调控双电层电渗流数值研究

在通道壁面垂直施加一个调控电场可以改变双电层电荷密度和Zeta电位势,实现对电渗流的调控.采用电场Poisson方程、动量守恒的Navier-Stokes方程、电解质离子输运的Nernst-Planck方程和液体混合反应的组分浓度输运方程,本文对微通道壁面离散布置调控电极的情况进行了数值分析.数值算例包括单电极、双电极...     

Numerical analysis of the thermal effect on electroosmotic flow and electrokinetic mass transport in microchannels

Joule heating is present in electrokinetically driven flow and mass transport in microfluidic systems. Nowadays, there is a trend of replacing costly glass-based microfluidic systems by the disposable, cheap polymer-based microfluidic systems. Due to poor thermal conductivity of polymer materials, the thermal management of the polymer-based microfluidic systems may become a problem. In this study, numerical analysis is presented for transient temperature development due to Joule heating and its effect on the electroosmotic flow (EOF) and mass species transport in microchannels. The proposed model includes the coupling Poisson-Boltzmann (P-B) equation, the modified Navier-Stokes (N-S) equations, the conjugate energy equation, and the mass species transport equation. The results show that the time development for both the electroosmotic flow field and the Joule heating induced temperature field are less than 1 s. The Joule heating induced temperature field is strongly dependent on channel size, electrolyte concentration, and applied electric field strength. The simulations reveal that the presence of the Joule heating can result in significantly different characteristics of the electroosmotic flow and electrokinetic mass transport in microchannels.     

Helmholtz-Smoluchowski velocity for viscoelastic electroosmotic flows

Many biofluids such as blood and DNA solutions are viscoelastic and exhibit extraordinary flow behaviors, not existing in Newtonian fluids. Adopting appropriate constitutive equations these exotic flow behaviors can be modeled and predicted reasonably using various numerical methods. However, the governing equations for viscoelastic flows are not easily solvable, especially for electroosmotic flows where the streamwise velocity varies rapidly from zero at the wall to a nearly uniform velocity at the outside of the very thin electric double layer. In the present investigation, we have devised a simple method to find the volumetric flow rate of viscoelastic electroosmotic flows through microchannels. It is based on the concept of the Helmholtz-Smoluchowski velocity which is widely adopted in the electroosmotic flows of Newtonian fluids. It is shown that the Helmholtz-Smoluchowski velocity for viscoelastic fluids can be found by solving a simple cubic algebraic equation. The volumetric flow rate obtained using this Helmholtz-Smoluchowski velocity is found to be almost the same as that obtained by solving the governing partial differential equations for various viscoelastic fluids.     

Reaction rates of heavy metal ions at goethite: relaxation experiments and modeling

In the present paper we extend our theory that calculates the fastest reaction step observable in suspensions containing charged microcrystals and heavy metal cations. The calculation requires the solution of the nonlinear Poisson-Boltzmann equation for nonsymmetric electrolytes plus the Nernst-Planck equation for transport of ions in electric fields. We find that the diffusional transport of ions to and from the surface is the rate-limiting process for our experimentally observed maximum rates. At low pH and low metal ion concentration the diffusion of metal ions is the rate-limiting step, whereas for high pH and high metal ion concentration the diffusion of the solvated protons controls the overall relaxation rate. The validity of this theory is checked for the reactions of Pb2+ and Cd2+ with goethite by means of pressure jump relaxation experiments over a wide range of temperature and pH. In all cases we observe fast processes (relaxation in the range of 10(3) s(-1)) in quantitative agreement with the theory, followed by slower processes, most probably caused by diffusion into the interior of the porous microcrystals.     

Electrokinetic transport through nanochannels

This article presents a numerical study of the electrokinetic transport phenomena (electroosmosis and electrophoresis) in a three-dimensional nanochannel with a circular cross-section. Due to the nanometer dimensions, the Boltzmann distribution of the ions is not valid in the nanochannels. Therefore, the conventional theories of electrokinetic flow through the microchannels such as Poisson-Boltzmann equation and Helmholtz-Smoluchowski slip velocity approach are no longer applicable. In the current study, a set of coupled partial differential equations including Poisson-Nernst-Plank equation, Navier-Stokes, and continuity equations is solved to find the electric potential field, ionic concentration field, and the velocity field in the three-dimensional nanochannel. The effects of surface electric charge and the radius of nanochannel on the electric potential, liquid flow, and ionic transport are investigated. Unlike the microchannels, the electric potential field, ionic concentration field, and velocity field are strongly size-dependent in nanochannels. The electric potential gradient along the nanochannel also depends on the surface electric charge of the nanochannel. More counter ions than the coions are transported through the nanochannel. The ionic concentration enrichment at the entrance and the exit of the nanochannel is completely evident from the simulation results. The study also shows that the flow velocity in the nanochannel is higher when the surface electric charge is stronger or the radius of the nanochannel is larger.     

