Analytical distribution function of relaxation times for porous electrodes and analysis of the distributions of time constants

Highlights

• An analytical distribution function of relaxation times of porous electrodes.

• Distributions of time constants corresponding to physical processes are revealed.

• Impedance reconstruction of transmission line models is demonstrated.

• Explicit treatment of low-frequency capacitance

Abstract

Based on the work by Boukamp (Boukamp, 2017), the method of Fuoss and Kirkwood (Fuoss and Kirkwood, 1941) is applied to derive an analytical distribution function of relaxation times for physics based porous electrode impedance cases. These impedance models are typically described by transcendental transfer functions. The porous electrode impedance treated here reflects a balance of the effective ionic and electronic impedances inside a porous electrode consisting of particles. Therefore, first the DFRT of the single particle interface impedance is derived. This includes treatment of charge transfer, double layer charging, solid state diffusion inside the particles, open-circuit voltage variations due to solid-state concentration, and insulating layers surrounding the particles. The resulting single particle DFRT relations are then incorporated into a mathematical description of the porous electrode DFRT. The results show that the DFRT of the porous electrode can be clearly separated into distributions of time constants corresponding to charge transfer, solid state diffusion and in case of intercalating particles, like in lithium-ion batteries, a third distribution of time constants is identified. A novelty of this work is the explicit treatment of the low-frequency capacitance and the resulting distribution of time constants in porous electrode systems. Analytical relations for the individual time constants are derived and reported. Since the ideal distribution of time constants can be represented by a series of R||C circuit elements, validation is performed by reconstruction of the impedance spectra, based on the analytical results.

Introduction

In recent years electrochemical impedance spectroscopy (EIS) has become an increasingly popular tool for the analysis of electrochemical systems like fuel cells and lithium-ion batteries. The general principles and applications of EIS are very well covered in literature [3,4]. Since the EIS method is essentially noninvasive, EIS has become an important method for performing fundamental investigations about the kinetic processes inside electrochemical cells. The interpretation of the measured data is usually accomplished by fitting the data with a suitable equivalent circuit model (EQCM) that is supposed to correlate to the physics of the system under investigation by means of a complex non-linear least squares fit [5]. Oftentimes the measured dynamical processes of the electrochemical system are happening in overlapping time scales leading to convoluted spectra. Inferring a suitable equivalent circuit model in such a case can be difficult, see Fig. 1 a), for a synthetically generated example of a convoluted spectrum. To help with the correct interpretation of the data, the method of transforming the measured impedance spectra into a distribution function of relaxation times (DFRT) has become popular during recent years [[6], [7], [8], [9], [10], [11]]. Important contributions have demonstrated the beneficial effect of performing the transformation and how it can help with the interpretation of the data [[12], [13], [14], [15], [16]]. The distribution function of relaxation times is defined in the following equation:

$$Z\left(\omega \right)=R_{\mathrm{\Omega }}+R_{\mathit{pol}}{\int }_{-\infty }^{+\infty }\frac{G\left(\tau \right)}{1+\mathit{i\omega \tau }}d\mathit{ln\tau }$$

Here Z(ω) is the impedance, R_Ω is the high-frequency direct current (DC) resistance, R_pol is the polarization resistance and G(τ) is the distribution function of relaxation times. In a general sense the distribution function G(τ) distributes the polarization resistance R_pol over an infinite number of R||C circuit elements, where the resistor is in parallel to a capacitor. Each R||C element corresponds to a unique time constant τ_i. Deriving G(τ) in Eq. (1) from an actual measurement is an ill-posed inverse problem which makes the transformation of measured impedance spectra into a DFRT challenging. Different methods exist to perform the transformation and these have been discussed in detail in [10]. Due to the nature of the applied mathematical and numerical procedures actual peak shape and peak occurrence can vary depending on the applied method. From this it follows that the user is required to be aware of the intricacies of the methods and the general physics of the system under investigation. One of the conclusions is that care must be taken in the interpretation of the DFRT. Several empirical impedance functions have a direct analytical representation in τ-space like the Hraviliak-Negami dispersion of which the Gerischer impedance is a special case [17]. One goal of the work presented here is to increase the number of analytical tools for the analysis of DFRT spectra. Applying the DFRT transformation to measured EIS spectra can help with analyzing contained information since, as shown in Fig. 1, the DFRT is able to deconvolve the spectra. As a result of the DFRT it will help with finding a suitable EQCM to analyse and quantify the measurement. For example, in Fig. 1 b), the resulting DFRT of the spectrum of Fig. 1 a) shows two gaussian shaped peaks, indicating that the spectrum contains two closely positioned time-constants. It is now possible to use this information as a starting point for developing a suitable EQCM consisting of two R||C or R||CPE-elements (constant phase element) to model the two time-constants. The area under the peaks corresponds to the resistance values respectively.

