Distribution of relaxation times: Foundations, methods, diagnostics, and prognosis for electrochemical systems

The influence of a distribution of relaxation times is studied in annealing experiments. A two-relaxation time model is proposed, which permits the calculation of the distribution of relaxation times from the crossover data of Spinner and Napolitano as well as that of Ritland. This model also characterizes the structure of any glass in terms of two parameters. Thus, quenched equilibrium as well as nonequilibrium glass were compared with rate cooled glasses with respect to their behavior upon further annealing as well as their conductivity at low temperatures with excellent agreement. Borosilicate crown glass was found to have a narrow distribution of relaxation times which is particular to associated liquid monomers rather than polymers. The results can be explained in terms of a topological model for the distribution of relaxation times. The temperature dependence of the viscosity is due to a true activation energy rather than a free volume effect.

The dynamical response of a linear viscoelastic solid, when a distribution of relaxation or retardation times is present, is studied in detail. It is shown how the internal friction or loss tangent can be expressed in terms of the expression for a standard anelastic solid, using a distribution which is related to the original distribution of relaxation or retardation times, introduced into the dynamical moduli or compliances, respectively. The internal friction is analyzed both as a function of temperature or the frequency and the concepts developed in the paper are applied to actual experimental data for the Snoek relaxation in Nb–O alloys.

Transient and steady‐state rheological data are reported for several anionic polystyrene solutions in tritolylphosphate (1. 6 < cM/ρMc < 7). Here c is the concentration of the solution, M is the molecular weight, ρ the density of the undiluted polymer, and Mc the molecular weight between entanglements as determined from zero‐shear viscosity. The polystyrene used had Mw = 410,000 and Mw/Mn < 1.06. Data are also given for solutions of polyisobutylene and poly(vinyl acetate) with larger Mw/Mn. The results give a critical strain γ′ ∝ c−1 such that linear viscoelastic behavior was obtained in a simple shear deformation with shear less than γ′. A simplified version of the constitutive equation of Bernstein, Kearsley, and Zapas is used with an empirical strain function F (γ) which contains γ′ as a parameter to discuss transient and steady‐state behavior in terms of the distribution of relaxation (or retardation) times determined for linear viscoelastic responce. Features of the dependence of the steady‐state viscosity ηk, recoverable compliance Rk, the first‐normal stress function Nk(1) on shear rate k are discussed in terms of F (γ) and the distribution of relaxation times to conclude that the latter plays a dominant role in the behavior observed in the range of k usually studied. The results predict that the reduced functions ηk/η0, Rk/R0, and Nk(1)/N0(1) should depend on η0R0k, and that the functional form depends markedly on the distribution of relaxation times, at least in the range η0R0k < 102. Comparison with the mechanistic model of Doi and Edwards shows a similar F (γ) but substantial differences in the reduced functions caused by a very narrow distribution of relaxation times in the model.

Analysis of composition fluctuation lifetimes in a critical oxide mixture by volume relaxation spectroscopy

• Supercapacitors can be modelled precisely using a dynamic equivalent circuit with a distribution of relaxation times. • Distribution of relaxation times provides an indicator of charge dynamics at the electrodes. • Both time dynamics (charging and self-discharging) and impedance spectroscopy can be studied within the model. Supercapacitors are often modelled using electrical equivalent circuits with a limited number of branches. However, the limited number of branches often cannot explain long-term dynamics, and one therefore has to resort to more computationally challenging basic models governing diffusion and drift of ions. Here, it is shown that consistent modelling of a supercapacitor can be done in a straightforward manner by introducing a dynamic equivalent circuit model that naturally allows a large number or a continuous distribution of time constants, both in time and frequency domains. Such a model can be used to explain the most common features of a supercapacitor in a consistent manner. In the time domain, it is shown that the time-dependent charging rate and the self-discharge of a supercapacitor can both be interpreted in this model with either a few or a continuous distribution of relaxation times. In the frequency domain, the impedance spectrum allows one to extract a distribution of relaxation times. The unified model presented here may help visualizing how the distribution of relaxation times or frequencies govern the behaviour of a supercapacitor under varying circumstances.

