电化学（中英文） ›› 2022, Vol. 28 ›› Issue (2): 2108471. doi: 10.13208/j.electrochem.210847
所属专题： “理论计算模拟”专题文章； iSAIEC 2023； “AI for Electrochemistry”专题文章
张露露1,#, 李琛坤2,#, 黄俊3,*(
)
收稿日期:2021-10-26
修回日期:2021-12-07
出版日期:2022-02-28
发布日期:2021-12-18
Lu-Lu Zhang1,#, Chen-Kun Li2,#, Jun Huang3,*()
Received:2021-10-26
Revised:2021-12-07
Published:2022-02-28
Online:2021-12-18
Contact:
*Tel: (+49)15906819204, E-mail:
About author:#Equal contribution.
摘要：
本文定位在一篇电化学双电层（EDL）理论建模方面入门级文章。我们首先简要介绍了EDL的基本特征,简述了EDL理论建模的发展历史,特别是D.C. Grahame之后近几十年的发展历史。然后,我们依次介绍了平衡状态和动态下不同复杂度的EDL模型。作为一篇入门级文章,我们尽可能详细地阐释理论模型的物理图像、假设、数学推导、形式分析、数值分析,并附上Matlab仿真代码。平衡状态下的模型包括Gouy-Chapman-Stern（GCS）模型,Bikerman-Poisson-Boltzmann（BPB）模型,和非对称离子尺寸模型。我们强调GCS模型和BPB模型在处理离子有限尺寸上存在一个微妙的不同。GCS模型通过人为引入Helmholtz平面来考虑离子有限尺寸,但在Helmholtz平面内及弥散层内却依然采用没有考虑离子尺寸效应的Poisson-Boltzmann理论,因而此处的离子浓度可以无限大。与之不同,BPB模型通过格子气体方法,能够自洽描述离子有限尺寸效应。不同以往直接采用Poisson-Nernst-Planck方程描述EDL动态行为,我们从EDL的巨势出发,运用基本的泛函分析方法,推导了一个考虑离子有限尺寸的EDL动态模型。这一理论方法拓展性好。读者可以根据研究对象的需要,建立不同复杂度的EDL动态模型。最后,我们基于EDL动态模型,推导了EDL的电化学阻抗谱理论模型,以试图向读者展示如何从一个时域物理模型出发,推导相应的阻抗谱物理模型。读者若想要踏进理论电化学这个美丽的花园,根据我们自己学习和研究的经验,一个可行的方式是拿起纸和笔来开始推导本文所介绍的这些模型。
张露露, 李琛坤, 黄俊. 平衡、非平衡、交流状态下电化学双电层建模的初学者指南[J]. 电化学（中英文）, 2022, 28(2): 2108471.
Lu-Lu Zhang, Chen-Kun Li, Jun Huang. A Beginners’ Guide to Modelling of Electric Double Layer under Equilibrium, Nonequilibrium and AC Conditions[J]. Journal of Electrochemistry, 2022, 28(2): 2108471.
Figure 1.
An electrochemical cell with two charged electrode/electrolyte interfaces. (color on line)
Figure 2.
Surface charging relation: (A) ϕM - ϕS < χ, (B) ϕM - ϕS = χ, (C) ϕM - ϕS > χ, and (D) with an additional potential drop Δχ at the surface due to chemisorption-induced surface dipole. In the presence of chemisorption, σM contains not only the excess charge on the electrode surface, but also the net charged carried on the charged adsorbates. (color on line)
Figure 3.
Key milestones of EDL modelling (color on line)
Figure 4.
Typical results of the GCS model, including the spatial distributions of (A) the electric potential and (B) the cation concentration, and the relationships between (C) the surface charge density and (D) the differential double-layer capacitance with the electrode potential. The inset in (A) illustrates the electric potential distribution within 2 nm near the electrode. The parameters for calculation are listed in Table 1. Matlab script of this model is provided in the supporting information. (color on line)
Figure 5.
Schematic illustration of the comparison between the GCS model and the BPB model. For the GCS model, the number density of particles at the HP and in the diffuse layer can be infinite for the point charge assumption. Correspondingly, we depict ions with dotted lines in the GCS model, c.f. solid lines for ions of finite size in the BPB model. (color on line)
Figure 6.
Typical results of the BPB model, including the spatial distributions of the electric potential and the anion concentration at a series of electrode potential in (A) and (B), and the relationships between (C) the surface charge density (D) the differential double-layer capacitance with the electrode potential. For the purpose of comparison, the results of the GCS model are shown in the black solid lines. The parameters for calculation are listed in Table 1. Matlab script of this model is provided in the supporting information. (color on line)
Figure 7.
