Broad Spectrum Of Relaxation Times Research Articles

Nanosecond solvation dynamics in a polymer electrolyte for lithium batteries.

Solvation dynamics critically affect charge transport. Spectroscopic experiments and computer simulations show that these dynamics in aqueous systems occur on a picosecond timescale. In the case of organic electrolytes, however, conflicting values ranging from 1 to several 100 picoseconds have been reported. We resolve this conflict by studying mixtures of an organic polymer and a lithium salt. Lithium ions coordinate with multiple polymer chains, resulting in temporary crosslinks. Relaxation of these crosslinks, detected by quasielastic neutron scattering, are directly related to solvation dynamics. Simulations reveal a broad spectrum of relaxation times. The average timescale for solvation dynamics in both experiment and simulation is one nanosecond. We present the direct measurement of ultraslow dynamics of solvation shell break-up in an electrolyte.

Requisites on material viscoelasticity for exceptional points in passive dynamical systems

Recent progress in non-Hermitian physics and the notion of exceptional point (EP) degeneracies in elastodynamics have led to the development of novel metamaterials for the control of elastic wave propagation, hypersensitive sensors, and actuators. The emergence of EPs in a parity-time symmetric system relies on judiciously engineered balanced gain and loss mechanisms. Creating gain requires complex circuits and amplification mechanisms, making engineering applications challenging. Here, we report strategies to achieve EPs in passive non-Hermitian elastodynamic systems with differential loss derived from viscoelastic materials. We compare different viscoelastic material models and show that the EP emerges only when the frequency-dependent loss-tangent of the viscoelastic material remains nearly constant in the frequency range of operation. This type of loss tangent occurs in materials that undergo stress-relaxation over a broad spectrum of relaxation times, for example, materials that follow the Kelvin–Voigt fractional derivative (KVFD) model. Using dynamic mechanical analysis, we show that a few common viscoelastic elastomers, such as polydimethylsiloxane and polyurethane rubber, follow the KVFD behavior such that the loss tangent becomes almost constant after a particular frequency. The material models we present and the demonstration of the potential of a widely available material system in creating EPs pave the way for developing non-Hermitian metamaterials with hypersensitivity to perturbations or enhanced emissivity.

Towards understanding the role of viscoelasticity in microstructural buckling in soft particulate composites

This work investigates the interplay between viscoelasticity and instabilities in soft particulate composites undergoing finite deformation. The composite is subjected to in-plane deformation at various constant strain rates, and experiences microstructural buckling upon exceeding the critical strain level. We characterize the dependence of the critical strain and wavelength on the applied strain rate through our numerical analysis.In the simulations, we employ the single and multiple-branch visco-hyperelastic models. We find that the critical strain and wavelength – characterized by the single-branch model – show a non-monotonic dependence on the strain rate, reaching a maximum at a specific strain rate. Remarkably, different buckling patterns (with different critical wavelengths) can be activated by changing strain rates. The space of admissible buckling modes widens in composites with higher instantaneous shear modulus. In the composites characterized by the multiple-branch model, the critical strain function exhibit multiple local maxima following a superposition of the single-branch responses. Typically, the branch with a larger relaxation time has a more significant effect on the critical strain. Moreover, the local maximum (of the critical strain function) is amplified by increasing the strain–energy factor of the corresponding branch term.Finally, we perform the experiments on the 3D-printed particulate soft composite characterized by a broad spectrum of relaxation times. The comparison of the experimental and simulation results demonstrates the ability of the numerical model to predict the critical buckling characteristics.

An efficient mode-of-deformation dependent rate-type constitutive relation for multi-modal cyclic loading of elastomers

We develop a 3D nonlinear viscoelastic model for filled elastomeric solids that exhibit good predictive capabilities across multiple deformation modes and strain rates using at most 11 parameters. Through the analysis-driven construction of the rate of dissipation within the rate-type thermodynamic framework of Rajagopal and Srinivasa (2000), we reduce the number of parameters and also introduce the mode-of-deformation-rate dependent viscosity (ηm(K3)) into the constitutive relations. The special form of ηm(K3) accounts for higher values of viscosity in tension as compared to that of other modes of deformation. The broad spectrum of relaxation times exhibited by the elastomers are characterized by categorizing it into short, medium, and long relaxations, each assumed to be associated with one of the three natural configurations. The strong mode-dependent response exhibited by HNBR50, where the compression–relaxation is faster than tension–relaxation, is predicted accurately only when all the natural configurations are active. In contrast, the response of NR is predicted by using just two natural configurations because the polymer molecules are restricted to two extremes of the relaxation spectrum as a consequence of the high affinity between carbon black and the polymer molecules. The entire model is implemented in Abaqus/Standard through the user subroutine UMAT that interacts with an external solver, DDASPK, which solves for the internal variables. We show that the analytical form for the consistent Jacobian can be derived, and establish the efficacy of the implementation by simulating non-homogeneous shear on a hockey puck geometry made of HNBR50 with a concave lateral surface. The simulation shows good agreement with experimental data.