Dielectric response of ( KBr) 1− x( KCN) x dilute compounds — distribution of relaxation times

In vivo measurements of T1 and T2 values in two experimental tumors growing in the legs of mice were made during tumor growth and after treatment of the tumor with either X-rays or cyclophosphamide. The T1 and T2 values were obtained by fitting the data to continuous distributions of relaxation times. This technique gives broad distributions of relaxation times which are characterized by a number of peaks with characteristic T1 and T2 values. Before treatment, the T1 and T2 values increased before a palpable tumor mass could be detected. The response to subcurative doses of either treatment method was a reduction in the T1 and T2 values and a parallel reduction in tumor weight. Although local recurrence was characterized by the same pattern of tumor growth as was observed before treatment, therapy was found to give higher relaxation time values than those measured in untreated tumors. The higher relaxation time values of tumor-bearing legs were the result of redistribution of the peaks in the distribution and not changes in the relaxation times of the individual peaks.

Anion exchange membrane water electrolysis (AEM-WE) stands out as a promising method for hydrogen production from renewable energy sources. Unlike proton exchange membrane water electrolysis (PEM-WE), AEM-WE offers the advantage of nonprecious metal catalysts due to mild alkaline conditions, while still enabling a compact cell design and operation under differential pressure. Remarkable performances of AEM-electrolysis cells have been demonstrated in literature, achieving current densities of up to 7.68 A·cm−2 [1], surpassing the record current density of PEM-WE at 6 A·cm−2 [2], both measured at 80 °C and 2 V. However, challenges related to efficiency and stability persist and demand effective solutions to unlock the full potential of AEM-WE. In order to achieve this, cell components, particularly the membrane, catalysts, and electrode structure have to be improved and advanced measurement techniques have to be applied.Different measurement tools serve to characterize, separate, and quantify the dynamic processes inherent in AEM-WE. Electrochemical impedance spectroscopy (EIS) serves as a valuable non-destructive, in-situ diagnostic tool for electrochemical cells, enabling the differentiation of various phenomena based on their respective relaxation times. Moreover, EIS enables the quantification of loss mechanisms thereby facilitating a targeted optimization of electrolysis components and the identification of weaknesses limiting long-time stability.Despite its utility, the broad use of EIS is hindered by its complex interpretation. The widely applied approach of fitting equivalent circuit models (ECMs) aims to extract quantitative data from the semi-circular shapes observed in Nyquist plots. However, the assumption of a physico-chemical meaningful ECM is challenging without prior knowledge of the number, size, and time constants of appearing loss mechanisms. Furthermore, the potential for multiple ECMs to fit one and the same experimental result introduces ambiguity, leading to potential misinterpretations of data.An alternative approach to analyse EIS spectra involves converting the measured data into the time (τ) domain and characterize the different mechanisms based on their characteristic relaxation times. The distribution of relaxation times (DRT) analysis has already been applied across diverse electrochemical systems, spanning from solid oxide fuel cells (SO-FC) and PEM fuel cells (PEM-FC) to PEM water electrolysis and AEM fuel cells (AEM-FC). However, to the best of the authors’ knowledge, DRT analysis of AEM-WE is still absent in literature.In this study we present a thorough investigation of an AEM-WE single-cell, employing a combination of electrochemical impedance spectroscopy (EIS) and the equivalent circuit model (ECM). Notably, we demonstrate the distribution of relaxation times (DRT) analysis at AEM-WE cells for the first time, utilizing a reversible hydrogen electrode (RHE) as a reference. Through half-cell EIS measurements and subsequent DRT spectra analysis, anodic and cathodic half-cell reactions are clearly identified and quantified. The DRT analysis differentiates five loss mechanisms within the AEM-WE system, encompassing the hydrogen evolution reaction, the oxygen evolution reaction, and ionic transport losses within the catalyst layers. By systematically varying operating parameters, we successfully attribute DRT peaks to their respective physicochemical origins. These findings provide valuable insights into the electrochemical processes within the AEM-WE single-cell, significantly advancing our understanding of underlying mechanisms.In fig. 1, we investigate the degradation of AEM-WE single cells through long-term experiments. Periodic EIS measurements enable an in-operando analysis of catalyst degradation. The impedance data is analysed using ECM and DRT. Ohmic contributions resulting from electric and ionic charge transport are distinguished from polarisation resistances arising from electrochemical reactions. A catalyst degradation is observable in fig 1b & c, where the low frequency resistance (LFR) in the Nyquist plot and the DRT peak area increases over time. However, fig. 1b also reveals the reduction of the high frequency resistance (HFR) resulting in an overall performance increase and voltage decrease.[1] N. Chen, S. Y. Paek, J. Y. Lee, J. H. Park, S. Y. Lee, Y. M. Lee, High-performance anion exchange membrane water electrolyzers with a current density of 7.68 a cm −2 and a durability of 1000 hours, Energy & Environmental Science 14 (12) (2021) 6338–6348. doi:10.1039/D1EE02642A.[2] M. Braig, R. Zeis, Distribution of relaxation times analysis of electrochemical hydrogen pump impedance spectra, Journal of Power Sources 576 (2023) 233203. doi:10.1016/j.jpowsour.2023.233203. Figure 1

