Fundamentals of Electrochemical Impedance Spectroscopy for Macrohomogeneous Porous Electrodes

doi:10.13208/j.electrochem.201126

Abstract

Abstract:

Electrochemical impedance spectroscopy (EIS) can be used to diagnose charge transfer reactions and mass transport in porous electrodes. The charge transfer reactions include interfacial charge accumulation and charge conduction as well as electrochemical reaction. In this paper, the complex phasor method is developed under the macrohomogeneous assumption to build an impedance model of porous electrodes for clarifying several vague expressions in the traditional approaches. The following researches are carried out: (1) Identifying characteristic parameters for the porous electrodes, including electrode electronic conductivity σ₁, electrolyte ionic conductivity σ₂, interface charge transfer conductivity g_ct, unit area interface capacitance C, solid phase diffusion coefficient D, rate constant k, electrode thickness d, characteristic hole depth L_p and unit volume surface area S_c; (2) elucidating characteristic output parameters for the impedance spectroscopic response, including field diffusion constant K, characteristic frequencies ω₀, ω₁, ω₂, ω₃, and ω_max for interface conduction reaction, finite field diffusion, redox reaction, pore diffusion and minimum characteristic pore size, respectively. In addition, the transition frequencies f_k₁ and f_k₂ from conduction to diffusion area and from diffusion to saturation area are also defined and studied respectively; (3) defining the parameters X and Z, herein, X = σ₁,Z = d、S_c, L_p, C, g_ct, D, k,which are responsible for the evolution trend of the characteristic parameters for impedance spectroscopic response, the competition effects of X and Z parameters coupled in charge transfer reaction are analyzed; (4) Further analyzing the competition effects of X and Z parameters coupled in the charge transfer reaction, the diverging frequencies f_XZ and f_XZ are phenomenologically defined. The locations of f_XZ and f_XZ can indicate the depth and breadth of the charge transfer reaction affected by the parameters X and Z. The non-existence of f_XZ and f_XZ indicates that the parameter X or Z can affect the charge transfer reaction over the whole frequency range. With the help of characteristic frequency and diverging frequency, the effects of electrode kinetic and microstructure parameters on the charge transfer reaction in porous electrodes are studied; on the other hand, the shape change and trend evolution of the impedance responses for porous electrodes are analyzed. The research results in this paper should be able to provide theoretical basis for system simulation and system identification of impedance spectroscopy, technical support for competitive analysis of charge transfer reaction in porous electrodes, and diagnostic tool for optimal design of electrochemical energy storage system.

Key words: porous electrode, charge transfer reaction, electrochemical impedance spectroscopy, characteristic frequencies, diverging frequencies

Xiang Li, Qiu-An Huang, Wei-Heng Li, Yu-Xuan Bai, Jia Wang, Yang Liu, Yu-Feng Zhao, Juan Wang, Jiu-Jun Zhang. Fundamentals of Electrochemical Impedance Spectroscopy for Macrohomogeneous Porous Electrodes[J]. Journal of Electrochemistry, 2021, 27(5): 467-497.

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Parameter	Default	References
Electrode thickness (d)	90 μm	[27, 28]
Unit area interface capacitance (C)	3×10^-5 F·cm^-2	[29-31]
Diffusion coefficient (D)	1×10^-13 cm²·s^-1	[32-34]
Characteristic hole depth (L_p)	90 nm	[29]
Unit volume surface area (S_c)	2×10⁴ cm^-1	[35,36]
Rate constant (k)	1×10^-7 cm·s^-1	[37,38]
Electronic conductivity (σ₁ )	5×10^-1 S·cm^-1	[39,40]
Electrolyte ionic conductivity (σ₂)	5×10^-3 S·cm^-1	[41-43]
Interface charge transfer conductivity (g_ct)	7.6 S·cm^-1	[29, 30]