Compared to this illustrative case the spectra of porous electrodes as found in polymer membrane fuel-cells (PEM-FC) and lithium-ion batteries (LIB) are highly convolved. Therefore, the interpretation of measured impedance data of these systems and development of a proper EQCM can be difficult. Using a synthetically generated spectrum of a porous electrode, see Fig. 1 c), and performing the DFRT analysis results in the DFRT seen in Fig. 1 d). The DFRT is itself severely more challenging to interpret. Counting peaks and assigning time-constants would result in an EQCM, that will likely generate an adequate fit to the data but interpretability of the EQCM in terms of the physics of the system under investigation would be more challenging. In a recent publication, Boukamp used the transformation method of Fuoss and Kirkwood [2] to derive an analytical DFRT for the fractal finite length Warburg diffusion impedance [1,16]. Based on his analysis he could demonstrate that a distributed impedance element like the finite length Warburg diffusion with transmissive boundary condition (FLW) is represented by a defined peak sequence in the τ-domain. Since porous electrodes inherently include the distribution of the interface impedance due to effective ionic and electronic conductivities throughout the thickness of the electrode, the investigation of an analytical DFRT can yield additional ways of analyzing measured impedance spectra and DFRT spectra of porous electrodes and help with finding suitable EQCM for fitting and interpreting the measurement data. This work aims at extending the presented method of [1] to porous electrode impedances with a focus on lithium-ion battery electrodes to provide an explanation of the observed peak-patterns as in Fig. 1, d), and how they might relate to the physics of a porous battery electrode. The following analysis extends the work presented in [1] and derives the analytical DFRT by applying the method from Fuoss and Kirkwood [2] to a well-known physics-based porous electrode impedance model and several interface impedance cases. The analysis presented here aims at explaining observed peak patterns of DFRT spectra of porous electrodes in terms of the physics of the system. It is therefore focussed on the analysis of a mathematical model of the porous electrode. Future work aims at extending and applying the approach to experimentally measured impedance spectra.

As shown in the work by Meyers et al., the porous electrode impedance model is comprised of two parts [5]. The first part is the interface impedance relation, Z_IF(ω), that describes the impedance of the processes occurring at the electrochemical interface of catalysts or active materials. The second part is the porous electrode impedance equation, which is a function of the interface impedance, Z_PE(ω, Z_IF), and is generally applicable to many electrochemical systems. The porous electrode impedance equation represents the generalized transfer function of a class of ordinary differential equations under a fixed set of boundary conditions [18]. The application to different systems requires knowledge of the specific interface impedance, Z_IF, of the system under investigation. Following this line of thought, the paper is structured as follows: In a first step the DFRT of the interface impedance is calculated. A battery active material with charge accumulation will serve as the example case. This requires the derivation of the DFRT of diffusion impedances with reflective boundary condition. This complements the results from [1], where a transmissive boundary condition was used. A detailed derivation of the diffusion impedance relations with regards to the boundary conditions can be found in [19] and a short summary is given in the appendix. In a second step the general DFRT equation of the porous electrode DFRT based on the impedance relation by Meyers is derived and analyzed for different interface impedance cases [5]. From the resulting mathematical description analytical relations can be derived. These relations describe the effect of the physical parameters of the porous electrode on the distribution of relaxation times and are presented for the first time in such a way to our knowledge.