Elsevier

Journal of Power Sources

Volume 403, 1 November 2018, Pages 184-191
Journal of Power Sources

Unique fitting of electrochemical impedance spectra by random walk Metropolis Hastings algorithm

https://doi.org/10.1016/j.jpowsour.2018.09.091Get rights and content

Highlights

  • We propose a random walk Metropolis Hastings algorithm to fit impedance spectra.

  • Parameter solution can escape from local minima by using the algorithm.

  • We successfully fit impedance spectra of (La,Li)TiO3 polycrystals in high accuracy.

Abstract

A curve fitting is the most important process in the analysis of electrochemical impedance spectra to evaluate the ionic conductivity of materials. To analyze the impedance spectra, a gradient method such as steepest descent has been used so far. However, the parameter solution by using the gradient method is often trapped into local minima, and the curve fitting strongly depends on the initial parameters. In this study, to avoid the local minima issue, we propose a random walk Metropolis Hastings algorithm to analyze impedance spectra, where we can provide a unique solution of the impedance spectra. As an example, we measured the solid-state oxide electrolyte of (La0.62Li0.15)TiO3, (La0.53Li0.40)TiO3 and (La0.31Li0.07)NbO3 polycrystal and we uniquely identify the respective lithium ion conductivity at the bulk and the grain boundary by using the random walk Metropolis Hastings algorithm. The present algorithm is free from the choice of the initial values of the fitting parameters and moreover the estimated accuracy of the Li-ion conductivity is better than 5%.

Introduction

Since the explosive popularization of smartphones and electric cars, it becomes inevitable to improve the performance of current electric power equipment such as lithium ion rechargeable battery (LIB). Recently, all solid-state LIB attracts much attention as a rechargeable battery with their prospective safety, and exploration of new materials and the evaluation of solid state electrolytes are actively carried out [[1], [2], [3], [4], [5]]. For the further development of LIB materials, it is prerequisite to precisely evaluate the Li-ion conductivity in the electrolytes [[6], [7], [8]], and electrochemical impedance spectroscopy plays a central role to estimate the electrochemical properties of solid materials such as conductivity and surface corrosion [[9], [10], [11], [12], [13], [14], [15], [16], [17]].

In electrochemical impedance spectroscopy, we measure the angular frequency dependence of the complex impedance in electrolytes [[18], [19], [20], [21], [22]]. The advantage of electrochemical impedance spectroscopy is that all elements of bulk and interface conductions appear in a single spectrum [[18], [19], [20], [21], [22]], and therefore we can identify the conduction for each elements from the single spectrum.

In LIB system, solid state electrolytes are mostly used in the form of polycrystals rather than a single crystal, and it is a central concern to evaluate the contribution of the grain boundaries (G.B.) to the total Li-ion conductivity because of their high resistivity for Li-ion conduction at the G.B [[23], [24], [25]]. However, in the electrochemical impedance analysis, it still remains an issue to extract G.B. contribution from the spectra, where we should perform curve fitting to determine the ionic conductivity for each element without any ambiguity. Historically, the gradient methods such as steepest descent has long been used to fit impedance spectra. A severe problem on the gradient method is that the parameter solution is often trapped into local minima, and the fitted result strongly depends on the initial value of the fitting parameters. Since the gradient method is an algorithm to find the parameters with a minimum difference between the model function and the fitted curve by only using the first derivative, most of the parameter solutions cannot get out of the local minima, suggesting that we will find the best solution under the initial condition (local minimum) rather than the unique solution (global minimum).

To identify the unique solution via fitting of impedance spectra without any trapping of local minima, Markov chain Monte Carlo method could be a potential algorithm, which is rapidly expanding in a field of data analysis [[26], [27], [28], [29], [30], [31], [32], [33], [34]]. For instance, in a physical science field, Nagata et al. have proposed to use the exchange Monte Carlo algorithm and demonstrated an excellent fit of an X-ray photoemission spectrum [30,31], where the used sophisticated algorithm is valid even when the number of parameters is unknown. In an electrochemical impedance spectrum, an equivalent circuit (model function) for fitting is solely determined by an experiment system, and we know the number of parameters in advance, suggesting that we can adapt a simpler algorithm rather than the exchange Monte Carlo algorithm.

