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Study of the Effect of AC Electric Potential Applied on Co-planar Microelectrode Array on Electroosmotic Flow in a Slit Microchannel by Naheed Ferdous Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN MECHANICAL ENGINEERING Department of Mechanical Engineering BANGLADESH UNIVERSITY OF ENGINEERING AND TECHNOLOGY Dhaka-1000, Bangladesh November, 2011 This thesis titled “Study of the Effect of AC Electric Potential Applied on Co-planar Microelectrode Array on Electroosmotic Flow in a Slit Microchannel”, submitted by Naheed Ferdous, Student No. 0409102031 Session April 2009 has been accepted as satisfactory in partial fulfillment of the requirement for the degree of MASTER OF SCIENCE IN MECHANICAL ENGINEERING on November 20, 2011. BOARD OF EXAMINERS Chairman _______________________ Dr. Noor Al Quddus Assistant Professor Department of Mechanical Engineering Bangladesh University of Engineering and Technology Dhaka-1000, Bangladesh. Member _______________________ Dr. Muhammad Mahbubul Alam Professor and Head Department of Mechanical Engineering Bangladesh University of Engineering and Technology (Ex-Officio) Dhaka-1000, Bangladesh. Member _______________________ Dr. Mohammad Ali Professor Department of Mechanical Engineering Bangladesh University of Engineering and Technology Dhaka-1000, Bangladesh. Member _______________________ (External) Dr. Dewan Hasan Ahmed Assistant Professor Department of Mechanical and Production Engineering Ahsanullah University of Science and Technology Tejgaon, Dhaka, Bangladesh. ii CANDIDATE’S DECLARATION It is hereby declared that this thesis or any part of it has not been submitted elsewhere for the award of any degree or diploma. Signature of the Candidate Naheed Ferdous iii CERTIFICATE OF RESEARCH This is to certify that the work presented in this thesis is carried out by the author under the supervision of Dr. Noor Al Quddus, Assistant Professor of the Department of Mechanical Engineering, Bangladesh University of Engineering and Technology, Dhaka, Bangladesh. _______________________ _______________________ Dr. Noor Al Quddus Naheed Ferdous iv Dedicated to My Parents v ACKNOWLEDGMENTS I want to express my gratefulness to my supervisor Dr. Noor Al Quddus who has guided me through the course of my graduate study and research. Without his guidance, support and suggestion this thesis was nearly unthinkable to complete. He has drawn my attention to various aspects of fluid mechanics and motivated me for the fundamental research work. I have benefitted enormously from our many discussions and his huge engagement with his students. I would like to thank my parents and my elder brother for giving me unquestionable support to continue my graduate study. I also want to express my gratitude to my husband for always giving me support and inspirations in the course of my graduate study. I give thanks to the lecturer Mahbubul Islam for his help in proceeding formally with my thesis. vi ABSTRACT Under the influence of an AC electric field, electrolytes on a planar microelectrodes exhibit steady fluid flow, termed as AC electroosmosis. A numerical model using finite element method has been developed to solve the electrokinetic flow parameters of the AC electroosmosis in a slit microchannel. A thin-layer, low-frequency, nonlinear analysis of the system is performed including Faradaic currents from electrochemical reactions at the electrodes. The non equilibrium model consists of Navier-Stokes, continuity, Nernst–Planck and Poisson equations. In the first step, transient simulations are carried out to obtain a homogeneous steady state stable periodic regime. The obtained stable period solutions are then analyzed to compute the time averaged net velocity and other characteristics of the AC electroosmotic flow. Net flow velocity and flow rate are observed for different geometric parameters, electric field parameters, electrode position, Debye length and effective thickness. Dependency of different parameters on the frequency of AC signal is analyzed for optimal design of AC electroosmotic micropump. Such mechanism can be used to transport, mix, separate, and manipulate various molecular or colloidal entities e.g. DNA, protein, polymers etc. in microfluidic chips. vii TABLE OF CONTENTS Item Page Title Page i Board of Examiners ii Candidate’s Declaration iii Certificate of Research iv Dedication v Acknowledgments vi Abstract vii Table of Contents viii Nomenclature x List of Table xii List of Figures xiii Introduction 1 1.1 Background and Motivation 1 1.2 Objective of the Thesis 3 1.3 Outline of the Thesis 4 Chapter 2 Literature Review 5 Chapter 3 Theoretical Description of the Model 12 3.1 Electric Double Layer 12 3.2 Electroosmosis 13 3.3 AC Electroosmosis 14 3.4 Model Geometry 16 3.5 Governing Equations (Equilibrium Model) 17 3.5.1 