Thickness-averaged model for numerical simulation of electroosmotic flow in three-dimensional microfluidic chips

Purpose The characteristics of fluid motions in micro-channel are strong fluid-wall surface interactions, high surface to volume ratio, extremely low Reynolds number laminar flow, surface roughness and wall surface or zeta potential. Due to zeta potential, an electrical double layer (EDL) is formed in the vicinity of the wall surface, namely, the stern layer (layer of immobile ions) and diffuse layer (layer of mobile ions). Hence, its competent designs demand more efficient micro-scale mixing mechanisms. This paper aims to therefore carry out numerical investigations of electro osmotic flow and mixing in a constricted microchannel by modifying the existing immersed boundary method. Design/methodology/approach The numerical solution of electro-osmotic flow is obtained by linking Navier–Stokes equation with Poisson and Nernst–Planck equation for electric field and transportation of ion, respectively. Fluids with different concentrations enter the microchannel and its mixing along its way is simulated by solving the governing equation specified for the concentration field. Both the electro-osmotic effects and channel constriction constitute a hybrid mixing technique, a combination of passive and active methods. In microchannels, the chief factors affecting the mixing efficiency were studied efficiently from results obtained numerically. Findings The results indicate that the mixing efficiency is influenced with a change in zeta potential (ζ), number of triangular obstacles, EDL thickness (λ). Mixing efficiency decreases with an increment in external electric field strength (Ex), Peclet number (Pe) and Reynolds number (Re). Mixing efficiency is increased from 28.2 to 50.2% with an increase in the number of triangular obstacles from 1 to 5. As the value of Re and Pe is decreased, the overall percentage increase in the mixing efficiency is 56.4% for the case of a mixing micro-channel constricted with five triangular obstacles. It is also vivid that as the EDL overlaps in the micro-channel, the mixing efficiency is 52.7% for the given zeta potential, Re and Pe values. The findings of this study may be useful in biomedical, biotechnological, drug delivery applications, cooling of microchips and deoxyribonucleic acid hybridization. Originality/value The process of mixing in microchannels is widely studied due to its application in various microfluidic devices like micro electromechanical systems and lab-on-a-chip devices. Hence, its competent designs demand more efficient micro-scale mixing mechanisms. The present study carries out numerical investigations by modifying the existing immersed boundary method, on pressure-driven electro osmotic flow and mixing in a constricted microchannel using the varied number of triangular obstacles by using a modified immersed boundary method. In microchannels, the theory of EDL combined with pressure-driven flow elucidates the electro-osmotic flow.

Chapter 4 - Single-Phase Electrokinetic Flow in Microchannels

Capillary zone electrophoresis (CZE) is attracting much attention (1-4) as a new separation technique that can complement high-performance liquid chromatography (HPLC). In CZE a migration channel of capillary dimensions is filled with electrolyte; a sample to be analyzed is injected at one end of the channel; and a high voltage is applied across the channel. When the electrolyte contacts the walls of the capillary, the inner surface of the capillary becomes charged, either through the ionization of surface groups on the capillary walls or through the adsorption of charged species from the electrolyte onto the inner surface. In either case, the electrolyte inside the capillary is no longer electroneutral but acquires a net charge, which may be positive or negative. Under the action of the applied electric field, the electrolyte moves toward one end of the capillary, and this movement is referred to as electroosmotic (electroendosmotic) flow. In addition to the bulk flow of the electrolyte, electrophoresis also takes place; that is, the applied electric field exerts a force on positively charged species to cause them to move to the negatively charged electrode and on negatively charged species to cause them to move to the positively charged electrode. As a result, the components in the injected sample separate into distinct zones, based on their different mobilities. However, in many cases, the rate of electrophoretic flow is typically less than the rate of electroosmotic flow. Consequently, species in the injected sample move in one direction-the direction of the electroosmotic flow-and thus the different species can be detected as each zone passes through some suitable detector located downstream from the capillary inlet. Clearly, a precise characterization of the electroosmotic flow is highly desirable not only for understanding CZE but also for optimizing the operation of CZE in analyzing a given sample. One way to measure the electroosmotic flow rate is to record the elution time of an injected uncharged marker solute, which will be carried through the capillary under the action of only electroosmotic flow (4-6). For this purpose it is necessary that the marker solute be truly neutral, that it have negligible interaction with the capillary walls, and that it be readily detected. Another way is to weigh the mass of electrolyte transferred from the capillary inlet to the capillary outlet over a timed interval (7). For this purpose, losses caused by evaporation must be eliminated and the use of a digital balance appears to be recommended. Both of these procedures have been demonstrated to give reliable measurements of the electroosmotic flow rate, provided that some care is taken. We describe here what we believe might be a simpler procedure for measuring the electroosmotic flow rate. I t is based on recording the time history of the current during CZE operation. Thus the new method requires no special type of injected solute or detector, and it can be used by anyone carrying out CZE separations.

