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