Electric field and ionic transport 17 3.5.2 Fluid flow 19 Necessity of Non-Equilibrium Model 19 Chapter 1 3.6 viii 3.7 Contribution of Faradaic Current 20 3.8 Non-Dimensionalization of the Equations 23 3.9 Boundary Conditions 24 Numerical Solution Methodology 26 4.1 Finite Element Method 26 4.2 Computer Implementation of the model 26 4.3 Mesh Sensitivity Analysis 27 4.4 Flowchart of the Overall Solution Methodology 28 Results and Discussion 29 5.1 Validation of the Model 29 5.2 Inclusion of Faradaic Current 32 5.3 Asymmetric vs Symmetric Electrodes 36 5.4 Velocity and Electric Field Characteristics 37 5.5 Effect of AC Frequency on Time Averaged Net Velocity and Flow Rate 46 5.6 Dependence of Time Averaged Net Velocity on Debye Length 48 and Compact Layer Relative Thickness 5.7 Effect of Electrode and Channel Geometry on Time Averaged 51 Net Velocity Chapter 4 Chapter 5 Conclusion and Recommendations 53 6.1 Concluding Remarks 53 6.2 Recommendation for Future Works 54 Chapter 6 55 References ix NOMENCLATURE Notation Definition A Amplitude [V] c Concentration [mol m-3] D Diffusivity [m2s-1] f Frequency [s-1] F The Faraday constant [96,485 Cmol-1] H Micro channel height [m] L Length of periodic segment [m] G Gap size [m] E Electrode size [m] n Normal unit vector p Pressure [Pa] R Molar gas constant [ 8.314 J K-1 mol -1] t Time [s] T Temperature [ 298.15 K] u Net velocity [ms-1 ] û Time averaged net velocity [ms-1] x, y Spatial coordinate [m] Electrolyte permittivity [Fm-1] Electric potential [V] Dynamic viscosity [PaS] D The Debye length [m], D RT 2 c0 F 2 q Space charge density [Cm-3 ] e Fundamental charge [ 1.6021765×10 −19 C] s Compact layer width [m] x Superscripts and Subscripts e electrode ~ dimensionless ± Either + or s OHP + 0 eq xi cation anion neutral Characteristic value LIST OF TABLES Table No. Table I: Page No. Dimensionless Parameters …….................................................................. 22 xii LIST OF FIGURES Fig no Fig 3.1 Fig 3.2 Structure of the electric double layer.................................................. Page 12 Geometry of a circular cylindrical microchannel used for modeling electroosmotic flow The principle of the AC electro osmotic flow above co-planar electrodes.......................................................................................... Electrostatic volume force induces the eddy formation (solid line) and the net flow (dashed line)............................................................. 13 Fig 3.5 Structure of the electric double layer (EDL) on an array of electrodes subjected to an AC potential........................................... 16 Fig 3.6 Scheme of one segment of the AC electroosmotic micropump. The dashed-dotted lines indicate the periodic boundary conditions......... 17 Fig 4.1 Mesh sensitivity analysis for time averaged net velocity................... 27 Fig 5.1 Model validation with analytical result............................................. 30 Fig 5.2 Schematic Diagram of the AC electroosmotic mircropump. The dashed dotted line indicates the periodic boundary conditions........ 30 Fig 5.3 Comparison of time averaged net velocity between Cervenka model and present model.............................................................................. 31 Fig 5.4 Effect of Faradaic current on net velocity. ...................................... 33 Fig 5.5 Net Velocity without Faradaic current............................................. 34 Fig 5.6 Net Velocity with Faradaic current.................................................... 35 Fig 5.7 Net velocity comparison for asymmetric and symmetric electrode arrangements...................................................................................... 36 Fig 5.8 Velocity stream lines. ....................................................................... 38 Fig 5.9 Velocity contour................................................................................. 38 Fig 5.10 The local orientation and the magnitude of the velocity vector by arrows............................................................................................... 39 Fig 5.11 Electric field and stream line.......................................................... 39 Fig 5.12 Plot of tangential field above electrodes.......................................... 40 Fig 3.3 Fig 3.4 xiii 14 15 Fig 5.13 Horizontal component of the velocity vector along a horizontal line 0.001µm above the electrodes.......................................................... 41 Fig 5.14 Horizontal component of the velocity vector along a vertical line at the channel outlet (far from the electrodes) ..................................... 42 Fig 5.15 Time averaged net velocity along a vertical line at the channel outlet (far from the electrodes) ........................................................ 