Thin electroosmotic flow (EOF) micropumps can generate flow in confined spaces such as lab-on-a-chip microsystems and implantable drug delivery devices. However, status quo methods for quantifying flow and other important parameters in EOF micropumps depend on microfluidic interconnects or fluorescent particle tracking: methods that can be complex and error-prone. Here, we present a novel connected droplet shape analysis (CDSA) technique that simplifies flow rate and zeta potential quantification in thin EOF micropumps. We also show that a pair of droplets connected by an EOF pump can function as a tunable convex lens system (TCLS). We developed a biocompatible and all polymer EOF micropump with an SU-8 substrate and poly(3,4-ethylenedioxythiophene) polystyrene sulfonate (PEDOT:PSS) electrodes. We microdrilled a channel through the electrode/SU-8/electrode layers to realize a monolithic EOF micropump. Then, we deposited a pinned droplet on each end of the microchannel so that it connected them. By controlling the EOF between the droplets and measuring the corresponding change in their shape, we quantified the nanoliter EOF rate and zeta potential at the interface of SU-8 with two liquids (deionized water and a l-glutamate neurotransmitter solution). When the droplet pair and pump were used as a TCLS, CDSA successfully predicted how the focal length would change when the pump drove fluid from one droplet to another. In summary, CDSA is a simple low-cost technique for EOF rate and zeta potential measurement, and a pair of droplets connected by an EOF micropump can function as a TCLS without any moving parts.

Velocity slip in electroosmotic flows (EOF) is investigated using an analytical solution of Poisson–Boltzmann and Stokes (PB-S) equations with slip correction, and the results are compared with the predictions from molecular dynamics (MD) simulations. Particularly, EOFs of NaCl solution in silicon nanochannels are simulated, and channel dimensions are picked to exhibit plug flow behavior in the bulk region. Electrochemical conditions are specified within the validity region of PB theory to avoid charge inversion and flow reversal effects. The analytical model shows normalization of the slip length (an interfacial property) with the Debye length determined by the ionic concentration, implying that the slip enhancement in EOFs is independent of the channel height. MD results are compared with the analytical solution using the apparent viscosity and slip lengths determined by force-driven flow simulations in charged silicon nanochannels. Onset of slip velocity within the electrical double layer and its effects on EOF are investigated as a function of the liquid–wall interaction strength. Good agreements between the analytical solutions and MD results validate PB-S equations in nanochannels as small as 3.5 nm. Results validate that slip enhancement in EOFs is independent of the channel height, which constitutes a theoretical basis for the slip-length measurements on hydrophobic dielectric surfaces using EOFs.

Characterizing the mainly incompressible and laminar flows of aqueous electrolyte solutions through channels with an overall dimension of O(1–100 μm) is of interest in a variety of microfluidics applications. Solid surfaces such as the channel wall become (usually negatively) charged due to direct ionization or dissociation of surface groups, where the charge is typically characterized by the wall zeta-potential ζw. The surface in turn attracts mobile counterions from the fluid to form a (usually positively) charged screening, or electric double, layer (EDL). An external electric field can therefore be used to “pump” fluids through microfluidic Labs-on-a-Chip (LOC) by driving the charged fluid in the EDL. The resulting electroosmotic flow (EOF) is uniform outside the EDL, which has a thickness less than 50 nm in most cases. This uniform flow results in a more favorable scaling of the volume flowrate with channel diameter for microchannels, and also has less convective dispersion than shear flows. Electroosmotic flow is, however, very sensitive to changes in ζw. Various studies have shown, for example, that adding multivalent counterions to a monovalent electrolyte solution can greatly change ζw through both electrostatic and chemical interactions, even leading to “charge inversion” where the zeta-potential changes its sign. Evanescent-wave particle velocimetry, which tracks the motion of colloidal fluorescent tracer particles illuminated by evanescent waves within ∼400 nm of the wall, was therefore used to study the flow of various aqueous monovalent electrolyte solutions with small amounts of divalent cations such as Mg++ driven by an electric field through channels with a minimum dimension of ∼30 μm. The technique measures both the velocity components parallel to the wall and the steady-state distribution of these near-wall tracers. In these experiments, the tracers are convected parallel to the wall by both the EOF and directly by the applied electric field via electrophoresis because the surfaces of the particles also become negatively charged when suspended in the electrolyte solution. The electrophoretic contribution to the measured particle velocity was determined by measuring the particle zeta-potential with light scattering, and subtracted from the particle velocity to determine the actual EOF velocity.

