Electrochemical impedance spectroscopy (EIS) is widely used in electrochemistry, energy sciences, biology, and beyond. Analyzing EIS data is crucial, but it often poses challenges because of the numerous possible equivalent circuit models, the need for accurate analytical models, the difficulties of nonlinear regression, and the necessity of managing large datasets within a unified framework. To overcome these challenges, non-parametric models, such as the distribution of relaxation times (DRT, also known as the distribution function of relaxation times, DFRT), have emerged as promising tools for EIS analysis. For example, the DRT can be used to generate equivalent circuit models, initialize regression parameters, provide a time-domain representation of EIS spectra, and identify electrochemical processes. However, mastering the DRT method poses challenges as it requires mathematical and programming proficiency, which may extend beyond experimentalists’ usual expertise. Post-inversion analysis of DRT data can be difficult, especially in accurately identifying electrochemical processes, leading to results that may not always meet expectations. This article examines non-parametric EIS analysis methods, outlining their strengths and limitations from theoretical, computational, and end-user perspectives, and provides guidelines for their future development. Moreover, insights from survey data emphasize the need to develop a large impedance database, akin to an impedance genome. In turn, software development should target one-click, fully automated DRT analysis for multidimensional EIS spectra interpretation, software validation, and reliability. Particularly, creating a collaborative ecosystem hinged on free software could promote innovation and catalyze the adoption of the DRT method throughout all fields that use impedance data.

Nowadays, yttria (Y3+)-stabilized ZrO2 (Y-TZP) is the most commonly used material in dental prosthetics. Y-TZP dental ceramics are mainly stabilized via the addition of 3 mol% yttrium oxide (Y2O3). These ceramics exhibit excellent mechanical properties, including high flexural strength, fracture toughness, elastic modulus, etc. Some manufacturers have recently introduced a new class of dental materials with multilayer composition with the aim of combining the advantages of adding more or less Y2O3 to the ceramic composition in one Y-TZP material. The flexural strength values of multilayer Y-TZP may vary depending on the dimensions of the specimen, layer distributions, and especially the layer exposed on the maximum tension side, i.e., loading configuration. Although previous studies have examined the flexural strength of separate Y-TZP layers, capturing the flexural strength of multilayer Y-TZP is still challenging. However, one should keep in mind that multilayer flexural strength is important for clinical indications. The objective of this study is to compare the flexural strength of three distinct multilayer translucent Y-TZP materials made up of layers with different Y3+ contents. Rectangular samples (2 mm × 2 mm × 16 mm) were prepared from CAD/CAM discs using the milling machine Programill PM7 (Ivoclar Vivadent AG). Milled bars were tested for flexural strength in a three-point bending test (ISO 6872:2015) using a universal testing machine (Inspekt Duo 5kN; Hegewald & Peschke, Nossen, Germany) at a crosshead speed of 0.5 mm/min. Representative samples of each type of material were selected for quantitative and qualitative analysis of the microstructure. Representative samples of each type of material were selected for structural, mechanical, and microstructural analyses.

The Electrochemical Spectroscopy Impedance (EIS) data are usually analyzed by solving different Complex Nonlinear Least Squares (CNLS) problems. These problems are generally solved by using the Levenberg-Marquardt algorithm (LMA), which is a self-adaptive regularization technique that applies the first derivatives values stored in the Jacobian matrix. The LMA convergence can be amplified by the application of both the limit tactics and the exact first derivative values. These exact values can be obtained by using the auto-matic differentiation (AD) algorithm embedded in PyTorch, which is an open-source machine learning frame-work. However, the joint application of AD, self-adapting regularization, and the limits has not been reported as a part of CNLS strategy yet. Herein, we have designed and developed a new CNLS strategy that applies the limits tactics, AD, and self -adapting regularization. Also, we claim that the new strategy can be straightforwardly employed as the exact first derivative values can be obtained by EEC and AD, which simplifies developing tasks. The results in this work clearly reveal that the LMA convergence is boosted when using the exact Jacobian matrix constructed by both the limits and AD. In addition to that, the new strategy decreases the impact of the poor starting param-eters. The tests in this work, conducted by analyzing ZARCs and FRACs synthetic data, show a superior perfor-mance of the newly developed CNLS strategy.