How reliable is distribution of relaxation times (DRT) analysis? A dual regression-classification perspective on DRT estimation, interpretation, and accuracy

The low‐temperature relaxation process of polycaprolactam, methyl‐substituted poly‐caprolactams, and other linear polyamides was studied by dielectric methods in the temperature range from – 140°C. to +50°C., and in the frequency range from 100 cps to 10 Mc./sec. In some of these polymers the temperature dependence of the complex shear modulus of elasticity was also measured at a frequency of about 1 cps. The methyl substituent provokes an increase of relaxation times of dielectric dispersion in poly‐γ‐methylcaprolactam and a decrease of relaxation times in poly‐ϵ‐methylcaprolactam. This decrease is ascribed to the increase in steepness of the potential barrier owing to a close attachment of dipole to the methyl‐substituted group. Irrespective of its position, the methyl substituent has been found in all cases to cause an extension of the distribution of relaxation times of shear modulus towards lower temperatures, as compared with the dispersion of nonsubstituted polyamides. The dependence of the widths of the distribution of dielectric relaxation times upon the temperature and the degree of crystalline arrangement have been discussed. For the copolymer of caprolactam. 7‐aminoheptanoic acid, and aminoundecanoic acid, which possess the same concentration of polar groups per unit of chain length as poly(hexamethylene sebacamide), a narrower distribution of relaxation times has been found. On the basis of ideas concerning low‐temperature dispersion it is possible to explain also higher relaxation times of low‐temperature dispersion occasioned by low molecular weight substances by an increase of the mean friction factor.

Physical ageing and the Johari–Goldstein relaxation in molecular glasses

An analysis of shear viscosity and ultrasonic measurements on a series of critical oxide mixtures shows that the supercritical, excess static viscosity is associated with a broadening in the distribution of structural relaxation times, rather than with an interaction between the long-wavelength components of the fluctuations and the shear flow processes. The broadening saturates near the critical point, and the excess static viscosity approaches a finite limit as a function of reduced temperature. These results are in disagreement with existing excess-viscosity theories. A new model is proposed which relates the distribution of relaxation times to microstructure in the melt resulting from the supercritical fluctuations in composition. It also limits the range of interaction associated with structural relaxation to a finite value. The resulting conclusions are consistent with the observed lack of divergence in the excess viscosity and width of the distribution of relaxation times at Tc.

