Fig.4 Model and simulation results. Convenient for discussion, current density (I, A) and η (V) overpotential are positively, η= 0.64 - E. Where E (V) is the cathode potential and is generally negative. The thermodynamic equilibrium potential of hydrogen peroxide generated by oxygen reduction under the conditions of PO2=1atm, pH=1.0, and 298.7K is 0.64v. Actual electrode reaction layer over potential at x = 0, η0 = 0.64 - E - IRL. R = RGDL + RE. RE is the volumetric resistance of the electrolyte and diaphragm. RGDL is the solid-phase volumetric resistance of the diffusion layer of the gas diffusion electrode. L is the thickness of electrolyte (i.e. the distance between the anode and cathode, cm). In Fig.4, the physical meanings and values of the symbols involved in the equation are shown in Table.2. (a) Schematic diagram of gas diffusion electrode section; (b) Schematic diagram of simplified model of reaction layer of gas diffusion electrode;(c) Section diagram of thin-layer plate electrode model;(d) Simulation results of oxygen concentration attenuation with depth x in the dry region under different current densities in the X direction, where:\(C_{g}=C_{g0}\exp\left(\frac{-x}{\text{\ L\ }_{\text{Dg}}}\right)\),\(\text{\ \ }\text{\ L\ }_{\text{Dg}}=\left(\frac{\text{nF}{\tilde{D}}_{g}C_{g0}}{2i^{0}S_{1}}\right)^{\frac{-1}{2}}\sinh^{\frac{-1}{2}}\left(\frac{\eta}{b}\right)\);(e) Simulation results of oxygen concentration attenuation with depth y in wet region under different current density in Y direction, where:\(C_{l}=C_{l0}\frac{\exp\left[\frac{\left(y-\right)}{L_{\text{Dl}}}\right]+exp\left[-\frac{\left(y-\right)}{L_{\text{Dl}}}\right]}{\exp\left(\frac{}{L_{\text{Dl}}}\right)+exp\left(-\frac{}{L_{\text{Dl}}}\right)}\), taking Δ=4Ldl; (f) Simulation results of attenuation of overpotential with depth x in the wet region under different current densities in the X direction, where:\(\eta=\eta_{0}\exp\left(-\frac{x}{L_{\Omega}}\right)\),\(L_{\Omega}=\left(\frac{b}{2i^{0}S_{1}{\tilde{\rho}}_{l}}\right)^{\frac{1}{2}}\ln\left[\frac{\tanh\left(\frac{\eta_{0}}{4b}\right)}{\tanh\left(\frac{\eta_{0}}{4b}-\frac{1}{4}\right)}\right]\), η0 is the electrode reaction layer overpotential at x = 0; (g) According to the derived polarization curve equation (2) and the experimental point, the Tafel value and the exchange current density are fitted. η0= a+ b´logI, due to η0/ b > 1, experimental data points in the region of the strong polarization, the Tafel slope obtained by fitting line value b, and exchange current density i0 obtained by the intercept; (h) The polarization curve is simulated based on the derived equation. Among them: the black line is the simulated ohmic polarization curve of electrolyte and diaphragm under the condition of 3.0cm electrolyte thickness, η=IREL. Fitting empirical equation RE=656*exp(-I/0.315)+29.1; the red line is the simulated polarization curve of electrochemical polarization and wet zone liquid resistance when the electrolyte thickness approaches zero,\(I=\sqrt{\frac{2i^{0}S_{1}b}{{\tilde{\rho}}_{l}}}\left[\exp\left(\frac{\eta_{0}}{2b}\right)-exp\left(-\frac{\eta_{0}}{2b}\right)\right]\);(i) histogram of over potential decomposition under different current densities. Among them: black column for electrochemical polarization (ηact, V), calculated by the equation (2), and red column says ohm polarization caused by wet area liquid resistance (ηohm,R, V), calculated by the equation ηohm,R0act. Blue column says ohm polarization caused by electrolyte and diaphragm (ηohm,E, V), calculated by the equation ηohm,E = IREL. The green point is the experimental point of cathode polarization curve corresponding to different current density. On the right side of the y coordinate for the cathode potential E, E=0.64 – (ηact+ ηohm,R+ ηohm,E).
Gas and liquid phase mass transfer resistance and solid and liquid phase resistance exist in the gas diffusion electrode used in industrial production, resulting in uneven current density and concentration polarization in the electrode, which makes some reaction surfaces in the reaction layer not fully utilized. However, the test results of trace samples in the electrochemical workstation can’t accurately reflect the complex pore structure and pore wall properties of the actual porous carbon electrode. Therefore, it is helpful for engineering amplification to establish a model for the polarization process of the actual gas diffusion electrode and describe the variation law of the actual polarization curves.
