Distribution Function Of Relaxation Times Research Articles

The Use of Simulation Models for Complex Description of Permittivity of Building Materials

Distribution function of relaxation time mathematically describes frequency dependence of the complex permittivity. Empirical function describing the complex permittivity in the frequency domain needs to be broken down to real and imaginary part, and thus separated. The permittivity is a function of the frequency of the electric field and describes behavior of technical dielectric in an alternating electric field. In this case the dielectric is a building material. The paper describes application of simulation models with variable distribution parameters and relaxation time.

Analyzing Capacitive Characteristics of Lithium-Ion Batteries Using a Distribution Function of the Differential Capacity

A generalization of the concept of the Distribution Function of Relaxation Times (DRT) for capacitive systems, the Distribution Function of the Differential Capacity (DDC) is presented. While the DRT describes correctly how different polarization processes build up after applying a current step, due to its underlying model structure it does not carry information about the intercalation of species into the electrode. The latter is described by the proposed DDC, which states at a given SOC (i) how much capacity there is, (ii) from how many components it is composed and (iii) how fast these components can be discharged.

Characterization of frequency dispersion in the impedance response of a distributed model from the mathematical properties of the distribution function of relaxation times

Heterogeneity and disorder univocally leads to distortion phenomena in the immittance response of a system. The analysis of this response in terms of a distribution of relaxation times (DRT) has become an important topic of basic and applied research. In this work we theoretically and numerically study the impact of the mathematical properties of a distribution function of relaxation times (DFRT) on the frequency dispersion displayed by the immittance response of a distributed model for an ideal dielectric system such as an ideally-polarized electrode, paying special attention to constant-phase-element (CPE) behavior. The analysis of the problem encompasses both explicit and implicit results reported in the literature, and a number of new findings. It is shown, for instance, that CPE exponent is upper bounded by 1, as found in experiments. Conditions to be fulfilled by the DFRT in order to give frequency dispersion are revealed, and CPE behavior is related to scale invariance. The extent of this capacitance dispersion on the frequency spectrum of the response is also addressed for both infinite and finite systems.

Correlation Between Microstructure Parameters and Performance Characteristics of Lithium-Ion Cathodes

Lithium-Ion Batteries receive growing attention because of their potentially high energy and power density, which are important requirements for automobile applications. Reports agreed that the positive electrode is hereby the performance limiting cell component. NxMyCz (nickel manganese cobalt spinel) is one of the most promising cathode materials, both due to its rather high operating voltage and to either high power or energy density [1]. Hereby, a compromise between high power (thin layers and, i.e., high porosity) and high energy density (thick layers and, i.e., low porosity) has to be reached, in particular for automotive applications. This demands for an extended understanding of the interplay between microstructure parameters and chemical stoichiometry of NMC cathodes on cell performance characteristics. In this study, two NMC - cathodes recovered from commercial high energy and high power cells are analyzed (i) by SEM and X-ray tomography and (ii) by Electrochemical Impedance spectroscopy (EIS). By evaluating the impedance spectra measured at 0 to 40°C and at SOC 0 to SOC 100 by the distribution function of relaxation times (DRT) [2], a physically motivated transition line model [3] was established. The microstructure parameters required for a quantitative analysis of the electrochemical loss processes were obtained by X-ray tomography data. Additionally, the exact composition is determined by energy dispersive X-ray spectroscopy (EDX) investigations The results confirm, that a NMC electrode custom tailored for high energy density is rather thick (70 µm) and made of large low-porous agglomerates of primary particles, as described in [4]. The significant difference between the NMC cathode of a high energy and a high power cell however is not, as expected, a significantly higher porosity, but a sophisticated combination of small carbon black particles and elongated carbon black fibers, enabling high discharge currents. The tortuosity of the electrode pores, which represent the transport pathways for the liquid electrolyte, is low for the high power cell, which raises the effective electrolyte conductivity significantly, when compared to the high energy cell [4]. Furthermore the high power cell features a special compound of two different NxMyCz active materials. By including all the results from EIS, DRT, X-ray tomography and EDX, an evaluation of the influence of composition and microstructure on performance characteristics of the NxMyCz cathodes will be presented.