The Levenberg-Marquardt algorithm (LMA) is arguably one of the best algorithms that can be used for solving Complex Nonlinear Least Squares (CNLS) problems.This process can be facilitated, e.g., by modifying the damping ($\lambda$) parameter update strategy, applying limits, and employing symbolic or analytical expressions to calculate the corresponding first derivatives (FD).In this work, we have demonstrated that the LMA convergence is strongly influenced by hyperparameter values that control the $\lambda$ update.The above claim was verified by using the "original" and "optimized" LMA hyperparameter values to solve several CNLS problems. The optimization of the LMA hyperparameter values was conducted by using machine learning tools. Furthermore, it was presented that the convergence properties of the optimized LMA were drastically improved. To be more precise, when using both the approximated FD and the limits, the optimized LMA fit successfully escaped the local minima, whilst the original one was stuck. Additionally, when using the limits and the exact FD obtained by automatic differentiation, the optimized (vs. original) LMA fit required fewer iterations. Overall, it was revealed that machine learning tools can be used to optimize LMA hyperparameter values, which in turn boost convergence properties.

Throughout the lifespan of a bridge, morphological changes in the riverbed affect the variable action-imposed loads on the structure. This emphasizes the need for accurate and reliable data that can be used in model-based projections targeted for the identification of risk associated with bridge failure induced by scour. The aim of this paper is to provide an analysis of scour depth estimation on large sand-bed rivers under the clear water regime, detect the most influential (i.e., explanatory) variables, and examine the relationship between them and scour depth as a response variable. A dataset used for the analysis was obtained from the United States Geological Survey’s extensive field database of local scour at bridge piers, i.e., the Pier-Scour Database (PSDB-2014). The original database was filtered to exclude the data that did not reflect large sand-bed rivers, and several influential variables were omitted by using the principal component analysis. This reduction process resulted in 10 influential variables that were used in multiple non-linear regression scour modeling (MNLR). Two MNLR models (i.e., non-dimensional and dimensional models) were prepared for scour estimation; however, the dimensional model slightly overperformed the other one. According to the Pearson correlation coefficients (r), the most influential variables for estimating scour depth were as follows: Effective pier width (r = 0.625), flow depth (r = 0.492), and critical and local velocity (r = 0.474 and r = 0.436), respectively. In the compounded hydraulic-sediment category, critical velocity had the greatest impact (i.e., the highest correlation coefficient) on scour depth in comparison to densimetric Froude and critical Froude numbers that were characterized by correlation coefficients of r = 0.427 and r = 0.323, respectively. The remaining four variables (local and critical bed shear stress, Froude number, and particle Reynolds number) exhibited a very weak correlation with scour depth, with r < 0.3.

The dental ceramic materials are constantly being developed due to their continuous clinical appli-cation in the field of esthetic dentistry. Glass ceramics (GC) materials are also of special interest for dental application due to their specific properties; and thus, they can be applied as crowns, veneers and small bridges. Purpose: However, due to a variety of different GC materials, it is of keen inter-est to inspect their morphology and ion-diffusion, which also governs aging properties. Material and methods: In this study, two different GC materials were processed, i.e., lithium silicate (LS-10) and lithium disilicate (LS-20). The aforementioned properties can be inspected by using impedance spec-troscopy (IS) and scanning electron microscopy (SEM). Results: SEM study suggested that LS-10 ma-terial is harder to mechanically process by computer-aided design/computer-aided manufacturing (CAD/CAM) technology. Furthermore, IS measurements showed that LS-20 (vs. LS-10) has more pro-nounced resistance properties. Conclusion: According to IS data, it was concluded that LS-20 (vs. LS -10) has more pronounced resistance properties that point out to hindered ion-diffusion and to bet-ter aging properties.