Tradeoff between mixing and transport for electroosmotic flow in heterogeneous microchannels with nonuniform surface potentials

Electroosmotic flow (EOF) is a phenomenon associated with the movement of an aqueous solution induced by the application of an electric field in microchannels. The characteristics of EOF depend on the nature of the surface potential, i.e., whether it is uniform or nonuniform. In this paper, a lattice Boltzmann model (LBM) combined with the Poisson-Boltzmann equation is used to simulate flow field in a rectangular microchannel with nonuniform (step change) surface potentials. The simulation results indicate that local circulations can occur near a heterogeneous region with nonuniform surface potentials, in agreement with those by other authors. Largest circulations, which imply a highest mixing efficiency due to convection and short-range diffusion, were found when the average surface potential is zero, regardless of whether the distribution of the heterogeneous patches is symmetric or asymmetric. In this work, we have illustrated that there is a trade-off between the mixing and liquid transport in EOF microfluidics. One should not simply focus on mixing and neglect liquid transport, as performed in the literature. Excellent mixing could lead to a poor transport of electroosmotic flow in microchannels.     

Frequency-dependent laminar electroosmotic flow in a closed-end rectangular microchannel

This article presents an analysis of the frequency- and time-dependent electroosmotic flow in a closed-end rectangular microchannel. An exact solution to the modified Navier-Stokes equation governing the ac electroosmotic flow field is obtained by using the Green's function formulation in combination with a complex variable approach. An analytical expression for the induced backpressure gradient is derived. With the Debye-Hückel approximation, the electrical double-layer potential distribution in the channel is obtained by analytically solving the linearized two-dimensional Poisson-Boltzmann equation. Since the counterparts of the flow rate and the electrical current are shown to be linearly proportional to the applied electric field and the pressure gradient, Onsager's principle of reciprocity is demonstrated for transient and ac electroosmotic flows. The time evolution of the electroosmotic flow and the effect of a frequency-dependent ac electric field on the oscillating electroosmotic flow in a closed-end rectangular microchannel are examined. Specifically, the induced pressure gradient is analyzed under effects of the channel dimension and the frequency of electric field. In addition, based on the Stokes second problem, the solution of the slip velocity approximation is presented for comparison with the results obtained from the analytical scheme developed in this study.     

Electroosmotic flow in a capillary annulus with high zeta potentials

The electroosmotic flow through an annulus is analyzed under the situation when the two cylindrical walls carry high zeta potentials. The analytical solutions for the electric potential profile and the electroosmotic flow field in the annulus are obtained by solving the Poisson-Boltzmann equation and the Stokes equation under an analytical scheme for the hyperbolic sine function. A mathematical expression for the average electroosmotic velocity is derived in a fashion similar to the Smoluchowski equation. Hence, a correction formula is introduced to modify the Smoluchowski equation, taking into account contributions due to the finite thickness of the electric double layer (EDL) and the geometry ratio-dependent correction. Specifically, under a circumstance when the two annular walls are oppositely charged, the flow direction can be determined from the sign of such correction formula, and there exists a zero-velocity plane inside the annulus. With the assumption of large electrokinetic diameters, the location of the zero-velocity plane can be estimated from the analytical expression for the velocity distribution. In addition, the characteristics of the electroosmotic flow through the annulus are discussed under the influences of the EDL parameters and geometric ratio of the inner radius to the outer radius of the annulus.     

EOF using the Ritz method: application to superelliptic microchannels

An efficient Ritz method is developed from the variational principle to solve the Poisson-Boltzmann equation under the Debye-Hückel approximation for studying the EOF in microchannels. The method is applied to the family of superelliptic cross sections which includes the elliptic channel and the rectangular channel as limiting cases. Several accurate tables presented are useful for design of electroosmotic channels, especially rectangular channels with rounded corners. It is shown how the flow rate Q is a sophisticated consequence of the nondimensional electrokinetic width K, the aspect ratio b as well as the superelliptic exponent n.     

Electroosmotic flow in microchannels with arbitrary geometry and arbitrary distribution of wall charge

General solutions are developed for direct current (DC) and alternating current (AC) electroosmotic flows in microfluidic channels with arbitrary cross-sectional geometry and arbitrary distribution of wall charge (zeta potential). The applied AC electric field can also be of arbitrary waveform. By proposing a nondimensional time scale varpi defined as the ratio of the diffusion time of momentum across the electric double-layer thickness to the period of the applied electric field, we demonstrate analytically that the Helmholtz-Smoluchowski electroosmotic velocity is an appropriate slip condition for AC electroosmotic flows in typical microfluidic applications. With this slip condition approach, electroosmotic flows in rectangular and asymmetric trapezoidal microchannels with nonuniform wall charge, as examples, are investigated. The unknown constants in the proposed general solutions are numerically determined with a least-squares method through matching the boundary conditions. We find that the wall charge affects significantly the electroosmotic flow while the channel geometry does not. Moreover, the flow feature is characterized by another nondimensional time scale Omega defined as the ratio of the diffusion time of momentum across the channel hydraulic radius to the period of the applied electric field. The onset of phase shift between AC electroosmotic velocity and applied electric field is also examined analytically.     