43 Fig 5.16 Horizontal component of the velocity vector along a vertical line at the midpoint between the electrodes................................................... 44 Fig 5.17 Time averaged net velocity along a vertical line at the midpoint between the electrodes ....................................................................... 45 Fig 5.18 Dependence of the time-averaged net velocity on the AC frequency............................................................................................ 46 Fig 5.19 Flow rate at channel outlet and midpoint between electrodes........ 47 Fig 5.20 Dependence of the time-averaged net velocity on Debye length ...... 48 Fig 5.21 Dependence on the compact layer thickness. .................................... 49 Fig 5.22 Flow reversal..................................................................................... 50 Fig 5.23 Effect of Microchannel height on time averaged net velocity. ........ 51 Fig 5.24 Effect of relative size of electrodes................................................... 52 xiv 1. INTRODUCTION 1.1 Background and Motivation A large family of microfluidic devices employs electrokinetic transport of liquids. The integration of mechanical components such as high-pressure pump or valve is not required in electrokinetically driven microfluidic chips. Instead of a pressure gradient, a gradient of electric potential is imposed over the microfluidic system. It can be provided by either DC or AC electric field. DC electroosmotic transport in microchannels has been studied intensively both theoretically [1] and experimentally [2-3]. Origin of the DC electroosmosis (DCEO) arises from coulombic interaction between an external DC electric field and a mobile electric charge localized at an electrolyte-dielectric interface. The use of the electroosmosis forced by a low amplitude AC electric field is a novel area investigated only for several years. The AC electroosmosis (ACEO) is mostly based on the coulombic interaction between a imposed AC electric field and a mobile electric charge temporally arising at electrolyte metal electrode interfaces. Increasing complexity of microfluidic devices enforces the need for integrated local fluid control in microchannels. Electric fields applied on microelectrodes can induce electrokinectic pumping of electrolytes. The electrically induced pumping systems are mostly based on DC electroosmotic phenomena that usually require high voltages. Integration of electrodes in microfluidic channels for standard DCEO pumping is seriously limited by undesirable effects related to high voltages, such as joule heating, significant changes of the electrolyte composition, and bubble formation. A promising alternative to the classical electroosmotic pumps relies on microeletrode fluidic systems that exploit AC signals with very low amplitude. The developments in microfluidics calls for a means to transport and mix fluid at microscale with reliability and control. Such microfluidic devices are instrumental to the realization or improvement of miniature bio- medical-chemical diagnostic kits [4-5], high performance liquid chromatographs, fuel cells, ion exchange devices [6], chip and micro-circuit cooling, biochips for drug screening etc. ACEO flows are receiving growing interest in microfluidic applications because the integration of microelectrodes into microchannels makes possible the local actuation of electrolytes by means of electric fields [7-8]. Recent developments in micro-fabrication and the technological promise of microfluidic "labs on a chip" have brought a renewed interest to the study of low Reynolds number flows [9]. The microscale mixing of miscible fluids must occur without the benefit of turbulence, by molecular diffusion alone. For extremely small devices, molecular diffusion is relatively rapid; however, in typical microfluidic devices, the mixing can be prohibitively long. Another limitation is that the pressure-driven flow rate through small channels decreases with the third or fourth power of channel size. For pumping, mixing, manipulating and separating on the micron length scale, many focuses on the use of surface phenomena, owing to the large 1 surface to volume ratios of typical microfluidic devices [10]. ACEO provides one of the most popular non-mechanical techniques in microfluidics. Electrical forces are used to transport, mix, separate, and manipulate various molecular or colloidal entities in microfluidic chips. This principle is utilized to separate proteins of different sizes or charges, DNA, bacteria, polymers and cells [11]. Under the influence of an AC electric field, electrolytes on a planar microelectrodes exhibit steady fluid flow, termed as AC electroosmosis. The flow has its origin in the interaction of the tangential component of the non-uniform field with the induced charge in the electric double layer on