Revealing the role of hydrogen bonding in polyurea with multiscale simulations

ABSTRACT Typically, polyurea is a polymer forming microphase-separated structures. Various coarse-grained (CG) simulation models have been developed to study the microphase structure and viscoelastic properties of polyurea. However, since the hydrogen atoms were modelled implicitly in the existing CG models, the anisotropic hydrogen bonding between urea groups in the hard domain might not be treated properly. Here, we introduce a hybrid all-atom/coarse-grained (AA/CG) model for polyurea, the CGU model, in which atomistic resolution are preserved for the urea groups. This model is proven to reproduce the results of the AA model, despite 24 times faster to run. By comparing the traditional CG model and the CGU model, we found that hydrogen bonding leads to a head-to-tail arrangement of the urea groups and a significant slowdown of the hard segment dynamics. Non-equilibrium simulations of stress relaxation process indicate that hydrogen bonding causes a broad relaxation time spectrum, which is consistent with experiment results.

Extremely slow settling behavior of particles in dilute wormlike micellar fluid with broad spectrum of relaxation times

We selected a 1.4 wt% Octadecyl trimethyl ammonium chloride (OTAC) concentration wormlike micellar fluid with broad relaxation time spectrum and another single relaxation Maxwell fluid of 4.0 wt% OTAC concentration with a constant weight ratio of OTAC to NaSal of 5:1. The shear viscosity, dynamic modulus as well as the extensional viscosity were tested. The steady and unsteady motion of spheres were investigated by high-speed video images in these two wormlike micellar fluids. Extremely slow settling velocity is found when smaller spheres with diameters of 3.2 and 4.0 mm settles in the 1.4 wt% OTAC solution. Stronger rebound of these spheres occurs in this fluid. Comparing with particle settling velocity in inelastic power law fluid, we concluded that this extremely slow sedimentation in the 1.4 wt% OTAC solution seems to be dominated by the slowest stress relaxation of this solution. The rheology test demonstrated that this wormlike micellar fluid has more tensile hardening, low viscosity and strong elasticity. The oscillation of sphere results from the elasticity of the entangled microstructures of wormlike micelles. For a Maxwell viscoelastic 4.0 wt% OTAC solution, settling behaviors of spheres agree with previous literatures that the flow becomes instability as the Wi number exceeds a critical value.

Sphere sedimentation in wormlike micelles: Effect of micellar relaxation spectrum and gradients in micellar extensions

We report experiments on the flow of wormlike micellar solutions based on cetylpyridinium chloride/sodium salicylate past a falling sphere via a combination of particle tracking velocimetry, particle image velocimetry, rheology, and flow induced birefringence (FIB). Recent studies have shown that beyond a critical extensional Deborah number, a falling sphere in wormlike micelles never reaches a constant terminal velocity; instead, it settles with an unsteady velocity. This behavior is linked to the wormlike micellar chain scission in the wake of the sphere. Similar instabilities in viscoelastic polymer solutions, where polymer chain scission is highly unlikely, are thought to be the results of a single-mode relaxation spectrum of the polymer chains or the asymmetry in the polymer chain extensions on the flanks of the falling sphere. In this paper, we examine the effect of micellar relaxation spectrum and gradients in micellar extensions on sphere instability in wormlike micelles over a wide range of flow parameters (10−2 < DeE < 40.7 and 10−6 < Re < 10). In wormlike micelles with a single-mode relaxation spectrum, the sphere instability occurs for DeE ≥ 2.6. However, for similar conditions (2.5 ≤ DeE ≤ 40.7 and 10−2 < Re < 10), spheres never exhibit unsteady motion in the wormlike micelles with a broad spectrum of relaxation times. This indicates the importance of the micellar relaxation spectrum on dynamics of sphere sedimentation in wormlike micelles. We show that a criterion based on the ratio of dissipated energy to the stored elastic energy of micelles can successfully describe the effect of micellar relaxation spectrum on sphere sedimentation dynamics. In addition, for conditions that give rise to sphere instability, FIB indicates that micellar extensions on sphere sideways are perfectly symmetric.