Unsteady electroosmosis in a microchannel with Poisson-Boltzmann charge distribution

The present study is concerned with unsteady electroosmotic flow (EOF) in a microchannel with the electric charge distribution described by the Poisson-Boltzmann (PB) equation. The nonlinear PB equation is solved by a systematic perturbation with respect to the parameter λ which measures the strength of the wall zeta potential relative to the thermal potential. In the small λ limits (λ<1), we recover the linearized PB equation - the Debye-Hückel approximation. The solutions obtained by using only three terms in the perturbation series are shown to be accurate with errors <1% for λ up to 2. The accurate solution to the PB equation is then used to solve the electrokinetic fluid transport equation for two types of unsteady flow: transient flow driven by a suddenly applied voltage and oscillatory flow driven by a time-harmonic voltage. The solution for the transient flow has important implications on EOF as an effective means for transporting electrolytes in microchannels with various electrokinetic widths. On the other hand, the solution for the oscillatory flow is shown to have important physical implications on EOF in mixing electrolytes in terms of the amplitude and phase of the resulting time-harmonic EOF rate, which depends on the applied frequency and the electrokinetic width of the microchannel as well as on the parameter λ.     

Transient analysis of electroosmotic flow in a slit microchannel

The transient aspects of electroosmotic flow in a slit microchannel are studied. Exact solutions for the electrical potential profile and the transient electroosmotic flow field are obtained by solving the complete Poisson-Boltzmann equation and the Navier-Stokes equation under an analytical approximation for the hyperbolic sine function. The characteristics of the transient electroosmotic flow are discussed under influences of the electric double layer and the geometric size of the microchannel.     

Flow behavior of periodical electroosmosis in microchannel for biochips

This paper presents an analytical solution for periodical electroosmotic flows in two-dimensional uniform microchannel based on Poisson-Boltzmann equations for electric double layer (EDL) and Navier-Stokes equation for incompressible viscous fluid. Analytical results indicate that the velocity of periodical electroosmosis strongly depends on Reynolds number Re=omegah(2)/nu, as well as on EDL properties and applied electric field. Slip velocity of EDL decreases as the Reynolds number increases. Electroosmotic velocity outside the EDL decreases, and lag phase angle of velocity increases as distance away from the channel wall increases. A wavelike velocity profile across the channel is found. An asymptotic solution for low Reynolds number is given in this paper. Periodical electroosmosis with low Reynolds has same velocity amplitude and a pluglike velocity profile as that of steady electroosmosis. Based on Debye-Hückel approximation, this paper also obtains a solution of periodical electroosmosis applicable to cases where the thickness of EDL is of the same order as half of channel width.     

Ion current rectification at nanopores in glass membranes

The origin of ion current rectification observed at conical-shaped nanopores in glass membranes immersed in KCl solutions has been investigated using finite-element simulations. The ion concentrations and fluxes (due to diffusion, migration, and electroosmotic convection) were determined by the simultaneous solution of the Nernst-Planck, Poisson, and Navier-Stokes equations for the two-ion (K+ and Cl-) system. Fixed surface charge on both the internal and external glass surfaces that define the pore structure was included to account for electric fields and nonuniform ion conductivity within the nanopores and electric fields in the external solution near the pore mouth. We demonstrate that previous observations of ion current rectification in conical-shaped glass nanopores are a consequence of the voltage-dependent solution conductivity in the vicinity of the pore mouth, both inside and outside of the pore. The simulations also demonstrate that current rectification is maximized at intermediate bulk ion concentrations, a combination of (i) the electrical screening of surface charge at high concentrations and (ii) a fixed number of charge-carrying ions in the pore at lower concentration, which are physical conditions where the voltage dependence of the conductivity disappears. In addition, we have quantitatively shown that electroosmotic flow gives rise to a significant but small contribution to current rectification.     