the electrode surfaces. The mechanism of AC electroosmotic driven fluid flow has recently been shown to be capable of producing unidirectional pumping of liquid on a microscale, on asymmetric pairs of coplanar microelectrodes as demonstrated experimentally [12], theoretically [13], and numerically [14]. The measurements have been analyzed as a function of applied electric field (potential, frequency), electrolyte conductivity, and position on the electrode. At low potentials, the predicted velocities are in reasonable agreement with the experiments. One can apply spatially inhomogeneous electric field, vary the geometry or electrical properties of the polarizable electrode surface etc. The directions seem promising to investigate numerically and pursue experimentally in real microfluidic devices. Coulombic polarization of the microelectrodes is usually considered in the theoretical models of AC electroosmotic transport. However, Faradaic currents can occur in such systems as a result of electrochemical reactions. Linear theoretical studies are used to measure fluid flow of electrolytic solutions induced by a traveling-wave potential applied to an array of coplanar inter digitated microelectrodes [15]. The effect of Faradaic currents on the electroosmotic slip velocity generated at the electrode/electrolyte interface is taken over the assumption of perfectly polarizable electrodes. The combined action of two experimentally relevant effects, i.e Faradaic currents from electrochemical reactions at the electrodes and differences in ion mobilities of the electrolyte is taken into account in the analyses of AC electroosmosis [16]. Theoretical results are also applied for analyzing the influence of Faradaic reactions on AC electroosmotic flows [17]. Electrokinetic flow induced by the pair of electrodes is numerically obtained using the model. A new prediction of the model is that, for certain values of the parameters, fluid flow can occur in opposite direction to that obtained in the absence of Faradaic reactions. A linear thindouble-layer analysis is employed where the electrical problem is decoupled from the mechanical one. In previous works, the AC electroosmotic motion has been analyzed theoretically under the assumption that only forces in the diffuse (Debye) layer are relevant. A thin-layer, lowfrequency, linear analysis is performed and the model is applied to the case of an electrolyte actuated by a traveling-wave signal. A steady liquid motion in opposite direction to the applied signal is predicted for some ranges of the parameters. This could serve as a partial explanation for the observed flow reversal in some experiments. Approximations such as negligible advective currents, linearization of Poisson-Nernst-Planck equations, thin layer 2 approximation etc are made for this solution. However, the need for a complete analysis which must be fully nonlinear is felt for explanation of the underlying mechanism of AC electroosmosis considering Faradaic currents. Motivated by this, a complete nonlinear model for low-frequency ACEO flow including Faradaic currents from electrochemical reactions at the electrodes has been developed in the present study. A two dimensional geometric model is utilized to obtain the solution for velocity distribution in a slit microchannel. Faradaic current is incorporated in the nonlinear (non equilibrium) analysis by using electrochemical reaction at electrode surface. The electric potential is solved assigning known potentials to the electrodes with sinusoidal function along with solving the double layer charge density using charge conservation equations. The motion of the bulk fluid is numerically calculated, using no slip condition at the electrode surface as the boundary condition. A commercial finite element package is used to obtain the solution. Horizontal component of velocity is calculated for one cycle when the system goes to a steady state from transient phase. Total velocity for different fraction of time in one steady state cycle is determined and time averaged net velocity is calculated for one complete cycle. Volumetric flow rate per unit width of microchannel is computed by integrating velocity. Responsive behavior of fluid velocity and flow rate in symmetric microelectrode array with the change of applied AC frequency, concentration of electrolyte i.e Debye length and geometric parameters of microchannel are investigated. 1.2 Objective of the Thesis The specific objectives of the research work are as follows: (a) To develop a 2-D finite element model employing nonlinear non equilibrium approach. (b) To study the electroosmotic effect when asymmetric AC electric potential is applied on symmetric electrodes (c) To investigate the effect of geometry, applied electric field, electrolyte concentration on the time averaged net velocity of electrolyte (d) To study the effect of Faradaic current on net flow for coplanar asymmetric and symmetric microelectrodes The interesting consequences of shape and field asymmetries, which generally lead to electroosmotic pumping or electrophoretic motion in AC fields will give some basic issues for the microfluidic devices. 