Analytical solution for the impedance of porous electrodes Introduction: To predict the dynamic model of battery behavior and diagnosis of quantities like actual capacity, state of charge and state of health, a clear understanding of transport properties and reaction processes occurring in porous electrodes is required and is a subject of permanent interest. These electrodes are under mixed control of electrode kinetics, mass transfer and ionic conduction in the electrodes. Electrochemical Impedance Spectroscopy (EIS) is a powerful means of analyzing the properties of such a complex systems. The basic experiment of EIS consists of applying a small, sinusoidal voltage or current signal to an electrochemical cell, measuring the system response with respect to the amplitude and the phase, determining the impedance of the system by complex division of ac voltage by ac current and repeating this for a certain range of different frequencies. However the limitation of the EIS studies has been the interpretation of the impedance spectra. The approach has been to fit data to electrical circuits composed of passive and frequency dependant circuit elements. More sophisticated approaches to interpret the data based on macroscopical models of the electrochemical devices numerically. Nevertheless the power of the simulation of impedance data from the analytical solution of the model equations is threefold. First is the accuracy second is the flexibility (new information maybe readily incorporated into the model) and the third is that the parameters used in the model to fit the impedance data are independently measurable and meaningful. Literature survey: Beginning with De Levie there exist a number of papers dealing with the theory of impedance of porous electrodes from various aspects. In analyzing the experimental data, most investigators used the modifications of the basic equivalent circuits involving the constant phase element (CPE) or a Warburg diffusion to account for mass transport in the electrodes. The impedance of the porous electrodes was modeled assuming that the electrode is composed of cylindrical pores of definite length and thickness, filled with electrolyte. In the presence of a constant current circulating to the electrode, potential and concentration gradients are formed inside the pores. Using this basic model the authors developed the analytical solution based on the limitations like, the absence of potential gradient, absence of concentration gradient, impedance independent of the distance within the pore. In case of both potential and concentration gradients only numerical solutions exists. Some authors used the branched transmission line equivalent circuit to describe the impedance. According to this theory the electrodes are considered to operate in contact with an electrolyte by the simultaneous transport of electronic and ionic species in the solid and liquid phase respectively. The solid phase in contact with conducting substrate provides a continuous path for transport of electrons. The electrolyte penetrates the void in the solid phase up to the substrate. The dimension of the both solid structural element and the liquid channels are considered to be quite small. The transport of charge carriers in both the phases is influenced by complex mechanisms of mass transport, double layer charging and the charge transfer occurring at the interface. Yet another approach is the impedance as developed by Paasch et al. considered an effective homogeneous mixture two phases in electrode region. Their paper presented a study that deals with macroscopically homogeneous porous electrodes, for which an analytical solution of the impedance was developed in the absence of the concentration gradient. The theory was further generalized by including an arbitrary time delay of the charge transfer process at the pore surface. Time delaying processes considered were the control of charge transfer by finite diffusion and control by both diffusion and charge storage on a molecular scale. Model development: Consider a planar porous electrode of thickness lp which interfaces with an electrolyte at one end and a metallic current collector at the other end. Both the material and the pores are interconnected and the electrode is macroscopically homogeneous. Electrode Geometry: Separator Li Metal Current Collector Electrode X=0 X= lp Assumptions: • One-dimensional model. • Diffusion coefficient and transference number are constant and do not vary with concentration. • Double layer capacitance is constant and does not vary with the solution phase potential and concentration. • Porosity is constant and uniform throughout the electrode. • Linear impedance response in considered. • The transport process in the solution phase is described by the concentrated solution theory. • Porous electrode theory is applied to develop the mass and charge balance in the electrode • Solid phase diffusion is neglected. • An irreversible reaction is considered at the solid liquid interface. • The thickness of the separator is considered very small. • The equations, boundary conditions and the solution are in terms of the perturbed variable. Material Balance in the solution phase taking into account the charge conservation in the electrode is given by aC ∂η ∂C ∂ 2C (1 − t+0 ) ε = ε Deff + a (1 − t+0 ) j + dl 2 F ∂t ∂t ∂x The right hand side of the above equation consists of the terms that accounts for the diffusion, the electrode kinetics, and the charge conservation in the porous electrode. The electrode kinetics is assumed for an irreversible electrochemical reaction and is given by the simple Butler-Volmer equation. α F α F jF = i0 exp a η − exp c η RT RT η = φ1 − φ2 − U U =0 U is the open circuit potential. The governing equation that describe the potential gradients in the electrode are given by the Ohms law in the solid phase and the solution phase as σ eff ∂φ1 ∂η = aFj + aCdl ∂x ∂t iapp + σ eff ∂φ1 ∂φ 2 RT κ eff = κ eff 2 + ∂x ∂x FC0 d ln f ± 0 ∂ ln C 1 + 1 − t+ ∂x d ln C C0 ( ) C0 is the initial concentration. The effective diffusion coefficient is related to the bulk diffusion coefficient as Deff = ε 0.5 D The effective solid phase conductivity and the solution phase conductivity are related to the bulk solid phase conductivity and the bulk solution phase conductivity as κ eff = ε 1.5κ σ eff = (1 − ε )1.5 σ The boundary conditions for the concentration are zero flux at the electrode-current collector interface and the concentration being zero at the center of the electrode due to cemetery. Similarly the boundary conditions for the overpotential is based on the fact that the current is carried by the solid phase at the electrode-current collector interface and the oeverpotential being zero at the center of the electrode due to cemetery. Determination of the impedance: Consider a sinusoidal current with amplitude of iapp applied to the porous electrode to perturb the system. The amplitude is very small so that it does not introduce a permanent change of the system from its equilibrium value. Impedance across a porous electrode is given by the potential drop across the porous electrode divided by the applied current. The potential drop across the electrode is given by ∆E = [φ1 ]x =l − [φ2 ]x =0 P Z= ∆E iapp The governing equations for the concentration and the overpotential are solved with the boundary conditions to obtain the concentration and overpotential as a function of the distance (x) from the electrode-electrolyte interface using Maple. This gives the potential drop across the electrode and thus the impedance of the system. Results and discussion: The analytical solution of the impedance in the laplace domain is given by Imp := 2 + p2 α1 β1 sinh( λ 1 ) cosh( λ 1 ) ( −β1 + α1 ) λ1 2 p1 β1 sinh( λ 1 ) cosh( λ 1 ) ( −β1 + α1 ) λ1 − − 2 p2 β1 α1 sinh( λ 2 ) ( −β1 + α1 ) cosh( λ 2 ) λ2 2 p1 α1 sinh( λ 2 ) ( −β1 + α1 ) cosh( λ 2 ) λ2 + 2 + 2 p1 λ1 , λ2 are the eigen values. They are functions of the S(=i*w) α1 λ2 1 = s B3 − B3 β1 = α := λ1 s B3 − 1 B3 β1 cosh( λ 1 ) ( −β1 + α1 ) β := − α1 ( −β1 + α1 ) cosh( λ 2 ) The functions B3, p1, p2 etc are constants that involve the properties of the materials and the electrolytes.The impedance plot for a set of parameter values are simulated using the model equations is given below 1 2 3 4 5 6 7 8 9 0.0011 10 10 0.0010 9 0.0009 8 0.0008 7 -ZIm(Ohm-cm2) Zim 0.0007 0.0006 6 0.0005 5 0.0004 4 0.0003 3 0.0002 2 0.0001 1 0.0000 0.998 0.999 1.000 1.001 Zre -ZReOhm-cm2) 1.002 1.003 Experimentally measured impedance: 50 1 2 3 4 5 6 7 8 9 cycle-1 cycle-150 cycle-300 40 10 10 9 8 7 30 6 zim -ZIm(Ohm) 20 5 4 10 3 2 0 1 0 50 100 150 200 250 300 zre ZReOhm) The simulation shows the behavior of the porous electrode due to both the concentration gradient and the potential gradient. As shown in the figure the impedance is given by two semicircles. The semicircle in the higher frequency region represents the charge transfer reaction coupled with the double layer capacitance at the interface. The semicircle in the lower frequency region is due to the mass transfer ie. the diffusion of the species due to the concentration gradients. The capacitive behavior of the impedance at very high frequencies gives rise to a 45-degree line. As the frequency is lowered, the behavior changes from capacitive to that of resistance and capacitance coupled together. The 45degree line in porous electrodes is often explained on the basis of the distribution of the capacitance at the solid-liquid interface due to the porous nature of the electrode. The impedance profile can also be explained in terms of the equivalent circuit as given below C CPE C R R Each semicircle is due to the parallel combination of the resistance and capacitance. The resistance and capacitance in the lower frequency region are due to the material resistance and the diffusion due to the concentration gradient. The resistance and capacitance at high frequencies is due to the charge transfer resistance and the distributed double layer capacitance. The time constant of each of the parallel combination of resistance and capacitance is given by RC. The value of each time constants should be comparable in order to observe both interfacial kinetics and mass transfer. Impedance of the carbon porous electrodes measured experimentally is shown below and exhibits a similar behavior. The experimental results could not fit using the model because the model does not include the solid phase diffusion and also the parameter values used to fit the experimental data were not accurate.
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