Communications in Nonlinear Science and Numerical Simulation
Coupled lattice Boltzmann method for simulating electrokinetic flows: A localized scheme for the Nernst–Plank model
Introduction
Recent remarkable developments in micro-/nano-fabrication techniques have prompted direct experimental observations of the electrokinetic phenomena important to micro-/nano-devices. The electrokinetic flow is a typical electrokinetic phenomenon, in which multiple transport processes, such as ionic diffusion, fluid flow, and electrostatic interaction, play important roles. In the thin layer adjacent to the interface between a charged solid wall and an electrolyte solution, referred to as the electrical double layer (EDL), the ion concentration is highly inhomogeneous [1], [2], [3], and the net charge within the EDL interacts with an externally applied electric field or pressure field, which causes various phenomena, such as electro-osmotic flow, streaming potential, and salt rejection from microchannels [4], [5], [6], [7], [8], [9], [10], [11], [12]. Although these phenomena have been known for over a century, active use of them in engineering applications has only recently attracted attention [13], [14], [15], [16], [17], [18], [19].
The most widely used model for analyzing electrokinetic flows consists of the Poisson–Boltzmann (PB) equation for the internal electrical potential and the Navier–Stokes equation for the flow field of electrolyte solutions. The ion concentration field is directly related to the electrical potential through the Boltzmann distribution, which is derived assuming thermodynamic equilibrium, in which the ionic distributions are not affected by the flows of the electrolyte solution. This model is pertinent under the condition that the external electric field is separate from the internal potential field and the system is in steady state. These conditions are satisfied in many applications, e.g., steady electro-osmotic flows through an infinitely long straight channel are correctly analyzed using this model. There are some important cases, however, which are not covered by the PB equation. For example, if the internal and external electric fields must be integrated into a single potential field because of strong nonlinearity of the electrical potential, it is not possible to assume a Boltzmann distribution with a unique reference (or bulk) potential. The PB equation should be replaced by the set of the Poisson equation and the Nernst–Planck equation for such cases, where the Nernst–Planck equation governs the transport processes of each ionic species [20], [21], [22], [23], [24], [25], [26], [27]. In the present paper, we propose a numerical framework to solve the model for electrokinetic flows based on the Nernst–Planck equation. More specifically, the model consists of the Poisson equation, the Nernst–Planck equation, and the Navier–Stokes equation. The method is also capable of analyzing transient behavior, which is not possible using existing methods that are based on the PB equation.
The numerical scheme for solving each governing equation is based on the lattice Boltzmann method (LBM). The LBM [28], [29] was originally developed as an alternative numerical method for solving Navier–Stoke-type equations and has been extended to solve the convection–diffusion-type equations [30], [31]. Although there are a vast number of alternative schemes for these equations associated with finite element and finite-difference methods, the LBM is nevertheless attractive because it is easy to use for programming and is compatible to parallel and GPU computing. In addition, when we consider the electrokinetic phenomena in complex morphology, such as ion transport in fuel cells [32], [33], [34], the LBM is a promising tool in view of the success in flows through porous media.
Several LBMs for analyzing electrokinetic flows have already been proposed [21], [22], [27], [35], [36], [37], [38], [39], and successfully applied, primarily to two-dimensional analysis. Most of these LBMs are based on the PB equation, whereas the method proposed by Wang and Kang [27] solves the Poisson, Nernst–Plank, and Navier–Stokes equations simultaneously and is capable of handling problems that are not covered by the PB approximation. However, the simplified Dirichlet boundary condition for the Nernst–Planck equation used in Ref. [27] still assumes the Boltzmann distribution near the boundary, which causes some problems when the external electric field is combined with the internal potential. Moreover, fully time-dependent problems are not solved accurately, because of the artificial flux across the boundary. In contrast, the method proposed by Capuani et al. [21] for fluid mixtures that covers the electrokinetic flows based on the Nernst–Plank equation as a special case [22], properly incorporates the original Neumann-type boundary condition. Although the locality of the original LBM is sacrificed because the scheme is based on the flux on the links between the lattice points, which is evaluated using the neighboring lattice points, the artificial flux across the boundary is avoided. Whereas in the method of Capuani et al. the LBM is supplemented with the discrete solution of the Poisson equation obtained using the finite-difference scheme rather than the LBM, in the method proposed herein, all the physical quantities are solved within the framework of the LBM. Particularly, the collision operator in the LBM for the Nernst–Planck equation is designed such that all the terms necessary for the ion flux are evaluated in a completely local manner, and communication with neighboring nodes is only through the streaming process. Since the method inherits the algorithmic simplicity of the original LBM, it is compatible with the massively parallel computing. The Neumann-type boundary condition at the solid/liquid interface is also imposed correctly, with employing the standard bounce-back procedure.
The remainder of the present paper is organized as follows. In Section 2, the governing equations for the electrical potential, ionic transports, and flows of electrolyte solutions are stated. The numerical scheme based on the LBM is presented in Section 3, including the detailed numerical procedure. In Section 4, two specific problems are considered and investigated numerically using the present LBM. An electro-osmotic flow through a channel of infinite length is considered in Section 4.1, and the LBM results are compared with the analytical solution. In Section 4.2, a comparison with the previous method is also made. A two-dimensional microchannel of finite length is then considered in Section 4.3. The electro-osmotic flow induced by an external electrical field is investigated, and the ends of the finite-length microchannel are found to cause resistance due to the local pressure gradient.
Section snippets
Governing equations
In this section, we state the model equations describing the behavior of an electrolyte solution. The scale of our interest ranges from several tens of nanometers to several micrometers, which is sufficiently larger than atomic scale. Therefore, we assume that the continuum descriptions of the transport phenomena are still valid. The flow of the electrolyte solution is then governed by the Navier–Stokes equation:where t is the time and is the
Lattice Boltzmann method
In this section, we present the LBM for solving the model equations for the electrokinetic flows. The set of equations consists of (i) the Poisson equation (4) with Eq. (5), (ii) the Nernst–Planck equation (3), and (iii) the Navier–Stokes equation (2) with Eqs. (1), (6). The lattice Boltzmann (LB) equations are defined for each of these equations. In Section 3.1, the LB equation commonly used for equations (i) through (iii) is described, and detailed definitions specific to each transport
Application to electro-osmotic flows in microchannels
In this section, we apply the proposed LBM to specific problems. We first consider a simple one-dimensional problem of electro-osmotic flows between two parallel plates. The numerical results will be compared with the analytical solution in order to validate the present algorithm. We next consider a two-dimensional microchannel of finite length with an entry and an exit. It will be shown that the presence of the entry and exit results in the occurrence of the pressure gradient that causes as a
Summary
In the present paper, a framework of numerical analysis for the electrokinetic model describing micro-/nano-flows has been proposed. The Nernst–Planck equation is adopted in order to incorporate the electrochemical migration and the convection of the electrolyte solution, in addition to the diffusion of the ion species. The governing equations are solved using the LBM. In the scheme for solving the Nernst–Planck equation described in Section 3.3, all the contributions to the ion flux are
Acknowledgment
The present work was partially supported by MEXT program “Elements Strategy Initiative to Form Core Research Center” (since 2012). (MEXT stands for Ministry of Education, Culture, Sports, Science and Technology, Japan.)
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