In this study, we propose a random walk Metropolis Hastings algorithm to uniquely fit the impedance spectrum with avoiding the local minima problem. On the basis of the present fitting algorithm, we demonstrate the unique fitting of the impedance spectra obtained from representative oxide electrolyte of (La0.62Li0.15)TiO3, (La0.53Li0.40)TiO3 and (La0.31Li0.07)NbO3 in a high accuracy.

Section snippets

Theoretical background

We denote yexp and yfit for the experimental and fitted curve data, respectively. Along the conventional manner, one expects to find the parameters with the minimum sum of the residual of yexp - yfit. In experiment, however, the data is always suffered by a noise, and the residual sum must be affected by the magnitude of noise. Since the noise is considered to be a white noise, we reasonably assume that the residual follows a normal distribution, and the probability density of the residual at

Experimental procedure

Li-poor and rich samples of (La0.62Li0.15)TiO3, (La0.53Li0.40)TiO3 and (La0.31Li0.07)NbO3 were synthesized via the solid phase method. To synthesize (La2/3-xLi3x)TiO3 (LLTO), the pure raw materials of Li2CO3, La(OH)3 and TiO2 were mixed by a ball milling and the mixture was annealed in air at 1170 K for 12 h. The calcined powder was pelletized and pressed at 246 MPa, and the pellet was then sintered in air at 1520 K for 12 h. To synthesize (La0.31Li0.07)NbO3 (LLNbO), the pure raw materials of Li

Results and discussion

To investigate the ionic conductivity of LLTO polycrystals, we performed impedance spectrum analysis of LLTO by using the random walk Metropolis Hastings algorithm. For comparison, we also performed spectrum analysis by the steepest descent method. Hereafter, we analyze the impedance spectrum of Li-poor (La0.62Li0.15)TiO3 to demonstrate the random walk Metropolis Hasting algorithm, and later we will discuss the spectra difference between Li-rich and Li-poor samples. To demonstrate the clear

Conclusion

In summary, we proposed the random walk Metropolis Hastings algorithm for the analysis of an electrochemical impedance spectrum. Unlike the gradient method, we can avoid the trap of the local minimum and find the unique (global minimum) solution of the fitting parameters. Using the present algorithm, we demonstrated the excellent fitting convergence of the impedance spectra in LLTO and LLNbO starting with different initial parameter sets. We also estimate the error range of the fitting

Acknowledgement

A part of this work was supported by the Research and Development Initiative for Scientific Innovation of New Generation Batteries (RISING2) project of the New Energy and Industrial Technology Development Organization (NEDO), Japan.

References (40)

  • J. Schnell et al.

    J. Power Sources

    (2018)
  • Z. Zhang et al.

    J. Alloy. Comp.

    (2018)
  • L. Chen et al.

    Nano Energy

    (2018)
  • A. Tron et al.

    J. Solid State Chem.

    (2018)
  • C. Sun et al.

    Nano Energy

    (2017)
  • D. Zhang et al.

    Electrochem. Acta

    (2018)
  • T. Teranishi et al.

    Solid State Ionics

    (2016)
  • F. Aguesse et al.

    Solid State Ionics

    (2015)
  • L. Wang et al.

    Int. J. Electrochem. Sci.

    (2012)
  • Y. Tong et al.

    Appl. Surf. Sci.

    (2017)
  • Z. Wang et al.

    Surface. Interfac.

    (2017)
  • K. Juttner

    Electrochem. Acta

    (1990)
  • Y. Inaguma et al.

    Solid State Commun.

    (1993)
  • Y. Inaguma et al.

    Solid State Ionics

    (1994)
  • D.V. Rubtsov et al.