Analyzing Capacitive Characteristics of Lithium-Ion Batteries Using a Distribution Function of the Differential Capacity

By analyzing the Distribution Function of Relaxation Times (DRT) of a measured impedance spectrum one can separate different loss processes and thereby study which physical mechanisms contribute how strongly to the overall polarization resistance. The method is based on the idea that certain parts of an impedance spectrum can be represented by a serial connection of parallel RC-Elements [1].The DRT has been successfully used for analyzing the polarization processes taking place in Solid Oxide Fuel Cells (SOFC) as well as in Li-Ion batteries [2][3]. For the latter however, the analysis had to be limited: Due to the structure of the underlying modelling approach using parallel RC-elements, only the non-capacitive region of the impedance could be analyzed using the DRT. Accordingly, information contained within the capacitive region could not be utilized optimally so far.For analyzing capacitive characteristics we therefore propose to adapt the concept of DRT to capacitive systems yielding a Distribution Function of Differential Capacities (DDC). By physical means this function states how much load is intercalated how fast into an electrode after applying a voltage step. Figure 1 further exemplifies the concept by showing (i) a simple model system consisting of three intercalation pathways. (ii) the non-measurable intercalation currents over time exhibiting the different time constants with which the partial capacity of different-sized particles can be utilized. (iii) the according shape of the DDC, which allows to quantify time-constants and capacity contributions of different-sized particles from simple impedance measurements.Hence, using the proposed DDC allows to easily obtain information regarding an electrodes capacitive characteristics and can therefore serve as a supporting tool for custom-tailoring modern electrode microstructures.A detailed motivation and derivation of the new method will be given and first results will be shown. REFERENCES[1] H. Schichlein, A.C. Müller, M. Voigts, A. Krügel, E. Ivers-Tiffée, “Deconvolution of electrochemical impedance spectra for the identification of electrode reaction mechanisms in solid oxide fuel cells”, J. Appl. Electrochem. 32, p. 875-882 (2002).[2] A. Leonide, V. Sonn, A. Weber and E. Ivers-Tiffée, “Evaluation and Modeling of the Cell Resistance in Anode-Supported Solid Oxide Fuel Cells”, J. Electrochem. Soc., 155 (1), p. B36-B41 (2008). [3] J. Illig, M. Ender, T. Chrobak, J. P. Schmidt, D. Klotz and E. Ivers-Tiffée, "Separation of Charge Transfer and Contact Resistance in LiFePO4-Cathodes by Impedance Modeling", J. Electrochem. Soc. 159, p. A952-A960 (2012). Figure 1

Approximability of impedance spectra by RC elements and implications for impedance analysis

Abstract A commonly used class of impedance spectra validity tests is based on fitting serially connected parallel coupled pairs of resistor and capacitor (RC elements) to a measured impedance spectrum. If the model approximates the spectrum well, the measurement is considered valid. It is considered invalid if approximation is poor. Despite being widely used, theoretical justification is still missing. It is not clear which electrochemical processes could be approximated by such a model and hence, for which processes a poor approximation of the spectrum truly indicates a false measurement, instead of merely a lack of generality of the model. The scope of this paper is to derive from a system theory point of view, which class of systems can be approximated by serially connected RC elements and from that to conclude for which electrochemical systems the mentioned class of validity tests is applicable. Moreover, the results will yield theoretical justification for generalizing the concept of distribution function of relaxation times (DRT) by using positive and negative RC elements so that its benefits can be utilized not only for strictly capacitive, but for any possible non-oscillating electrochemical system.