Analysis of rotation-driven electrokinetic flow in microscale gap regions of rotating disk systems

In the present study, a novel theoretical model is developed for the analysis of rotating thermal-fluid flow characteristics in the presence of electrokinetic effects in the microscale gap region between two parallel disks under specified electrostatic, rotational, and thermal boundary conditions. The major flow configuration considered is a rotor-stator disk system. Axisymmetric Navier-Stokes equations with consideration of electric body force stemming from streaming potential are employed in the momentum balance. Variations of the fluid viscosity and permittivity with the local fluid temperature are considered. Between two disks, the axial distribution of the electric potential is determined by the Poisson equation with the concentration distributions of positive and negative ions obtained from Nernst-Planck equations for convection-diffusion of the ions in the flow field. Effects of disk rotation and electrostatic and thermal conditions on the electrokinetic flow and thermal characteristics are investigated. The electrohydrodynamic mechanisms are addressed with an interpretation of the coupling nature of the electric and flow fields. Finally, solutions with electric potential determined by employing nonlinear or linearized Poisson-Boltzmann equation and/or invoking assumptions of constant properties are compared with the predictions of the present model for justification of various levels of approximation in solution of the electrothermal flow behaviors in rotating microfluidic systems.     

Analysis of self-electrophoretic motion of a spherical particle in a nanotube: effect of nonuniform surface charge density

Autonomous motions of a spherical nanoparticle in a nanotube filled with an electrolyte solution were investigated using a continuum theory, which consisted of the Nernst-Planck equations for the ionic concentrations, the Poisson equation for the electric potential in the solution, and the Stokes equation for the hydrodynamic field. Contrary to the usual electrophoresis, in which an external electric field is imposed to direct the motion of charged particles, the autonomous motion originates from the self-generated electric field due to the ionic concentration polarization of the liquid medium surrounding an asymmetrically charged particle. In addition to the particle motion, the interaction between the electric field generated and the free charges of the polarized solution induces electroosmotic flows. These autonomous motions of the fluid as well as the particle were examined with focus on the effects of the surface-charge distribution of the particle, the size of the nanotube, and the thickness of the electric double layer, which affected the direction and the speed of the particle significantly.     

Numerical modeling of Joule heating-induced temperature gradient focusing in microfluidic channels

Temperature gradient focusing (TGF) is a recently developed technique for spatially focusing and separating ionic analytes in microchannels. The temperature gradient required for TGF can be generated either by an imposed temperature gradient or by Joule heating resulting from an applied electric field that also drives the flow. In this study, a comprehensive numerical model describing the Joule heating induced temperature development and TGF is developed. The model consists of a set of governing equations including the Poisson-Boltzmann equation, the Laplace equation, the Navier-Stokes equations, the energy equations and the mass transport equation. As the thermophysical and electrical properties including the liquid dielectric constant, viscosity, and electric conductivity are temperature-dependent, these governing equations are coupled, and therefore the coupled governing equations are solved numerically by using a CFD-based numerical method. The numerical simulations agree well with the experimental results, suggesting the valid mathematical model presented in this study.     

Electrokinetic flow in a capillary with a charge-regulating surface polymer layer

An analytical study of the steady electrokinetic flow in a long uniform capillary tube or slit is presented. The inside wall of the capillary is covered by a layer of adsorbed or covalently bound charge-regulating polymer in equilibrium with the ambient electrolyte solution. In this solvent-permeable and ion-penetrable surface polyelectrolyte layer, ionogenic functional groups and frictional segments are assumed to distribute at uniform densities. The electrical potential and space charge density distributions in the cross section of the capillary are obtained by solving the linearized Poisson-Boltzmann equation. The fluid velocity profile due to the application of an electric field and a pressure gradient through the capillary is obtained from the analytical solution of a modified Navier-Stokes/Brinkman equation. Explicit formulas for the electroosmotic velocity, the average fluid velocity and electric current density on the cross section, and the streaming potential in the capillary are also derived. The results demonstrate that the direction of the electroosmotic flow and the magnitudes of the fluid velocity and electric current density are dominated by the fixed charge density inside the surface polymer layer, which is determined by the regulation characteristics such as the dissociation equilibrium constants of the ionogenic functional groups in the surface layer and the concentration of the potential-determining ions in the bulk solution.     

纳滤膜对电解质溶液分离特性的理论研究(II): 混合电解质溶液

假定纳滤膜具有狭缝状孔, 使用纯水透过系数、膜孔径及膜表面电势来表征纳滤膜的分离特征, 用流体力学半径和无限稀释扩散系数表征了离子特性. 采用扩展Nernst-Planck方程、Donnan平衡模型和Poisson-Boltzmann理论描述了混合电解质溶液中离子在膜孔内的传递现象, 计算了三种商用纳滤膜(ESNA1-LF, ESNA1和LES90)对同阴离子、同阳离子和含四种离子的混合电解质体系中离子的截留率, 并与实验数据进行了比较. 计算结果表明, 电解质溶液中离子在纳滤膜孔内传递的主要机理是离子的扩散和电迁移, 纳滤膜对混合电解质溶液中离子的分离效果主要由空间位阻和静电效应决定. 该模型在低浓度时对含一价离子的混合电解质溶液通过纳滤膜的截留率计算结果比较准确, 但对高浓度或含高价离子的混合电解质溶液则偏差较大.