Theoretical study on potential distribution and electroosmotic flow velocity in microscale channel

A developed mathematical model for calculating potential distribution inside the electrical double layer is explored in this paper based on the Poisson-Boltzmann equation. By modifying the ion concentration, we numerically simulated the potential profile inside the actual electrical double layer according to the zeta potential. Then a theoretical analysis on the streamwise electroosmotic velocity in microscale channel is presented. Furthermore, the expression of the electroosmotic velocity is significantly suppressed after considering the Helmboltz-Smolucbowski equation boundary conditions. The results show that the calculated electroosmotic values basically agree with the experimental ones. Therefore, this provides the data for micro- and nano-channels’ electrophoretic transport, as well as separation of neutral and charged electrolyte.     

Frequency dependence of gold nanoparticle superassembly by dielectrophoresis

Dielectrophoresis is an effective method for capturing nanoparticles and assembling them into nanostructures. The frequency of the dielectrophoretic alternating current (ac) electric field greatly influences the morphology of resultant nanoparticle assemblies. In this study, frequency regimes associated with specific gold nanoparticle assembly morphologies were identified. Gold nanoparticles suspended in water were captured by microelectrodes at different electric field frequencies onto thin silicon nitride membranes. The resultant assemblies were examined by transmission electron microscopy. For this system, the major frequency-dependent influence on morphology appears to arise not from the Clausius-Mossotti factor of the dielectrophoretic force itself, but instead from ac electroosmotic fluid flow and the influence of the electrical double layer at the electrode-solution interface. Frequency regimes of technological interest include those forming one-dimensional nanoparticle chains, microwires, combinations of microwires and nanoparticle chains suitable for nanogap electrode formation, and dense three-dimensional assemblies with very high surface area.     

Generalized model for time periodic electroosmotic flows with overlapping electrical double layers

In this article, an analytical model is devised for analyzing time periodic electroosmotic flows through nanochannels within the continuum regime, without presuming the validity of the Boltzmann distribution of ionic charges. The charge density distributions are obtained from the conservation considerations of the individual ionic species and other thermochemical constraints and are subsequently utilized to derive the potential distribution within the electrical double layer (EDL). This, coupled with the Navier-Stokes equation, yields a closed-form expression of the time-dependent velocity field that is valid under overlapped EDL conditions. This expression is first validated in asymptotic limits of thin EDLs, for which closed form expressions have been benchmarked in the literature. Further analyses are carried out to bring out the influences of the frequency of the electrical field on the electroosmotic flow features in the presence of overlapped EDLs.     

毛细管电色谱中双电层叠加的研究

通过流动相中电解质浓度对毛细管电色谱柱效能的流动相平均线速度的影响，研究了ＣＥＣ中双电层叠加现象。提出选择合适电解质组成的浓度及在制备色谱柱过程中避免细小和破碎的固定相颗粒进入柱中，以有效地减小双电层叠加作用。     

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.     

Liquid transport in rectangular microchannels by electroosmotic pumping

The characteristics of electroosmotic flow in rectangular microchannels were investigated in this paper. A 2D Poisson–Boltzmann equation and the 2D momentum equation were used to model the electric double layer field and the flow field. The numerical solutions show significant influences of the channel cross-section geometry (i.e. the aspect ratio) on the velocity field and the volumetric flow rate. Also, the numerical simulation of the electroosmotic flow reveals how the velocity field and the volumetric flow rate depend on the ionic concentration, zeta potential, channel size and the applied electrical field strength.     

DC-electrokinetic motion of colloidal cylinder(s) in the vicinity of a conducting wall

The boundary effects on DC-electrokinetic behavior of colloidal cylinder(s) in the vicinity of a conducting wall is investigated through a computational model. The contribution of the hydrodynamic drag, gravity, electrokinetic (i.e., electrophoretic and dielectrophoretic), and colloidal forces (i.e., forces due to the electrical double layer and van der Waals interactions) are incorporated in the model. The contribution of electrokinetic and colloidal forces are included by introducing the resulting forces as an external force acting on the particle(s). The colloidal forces are implemented with the prescribed expressions from the literature, and the electrokinetic force is obtained by integrating the corresponding Maxwell stress tensor over the particles' surfaces. The electrokinetic slip-velocity together with the thin electrical double layer assumption is applied on the surfaces. The position and velocity of the particles and the resulting electric and flow fields are obtained and the physical insight for the behavior of the colloidal cylinders are discussed in conjunction with the experimental observations in the literature.     