    J. Magn. Reson.

    (2007)
  • K. Nagata et al.

    Neural Network.

    (2008)
  • K. Nagata et al.

    Neural Network.

    (2012)
  • S. Sasano et al.

    Appl. Phys. Exp.

    (2017)
  • X. Gao et al.

    Chem. Mater.

    (2013)
  • M.S. Whittingham et al.

    Solid State Ionics

    (2018)
  • Cited by (20)

    • Formation of La-rich tysonite nano-precipitates in fluorite Ba<inf>0.6</inf>La<inf>0.4</inf>F<inf>2.4</inf>

      2023, Journal of Power Sources
      Citation Excerpt :

      To evaluate the activation energy for the F− ion conductivity, we measured temperature dependence of the impedance spectra in the range of 248 K–466 K. The impedance spectra were precisely evaluated by the random walk Metropolis-Hastings algorithm [17–19]. The BLF sample was gently crushed in ethanol and dispersed on a holy amorphous carbon grid for S/TEM observation.

    • A porous ceramic separator prepared from natural minerals: Research on the mechanism of high liquid absorption and electrochemical properties of mineral material separator

      2021, Materials Chemistry and Physics
      Citation Excerpt :

      Fig. 6b shows the fitting graph of EIS curve of cell assembled with PCS-50 after 100 cycles. The components of the fitting circuit include the interface impedance (Rb) caused by the battery case, electrolyte, separator and other components, the double electric layer capacitance (CPE) caused by the interface charge, the impedance caused by defects and substance diffusion (Rct), capacitive element impedance caused by periodic changes in concentration (W) [26]. Rct values of cell with PCS-50 before and after 100 cycles are 236.5Ω and 703.2Ω, respectively.

    • Sampling methods for solving Bayesian model updating problems: A tutorial

      2021, Mechanical Systems and Signal Processing
      Citation Excerpt :

      The MH algorithm has been implemented in numerous areas of research. For instance, to predict precipitation behaviours in Nickel-Titanium alloys via Bayesian probability [129]; to analyse an electrochemical impedance spectra and estimate the conductivity of a Lithium ion within a solid-state oxide electrolyte [130]; to predict and quantify the uncertainty associated with the forecasts for daily river flow rate of Zhujiachuan River [131]; and to sample classical thermal states from one-dimensional Bose–Einstein quasi-condensates under the classical fields approximation [132]; to update the finite element model of a concrete structure [133]; to quantify the uncertainty associated with the joint model parameters of a stochastic generic joint model [51]; to perform joint input-state-parameter estimation for wave loading [52]; to perform Bayesian system identification of dynamical systems [134]; and to perform Bayesian model identification of higher-order frequency response functions of structures [45]. The TMCMC sampler has already been applied in different fields.

    • Room temperature fluoride ion conductivity in defective β-KSb<inf>1-δ</inf>F<inf>4-3δ</inf> polycrystals

      2021, Journal of Power Sources
      Citation Excerpt :

      To evaluate the activation energy for the F− ion conductivity, we measured temperature dependence of the impedance spectrum in the range of 300–328 K. The impedance spectra were precisely evaluated by the random walk Metropolis-Hastings algorithm [29,30]. We aquired mass spectra during heat treatment by quadrupole mass (qmass) analyzer (PrismaPro QMG250, Pfeiffer Vacuum), where the pressure in the furnace was below 3 × 10−4 Pa.

    • Dielectric relaxation of neodymium chloride in water and in methanol

      2020, Journal of Molecular Liquids
      Citation Excerpt :

      In the first step, the Metropolis-Hastings algorithm [51] was applied to correct for the electrode polarization (EP) effect and to estimate both the specific conductance (κ) and the permittivity at the high-frequency limit (ϵ∞). The Metropolis-Hastings algorithm [51], a Markov Chain Monte Carlo method, is easy to implement and does not show a significant dependence on the initial estimate of the parameters [52]. It was also used to obtain relaxation time distributions [53].

    View all citing articles on Scopus
    View full text