Performance and Degradation of Solid Oxide Electrolysis Cells in Stack

A Solid Oxide Electrolysis (SOE) short stack consisting of anode-supported cells (ASCs in fuel cell mode) was assembled in JÜLICH’s F10-design. ASCs are based on Ni/8YSZ (8 mol-% yttria-stabilized zirconia) with an LSCF air electrode (La0.58Sr0.4Co0.2Fe0.8O3-δ) and 8YSZ electrolyte. A gadolinium-doped ceria (GDC) (Ce0.8Gd0.2O1.9) barrier layer was deposited between 8YSZ and LSCF by means of physical vapor deposition (PVD). The stack was mainly characterized in a furnace environment in both fuel cell and electrolysis modes, with 50% humidified H2 at 700 and 800 °C. An endothermic long-term electrolysis operation was carried out at 800 °C with a current density of -0.5 Acm-2 and steam conversion rate of 50%. After 2300 h of operation, the stack showed an average voltage degradation rate of 0.7%/kh. Electrochemical impedance spectroscopy (EIS) and analysis of the distribution function of relaxation times (DRT) showed that the degradation was primarily due to the increase in ohmic resistance.

Phase interface effects in the total enthalpy-based lattice Boltzmann model for solid–liquid phase change

In this paper, phase interface effects, including the differences in thermophysical properties between solid and liquid phases and the numerical diffusion across phase interface, are investigated for the recently developed total enthalpy-based lattice Boltzmann model for solid–liquid phase change, which has high computational efficiency by avoiding iteration procedure and linear equation system solving. For the differences in thermophysical properties (thermal conductivity and specific heat) between solid and liquid phases, a novel reference specific heat is introduced to improve the total enthalpy-based lattice Boltzmann model, which makes the thermal conductivity and specific heat decoupled. Therefore, the differences in thermal conductivity and specific heat can be handled by the dimensionless relaxation time and equilibrium distribution function, respectively. As for the numerical diffusion across phase interface, it is revealed for the first time and found to be induced by solid–liquid phase change. To reduce such numerical diffusion, multiple-relaxation-time collision scheme is exploited, and a special value (one fourth) for the so-called “magic” parameter, a combination of two relaxation parameters, is found. Numerical tests show that the differences in thermophysical properties can be correctly handled and the numerical diffusion across phase interface can be dramatically reduced. Finally, theoretical analyses are carried out to offer insights into the roles of the reference specific heat and “magic” parameter in the treatments of phase interface effects.

Performance and Degradation of Solid Oxide Electrolysis Cells in Stack

A Solid Oxide Electrolysis (SOE) short stack consisting of anode-supported cells (ASCs in fuel cell mode) was assembled in JÜLICH's F10-design. ASCs are based on Ni/8YSZ (8 mol-% yttria-stabilized zirconia) with an LSCF air electrode (La0.58Sr0.4Co0.2Fe0.8O3-δ) and 8YSZ electrolyte. A gadolinium-doped ceria (GDC) (Ce0.8Gd0.2O1.9) barrier layer was deposited between 8YSZ and LSCF by means of physical vapor deposition (PVD). The stack was mainly characterized in a furnace environment in both fuel cell and electrolysis modes, with 50% humidified H2 at 700 and 800°C. An endothermic long-term electrolysis operation was carried out at 800°C with a current density of −0.5 Acm−2 and steam conversion rate of 50%. After 2300 h of operation, the stack showed an average voltage degradation rate of 0.7%/kh. Electrochemical impedance spectroscopy (EIS) and analysis of the distribution function of relaxation times (DRT) showed that the degradation was primarily due to the increase in ohmic resistance.