Thermal behavior, structural relaxation and magnetic study of a new Hf-microalloyed Co-based glassy alloy with high thermal stability

In the present work, the influence of a minor Hf addition on the atomic structure, crystallization behavior, thermal stability and magnetic properties of a Co-based metallic glass was studied. Thermal analysis indicates that the thermal stability of new glassy ribbons microalloyed with 2.5 at.% Hf is notably enhanced through increasing the incubation time prior to devitrification and enlarging the width of the supercooled liquid region from 72 K to 96 K. Magnetic studies reveal that the new glass exhibits an excellent soft magnetic behavior, i.e., a very low coercivity of 0.26 A/m in the relaxed state, and a comparable saturation magnetization as the Hf-free ribbon. Structural relaxation of the Hf-containing alloy upon isothermal annealing below the glass transition temperature, Tg, was investigated by differential scanning calorimetry (DSC) and high-energy synchrotron X-ray diffraction (XRD). The evolution of the recovered enthalpy with annealing time can be expressed by the Kohlrausch-Williams-Watts (KWW) exponential function with a Kohlrausch exponent of 0.88, indicating a broad spectrum of relaxation times. Analysis of the reduced pair correlation functions, G(r), reveals volume shrinkage upon annealing caused by elimination of liquid-like sites including free-volume and anti-free-volume according to the shift in the positions of the G(r) maxima in the medium-range scale. The influence of structural relaxation on the variation of the Curie temperature and the coercivity of the new Hf-microalloyed glassy ribbon is discussed. A faster evolution of the Curie temperature with annealing time compared to the coercivity indicates a preferential dependence of the former on the chemical short-range order (SRO).

Dynamic cross-correlations between entangled biofilaments as they diffuse

Entanglement in polymer and biological physics involves a state in which linear interthreaded macromolecules in isotropic liquids diffuse in a spatially anisotropic manner beyond a characteristic mesoscopic time and length scale (tube diameter). The physical reason is that linear macromolecules become transiently localized in directions transverse to their backbone but diffuse with relative ease parallel to it. Within the resulting broad spectrum of relaxation times there is an extended period before the longest relaxation time when filaments occupy a time-averaged cylindrical space of near-constant density. Here we show its implication with experiments based on fluorescence tracking of dilutely labeled macromolecules. The entangled pairs of aqueous F-actin biofilaments diffuse with separation-dependent dynamic cross-correlations that exceed those expected from continuum hydrodynamics up to strikingly large spatial distances of ≈15 µm, which is more than 104 times the size of the solvent water molecules in which they are dissolved, and is more than 50 times the dynamic tube diameter, but is almost equal to the filament length. Modeling this entangled system as a collection of rigid rods, we present a statistical mechanical theory that predicts these long-range dynamic correlations as an emergent consequence of an effective long-range interpolymer repulsion due to the de Gennes correlation hole, which is a combined consequence of chain connectivity and uncrossability. The key physical assumption needed to make theory and experiment agree is that solutions of entangled biofilaments localized in tubes that are effectively dynamically incompressible over the relevant intermediate time and length scales.

βrelaxation and low-temperature aging in a Au-based bulk metallic glass: From elastic properties to atomic-scale structure