Typical results of the BPB model with the asymmetric size effect, including the relationships between (A) the surface charge density and (B) the differential double-layer capacitance with the electrode potential. The results of the symmetric BPB model are shown in the blue circles, as γ+ = γ- = 1. The results at γ+ = 1 and γ- = 3 are shown in the red solid lines, while the results at γ+ =3 and γ- = 1 are shown in the yellow solid lines. The grey circles represent the results at γ+ = γ- = 3. The parameters for calculation are listed in Table 1. Matlab script of this model is provided in the supporting information. (color on line)
Table 1
List of the Model Parameters
| Symbol (unit) | Value | Physical significance | Note |
|---|---|---|---|
| Constants | |||
| kB(J·K-1) | 1.381 × 10-23 | Boltzmann constant | |
| T(K) | 298.15 | Absolute temperature | |
| h(J·s) | 6.626 × 10-34 | Planck constant | |
| e0(C) | 1.602 × 10-19 | Elementary charge | |
| NA(mol-1) | 6.022 × 1023 | Avogadro constant | |
| $\epsilon_{0}$(F·m-1) | 8.854 × 10-12 | Vacuum permittivity | |
| F(C·mol-1) | 96485 | Faraday constant | |
| R(J·K-1·mol-1) | 8.314 | Gas constant | |
| Solution properties | |||
| $\epsilon_{HP}$(F·m-1) | 6$\epsilon_{0}$ | Dielectric permittivity of the space between the electrode and the HP | Ref [56] |
| $\epsilon_{S}$(F·m-1) | 78.5$\epsilon_{0}$ | Bulk dielectric permittivity of the water solvent medium | |
| δHP(nm) | 0.4125 | Distance from the electrode to the HP, calculated by 1.5 d with the diameter of water d = 0.275 nm. | |
| D(m2·s-1) | 1 × 10-9 | Diffusion coefficient | |
| cb(mol·m-3) | 1 | Concentration of total cations (anions) in the bulk solution | |
| cA0,cB0,cH+0(mol·m-3) | 1 | Concentrations of B, A and H+ under standard conditions | |
| Electrode properties | |||
| ϕpzc(VSHE) | 0.3 | Potential of zero charge | Estimated |
| aM(Å) | 3.5 | Lattice constant of the electrode | Estimated |
| nM (m-2) | 4.713 × 1018 | Areal number density of M(111), calculated by ($\sqrt{3}$aM2)-1 | |
| Reaction properties | |||
| E00(VSHE) | 0.6 | Equilibrium potential of the reaction at standard state | Estimated |
| ΔGa00(eV) | 0.4 | Activation energy of the reaction at standard equilibrium state | Estimated |
Figure 8.
Typical results of the PNP theory, including the current density varying with the electrode potential, and the distributions of the concentration of A at 0.1 s, 1 s and 5 s at 0.4 VSHE. The spatial range for calculation is 100 μm from the HP, and the time duration for calculation is 5 s. The parameters for calculation are listed in Table 1. Matlab script of this model is provided in the supporting information.
Figure 9.
ECM in the electrochemical system. Cdl is an interfacial capacitance, associated with the double-layer charging process, Rct is a resistance, associated with the charge transfer reactions process, W describes the diffusion of species involved in the charge transfer reactions, Rs is the electrolyte solution resistance, associated with the migration process in the bulk solution.
Figure 10.
(A) A simple RC electrical circuit; (B) The Nyquist plot; (C) The Bode plot of amplitude; (D) The Bode plot of phase angle. The parameters used for calculation are as follows, R0 = R = 1 Ω, C = 0.5 F, and the frequency range: 1 × 10-4 Hz to 1 × 104 Hz. Matlab script of this model is provided in the supporting information. (color on line)pedance amplitude and the frequency is shown in Figure 10(C). At very low frequencies, the amplitude of im-pedance is equal to R + R0. At very high frequencies, the amplitude of impedance approaches R0. Figure 10(D) shows how the phase angle varies with frequency. There is only a characteristic frequency at 1/RC which corresponds to the peak in the Nyquist plot. Notably, the peak frequency in the Bode plot deviates from 1/RC[57].
Figure 11.
Nyquist plots of simplified impedance in different frequency ranges. (A) Full frequency range, 1×106 Hz ~ 1×10-4 Hz; (B) ωnd > ωctnd, 1×106 Hz ~ 1×105 Hz; (C) ωdnd < ωnd < ωctnd, 1×104 Hz ~ 10 Hz; (D) ωnd < ωdnd, 1 Hz ~ 1×10-4 Hz. Parameters are c0 = 100 mol·m-3, k0 = 3×10-4 mol·m-2·s-1, D = 1×10-10 m2·s-1. Matlab script of this model is provided in the supporting information. (color on line)
Figure 12.
Comparison between the AFT-calculated and the analytical impedance expressed in Eq. (127). Parameters used in calculation are as follows, c0 = 100 mol·m-3, k0 = 4.45× 10-5 mol·m-2·s-1, D = 3.2×10-11 m2·s-1, frequency range: 4.22×10-4 ~ 5×105 Hz, sampling frequency: 100*(excitation frequency), sampling duration: the reciprocal of sampling frequency. Matlab script of this model is provided in the supporting information.
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