σ₁(S·cm^-1)	1σ₀	10σ₀	100σ₀
K(cm²·s^-1)	0.0042	0.0076	0.0083
ω_max(rad·s^-1)	5×10⁷	9×10⁷	1×10⁸
ω₀(rad·s^-1)	22	22	22
ω₁(rad·s^-1)	51	94	102
ω₂(rad·s^-1)	0.1	0.1	0.1
ω₃(rad·s^-1)	0.0012	0.0012	0.0012
f_k₁(Hz)	0.195	0.195	0.195
f_k₂(Hz)	0.001	0.001	0.001

g_ct(S·cm^-1)	σ₁ =1σ₀			σ₁ =10σ₀			σ₁ =100σ₀
g_ct(S·cm^-1)	1g₀	2g₀	10g₀	1g₀	2g₀	10g₀	1g₀	2g₀	10g₀
K(cm²·s^-1)	0.0042	0.0042	0.0042	0.0076	0.0076	0.0076	0.0083	0.0083	0.0083
ω_max(rad·s^-1)	5×10⁷	5×10⁷	5×10⁷	9×10⁷	9×10⁷	9×10⁷	1×10⁸	1×10⁸	1×10⁸
ω₀(rad·s^-1)	11	22	111	11	22	111	11	22	111
ω₁(rad·s^-1)	51	51	51	94	94	94	102	102	102
ω₂(rad·s^-1)	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1
ω₃(rad·s^-1)	0.0012	0.0012	0.0012	0.0012	0.0012	0.0012	0.0012	0.0012	0.0012
f_k₁(Hz)	0.119	0.195	0.557	0.119	0.195	0.557	0.119	0.195	0.557
f_k₂(Hz)	0.001	0.001	0.001	0.001	0.001	0.001	0.001	0.001	0.001

C(F·cm^-2)	σ₁=1σ₀			σ₁=10σ₀			σ₁=100σ₀
C(F·cm^-2)	1C₀	6C₀	36C₀	1C₀	6C₀	36C₀	1C₀	6C₀	36C₀
K(cm²·s^-1)	0.025	0.0042	0.0007	0.045	0.0076	0.0013	0.05	0.0083	0.0014
ω_max(rad·s^-1)	3×10⁸	5×10⁷	8×10⁶	5.6×10⁸	9×10⁷	1.5×10⁷	6×10⁸	1×10⁸	1.7×10⁷
ω₀(rad·s^-1)	133	22	4	133	22	4	133	22	4
ω₁(rad·s^-1)	309	51	9	561	94	16	611	102	17
ω₂(rad·s^-1)	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1
ω₃(rad·s^-1)	0.0012	0.0012	0.0012	0.0012	0.0012	0.0012	0.0012	0.0012	0.0012
f_k₁(Hz)	0.59	0.195	0.05	0.59	0.195	0.05	0.59	0.195	0.05
f_k₂(Hz)	0.001	0.001	0.001	0.001	0.001	0.001	0.001	0.001	0.001

D(cm²·s^-1)	σ₁=1σ₀			σ₁=10σ₀			σ₁=100σ₀
D(cm²·s^-1)	1D₀	10D₀	100D₀	1D₀	10D₀	100D₀	1D₀	10D₀	100D₀
K(cm²·s^-1)	0.0042	0.0042	0.0042	0.0076	0.0076	0.0076	0.0083	0.0083	0.0083
ω_max(rad·s^-1)	5×10⁷	5×10⁷	5×10⁷	9×10⁷	9×10⁷	9×10⁷	1×10⁸	1×10⁸	1×10⁸
ω₀(rad·s^-1)	22	22	22	22	22	22	22	22	22
ω₁(rad·s^-1)	51	51	51	94	94	94	102	102	102
ω₂(rad·s^-1)	0.1	0.01	0.001	0.1	0.01	0.001	0.1	0.01	0.001
ω₃(rad·s^-1)	0.0012	0.012	0.12	0.0012	0.012	0.12	0.0012	0.012	0.12
f_k₁(Hz)	0.195	0.087	0.054	0.195	0.087	0.054	0.195	0.087	0.054
f_k₂(Hz)	0.001	0.011	——	0.001	0.011	——	0.001	0.011	——