We use perturbation methods to analyze the “asymmetric rectified electric field (AREF)” generated when an oscillating voltage is applied across a model electrochemical cell consisting of a binary, asymmetric electrolyte bounded by planar, parallel, blocking electrodes. The AREF refers to the steady component of the electric potential gradient within the electrolyte, as discovered via numerics by Hashemi Amrei et. al. (Phys Rev Lett 121(18):185504). We adopt the Poisson–Nernst–Planck framework for ion transport in dilute electrolytes, taking into account unequal ionic diffusivities. We consider the mathematically singular, and practically relevant, limit of thin Debye layers, \(1/(\kappa L) = \epsilon \rightarrow 0 \), where \(\kappa ^{-1}\) is the Debye length, and L is the length of the half-cell. The dynamics of the electric potential and ionic strength in the “bulk” electrolyte (i.e., outside the Debye layers) are solved subject to effective boundary conditions obtained from consideration of the Debye-scale transport. We specifically analyze the case when the applied voltage has a frequency comparable to the inverse bulk ion diffusion time scale, \(\omega = {\mathcal {O}}(D_A/L^2)\), where \(D_A = 2D_+ D_-/(D_+ + D_-)\) is the ambipolar diffusivity, and \(D_{\pm }\) are the ionic diffusivities. In this regime, the AREF extends throughout the bulk of the cell, varying on a lengthscale proportional to \( \sqrt{D_A /\omega }\), and has a magnitude of \({\mathcal {O}}(\epsilon ^2 k_B T /(L e))\) to leading order in \(\epsilon \). Here, \(k_B\) is the Boltzmann constant, T is temperature, and e is the charge on a proton. We obtain an analytical approximation for the AREF at weak voltages, \(V_0 \ll k_B T/e\), where \(V_0\) is the amplitude of the voltage, for which the AREF is \({\mathcal {O}}(\epsilon ^2 V_0^2 e /(k_B T L))\). Our asymptotic scheme is also used to calculate a numerical approximation to the AREF that is valid up to logarithmically large voltages, \(V_0 = {\mathcal {O}}((k_B T/e)\ln (1/\epsilon ))\). The existence of an AREF implies that a charged colloidal particle undergoes net electrophoretic motion under the applied oscillatory voltage. Additionally, a gradient in the bulk ionic strength, caused by the difference in ionic diffusivities, leads to rectified diffusiophoretic particle motion. Here, we predict the electrophoretic and diffusiophoretic velocities for a rigid, spherical, colloidal particle. The diffusiophoretic velocity is comparable in magnitude to the electrophoretic velocity, and can thus affect particle motion in an AREF significantly.

我们使用微扰方法来分析当振荡电压施加在模型电化学电池上时产生的“非对称整流电场 (AREF)”，该电池由以平面、平行、阻塞电极为边界的二元非对称电解质组成。AREF 是指电解质内电势梯度的稳定分量，正如 Hashemi Amrei 等人通过数值发现的那样。阿尔。(Phys Rev Lett 121(18):185504)。我们采用 Poisson-Nernst-Planck 框架进行稀电解质中的离子传输，同时考虑到不等离子扩散系数。我们考虑薄德拜层的数学上奇异且实际相关的极限\(1/(\kappa L) = \epsilon \rightarrow 0 \)，其中\(\kappa ^{-1}\)是德拜长度, 和L是半电池的长度。“体”电解质（即，德拜层外）的电势和离子强度的动力学根据考虑德拜尺度传输获得的有效边界条件进行求解。我们专门分析了当施加的电压具有与反向体离子扩散时间尺度相当的频率时的情况，\(\omega = {\mathcal {O}}(D_A/L^2)\)，其中\(D_A = 2D_ + D_-/(D_+ + D_-)\)是双极扩散系数，\(D_{\pm }\)是离子扩散系数。在这种情况下，AREF 延伸到整个单元格，在与\( \sqrt{D_A /\omega }\)成比例的长度尺度上变化，并且大小为\({\mathcal {O}}(\epsilon ^2 k_B T /(L e))\)到\(\epsilon \) 中的前导顺序。这里，\(k_B\)是玻尔兹曼常数，T是温度，e是质子上的电荷。我们获得了弱电压下AREF 的解析近似值\(V_0 \ll k_B T/e\)，其中\(V_0\)是电压的幅度，其中 AREF 为\({\mathcal {O} }(\epsilon ^2 V_0^2 e /(k_B TL))\)。我们的渐近方案还用于计算 AREF 的数值近似值，该近似值在对数大电压下有效，\(V_0 = {\mathcal {O}}((k_B T/e)\ln (1/\epsilon )) \). AREF 的存在意味着带电胶体粒子在施加的振荡电压下经历净电泳运动。此外，由离子扩散率的差异引起的体离子强度梯度导致修正的扩散泳粒子运动。在这里，我们预测了刚性球形胶体颗粒的电泳和扩散速度。扩散泳速度在幅度上与电泳速度相当，因此可以显着影响 AREF 中的粒子运动。