The GDE reaction layer represented in Fig.4a is simplified into two structural regions by using the ”thin-layer plate model”. The simplified model of the electrode reaction layer is shown in Fig.4b. One is the ”dry zone”, which consists of hydrophobic components and their surrounding pores. The other is the ”wet zone”, which consists of the electrolyte and the catalyst aggregates soaked in it. These two regions exist in a ”thin and long” form and form a continuous network of staggered arrangements. Assuming that the z-axis direction electrode is uniform, the schematic diagram of the partially enlarged dry and wet area of XY section is shown in Fig.4c.The X axis reaction zone length is equal to d (μm), and Y direction electrode reaction thickness in the wet area is 2Δ(μm).
For the ideal planar electrode with smooth surface, the reaction layer is very thin. The catalyst aggregates are small, and it is evenly mixed with the binder. This situation is similar to the electrochemical workstation, with only electrochemical polarization, and the relationship between current and potential can be described by Bulter-Volmer equation (2).
\(I=2i^{0}S_{1}\text{dsinh}\operatorname{}\ \) (2)
However, for the thin-layer electrode model, in addition to the electrochemical polarization caused by the reaction in the reaction zone, there may be three kinds of polarization, namely (1) the attenuation of the oxygen concentration in the dry zone in the x direction, and (2) the concentration polarization of the dissolved oxygen in the y direction caused by the mass transfer resistance in the liquid phase.(3) changes in electromotive force and current density in the x direction caused by liquid phase resistance. In the following, we will evaluate the importance of each polarization by using the method of characteristic reaction depth Lx (μm), which is the depth at which the concentration of particles (including oxygen molecules, electrons, etc.) drops to 1/e of the initial value. The characteristics of the electrode reaction depth is not only related to electrode properties (such as catalytic activity of the electrode, specific surface area, pore size distribution, liquid phase resistance, gas solubility, diffusion coefficient, etc.), also affected by the over-potential η0 of electrode surface. The detailed derivation of the characteristic reaction depth equation is shown in Supporting information, Part 4.
We calculate the characteristic reaction depth Lx under different current densities, as shown in Fig4d, e and f. Considering that industrial applications are generally carried out under strong current and strong polarization conditions, we discuss the characteristic reaction depth Lx under the condition of η0 = 1.78V, I = 80mA•cm-2.
\(L_{\text{Dg}}\)=546μm, this means that in the dry region, it takes 546μm for the gas phase oxygen concentration to decay in the x direction to 1/e(=36.8%) of the initial value, whereas in the wet region, the d value is about 50μm. Therefore, the polarization caused by gas phase oxygen concentration in the dry region can be neglected.
\(L_{\text{Dl}}\)=0.90μm, this means that under strong polarization conditions, when the concentration of dissolved oxygen caused by mass transfer resistance in the liquid phase attenuates to 36.8% of the initial value, a depth of 0.90μm is required in the y direction in the wet zone. Considering that the distribution range of hierarchical pore aperture in the self-made GDE ranges from 2nm to 5μm, the pore diameter is mainly concentrated in the range of 20-80nm, and there are few holes larger than 1μm. Therefore, the concentration polarization of dissolved oxygen in most reaction zones can also be neglected. The simulation results also show that although large holes are beneficial to mass transfer, the reactions in the channel larger than 1μm are concentrated on the surface. Oxygen transport in both dry and wet zones is not a speed- control step affecting the reaction rate. The reaction rate can be increased by increasing the electrochemical active surface area.
Shown in such as Fig. 4f, in greater polarization (η/b>5), the actual characteristics reaction depth attenuates quickly. This means that in the x direction, the greater the overpotential, the shorter the length of the electromotive force caused by the liquid phase resistance attenuation to zero. Therefore, the effective reaction thickness d in the reaction layer will decrease with the increase of potential, and the GDE will eventually degenerate into a plate electrode. This suggests that moderate current density should be considered to improve the effective surface area of the electrode when optimizing the process conditions of hydrogen peroxide preparation.
Based on the theoretical analysis of the above polarization factors, electrochemical polarization and liquid phase resistance polarization are the main components in the electrode. Therefore, the polarization curve equation derived is
\(I=\sqrt{\frac{2i^{0}S_{1}b}{{\tilde{\rho}}_{l}}}\left[\exp\left(\frac{\eta_{0}}{2b}\right)-exp\left(-\frac{\eta_{0}}{2b}\right)\right]\)(3)
The detailed derivation of the equation (3) is shown in Supporting information, Part 4. Next, we use the equation to fit the polarization curve measured in the experiment. The fitting results i0 and b obtained are shown in Fig. 4g. The fitting parameters are listed in Table.2.
Table.2 Equation symbols, physical meanings and values