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,R =η0-ηact. 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