Glass transition processes have often been explained in terms of wide distributions of relaxation times. By means of a simple stochastic model we here show how dynamic heterogeneity is the key to the emergence of the glass transition. A non-Markovian model representing a small open region of the amorphous material was previously shown to reproduce the time and thermal characteristic behavior of supercooled liquids and glasses. Due to the interaction of the open regions with their environment, the temperature dependence of the equilibrium relaxation times differs from the featureless behavior of the relaxation times of closed regions, whose static disorder does not lead to a glass transition, even with wider distributions of equilibrium relaxation times. The dynamic heterogeneity of the open region produces a glass transition between two different regimes: a faster-than-Arrhenius and non-diverging growth of the supercooled liquid relaxation times and an average Arrhenius behavior of the ideal glass. The Kovacs’ expansion gap was studied by evaluating the nonequilibrium distribution of relaxation times after the temperature quenches.

Electrochemical impedance spectroscopy (EIS) is the experimental technique of choice for studying the properties of various electrochemical systems, including batteries, fuel cells, and supercapacitors [1]. In particular, EIS data collected from supercapacitors was successfully analyzed with the distribution of relaxation times (DRT) to elucidate charging and discharging mechanisms at the electrodes [2]. The DRT method is especially appealing as it represents a non-parametric alternative to the standard equivalent circuits and physical models [3].The DRT method relies on the assumption that a blip of current linearly results in a voltage that decays with a characteristic timescale [4]. Deconvolving the DRT from measured impedances allows the identification of these characteristic timescales to characterize and optimize the performance of supercapacitors. Nevertheless, due to diffusional and mass-transport-related phenomena, the imaginary part of the impedance of supercapacitors generally diverges at low frequencies, which renders the DRT method less accurate [5]. In fact, the impedance modelled with the DRT tends to a finite real value at low frequencies. In other words, the DRT method is less suitable to identify timescales larger than a few seconds. To overcome this limitation, the distribution of capacitive times (DCT) was proposed by Brug et al. [6]. The DCT method models the admittance, i.e., the inverse of the impedance, in a similar fashion as the DRT describes the impedance. Recently, our group derived the theory underlying the DCT method to facilitate its application and enrich the framework of non-parametric distributions for EIS data analysis [7]. We also used artificial and real battery data to showcase the potential of the DCT method.In this presentation, we will first summarize the main properties of the DCT, and in particular highlight the differences between the DCT and the DRT. Then, we will illustrate how to use the DCT to analyze real data from an electric double-layer supercapacitor with porous carbon electrodes. Moreover, we will explain how to recover the DCT from the experimental impedance, before showing how to compute physical parameters from the DCT spectrum. Overall, we envision that the DCT method will deepen the understanding of EIS data measured from supercapacitors.

The effect of local heterogeneity on the distribution of dielectric relaxation times was studied for concentrated solutions of poly(vinyl acetate) (PVAc) in 1-methylnaphthalene (MN) and those of poly(vinyl octanoate) (PVOc) in toluene (Tol). The half-widths A of the dielectric loss curves for the primary processes of those systems are compared with those of the PVAc/Tol system reported previously. The data indicate that A decreases in order of PVAc/Tol, PVAc/MN, and PVOc/Tol systems if compared at the same temperature T/Tg where Tg denotes the glass transition temperature of solutions. The half-width of each solution increases with decreasing temperature. The loss curve is calculated by assuming a Gaussian distribution of the local concentration Φ which results in the distribution of relaxation times g(τ). The calculated loss curves agree fairly well with the observed ones. The broadening behavior is also explained by assuming that A is proportional to the amplitude of the local concentration fluctuation ΔΦ times the slope of the Φ dependence curve of the average relaxation time r. This assumption leads to a linear relationship between A and ΔΦ/(T - T 0 ) 2 where To is the Vogel critical temperature. Using this relation, we have attempted to estimate ΔΦ. Small-angle X-ray scattering was also measured on PVAc/ MN solutions and found that the scattering intensity was lower than that for PVAc/Tol. This is consistent with the fact that A of PVAc/MN solutions is narrower than PVAc/Tol solutions.