The slow $\ensuremath{\beta}$ relaxation is understood to be a universal feature of glassy dynamics. Its presence in bulk metallic glasses (BMGs) is evidence of a broad relaxation time spectrum that extends to deep within the glassy state. Despite the breadth of research devoted to this phenomenon, its microscopic origin is still not fully understood. The low-temperature aging behavior and atomic structural rearrangements of a ${\mathrm{Au}}_{49}$${\mathrm{Cu}}_{26.9}$${\mathrm{Si}}_{16.3}$${\mathrm{Ag}}_{5.5}$${\mathrm{Pd}}_{2.3}$ BMG are investigated in the regime of the slow $\ensuremath{\beta}$ relaxation by employing an ensemble of experimental techniques such as high-intensity synchrotron x-ray scattering, modulated differential scanning calorimetry (MDSC), dynamic mechanical analysis (DMA), impulse excitation, and dilatometry. Evidence of a distinct slow $\ensuremath{\beta}$-relaxation regime is seen in the form of (1) an excess wing of the DMA loss modulus beginning at $\ensuremath{\sim}$50 ${}^{\ensuremath{\circ}}$C, (2) a crossover effect of elastic modulus with isothermal aging at 50${\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{\circ}}$C, and (3) a broad, nonreversing and largely irreversible sub-${T}_{g}$ endotherm in the MDSC results. Atomic rearrangements occurring at the onset of the measured slow $\ensuremath{\beta}$-relaxation temperature regime were found to be confined mainly to the short-range order length scale while no significant atomic rearrangements occur on the length scale of the medium-range order. Furthermore, evidence is presented that suggests the crossover effect in Young's modulus is due to the evolution of chemical short-range order. These results support the emergent picture of a dynamically heterogeneous glassy structure, in which low-temperature relaxation occurs through atomic rearrangements confined mostly to the short-range order length scale.

Protein dynamics to optimize and control bacterial photosynthesis

Proteins function by sampling conformational sub-states within a given fold. How this configurational flexibility and the associated protein dynamics affect the rates of chemical reactions are open questions. The difficulty in exploring this issue arises in part from the need to identify the relevant nuclear modes affecting the reaction rate for each characteristic time-scale of the reaction. Proteins as reaction media display a hierarchy of such nuclear modes, of increasingly collective character, that produce both a broad spectrum of static fluctuations and a broad spectrum of relaxation times. In order to understand the effect of protein dynamics on reaction rates, we have chosen to study a sub-nanosecond electron transfer reaction between the bacteriopheophytin and primary quinone cofactors of the photosynthetic bacterial reaction center. We show that dynamics affects the activation barrier of the reaction through a dynamical restriction of the configurational space sampled by the protein–water solvent on the reaction time-scale. The modes which become dynamically arrested on the reaction time-scale of hundreds of picoseconds are related to elastic motions of the protein that are strongly coupled to the hydration layer of water. Several mechanistic consequences for protein electron transfer emerge from this picture. Importantly, energy parameters used to define the activation barrier of electron transfer reactions lose their direct connection to equilibrium thermodynamics and become dependent in a very direct way on the relative magnitudes of the reaction and nuclear reorganization time-scales. As a result, the energetics of protein electron transfer need to be defined on each specific reaction time-scale. This perspective offers a mechanism to optimize protein electron transfer by tuning the reaction rate to the relaxation spectrum of the reaction coordinate.

Molecular Exchange in Ordered Diblock Copolymer Micelles

Molecular exchange kinetics of diblock copolymers forming spherical micelles packed on a body-centered cubic (bcc) lattice were investigated using time-resolved small-angle neutron scattering (TR-SANS). Ordered arrays of polystyrene spheres were prepared by mixing 15 vol % poly(styrene-b-ethylene-alt-propylene) (PS−PEP) in squalane, a highly selective solvent for the PEP block. Two pairs of diblock copolymers were examined, characterized by two different PS core block molecular weights (⟨MPS⟩ = 26500 and 42800). Each pair contained two nearly identical diblock copolymers, one with a deuterated PS block (dPS−PEP) and the other with a protonated PS block (hPS−PEP). Protonated and deuterated squalane were blended to achieve a contrast-matched condition with uniformly mixed dPS and hPS (50/50 by volume) core chains. Beginning with a statistically random array of pure dPS and hPS cores distributed on the bcc lattice, molecular exchange was monitored at multiple temperatures by TR-SANS. Exchange of deuterated and protonated chains results in a decay in SANS intensity, which yields a kinetic function revealing a broad spectrum of relaxation times. These results are qualitatively consistent with our previously reported findings for molecular exchange between PS−PEP micelles in a dilute (1 vol %) squalane solution. However, the measured characteristic time constant for the concentrated, ordered system is more than an order of magnitude slower than in the dilute, disordered dispersion.

