Biesheuvel P M, Fu Yeqing, Bazant Martin Z
Department of Environmental Technology, Wageningen University, Wageningen, The Netherlands.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Jun;83(6 Pt 1):061507. doi: 10.1103/PhysRevE.83.061507. Epub 2011 Jun 23.
Porous electrodes instead of flat electrodes are widely used in electrochemical systems to boost storage capacities for ions and electrons, to improve the transport of mass and charge, and to enhance reaction rates. Existing porous electrode theories make a number of simplifying assumptions: (i) The charge-transfer rate is assumed to depend only on the local electrostatic potential difference between the electrode matrix and the pore solution, without considering the structure of the double layer (DL) formed in between; (ii) the charge-transfer rate is generally equated with the salt-transfer rate not only at the nanoscale of the matrix-pore interface, but also at the macroscopic scale of transport through the electrode pores. In this paper, we extend porous electrode theory by including the generalized Frumkin-Butler-Volmer model of Faradaic reaction kinetics, which postulates charge transfer across the molecular Stern layer located in between the electron-conducting matrix phase and the plane of closest approach for the ions in the diffuse part of the DL. This is an elegant and purely local description of the charge-transfer rate, which self-consistently determines the surface charge and does not require consideration of reference electrodes or comparison with a global equilibrium. For the description of the DLs, we consider the two natural limits: (i) the classical Gouy-Chapman-Stern model for thin DLs compared to the macroscopic pore dimensions, e.g., for high-porosity metallic foams (macropores >50 nm) and (ii) a modified Donnan model for strongly overlapping DLs, e.g., for porous activated carbon particles (micropores <2 nm). Our theory is valid for electrolytes where both ions are mobile, and it accounts for voltage and concentration differences not only on the macroscopic scale of the full electrode, but also on the local scale of the DL. The model is simple enough to allow us to derive analytical approximations for the steady-state and early transients. We also present numerical solutions to validate the analysis and to illustrate the evolution of ion densities, pore potential, surface charge, and reaction rates in response to an applied voltage.
在电化学系统中，多孔电极而非平板电极被广泛应用，以提高离子和电子的存储容量，改善质量和电荷传输，并提高反应速率。现有的多孔电极理论做了一些简化假设：（i）假设电荷转移速率仅取决于电极基体与孔溶液之间的局部静电势差，而不考虑其间形成的双电层（DL）结构；（ii）不仅在基体 - 孔界面的纳米尺度上，而且在通过电极孔的宏观传输尺度上，电荷转移速率通常等同于盐转移速率。在本文中，我们通过纳入法拉第反应动力学的广义弗鲁姆金 - 巴特勒 - 伏尔默模型来扩展多孔电极理论，该模型假定电荷转移发生在位于电子传导基体相和DL扩散部分中离子最接近平面之间的分子斯特恩层上。这是对电荷转移速率的一种简洁且纯粹局部的描述，它自洽地确定表面电荷，并且不需要考虑参比电极或与全局平衡进行比较。对于DL的描述，我们考虑两种自然极限情况：（i）与宏观孔尺寸相比薄DL的经典古伊 - 查普曼 - 斯特恩模型，例如对于高孔隙率金属泡沫（大孔>50 nm）；（ii）对于强重叠DL的修正唐南模型，例如对于多孔活性炭颗粒（微孔<2 nm）。我们的理论适用于两种离子都可移动的电解质，并且它不仅考虑了整个电极宏观尺度上的电压和浓度差异，还考虑了DL局部尺度上的差异。该模型足够简单，使我们能够推导出稳态和早期瞬态的解析近似值。我们还给出了数值解，以验证分析结果，并说明响应于施加电压时离子密度、孔电势、表面电荷和反应速率的演变。