Electrophoresis of solid particles at large Peclet numbers

A theory of concentration polarization of a thin electrical double layer (DL) on a spherical particle is developed for the regime of large Peclet numbers which is realized in strong electric fields. In this regime, the concentration field arising outside DL is estimated under influence of diffusion and convection. According to the theory developed, polarization of DL at large Peclet numbers causes a change in the Stern potential, the formation of a dipole moment and the long-range potential. The diffuse layer deviates strongly from spherical symmetry and electroneutrality, and the screen of the surface charge is provided not only by the diffuse atmosphere but also by the charge induced in the convective-diffusion layer. The effect of electric field on the induced charge gives rise to the additional electroosmotic slip, that was called "secondary electroosmosis". Thus, a nonlinear additional term for the Smoluchowski formula of electrophoretic velocity is based on the changes of zeta-potential and on the secondary electroosmotic slip. The comparison of theory with experimental results revealed considerable fitting.     

Control of electroosmotic flow by polymer coating: effects of the electrical double layer

We report on the molecular dynamics simulations of electroosmotic flow control by polymer coating. We show that polymer coating modulates the flow by rendering drag to fluids and by changing the ion distribution and ion-surface interactions in the electrical double layer. Because of the latter two effects, the polymer coating can even enhance the flow under certain conditions. Identifying the effects of these processes is crucial for the rational design of polymer coating for electroosmotic flow control.     

Electrostatic potential and electroosmotic flow in a cylindrical capillary filled with symmetric electrolyte: analytic solutions in thin double layer approximation

The electrostatic potential in a capillary filled with electrolyte is derived by solving the nonlinear Poisson-Boltzmann equation using the method of matched asymptotic expansions. This approach allows obtaining an analytical result for arbitrary high wall potential if the double layer thickness is smaller than the capillary radius. The derived expression for the electrostatic potential is compared to numerical solutions of the Poisson-Boltzmann equation and it is shown that the agreement is excellent for capillaries with radii greater or equal to four times the electrical double layer thickness. The knowledge of the electrostatic potential distribution inside the capillary enables the derivation of the electroosmotic velocity flow profile in an analytical form. The obtained results are applicable to capillaries with radii ranging from nanometers to micrometers depending on the ionic strength of the solution.     

A linear analysis of the effect of Faradaic currents on traveling-wave electroosmosis

Net fluid flow of electrolytic solutions induced by a traveling-wave potential applied to an array of co-planar interdigitated microelectrodes has been reported. At low applied voltages the flow is driven in the direction of the traveling-wave potential, as expected by linear and weakly nonlinear theoretical studies. The flow is driven at the surfaces of the electrodes by electrical forces acting in the diffuse electrical double layer. The pumping mechanism has been analyzed theoretically under the assumption of perfectly polarizable electrodes. Here we extend these studies to include the effect of Faradaic currents on the electroosmotic slip velocity generated at the electrode/electrolyte interface. We integrate the electrokinetic equations under the thin-double-layer and low-potential approximations. Finally, we analyze the pumping of electrolyte induced by a traveling-wave signal applied to a microelectrode array using this linear model.     

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.     

Numerical analysis of electroosmotic flow in dense regular and random arrays of impermeable, nonconducting spheres

We present a numerical scheme for analyzing steady-state isothermal electroosmotic flow (EOF) in three-dimensional random porous media, involving solution of the coupled Poisson, Nernst-Planck, and Navier-Stokes equations. While traditional finite-difference methods were used to resolve the Poisson-Nernst-Planck problem, the (electro)hydrodynamics has been addressed with high efficiency using the lattice-Boltzmann method. The developed model allows simulation of electrokinetic transport under most general conditions, including arbitrary value and distribution of electrokinetic potential at the solid-liquid interface, electrolyte composition, and pore space morphology. The approach provides quantitative information on a spatial distribution of simulated velocities. This feature was utilized to characterize EOF fields in regular and random, confined and bulk packings of hard (i.e., impermeable, nonconducting) spheres. Important aspects of pore space morphology (sphere size distribution), surface heterogeneity (mismatch in electrokinetic potentials at confining wall and sphere surface), and fluid phase properties (electrical double layer thickness) were investigated with respect to their influence on the EOF dynamics over microscopic and macroscopic spatial domains. Most important is the observation of a generally nonuniform pore-level EOF velocity profile in the sphere packings (even in the thin double layer limit) which is caused by pore space morphology and which is in contrast to the pluglike velocity distribution in a single, straight capillary under the same conditions.     

