A thin double layer analysis of asymmetric rectified electric fields (AREFs)

• Published in: Journal of engineering mathematics Vol. 129; no. 1
• Main Authors: Balu, Bhavya, Khair, Aditya S.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.08.2021; Springer Nature B.V
• Subjects: Applications of Mathematics; Approximation; Asymmetry; Boundary conditions; Computational Mathematics and Numerical Analysis; Debye length; Electric fields; Electric potential; Electrochemical cells; Electrolytes; Electrophoresis; Ion diffusion; Ion transport; Ions; Mathematical analysis; Mathematical and Computational Engineering; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Particle motion; Perturbation methods; Potential gradient; Practical Asymptotics VII; Theoretical and Applied Mechanics; Thin films; Voltage; Ion transport; Electrophoresis; Electric field; Debye layer; Electrolyte
• ISSN: 0022-0833; 1573-2703
• DOI: 10.1007/s10665-021-10139-x

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 / ( κ L ) = ϵ → 0 , where κ - 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, ω = O ( D A / L 2 ) , where D A = 2 D + D - / ( D + + D - ) is the ambipolar diffusivity, and D ± are the ionic diffusivities. In this regime, the AREF extends throughout the bulk of the cell, varying on a lengthscale proportional to D A / ω , and has a magnitude of O ( ϵ 2 k B T / ( L e ) ) to leading order in ϵ . 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 ≪ k B T / e , where V 0 is the amplitude of the voltage, for which the AREF is O ( ϵ 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 = O ( ( k B T / e ) ln ( 1 / ϵ ) ) . 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.