Investigation of Oxygen Reduction Reaction on La0.1Sr0.9Co0.8Fe0.2O3-δ Electrode by Electrochemical Impedance Spectroscopy

The oxygen reduction reaction (ORR) on La0.1Sr0.9Co0.8Fe0.2O3-δ (LSCF1982) electrode is investigated by electrochemical impedance spectroscopy (EIS) in 550–650°C range under different oxygen partial pressures (pO2). The distribution function of relaxation time (DRT) analysis of EIS data enabled us to distinguish 3 individual sub-processes contributing toward the overall electrode polarization during ORR. Based on these results, oxygen chemical diffusivity (Dchem) and surface exchange coefficient (Ksur) values are obtained under open circuit voltage (OCV) conditions by Gerischer impedance model. In 500–650°C range, Ksur values ranges between 10−5−10−8 cm⋅s−1 and Dchem values are in 10−4−10−6 cm2⋅s−1 range which, due to the higher ionic conductivity of LSCF1982, is relatively higher than those for similar mixed-conducting LaxSr1−xCo1−yFeyO3−δ (LSCF). The values of activation overpotential are separated under the fixed current load, which are well-fitted to the Butler-Volmer equation. Under the current load, the variation of exchange current density (i0) with pO2 indicated that at the high temperature the total reaction-rate is determined by charge transfer reaction, but when temperature is decreased the reaction-rate is determined by the gas diffusion reaction.

Calculating relaxation time distribution function from power spectrum based on inverse integral transformation method

Abstract A novel method is presented for obtaining the distribution function of relaxation times G ( τ ) from power spectrum 1 / f α ( 1 ≤ α ≤ 2 ) . It is derived using McWhorter model and its inverse Stieltjes transform. Unlike the pre-assumed conventional g ( τ ) distribution, the extracted G ( τ ) has a peak whose width increases as the slope of the power spectrum α decreases. The peak position determines the dominant time constant of the system. Our method is unique because the distribution function is directly extracted from the measured power spectrum. We then demonstrate the validity of this method in the analysis of noise in transistor.

Fourier transform distribution function of relaxation times; application and limitations

A simple Fourier transform (FT) method is presented for obtaining a Distribution Function of Relaxation Times (DFRT) for electrochemical impedance spectroscopy (EIS) data. By using a special data extension procedure the FT is performed over the range from -∞≤lnω≤+∞. The integration procedure is analytic in the end regions, otherwise a quadratic interpolation is used. For the necessary windowing of the first FT result, a special Tanh window function is used, which shows better results than the well-known Hann window function. An alternative approach, based on a complex nonlinear least squares (CNLS) fit of a linear sequence of (RQ)’s, provides the DFRT directly, but often significantly deviating from the FT-DFRT. With several examples the usefulness of a DFRT in impedance analysis is tested. The main conclusion is that the visibility in the DFRT of separate dispersive contributions strongly depends on the associated resistance and the width of the distribution. ‘Near equal strength’ dispersive contributions show up in the DFRT, where the area under the DFRT-peak represents the ‘strength’ or resistance. But small peaks with a large FWHM are lost, although these can easily be identified in a CNLS-fit.A comparison between the finite length Warburg (FLW) and the almost identical Gerischer impedance shows a clear difference in the DFRT. Indications are found that the characteristic Gerischer DFRT might be distinguishable in real measurement DFRT's.In cooperation with an external group a brief test with an adaptive genetic evolution method showed promising results for a direct definition of a DFRT. In this process the reconstructed impedance is matched to the actual measurement.

Characterization of Na1.3Al0.3Zr1.7(PO4)3 solid electrolyte ceramics by impedance spectroscopy

The NASICON-type Na1.3Al0.3Zr1.7(PO4)3 compound has been synthesized by Pechini method. The investigation of the electrical properties was performed in 10Hz–3·109Hz and 300–500K frequency and temperature ranges by means of impedance spectroscopy. The results allowed us to explore the properties of dispersion of electrical properties attributed to charge carrier relaxation. The distribution of relaxation times function was obtained for experimental data by numerically solving a set of integral equations. To reveal the influence of compositional variations on electrical properties the results were compared with the ones of Na1.3Al0.3Ti1.7(PO4)3.