Influence of confinement on molecular and director reorientational dynamics of liquid crystal

We present the qualitative results of investigations of influence of confinement of liquid crystal in random porous medium on relaxation due to molecular reorientations investigated by dielectric spectroscopy and fluctuations of director orientations (collective relaxation) investigated by photon correlation spectroscopy. The relaxation times of the process due to the molecular rotations in deeply supercooled state of confined liquid crystal were slower than at the temperatures corresponding to nematic phase by a factor of 10 6. This slowing down was accompanied by anomalous broadening of the dielectric spectra. We attribute this qualitative change in molecular relaxation to coupling of molecules to the pore walls and among themselves via molecule-wall interactions. In the nematic phase of bulk liquid crystals, the relaxation process due to director orientation fluctuations (collective mode) was single exponential. The dynamics of this process was also drastically modified by random confinement. The slow relaxation process, which does not exist in the bulk liquid crystal, and a broad spectrum of relaxation times, appear for confined liquid crystal. These results on the influence of confinement on relaxation properties of liquid crystal were complimented by differential scanning calorimetry experiments.

Theory of relaxation spectra for two identical interpenetrating polymer networks

Viscoelastic models for the description of the relaxation characteristics of two identical swollen interpenetrating polymer networks with different topologies moving against the background of an external viscous medium are considered. Two dynamic models that differ in the character of mutual interaction between network junctions are proposed. According to the first model, viscoelastic interaction is assumed to be constant and provided by the entanglements between a junction of one network with eight symmetrically arranged junctions of the other network. The second model involves (i) the predominant interaction between multiple-network junctions most closely located owing to entanglements and (ii) a weaker interaction with more distant junctions of neighboring cells. For the systems composed of two interpenetrating networks, relaxation-time spectra and average inverse relaxation-time spectra are compared with the corresponding spectra and characteristic times for individual noninteracting regular networks. Both models can involve two branches of the relaxation spectrum. One branch is a collective branch corresponding to the motion of the double network, whose parameters are controlled by the constants of elasticity of each of the interacting networks as well as by the effective mutual viscoelastic interactions between networks. This low-frequency branch is characterized by a broad spectrum of relaxation times. The second branch is a high-frequency branch that is primarily provided by mutual local motions of two interacting networks. This branch is characterized by a comparatively narrow relaxation spectrum and depends on quasielastic constants, which describe network entanglements, and on the characteristic elasticity of each network. The second branch does not involve any infinitely long relaxation times for infinitely continuous networks.

Model Theory of Visco‐Elastic Properties and Relaxation Spectra of Two Different Interpenetrating Polymer Networks

AbstractDynamic viscoelastic models of the system of two different interpenetrating polymer networks with different topology and type of interactions were used for calculating spectra of relaxation times of the system under consideration. It was shown, that two branches of the relaxation spectrum appear for two models of interpenetrating networks with different components. One of the branches is the branch of the collective motion of double network consisting of two initial interacting networks. Parameters of this branch of relaxation spectrum are defined by both own elastic constants of each of interacting networks and by effective quasi‐elastic interactions between two networks. This branch is the low frequency one and is described by broad relaxation time spectrum. The second branch is the high frequency one and characterizes mutual local motions of two interacting networks. The relaxation spectrum of this branch is comparatively narrow and depends on the quasi‐elastic constants and mutual friction which is defined by the entanglements of the networks and by its effective rigidity. The second branch does not contain extremely large relaxation times for infinitely large networks.

Deformation of a viscoelastic drop in planar extensional flows of a Newtonian fluid

This communication presents experimental data for steady and transient deformations of a purely elastic drop (PIB/PB Boger fluid) suspended in a Newtonian fluid (PDMS) undergoing a planar extensional flow produced by a four-roll mill. The viscoelastic effects alter the steady drop shape from being ellipsoidal to drop shapes with more blunt ends. The results reflect a balance between the direct tensile stress contribution of the viscoelastic fluid to the normal stress balance, and modifications of the viscous (i.e. Newtonian) stress and pressure due to viscoelastic changes in the flow. This is qualitatively consistent with the conclusion from recent simulation results [S. Ramaswamy, L.G. Leal, The deformation of a viscoelastic drop subjected to steady uniaxial extensional flow of a Newtonian fluid, Journal of Non-Newtonian Fluid Mechanics 85 (2–3) (1999) 127–163]. There is no overshoot of the drop deformation upon start-up of the flow, and a relative insensitivity to the Deborah number. However, when the external flow is turned off, there is a broad spectrum of relaxation times for the drop to return to a spherical shape, between the characteristic capillary timescale for a Newtonian fluid, aμ/ σ, to the longer polymer relaxation time, τ R .