Nowadays, Electrochemical Impedance Spectroscopy is attracting more attention due to an increasing production of power sources. One of highly popular tools to diagnose diverse power sources is Distribution Function of Relaxation Times (DRT). Because of that, there are numerous approaches to extract DRT from impedance data. The majority of them are based on the numerical approximation of integral. However, herein we have applied an analytical approximation of the EIS integral. For the first time, we have employed Levenberg-Marquardt algorithm (LMA) to extract the applicable DRT from impedance data by using the Jacobian matrix that was obtained without any discretization errors. Although LMA was previously used to fit EIS data by DRT characteristics, the DRT profile was not applicable due to discretization errors. In this work, LMA was applied as it has an automatic update of the regularization (λ) parameter. The tests conducted in this work have shown that LMA is capable of extracting DRT from ZARC and FRAC synthetic data.

Electrochemical Impedance Spectroscopy (EIS) is a powerful tool for the analysis of different power sources and various materials. One of the methods used for studying EIS data is the distribution function of relaxation times (DRT). EIS data can be converted into a Fredholm integral of the first kind; and DRT extraction is known to be an inverse ill-posed problem. Herein, a new strategy to extract DRT by applying the Levenberg-Marquardt algorithm (LMA) is proposed. The Jacobian matrix appearing in LMA is partially numerically approximated by applying the radial basis function as a basis for the discretization. DRT data are smoothed by the application of the finite difference matrix and the negative values are avoided by the limits application. The tests conducted with ZARCs/FRACs synthetic data show that the extracted DRT profiles correspond well to their analytical counterparts. The application of LMA in solving Fredholm integral equation of the first kind (i.e., DRT extraction) resulted in the automatic tuning of the regularization parameter. The aforementioned findings show that by modifying LMA it is possible to both solve the Fredholm integral equation of the first kind in a completely data-driven way and to obtain the applicable DRT data for general EIS study.

Zirconia (ZrO2), a widely known material with an extensive range of applications, is especially suitable for dental applications. This kind of dental material is produced in the form of blocks or discs (mostly discs-depending on CAD/CAM machines) by cold isostatic pressing (CIP). Such discs are subsequently milled by CAM/CAD technology into a desirable form. Due to the application of CIP, the resulting discs consist of different yttria-stabilized tetragonal zirconia polycrystal (Y-TZP) powders, additives and pigments. The diverse composition of the discs (e.g., different Y3+ content) governs material properties, such as hardness, toughness and translucence. The aim of this work was to investigate the impact of Y3+ addition on the grains and grain boundaries, on the ZrO2 phases lattice parameter values and on the electrical equivalent circuit parameters of the prepared Y-TZP samples. The disc-shaped samples were prepared by using CAM/CAD technology. It was observed that the grain size and the grain density were increased by Y3+ addition. The sample with the lowest Y3+ content was characterized by the highest portion of the tetragonal phase, whilst the disc with the highest Y3+ addition consisted mainly of the cubic phase. It was also observed that at the higher Y3+ ion concentrations, these ions mainly incorporated the tetragonal phase. Furthermore, conductivity investigations showed that the resistivity of the grains in the samples with the higher Y3+ concentrations was decreased as these ions were mainly trapped in the grain boundary. On the other hand, the Y3+ trapping increased the capacitance of the grain boundary.

Electrochemical impedance spectroscopy (EIS) is an important electrochemical technique that is used to detect changes and ongoing processes in a given material. The main challenge of EIS is interpreting the collected measurements, which can be performed in several ways. This article focuses on the electrical equivalent circuit (EEC) approach and uses grammatical evolution to automatically construct an EEC that produces an AC response that corresponds to one obtained by the measured electrochemical process(es). For fitting purposes, synthetic measurements and data from measurements in a realistic environment were used. In order to be able to faithfully fit realistic data from measurements, a new circuit element (ZARC) had to be implemented and integrated into the SPICE simulator, which was used for evaluating EECs. Not only is the presented approach able to automatically (i.e., with almost no user input) produce a more than satisfactory EEC for each of the datasets, but it also can also generate completely new EEC configurations. These new configurations may help researchers to find some new, previously overlooked ongoing electrochemical processes.