Solid-state nanopores constitute a versatile platform for study of ion transport in nanoconfinement. The electrical double layer (EDL) plays a vital role in such nanoconfinements, but effects of induced surface charge on the EDL in the presence of an external transmembrane electric field are yet to be characterized. Here, the formation of induced charge on the nanopore sidewall surface and its effects, via modulation of the EDL and electroosmotic flow, on the ionic current are elucidated using a novel experimental setup with solid-state truncated-pyramidal nanopores. This study consists of three complementary approaches, i.e., an analytical model for induced surface charge, numerical simulation of induced surface charge, electroosmotic flow, and ionic current, and experimental validation with respect to the ionic current. The induced surface charge is generated by polarization in the dielectric membrane as a response to the applied electric field. This charge generation results in a nonuniform density of surface charge along the nanopore sidewall. It further causes ions in the electrolyte to redistribute, leading to a massive accumulation of single-polarity ions in the EDL and their counterions near the smaller opening of the nanopore. It also alters electrohydrodynamic properties in the nanopore, giving rise to the formation of electroosmotic vortexes in the vicinity of the smaller opening of the nanopore. Finally, the pattern of the electroosmotic flow can significantly influence the transport properties of the nanopore.

This paper presents a theoretical model for the electro-osmotic flow (EOF) of semi-dilute polyelectrolyte (PE) solutions in nanochannels. We use mean-field theories to describe the properties of electric double layer and viscosity of PE solutions that are prerequisites for constructing the EOF model. The EOF model is validated via a good match to the existing experimental results. Based on the validated EOF model, we conduct a comprehensive analysis of EOF of semi-dilute PE solutions in nanochannels. First, we observe considerable EOF of PE solutions in the uncharged nanochannels, which is in stark contrast to EOF of simple electrolyte solutions. The analyses show that the EOF of PE solutions in uncharged nanochannels is triggered by the external electric field acting on the near-wall non-electroneutral regions resulting from the confinement-induced inhomogeneous distribution of PE monomers. Although the solutions are electroneutral as a whole, the presence of local non-electroneutral regions and the mismatch between non-electroneutral regions and high-viscosity regions lead to the net EOF in uncharged nanochannels. Furthermore, we reveal that the EOF mobility $\mu _{{eof}}$ in uncharged nanochannels exhibits a scaling law $\mu _{{eof}} \propto a^{-0.44}$ (wherein $a$ denotes monomer Kuhn length) and is inversely proportional to the PE chain length, while it decreases nonlinearly with the charge fraction of the PE chains. Moreover, the EOF mobility reaches its maximum at specific bulk monomer concentration, and increases with the nanochannel height before converging to that under no confinement. Second, we analyse the EOF of PE solutions in nanochannels with various wall effects, such as surface charge density, slip length and adsorption length. When the surface charge is absent, the adsorption length significantly influences the direction and magnitude of the EOF, whereas the slip length has no effect. When the wall becomes increasingly charged, the influence of adsorption length on EOF gradually diminishes, while the importance of the slip length progressively intensifies and the EOF is highly influenced by the co-action of various wall effects in a complicated manner. When the surface wall is oppositely charged to polymer monomers, the EOF mobility varies nonlinearly with the surface charge density, while a zero net flow of EOF followed by a direction reversal is discovered when the wall is likely charged to polymer monomers.

Carbon nanotube membranes (CNMs) were prepared by doing chemical vapor deposition of carbon within the pores of a microporous alumina template. Electroosmotic flow (EOF) was driven across the CNMs by allowing the membrane to separate two electrolyte solutions and using an electrode in each solution to pass a constant ionic current through the nanotubes. EOF was investigated by measuring the flux of a probe molecule (phenol) across the CNM. The as-synthesized CNMs have anionic surface charge, and EOF is in the direction of cation migration across the membrane. Measurements of the rate of EOF as a function of applied transmembrane current provided the zeta potential. The effect of pH on zeta provided the pK(a) for the surface acidic sites responsible for this anionic charge; the acidic-site density was also determined. An electrochemical derivatization method was used to attach carboxylate groups to the nanotube walls; this enhances the anionic surface charge density, resulting in a corresponding increase in the EOF rate. Electrochemical derivatization was also used to attach cationic ammonium sites to the nanotube walls to yield CNMs that show EOF in the opposite direction of the as-synthesized or carboxylated membranes.

The differential migration of ions in an applied electric field is the basis for separation of chemical species by capillary electrophoresis. Axial diffusion of the concentration peak limits the separation efficiency. Electromigration dispersion is observed when the concentration of sample ions is comparable to that of the background ions. Under such conditions, the local electrical conductivity is significantly altered in the sample zone making the electric field, and therefore, the ion migration velocity concentration dependent. The resulting nonlinear wave exhibits shock like features, and, under certain simplifying assumptions, is described by Burgers' equation (S. Ghosal and Z. Chen Bull. Math. Biol. 201072, pg. 2047). In this paper, we consider the more general situation where the walls of the separation channel may have a non-zero zeta potential and are therefore able to sustain an electro-osmotic bulk flow. The main result is a one dimensional nonlinear advection diffusion equation for the area averaged concentration. This homogenized equation accounts for the Taylor-Aris dispersion resulting from the variation in the electro-osmotic slip velocity along the wall. It is shown that in a certain range of parameters, the electro-osmotic flow can actually reduce the total dispersion by delaying the formation of a concentration shock. However, if the electro-osmotic flow is sufficiently high, the total dispersion is increased because of the Taylor-Aris contribution.