Charge carrier relaxation and phase transition in scandium stabilized zirconia ceramics

Structural, electrical properties and charge carriers relaxation phenomena of 10mol% Sc2O3–90mol% ZrO2 ceramics have been investigated in frequency range 1Hz to 10GHz and temperature interval from room temperature to 900K. Temperature dependencies of electrical conductivity and dielectric permittivity of the compound showed an abrupt jumps in temperature range from 790 to 820K. Differential thermal analysis and X-ray diffraction studies confirmed that the ceramics undergoes a structural phase transition from rhombohedral to cubic in this temperature region. Grain boundary contribution to total electrical conductivity was not noticeable in impedance spectra of low temperature phase, while it shows up at high temperature phase. The impedance spectra were also analyzed in terms of distribution of relaxation times by using Tikhonov regularization method. The distribution of charge carrier relaxation times in grains tend to narrow with increasing temperature in the temperature ranges far from phase transition, while a significant broadening of the distribution was observed in the temperature range of phase transition. The separation of grain and grain boundary contributions to total conductivity was enabled due to observation of corresponding peaks in obtained distribution of relaxation times function. The behavior of grain boundary conductivity around the phase transition temperature region revealed that crystalline structure of the crystallites affects electrical properties of grain boundary medium.

Correlation of carrier localization with relaxation time distribution and electrical conductivity relaxation in silver-nanoparticle-embedded moderately doped polypyrrole nanostructures

The electrical conductivity relaxation in moderately doped polypyrrole and its nanocomposites reinforced with different proportion of silver nanoparticles was investigated in both frequency and time domain. An analytical distribution function of relaxation times is constructed from the results obtained in the frequency domain formalism and is used to evaluate the Kohlrausch-Williams-Watts (KWW) type decay function in the time domain. The thermal evolution of different relaxation parameters was analyzed. The temperature-dependent dc electrical conductivity, estimated from the average conductivity relaxation time is observed to depend strongly on the nanoparticle loading and follows Mott three-dimensional variable range hopping (VRH) conduction mechanism. The extent of charge carrier localization calculated from the VRH mechanism is well correlated to the evidences obtained from the structural characterizations of different nanostructured samples.

Charge carrier relaxation in solid VO•• conductors

Abstract The electrical properties (impedance and electric conductivity) of various VO•• conductors have been investigated in broad frequency (from 10 Hz to 3 GHz) range and wide temperature (from ambient to 1000 K) interval by impedance spectroscopy methods. Two different impedance spectroscopy techniques were used for measurements of electrical properties. Investigations in low frequency range (from 10 Hz to 2 MHz) were performed by 2-electrode method. Corresponding investigations in the high frequency range (from 0.3 MHz to 3 GHz) were based on coaxial waveguide technique. Results allowed us to consistently explore peculiarities of dispersion of electrical properties attributed to charge carrier relaxation. Lightly depressed semicircular shape of relative impedance arcs was observed in complex plane plots, which was assumed as a consequence of existence of a distribution of relaxation times. This phenomenon was studied in more detail examining experimental data and numerically solving integral equation (Fredholm integral equation of first kind) for obtaining the distribution of relaxation times (DRT) function. Analysis of measured data revealed minor DRT function narrowing with temperature increase for all the investigated compounds.

Stability and error analysis of the polarization estimation inverse problem for microbial fuel cells

Determining parameters which describe the performance of a microbial fuel cell requires the solution of an inverse problem. Two formulations have been presented in the literature: a convolutional approach or a direct quadrature approach. A complete study and analysis of the direct quadrature method, which leads to two systems for the unknown signal given measured complex data, known as the distribution function of relaxation times, is presented. A theoretical analysis justifies the minimal range of integration that is appropriate for the quadrature and suggests that the systems should be combined giving an overdetermined system that is not well posed but not as ill-posed as either system considered separately. All measures of ill-posedness support using the combined data when the level of error in both components of the complex measurements is equivalent. Tikhonov regularization for the filtered singular value and truncated singular value decomposition are used to find solutions of the underlying model system. Given such solutions the application requires the determination of the model parameters that define the signal, among which are the location and peaks of the individual processes of the cell. A nonlinear data fitting approach is presented which consistently estimates these parameters. Simulations support the use of the combined systems for finding the underlying distribution function of relaxation times and the subsequent nonlinear data fitting to these curves. The approach is also illustrated for measured practical data, demonstrating that without the theoretical analysis incorrect conclusions on the underlying physical system would arise. This work justifies the use of Tikhonov regularization combined with nonlinear data fitting for finding reliable solutions for the specific model, when the signal is comprised of a mixture of signals from a small number of processes.