Transient electroosmotic flow induced by DC or AC electric fields in a curved microtube

This study investigates transient electroosmotic flow in a rectangular curved microtube in which the fluid is driven by the application of an external DC or AC electric field. The resultant flow-field evolutions within the microtube are simulated using the backwards-Euler time-stepping numerical method to clarify the relationship between the changes in the axial-flow velocity and the intensity of the applied electric field. When the electric field is initially applied or varies, the fluid within the double layer responds virtually immediately, and the axial velocity within the double layer tends to follow the varying intensity of the applied electric field. The greatest net charge density exists at the corners of the microtube as a result of the overlapping electrical double layers of the two walls. It results in local maximum or minimum axial velocities in the corners during increasing or decreasing applied electric field intensity in either the positive or negative direction. As the fluid within the double layer starts to move, the bulk fluid is gradually dragged into motion through the diffusion of momentum from the double layer. A finite time is required for the full momentum of the double layer to diffuse to the bulk fluid; hence, a certain phase shift between the applied electric field and the flow response is inevitable. The patterns of the axial velocity contours during the transient evolution are investigated in this study. It is found that these patterns are determined by the efficiency of momentum diffusion from the double layer to the central region of the microtube.     

Investigation of the electroosmotic flow effect on the efficiency of capillary electrophoresis.

The influence of the electroosmotic flow profile on the efficiency and resolution of capillary electrophoresis is studied. The mathematical model is formulated and the set of equations is solved numerically. The results of the analysis are applicable to a wide range of buffer concentrations and capillary diameters. The temperature dependence of electrophoretic mobility, viscosity and thermal conductivity and the dependence of electrical conductivity on temperature and ion concentration in the double layer are taken into account. It is shown that there exists a region of buffer concentrations and capillary diameters where the influence of the electroosmotic flow profile on the efficiency and resolution is much greater than that of the temperature dependence of the electrophoretic mobility. The results are especially essential for small buffer concentrations or capillary diameters comparable with the double electrical layer thickness.     

Transient electrophoresis of dielectric spheres

The dynamic electrophoretic response of a spherical dielectric particle suspended in an electrolyte solution to a step change in the applied electrics field is analytically studied. The electrical double layer surrounding the particle may have either a small but finite thickness or a very large thickness relative to the particle radius. For the case of electrophoresis of a particle with a thin double layer, the local electroosmotic velocity at the outer edge of the double layer evolving with time after the external field is imposed is used as an apparent slip boundary condition at the particle surface so that the unsteady equation of motion for the fluid flow outside the double layer is solved. Closed-form formulas for the transient electrophoretic mobility of the particle are derived as functions of relevant parameters. The results demonstrate that, when the double layer surrounding the particle is relatively thin, the normalized electrophoretic mobility at a given dimensionless time decreases monotonically with a decrease in the parameter kappaa, where kappa(-1) is the Debye screening length and a is the particle radius. When the double layer of the particle is relatively thick, the particle mobility can have magnitudes comparable to those for a particle with a thin double layer in the initial stage, but will become much smaller afterward. In general, the effect of the relaxation time for transient electrophoresis is negligible, regardless of the value of kappaa.     

Studies on electroosmotic and electrophoretic behavior of cholesterol in aqueous solutions

Electrical properties of cholesterol interfaces have been investigated. For this purpose electroosmotic and hydrodynamic permeation of water, sodium chloride, barium chloride, aluminum chloride, and urea solutions across a cholesterol plug have been investigated. Dependence of electroosmotic permeability on concentration of electrolytes has also been studied. Electrophoretic mobility of cholesterol particles of known size distribution and dispersed in solutions of varying concentrations of electrolytes and urea has also been studied. The data have been used to estimate ζ potentials in order to have a plausible picture of the electrical double layer at the cholesterol/ solution interfaces.     

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.     

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.     

Electroosmotic flow in a water column surrounded by an immiscible liquid

In this paper, we conducted numerical simulation of the electroosmotic flow in a column of an aqueous solution surrounded by an immiscible liquid. While governing equations in this case are the same as that in the electroosmotic flow through a microchannel with solid walls, the main difference is the types of interfacial boundary conditions. The effects of electric double layer (EDL) and surface charge (SC) are considered to apply the most realistic model for the velocity boundary condition at the interface of the two fluids. Effects on the flow field of ?-potential and viscosity ratio of the two fluids were investigated. Similar to the electroosmotic flow in microchannels, an approximately flat velocity profile exists in the aqueous solution. In the immiscible fluid phase, the velocity decreases to zero from the interface toward the immiscible fluid phase. The velocity in both phases increases with ?-potential at the interface of the two fluids. The higher values of ?-potential also increase the slip velocity at the interface of the two fluids. For the same applied electric field and the same ?-potential at the interface of the two fluids, the more viscous immiscible fluid, the slower the system moves. The viscosity of the immiscible fluid phase also affects the flatness of the velocity profile in the aqueous solution.