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Electrochemical Impedance Spectroscopy in a Fuel Cell
Introduction
The electrochemical (AC) impedance technique is based on the analysis of current response to small sinusoidal perturbation in potential. This perturbation is applied for a given frequency range, from hundreds of kHz down to the mHz scale. The variation in frequency response makes it possible to separate processes and mechanisms in the electrochemical phenomena that is occurring at electrode surfaces. In particular, AC impedance or impedance spectroscopy is utilized to separate slow processes from fast, by analyzing the phase shift, while the impedance absolute value can also give a lot of information about electrochemical systems.
Electrochemical impedance spectroscopy (EIS) is used in all areas of electrochemistry such as fuel cells, batteries, electrolytic processes, and corrosion. Information gathered from such studies helps in extracting values for important design parameters in electrochemical processes, such as effective transport properties, reaction kinetics and mechanisms, and Ohmic losses (Ref. 1 and Ref. 2).
This example looks specifically at the AC impedance of a porous cathode of a generic fuel cell in 1D.
At open circuit and for high electrode conductivities, the model has been validated against the analytical solution presented in Ref. 3.
Model Definition
The geometry is set up as a 1D model, according to Figure 1. The geometry represents a fuel cell half cell and consists of two domains: a membrane and a gas diffusion electrode (GDE) domain.
Figure 1: Half cell 1D geometry.
The Hydrogen Fuel Cell interface is used to model a secondary current distribution along with a Butler-Volmer expression to describe the fuel cell cathode kinetics in the GDE.
The electrolyte potential at the outer membrane boundary is set to zero. At the outer GDE boundary the electrode current density is set to −0.1 A/cm2. A harmonic perturbation of 5 mA/cm2 is also applied to this boundary.
The model is solved in two steps. The first step solves for the stationary solution, in a second step solves for the frequency domain, for a range of frequencies from 1 to 105 Hz. Additionally, a parametric sweep is set up for the electrical conductivity of the GDE.
Results and Discussion
Figure 2 shows the Nyquist plot of the impedance for the two different values for the electrical conductivity in the GDE.
Figure 2: Nyquist plot.
At high frequencies, the double layer contributes with very low impedance, which means that the alternating current is transferred between the electrode and the electrolyte at a position close to the free electrolyte-electrode interface. The real part of the impedance, given by the Ohmic losses in the free electrolyte, therefore dominates the total impedance at high frequencies. For electrodes of high electronic conductivity, compared to the electrolyte, the Nyquist plot can be used to extract the pure Ohmic resistance in the electrolyte during measurements. Extrapolating the impedance from the frequency of 50 kHz to the intersection with the x-axis in Figure 2, gives a value for the real impedance of around 0.2 Ωcm2.
The diameter of the semicircle in the Nyquist plot gives a measure of the kinetic resistance in the electrode. If the performance of the electrode is limited by the electrode kinetics, and the current distribution is fairly uniform along the thickness of the electrode, the diameter gives the Tafel slope divided by the current density (Ref. 1). The expected value of the diameter, at a steady state current of 0.1 A/cm2 in a unit square, is according to the following:
The value obtained from the simulation is 0.64 Ωcm2. The deviation from the value in the above calculation is due to the fact that the current distribution is not perfectly uniform in our simulation.
Figure 2 also shows that the real part of the impedance at high frequency increases from around 0.2 Ωcm2 to 0.3 Ωcm2 when lowering the electrode conductivity. At very low frequencies the difference in the real part impedance is lower than for high frequencies, indicating that the current distribution in the electrode is more uniform.
In a detailed study of the electrode, the current density is varied from small to large values and the EIS simulations are compared at each single current density. The model should be able to describe the qualitative and quantitative changes in both impedance spectra and steady-state performance along the whole polarization of the electrode.
When including spatial dependencies in more dimensions, the effects of nonuniform current densities, and their influence on the impedance spectra, become even more intricate.
References
1. C. Lagergren, Electrochemical Performance of Porous MCFC Cathodes, Doctoral Thesis, Department of Chemical Engineering and Technology, Applied Electrochemistry, Kungliga Tekniska Högskolan, Stockholm, Sweden, 1997.
2. F. Jaouen, Electrochemical Characterization of Porous Cathodes in the Polymer Electrolyte Fuel Cell, Doctoral Thesis, Department of Chemical Engineering and Technology, Applied Electrochemistry, Kungliga Tekniska Högskolan, Stockholm, Sweden, 2003.
3. G. Lindbergh, “Experimental determination of the effective electrolyte conductivity in porous lead electrodes in the lead-acid battery,” Electrochimica Acta, vol. 42, no. 8, pp. 1239–1246, 1997.
Application Library path: Fuel_Cell_and_Electrolyzer_Module/Fuel_Cells/ac_fuel_cell
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  1D.
2
In the Select Physics tree, select Electrochemistry > Hydrogen Fuel Cells > Generic (fc).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select Preset Studies for Selected Physics Interfaces > AC Impedance, Stationary.
6
Click  Done.
Global Definitions
Load model parameters from a text file.
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file ac_fuel_cell_parameters.txt.
Geometry 1
Create the geometry consisting of two intervals, the membrane and the oxygen gas diffusion electrode (O2 GDE).