3 1.3 Outline of the Thesis A detail literature review is provided in Chapter 2. The theoretical description of the model is described in Chapter 3. The slit microchannel geometric configuration, governing equations for fluid flow, ion transport, electric field, boundary conditions, non-dimensionalization of the model are described in this chapter. The origin of the electrokinetic transport and the principle of AC electroosmotic micropumps are also shortly described. The multiphysical model of electro-microfluidic systems based on the Poisson-Nernst-Planck-Navier-Stokes approach and its possible simplifications is briefly described in this section. In Chapter 4, numerical solution methodology of the model is described. A brief description of the procedure involved in finite element analysis is provided. A flowchart depicts the overall solution methodology. Chapter 5 presents the simulation results. First the model is validated by comparing the calculated electroosmotic velocity of fluid flowing through a channel, against analytical and numerical results found in the literature. In the later sections, the values of the time averaged net velocity and flow rate are presented for different AC frequency values, electrolyte concentration, geometric parameters of microchannel. In Chapter 6, the conclusion and recommendations for future works are presented. 4 2. LITERATUE REVIEW Under the influence of an AC electric field, electrolytes on planar microelectrodes exhibit fluid flow. The nonuniform electric field generated by the electrodes interacts with the suspending fluid through a number of mechanisms giving rise to body forces and fluid flow. The mechanism responsible for this motion has been termed AC electroosmosis (ACEO). In other models, it is a continuous flow driven by the interaction of the oscillating electric field and the charge at the diffuse double layer on the electrodes. Experiments were performed to find optimal conditions for obtaining particle and bacterial assembly lines on electrodes by ACEO and preliminary results showed good resolution at a concentration of 104 bacteria/ml, indicating that combining ACEO with impedance measurement can improve the sensitivity of particle electrical detection [4]. Due to different bacterial impedance signatures in tap water and PBS buffers, experiments were performed to find the optimal voltage and frequency ranges such that a trapping converging flow exists on the electrodes and the assembled cells exhibit sensitive impedance spectrum signatures [5]. Two types of ACEO devices, in the configurations of planar interdigitated electrodes and parallel plate electrodes, and a biased ACEO technique were studied, which provided added flexibility in particle manipulation and line assembly [6]. A real-time particle concentration technique using a novel electrokinetic method was used to bridge the gap between the detectable level and infectious level of bacterial solutions [7]. The dynamics of a particle with both polarizability and net charges in a non-uniform AC electric trapping field was investigated through the study of AC electrokinetics and AC electrophoretic (ACEP) phenomena [8]. Velocity was measured for a new type of on-chip micro-pump that exploits the AC electro-kinetic forces acting in the volume of a fluid in the presence of a temperature gradient [9]. Covalent surface modification techniques, in particular surface oxidation procedures, was employed as a mean to modify polymer microfluidic channels for the purpose of modulating microflow [10]. Due to the fact that skin electrical resistance can be controlled by an alternating current (AC) electric field, a human epidermal membrane was modeled to assess the effects of AC voltage and frequency and direct current (DC) offset on the flux of neutral and ionic model permeants [11]. The velocity of fluid flow on microelectrodes at frequencies below the charge relaxation frequency of the electrolyte was shown by detailed experimental measurements [12]. The velocity of latex tracer particles was measured as a function of applied signal frequency and potential, electrolyte conductivity, and position on the electrode surface. The data were discussed in terms of a linear model of AC electroosmosis: the interaction of the nonuniform AC field and the induced electrical double layer. The fluid flow was predominant at frequencies of the order of the relaxation frequency of the electrode electrolyte system. A theoretical approach was developed to this problem using a 5 linear double layer analysis [13]. The