Diffusing wave spectroscopy in Maxwellian fluids

We present a critical assessment of the diffusing wave spectroscopy (DWS) technique for obtaining the characteristic lengths and for measuring the loss and storage moduli of a reasonable well-known wormlike micelle (WM) system. For this purpose, we tracked the Brownian motion of particles using DWS embedded in a Maxwellian fluid constituted by a wormlike micellar solution made of cetyltrimethylammonium bromide (CTAB), sodium salicylate (NaSal), and water. We found that the motion of particles was governed by the viscosity of the solvent at short times and by the stress relaxation mechanisms of the giant micelles at longer times. From the time evolution of the mean square displacement of particles, we could obtain for the WM solution the cage size where each particle is harmonically bound at short times, the long-time diffusion coefficient, and experimental values for the exponent that accounts for the broad spectrum of relaxation times at the plateau onset time found in the (deltar2(t)) vs. time curves. In addition, from the (deltar2(t)) vs. time curves, we obtained G'(omega) and G"(omega) for the WM solutions. All the DWS microreological information allowed us to estimate the characteristic lengths of the WM network. We compare our DWS microrheological results and characteristic lengths with those obtained with mechanical rheometers at different NaSal/CTAB concentration ratios and temperatures.

Energy relaxation in disordered charge and spin density waves

We investigate collective effects in the strong pinning model of disordered charge and spin density waves (CDWs and SDWs) in connection with heat relaxation experiments. We discuss the classical and quantum limits that contribute to two distinct contribution to the specific heat (a $C_v \sim T^{-2}$ contribution and a $C_v \sim T^{\alpha}$ contribution respectively), with two different types of disorder (strong pinning versus substitutional impurities). From the calculation of the two level system energy splitting distribution in the classical limit we find no slow relaxation in the commensurate case and a broad spectrum of relaxation times in the incommensurate case. In the commensurate case quantum effects restore a non vanishing energy relaxation, and generate stronger disorder effects in incommensurate systems. For substitutional disorder we obtain Friedel oscillations of bound states close to the Fermi energy. With negligible interchain couplings this explains the power-law specific heat $C_v \sim T^{\alpha}$ observed in experiments on CDWs and SDWs combined to the power-law susceptibility $\chi(T)\sim T^{-1+\alpha}$ observed in the CDW o-TaS$_3$.

Regeneration of thixotropic magnetic gels studied by mechanical spectroscopy: the effect ofthe pH

Regeneration of thixotropic gels resulting from aggregation of positively charged magneticnanoparticles is here investigated by mechanical spectroscopy. The frequency independenceof the ratio ( and being respectively the viscous and the elastic moduli) at the gel point seenas a liquid–solid transition allows the determination of the gelation timetg and of the power law exponent on the angular frequency for both moduli (, ). This behaviour is similar to the one observed for classical chemical andphysical gels. It implies a broad relaxation time spectrum which is compatiblewith previous dynamic magneto-optical birefringence measurements. The value oftg decreases exponentially with an increase of the pH that controls the interactionsbetween particles through their surface density of charges. In contrast, the relaxationexponent is found to be nearly insensitive to the pH. A fractal dimension of2.07 ± 0.06 is deduced from thepower law exponent Δ at the gelation time.

Slow relaxation experiments in disordered charge and spin density waves: collective dynamics of randomly distributed solitons

We show that the dynamics of disordered charge density waves (CDWs) and spin density waves (SDWs) is a collective phenomenon. The very low temperature specific heat relaxation experiments are characterized by: (i) “interrupted” ageing (meaning that there is a maximal relaxation time); and (ii) a broad power-law spectrum of relaxation times which is the signature of a collective phenomenon. We propose a random energy model that can reproduce these two observations and from which it is possible to obtain an estimate of the glass cross-over temperature (typically T g≃ 100-200 mK). The broad relaxation time spectrum can also be obtained from the solutions of two microscopic models involving randomly distributed solitons. The collective behavior is similar to domain growth dynamics in the presence of disorder and can be described by the dynamical renormalization group that was proposed recently for the one dimensional random field Ising model [D.S. Fisher, P. Le Doussal, C. Monthus, Phys. Rev. Lett. 80, 3539 (1998)]. The typical relaxation time scales like ∼τexp(T g/T). The glass cross-over temperature Tg related to correlations among solitons is equal to the average energy barrier and scales like T g∼ 2xξΔ. x is the concentration of defects, ξ the correlation length of the CDW or SDW and Δ the charge or spin gap.