Identification of ongoing processes in solid oxide fuel cells (SOFC) enables both optimizing the operating environment and prolonging the lifetime of SOFC. The Levenberg–Marquardt algorithm (LMA) is commonly used in the characterization of unknown electrochemical processes within SOFC by extracting equivalent electrical circuit (EEC) parameter values from electrochemical impedance spectroscopy (EIS) data. LMA is an iteration optimization algorithm regularly applied to solve complex nonlinear least square (CNLS) problems. The LMA convergence can be boosted by the application of an ordinary limit strategy, which avoids the occurrence of off-limit values during the fit. However, to additionally improve LMA descent properties and to discard the problem of a poor initial parameters choice, it is necessary to modify the ordinary limit strategy. In this work, we designed a new automatic update (i.e., adaptive) limit strategy whose purpose is to reduce the impact of a poor initial parameter choice. Consequently, the adaptive limit strategy was embedded in a newly developed EIS fitting engine. To demonstrate that the new adaptive (vs. ordinary) limit strategy is superior, we used it to solve several CNLS problems. The applicability of the adaptive limit strategy was also validated by analyzing experimental EIS data collected by using industrial-scale SOFCs.

Over the last decade, zirconia (ZrO2)-based ceramic materials have become more applicable to modern dental medicine due to the sustained development of diverse computer-aided design/computer-aided manufacturing (CAD/CAM) systems. However, before the cementation and clinical application, the freshly prepared zirconia material (e.g., crowns) has to be processed by sandblasting in the dental laboratory. In this work, the impact of the sandblasting on the zirconia is monitored as changes in morphology (i.e., grains and cracks), and the presence of impurities might result in a poor adhesive bonding with cement. The sandblasting is conducted by using Al2O3 powder (25, 50, 110 and 125 µm) under various amounts of air-abrasion pressure (0.1, 0.2, 0.4 and 0.6 MPa). There has been much interest in both the determination of the impact of the sandblasting on the zirconia phase transformations and conductivity. Morphology changes are observed by using Scanning Electron Microscope (SEM), the conductivity is measured by Impedance Spectroscopy (IS), and the phase transformation is observed by using Powder X-Ray Diffraction (PXRD). The results imply that even the application of the lowest amount of air-abrasion pressure and the smallest Al2O3 powder size yields a morphology change, a phase transformation and a material contamination.

Determination of the distribution function of relaxation times (DFRT) is an approach that gives us more detailed insight into system processes, which are not observable by simple electrochemical impedance spectroscopy (EIS) measurements. DFRT maps EIS data into a function containing the timescale characteristics of the system under consideration. The extraction of such characteristics from noisy EIS measurements can be described by Fredholm integral equation of the first kind that is known to be ill-posed and can be treated only with regularization techniques. Moreover, since only a finite number of EIS data may actually be obtained, the above-mentioned equation appears as after application of a collocation method that needs to be combined with the regularization. In the present study, we discuss how a regularized collocation of DFRT problem can be implemented such that all appearing quantities allow symbolic computations as sums of table integrals. The proposed implementation of the regularized collocation is treated as a multi-parameter regularization. Another contribution of the present work is the adjustment of the previously proposed multiple parameter choice strategy to the context of DFRT problem. The resulting strategy is based on the aggregation of all computed regularized approximants, and can be in principle used in synergy with other methods for solving DFRT problem. We also report the results from the experiments that apply the synthetic data showing that the proposed technique successfully reproduced known exact DFRT. The data obtained by our techniques is also compared to data obtained by well-known DFRT software (DRTtools).

