Stout Robert F, Khair Aditya S
Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA.
Phys Rev E. 2017 Aug;96(2-1):022604. doi: 10.1103/PhysRevE.96.022604. Epub 2017 Aug 7.
Thermoelectrics are increasingly being studied as promising electrical generators in the ongoing search for alternative energy sources. In particular, recent experimental work has examined thermoelectric materials containing ionic charge carriers; however, the majority of mathematical modeling has been focused on their steady-state behavior. Here, we determine the time scales over which the diffuse charge dynamics in ionic thermoelectrochemical systems occur by analyzing the simplest model thermoelectric cell: a binary electrolyte between two parallel, blocking electrodes. We consider the application of a temperature gradient across the device while the electrodes remain electrically isolated from each other. This results in a net voltage, called the thermovoltage, via the Seebeck effect. At the same time, the Soret effect results in migration of the ions toward the cold electrode. The charge dynamics are described mathematically by the Poisson-Nernst-Planck equations for dilute solutions, in which the ion flux is driven by electromigration, Brownian diffusion, and thermal diffusion under a temperature gradient. The temperature evolves according to the heat equation. This nonlinear set of equations is linearized in the (experimentally relevant) limit of a "weak" temperature gradient. From this, we show that the time scale on which the thermovoltage develops is the Debye time, 1/Dκ^{2}, where D is the Brownian diffusion coefficient of both ion species, and κ^{-1} is the Debye length. However, the concentration gradient due to the Soret effect develops on the bulk diffusion time, L^{2}/D, where L is the distance between the electrodes. For thin diffuse layers, which is the condition under which most real devices operate, the Debye time is orders of magnitude less than the diffusion time. Therefore, rather surprisingly, the majority of ion motion occurs after the steady thermovoltage has developed. Moreover, the dynamics are independent of the thermal diffusion coefficients, which simply set the magnitude of the steady-state thermovoltage.
在持续寻找替代能源的过程中，热电材料作为有前景的发电机正越来越多地被研究。特别是，最近的实验工作研究了含有离子电荷载流子的热电材料；然而，大多数数学建模都集中在它们的稳态行为上。在这里，我们通过分析最简单的模型热电池：两个平行的阻塞电极之间的二元电解质，来确定离子热电化学系统中扩散电荷动力学发生的时间尺度。我们考虑在器件上施加温度梯度，而电极彼此保持电隔离。这通过塞贝克效应产生一个净电压，称为热电压。同时，索雷特效应导致离子向冷电极迁移。电荷动力学由稀溶液的泊松 - 能斯特 - 普朗克方程数学描述，其中离子通量由电迁移、布朗扩散和温度梯度下的热扩散驱动。温度根据热方程演化。这组非线性方程在“弱”温度梯度的（实验相关）极限下被线性化。由此，我们表明热电压发展的时间尺度是德拜时间，(1 / Dκ^{2})，其中(D)是两种离子物种的布朗扩散系数，(κ^{-1})是德拜长度。然而，由于索雷特效应引起的浓度梯度在体扩散时间(L^{2} / D)上发展，其中(L)是电极之间的距离。对于大多数实际器件运行的薄扩散层条件，德拜时间比扩散时间小几个数量级。因此，相当令人惊讶的是，大多数离子运动发生在稳定热电压发展之后。此外，动力学与热扩散系数无关，热扩散系数仅设定稳态热电压的大小。
