双电层电容器的瞬态动力学：Debye-Falkenhagen近似下的精确表达式
Transient dynamics of electric double-layer capacitors: Exact expressions within the Debye-Falkenhagen approximation
摘要：
我们重温了一个经典的理论电化学问题：双电层电容器（EDLC）受到一个小的，突然施加的外部电位的响应。我们求解Debye-Falkenhagen方程以获得关键EDLC量的精确表达式：离子电荷密度、离子电流密度和电场。与以前的工作相比，我们的结果不限于这些量的长时间渐近性。解的形式是无穷和，其连续项都随着弛豫时间的缩短而指数衰减。重要的是，这组弛豫时间在所有上述EDLC量中是相同的;这一属性是基于物理基础而要求的，但通常在近似方案中无法实现。决定长时间衰减的最大弛豫时间尺度$τ_{1}$的标度与早期的结果一致：根据德拜长度$λ_{D}$和电极间距$2L$，$L\ggλ_{D}$相当于$τ_{1}\simeqλ_{D} L/D$，$L\llλ_{D}$相当于$τ_{1} \simeq 4 L^2/(π^2 D)$（$D$是离子扩散率）。
英文摘要：
We revisit a classical problem of theoretical electrochemistry: the response of an electric double layer capacitor (EDLC) subject to a small, suddenly applied external potential. We solve the Debye-Falkenhagen equation to obtain exact expressions for key EDLC quantities: the ionic charge density, the ionic current density, and the electric field. In contrast to earlier works, our results are not restricted to the long-time asymptotics of those quantities. The solutions take the form of infinite sums whose successive terms all decay exponentially with increasingly short relaxation times. Importantly, this set of relaxation times is the same among all aforementioned EDLC quantities; this property is demanded on physical grounds but not generally achieved within approximation schemes. The scaling of the largest relaxation timescale $τ_{1}$, that determines the long-time decay, is in accordance with earlier results: depending on the Debye length, $λ_{D}$, and the electrode separation, $2L$, it amounts to $τ_{1}\simeqλ_{D} L/D$ for $L\ggλ_{D}$, and $τ_{1} \simeq 4 L^2/(π^2 D)$ for $L\llλ_{D}$, respectively (with $D$ being the ionic diffusivity).
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