Abstract This paper is inspired by recently proposed approach for interpreting data of Electrochemical Impedance Spectroscopy (EIS) in terms of Distribution of Diffusion Times (DDT). Such an interpretation requires to solve a Fredholm integral equation of the first kind, which may have a non-square-integrable kernel. We consider a class of equations with above-mentioned peculiarity and propose to regularize them in weighted functional spaces. One more issue associated with DDT-problem is that EIS data are available only for a finite number of frequencies. Therefore, a regularization should unavoidably be combined with a collocation. In this paper we analyze a regularized collocation in weighted spaces and propose a scheme for its numerical implementation. The performance of the proposed scheme is illustrated by numerical experiments with synthetic data mimicking EIS measurements.

In this study, two different dental ceramics, based on zirconia-reinforced lithium-silicate (LS1) glass-ceramics (Celtra Duo, Dentsply Sirona, Bensheim, Germany) and lithium disilicate (LS2) ceramics (IPS e.max CAD, Ivoclar, Vivadent, Schaan, Liechtenstein) were examined. They were tested prior to and after the crystallization by sintering in the dental furnace. Additionally, the impact of ageing on ceramic degradability was investigated by immersing it in 4% acetic acid at 80 °C for 16 h. The degradability of the materials was monitored by Impedance Spectroscopy (IS), X-Ray Powder Diffraction (XRPD), and Field Emission Scanning Electron Microscope (FE-SEM) techniques. It was detected that LS2 (vs. LS1) samples had a lower conductivity, which can be explained by reduced portions of structural defects. XRPD analyses also showed that the ageing increased the portion of defects in ceramics, which facilitated the ion diffusion and degradation of samples. To summarize, this study suggests that the non-destructive IS technique can be employed to probe the ageing properties of the investigated LS1 and LS2 ceramics materials.

We have synthesized polyaniline/ZnO-based (PANI/ZnO) multiphase composite photocatalysts from acid media by a newly proposed two-step in situ polymerization. The first step of synthesis yielded PANI salt required for the PANI/ZnO synergistic effect. In the second step, the aniline oxidation continued, without ZnO dissolution, and it produced PANI base. Thus, both PANI salt and base phases in the composites were detected by FTIR and UV/Vis, while the presence of both ZnO and PANI polymer was confirmed by XRD. Additionally, XRD also showed Zn5(OH)8(NO3)2·2H2O and Zn(SO4)(H2O) phases in PANI/ZnO-based multiphase composites. Furthermore, the impact of the synthesis conditions on the morphology of the composites was investigated by FE-SEM. The images displayed that ZnO particles were encapsulated in PANI sheets that were formed by the aniline oligomers. Photocatalytic evaluation of PANI/ZnO-based catalysts (i.e., degradation of Acid Blue 25 dye) was conducted and the obtained results confirmed that all the studied composites experienced the PANI/ZnO synergistic effect. It was observed that the best photocatalytic properties were held by the PANI/ZnO_2 sample due to its optimal particle size.

The Levenberg-Marquardt algorithm (LMA) is generally used to solve diverse complex nonlinear least square (CNLS) problems and is one of the most used algorithms to extract equivalent electrochemical circuit (EEC) parameters from electrochemical impedance spectroscopy (EIS) data. It is a well-known fact that the convergence properties of the algorithm can be boosted by applying limits on EEC parameter values. However, when EEC parameter values are low (i.e., of the order of magnitude of 10−4 or smaller), the applied limits increase the first derivatives approximation errors which occur when using a numerical Jacobian matrix. In this work, we discuss the importance of the Jacobian matrix in LMA and propose a design of a new EIS fitting engine. The new engine is based on a novel fitting scheme using limits and a symbolic Jacobian matrix instead of the numerical one, i.e. a strategy that has not yet been reported in any EIS study. We show that using a symbolic Jacobian matrix the algorithm convergence is superior to the one with a numerical Jacobian matrix. We also investigate how to improve poor convergence properties when we still have to use a numerical Jacobian matrix when analytic derivatives are not available.

