed within a framework of Edwards's statistical mechanics. The functional forms of the activation barrier between admissible jammed folding configurations of the crumpling network under axial compression are derived. It is shown that relaxation kinetics can be mapped to activated dynamics of depinning and creep of elastic interface in a disordered medium.