Stress relaxation behavior of (Ala-Gly-Pro-Arg-Gly-Glu-4Hyp-Gly-Pro-) gelatin hydrogels under electric field: Time-electric field superposition

An investigation has been conducted on stress relaxation functions and the corresponding relaxation time distribution functions of gelatin hydrogels for the effects of degree of crosslinking and the applied electric field strength. The experimental shift factors can be thus obtained from either the stress relaxation functions or the storage and loss moduli. Both approaches yield numerically the same shift factor values which successfully allow the time-electric field superposition of various related functions. The modified shift factors are obtained from the effective relaxation times and the material constant (Cτ). The material constant critically depends on the degree of crosslinking and weakly on the reference electric field strength used in the superposition. The relaxation time distribution clearly indicates two modes of relaxation under an electric field. The fast mode of relaxation, which occurs only under an electric field, can be attributed to the relaxation of protons present in the gelatin hydrogels.

Estimation of petrophysical and fluid properties using integral transforms in nuclear magnetic resonance

In the past decade, low-field NMR relaxation and diffusion measurements in grossly inhomogeneous fields have been used to characterize properties of porous media, e.g., porosity and permeability. Pulse sequences such as CPMG, inversion and saturation recovery as well as diffusion editing have been used to estimate distribution functions of relaxation times and diffusion. Linear functionals of these distribution functions have been used to predict petro-physical and fluid properties like permeability, viscosity, fluid typing, etc. This paper describes an analysis method using integral transforms to directly compute linear functionals of the distributions of relaxation times and diffusion without first computing the distributions from the measured magnetization data. Different linear functionals of the distribution function can be obtained by choosing appropriate kernels in the integral transforms. There are two significant advantages of this approach over the traditional algorithm involving inversion of the distribution function from the measured data. First, it is a direct linear transform of the data. Thus, in contrast to the traditional analysis which involves inversion of an ill-conditioned, non-linear problem, the estimates from this new method are more accurate. Second, the uncertainty in the linear functional can be obtained in a straight-forward manner as a function of the signal-to-noise ratio (SNR) in the measured data. We demonstrate the performance of this method on simulated data.

Electrical properties of YSZ and CaSZ single crystals

An impedance and electric conductivity of 10mol%Y2O3–90mol%ZrO2 and 15mol%CaO–85mol%ZrO2 single crystals were investigated in frequency range from 10Hz to 3GHz at temperatures from room temperature to 970K by impedance spectroscopy methods. Investigations in the high frequency range were performed by the measurement of S-parameter matrix for a 2-port network of a coaxial waveguide and a sample, inserted in the gap of a central conductor. At the low frequencies 2- and 4-electrode methods were used for accurate measurements of the impedance and electrical conductivity. The quality of the experimental data was verified using Kramers–Kronig transforms. The obtained frequency dispersion of the electrical properties related to charge carrier relaxation in the electrical field was examined in terms of distribution of relaxation times (DRT). DRT function was obtained by numerically solving the Fredholm integral equation of first kind with newly developed Matlab application. Analysis of the experimental data has revealed slight spreading of the relaxation time distribution with temperature decrease for both single crystals. The temperature dependencies of the relaxation time corresponding to the DRT function peak showed Arrhenius behavior with activation energy close to that of electrical conductivity.