Nelder-Mead "simplex" algorithm (NMA) is a derivative-free algorithm that can be used to solve complex nonlinear least-squared (CNLS) problems. NMA can fit equivalent electrical circuit models to noisy electrochemical impedance spectroscopy (EIS) data. The ability of NMA simplex to adapt itself onto the local landscape of mathematical functions is governed by the simplex steps (reflection, expansion, inside and outside contraction and shrink steps). However, according to EIS literature, a method to compute the simplex step efficiency has not been reported yet. Herein, we provide and recommend a new method to compute the simplex step efficiency. The new method was used to evaluate the adaptive and standard NMA modifications (SNMA and ANMA). Be advised that by using the new method we detected an unknown property of a more successful ANMA (vs. SNMA), i.e. the application of adaptive parameters decreased only the inside contraction step efficiency. However, the aforementioned drawback was resolved by modifying and boosting the existing ANMA, which was a process that demonstrated the usefulness of the new method. Thereby, SNMA, ANMA, and the modified ANMA fitting engines were embedded in the free (MIT licensed) software by using Python programming language. (C) 2019 Elsevier B.V. All rights reserved.

Majority of derivative-free fitting engines applied in electrochemical impedance spectroscopy (EIS) study utilize the original Nelder and Mead algorithm. However, the original algorithm cannot adapt itself to the size of the problem as it applies the standard choice of parameters. The inability to adapt increases simplex distortions that disorient search direction; and consequently, fit can finish in local minima. Note that the application of the adaptive choice of parameters can resolve the aforementioned drawbacks. However, the impact of the adaptive parameters choice on an ability to escape from local minima while solving complex nonlinear least squares problem has not been clarified yet. This paper recommends a design of a new EIS fitting engine that uses both standard and adaptive choice of parameters. The application of adaptive (vs. standard) choice of parameters prevented sudden simplex distortions and preserved the search direction; and thus, algorithm escaped from local minima up to ≈ 5.6 times more frequent. Additionally, the new adaptive technique (vs. Levenberg-Marquardt algorithm) avoided local minima up to 3 times more efficiently. Consequently, the new engine was embedded into MIT licensed software by using Python programming language.

Polyaniline/zinc oxide (PANI/ZnO) composite photocatalysts were prepared from neutral media by in situ chemical oxidation of aniline (ANI) in the presence of different amounts of diethylene glycol (DEG). The PANI/ZnO composite photocatalysts were synthesized to efficiently remove organic dye (acid blue, AB25) from model wastewater. The PANI/ZnO composite photocatalysts were studied with the intention of efficient removal of organic dye (acid blue, AB25) from wastewater to obtain low-cost heterogeneous catalysts that offer high catalytic activity and stability. The conductive PANI polymer, which absorbs Vis irradiation, was used in this work as ZnO absorbs only ultraviolet (UV) irradiation; thus, the composite photocatalysts’ activity was broadened into the Vis region. Characterization of the composite photocatalysts was done by Fourier transform infrared spectroscopy, X-ray diffraction, thermogravimetric analysis, scanning electron microscopy, electric conductivity, UV-Vis spectroscopy, and by specific surface area (SBET) measurements. The composites’ photocatalytic activity under solar irradiation was validated by monitoring degradation of the AB25 dye. This study presented that it was possible both to prepare PANI and to prevent ZnO dissolution if in situ polymerization starts from neutral media with the addition of DEG. Additionally, efficient removal of AB25 dye, about 90% in 60 min, was achieved. The first-order rate constants of the photodegradation of AB25 by PANI/ZnO 0.02/0.024/0.04 DEG (and pure ZnO)) were computed to be 0.0272/0.0281/0.0325 (and 0.0062) min−1, indicating that the morphology and surface of the photocatalysts have significantly influenced the catalytic activity.