Interval 1 (i1)
1
In the Model Builder window, under Component 1 (comp1) right-click Geometry 1 and choose Interval.
2
In the Settings window for Interval, locate the Interval section.
3
From the Specify list, choose Interval lengths.
4
In the table, enter the following settings:
Lengths (m)
L
L
5
Click  Build All Objects.
Hydrogen Fuel Cell (fc)
Set up the electrochemical model. Start with adding the relevant domain nodes.
Membrane 1
1
In the Physics toolbar, click  Domains and choose Membrane.
2
Select Domain 1 only.
O2 Gas Diffusion Electrode 1
1
In the Physics toolbar, click  Domains and choose O2 Gas Diffusion Electrode.
2
Select Domain 2 only.
Electrolyte Phase 1
Set up the electrolyte conductivity in the Electrolyte Phase node.
1
In the Model Builder window, click Electrolyte Phase 1.
2
In the Settings window for Electrolyte Phase, locate the Electrolyte Charge Transport section.
3
From the σl list, choose User defined. In the associated text field, type sigmal.
O2 Gas Diffusion Electrode 1
Set up the electric conductivity and the effective electrolyte conductivity correction in the O2 Gas Diffusion Electrode node. The details of electrode kinetics and double layer are set in the child nodes.
1
In the Model Builder window, click O2 Gas Diffusion Electrode 1.
2
In the Settings window for O2 Gas Diffusion Electrode, locate the Electrode Charge Transport section.
3
In the σs text field, type k.
The electric conductivity of the O2 GDE will be used later when setting up a parametric sweep.
4
Locate the Effective Electrolyte Charge Transport section. From the Effective conductivity correction list, choose User defined. In the fl text field, type fl.
O2 Gas Diffusion Electrode Reaction 1
1
In the Model Builder window, click O2 Gas Diffusion Electrode Reaction 1.
2
In the Settings window for O2 Gas Diffusion Electrode Reaction, locate the Equilibrium Potential section.
3
From the Eeq list, choose User defined. In the associated text field, type Eeq.
4
Locate the Electrode Kinetics section. In the i0 text field, type i0.
5
Locate the Active Specific Surface Area section. In the av text field, type av.
O2 Gas Diffusion Electrode 1
In the Model Builder window, click O2 Gas Diffusion Electrode 1.
Porous Matrix Double Layer Capacitance 1
1
In the Physics toolbar, click  Attributes and choose Porous Matrix Double Layer Capacitance.
2
In the Settings window for Porous Matrix Double Layer Capacitance, locate the Porous Matrix Double Layer Capacitance section.
3
In the Cdl text field, type Cdl.
4
In the av,dl text field, type avdl.
Electrolyte Phase 1
Finally, set up the boundary conditions.
1
In the Model Builder window, under Component 1 (comp1) > Hydrogen Fuel Cell (fc) click Electrolyte Phase 1.
Electrolyte Potential 1
1
In the Physics toolbar, click  Attributes and choose Electrolyte Potential.
2
Select Boundary 1 only.
Electronic Conducting Phase 1
In the Model Builder window, under Component 1 (comp1) > Hydrogen Fuel Cell (fc) click Electronic Conducting Phase 1.
Electrode Current 1
1
In the Physics toolbar, click  Attributes and choose Electrode Current.
2
Select Boundary 3 only.
3
In the Settings window for Electrode Current, locate the Electrode Current section.
4
From the list, choose Average current density.
5
In the is,average text field, type iavg.
6
Click to expand the Harmonic Perturbation section. Add a harmonic perturbation to be used in the frequency domain study.
7
Select the Include harmonic perturbation checkbox.
8
In the Δis,average text field, type delta_iavg.
Global Definitions
Default Model Inputs
Set up the temperature value used in the entire model.
1
In the Model Builder window, under Global Definitions click Default Model Inputs.
2
In the Settings window for Default Model Inputs, locate the Browse Model Inputs section.
3
In the tree, select General > Temperature (K) - minput.T.
4
Find the Expression for remaining selection subsection. In the Temperature text field, type T.
Study 1
Add a parametric sweep to solve for two different electric conductivities of the O2 GDE.
Parametric Sweep
1
In the Study toolbar, click  Parametric Sweep.
2
In the Settings window for Parametric Sweep, locate the Study Settings section.
3
Click  Add.
4
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
k (Electric conductivity in porous electrode)
10 1000
S/m
Step 2: Frequency-Domain Perturbation
Set up the frequency range of the solver.
1
In the Model Builder window, click Step 2: Frequency-Domain Perturbation.
2
In the Settings window for Frequency-Domain Perturbation, locate the Study Settings section.
3
In the Frequencies text field, type 10^range(0,0.05,4.95) 10^range(5,0.5,7).
Step 1: Stationary
The problem is now ready for solving. Lower the relative tolerance of the first solver step in the default solver sequence. This will improve the accuracy of the static solution for which the frequency perturbation study step is run. Then solve the problem.
1
In the Model Builder window, click Step 1: Stationary.
2
In the Settings window for Stationary, locate the Study Settings section.
3
From the Tolerance list, choose User controlled.
4
In the Relative tolerance text field, type 1e-5.
5
In the Study toolbar, click  Compute.
Results
Impedance with Respect to Ground, Nyquist (fc)
In the Settings window for 1D Plot Group, click  Plot.