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

Electroosmotic flow (EOF) is one of the most important techniques in a microfluidic system. Many microfluidic devices are made from a combination of different materials, and thus asymmetric electrochemical boundary conditions should be applied for the reasonable analysis of the EOF. In this study, the EOF of power-law fluids in a slit microchannel with different zeta potentials at the top and bottom walls are studied analytically. The flow is assumed to be steady, fully developed, and unidirectional with no applied pressure. The continuity equation, the Cauchy momentum equation, and the linearized Poisson-Boltzmann equation are solved for the velocity field. The exact solutions of the velocity distribution are obtained in terms of the Appell’s first hypergeometric functions. The velocity distributions are investigated and discussed as a function of the fluid behavior index, Debye length, and the difference in the zeta potential between the top and bottom.

We perform a systematic study on the modulation of electroosmotic flow (EOF), tuning the selectivity using electrolyte ions and hydrodynamic dispersion of the solute band across the soft nanochannel. The supporting walls of the channel are considered to be hydrophobic and bear non-zero surface charge. For such a channel, the inner side of the supporting rigid walls of the channel are coated with a soft polyelectrolyte layer (PEL). The inhomogeneous distribution of monomers and accompanying volume charge within the PEL is modelled via soft-step function. The dielectric permittivity of the PEL and electrolyte solution are in general different, which in turn leads to the ion partitioning effect. The impact of ion steric effects due to finite sized ions is further accounted through the modified ion activity coefficient. To model the EOF modulation considering the combined impact of the ion steric and ion partitioning effects as well as inhomogeneous distribution of monomers across the PEL, we adopt the modified Poisson-Boltzmann equation as the governing equation for electrostatic potential. The Debye-Bueche model is adopted to study the flow field across the PEL and the Stokes equation governs the EOF outside the PEL. In order to study the impact of the modulated EOF field on the dispersion of uncharged solution, we adopt three different models, i.e., a general 2D convective-diffusion model as well as cross-sectional averaged dispersion models due to Gill and late-time Taylor and Aris. Going beyond the widely employed Debye-Hückel approximation and uniform distribution of the monomer as well as accompanying volume charge, we find the results for the electric double layer (EDL) potential, EOF field and averaged throughput, by tuning the ion selectivity, etc., which is sufficient to analyze the transport of ionized liquid across the channel. The numerical results are supplemented with analytical results for the EDL potential as well as the EOF field under various limiting situations. Besides, we have further shown the impact of the modulated EOF field on the solute dispersion process. We have presented results that highlight the impact of parameters related to EOF field modulation, on solute dispersion governed by a convective-diffusive process, as well as obtaining the results for an effective dispersion coefficient. The dispersion models under the modulated EOF field adopted in the present study can thus be applied to study the dispersion process in engineered microdevices.

Growth of electroosmotic circulation (EOC) of a liquid in a closed capillary, which is generated after the application of an electric field between the two electrodes which cover the ends of the silica capillary, is examined numerically. The applied electric field accelerates the excess cations in the electric double layer on the inner surface of the capillary. The occurrence of collisions between the accelerated cations and the neutral molecules induces electroosmotic flow (EOF). A forward flow, caused by the EOF, initiates the EOC. The EOC generates a uniform cross-sectional force to create a return flow for conserving zero total flux, and approaches a steady parabolic flow profile, which is similar to the flow profile of the Poiseuille flow. Flow spectra reveal that the forward and return flows enhance each other under zero total flux and accelerate the growth of the EOC, which is more rapid than that of the EOF and the Poiseuille flow.

A numerical study is presented for the electroosmotic flow (EOF) at the cross region in microfluidic chips. The distributions of the electric potential due to the electric double layer (EDL) and the external electric field are discussed and the calculation of the latter can give rough speculations on the flow tendencies in the channels during various operation modes. Simplification of the two-dimensional Navier-Stokes (N-S) equations is obtained by focusing on the solution of interior flows, and the numerical calculation results show good agreement with the experimental images. The sample leakage to the separation channel during the "float" sampling proved to be caused not only by the sample diffusion, but also by the weak extension of the sampling electric field. It is also verified that with suitable voltage configuration, the "pinch" sampling mode is better than the "float" mode in sample plug control.