theoretical results were compared with the experiments, and a good correlation was found. ACEO was studied experimentally and theoretically using linear analysis: Potential drop across the double layer at the surface of the electrodes was calculated numerically using a linear double layer model [14]. Experimental observations of the fluid flow profile were pre obtained by superimposing images of particle movement in a plane normal to the electrode surface. These experimental streamlines demonstrated that the fluid flow was driven at the surface of the electrodes. Experimental measurements of the impedance of the electrical double layer on the electrodes were also reported and was used in numerical analysis. The AC electroosmotic flow at the surface of the electrodes was calculated using the HelmholtzSmoluchowski formula. The bulk fluid flow driven by this surface velocity was numerically calculated as a function of frequency and good agreement was found between the numerical and experimental streamlines. Net fluid flow of electrolytic solutions induced by a traveling-wave potential applied to an array of co-planar microelectrodes was reported in literature [15]. At low applied voltages the flow was driven in the direction of the traveling-wave potential, as expected by linear and weakly nonlinear theoretical studies. The flow was 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, the study was extended to include the effect of Faradaic currents on the electroosmotic slip velocity generated at the electrode/electrolyte interface. The electrokinetic equations was integrated under the thin-double-layer and low-potential approximations. Finally, the pumping of electrolyte induced by a traveling-wave signal applied to a microelectrode array was analyzed using this linear model. Previous analyses of AC electroosmosis was extended [16] to account for the combined action of two experimentally relevant effects: (i) Faradaic currents from electrochemical reactions at the electrodes, and (ii) differences in ion mobilities of the electrolyte. In previous works, the AC electroosmotic motion has been analyzed theoretically under the assumption that only forces in the diffuse (Debye) layer are relevant. Here, it was shown that different ion mobilities of a 1-1 aqueous solution made the charged zone to expand from the Debye layer to include the diffusion layer. The Faradaic currents were included later and, as an attempt to explore both factors simultaneously, a thin-layer, low-frequency, linear analysis of the system was performed. Finally, the model was also applied to the case of an electrolyte actuated by a traveling-wave signal. A steady liquid motion in opposite direction to the applied signal was predicted for some ranges of the parameters. The simplest microelectrode structure was chosen to analyze the electrokinetic flow induced by a pair of coplanar symmetric microelectrodes[17]. The theoretical results were applied for analyzing the influence of Faradaic reactions on the flows. The liquid was assumed to be a 1:1 electrolyte with ions of different diffusivities. For simplicity, in the model only the cations were responsible for the Faradaic reactions while the anions did not react at the 6 electrodes. The prediction of the model was that, for certain values of the parameters, fluid flow can occur in opposite direction to that obtained in the absence of Faradaic reactions. The general phenomenon of ‘induced-charge electroosmosis’ (ICEO) was described as the nonlinear electroosmotic slip that occurs when an applied field acts on the ionic charge it induces around a polarizable surface. Motivated by a simple physical picture, ICEO flows were calculated around conducting cylinders in steady (DC), oscillatory (AC), and suddenly applied electric fields [18]. The system represented perhaps the clearest example of nonlinear electrokinetic phenomena. This physically motivated approach was complemented and verified using a matched asymptotic expansion to the electrokinetic equations in the thindouble-layer and low-potential limits. ICEO slip velocities vary as u s ∝E20L, where E0 is the field strength and L is a geometric length scale, and are set up on a time scale τc =λDL/D, where λD is the screening length and D is the ionic diffusion constant. ICEO microfluidic pumps and mixers were proposed and analyzed under low applied potentials. Similar flows around metallic colloids with fixed total charge have been described in the Russian literature. ICEO flows around conductors with fixed potential, on the other hand, have no colloidal analogue and offer further possibilities for microfluidic applications. A physical description of ICEO and the nonlinear electrokinetic slip at a polarizable surface were given in the context of some new techniques for microfluidic pumping and mixing [19]. ICEO generalizes ‘‘AC electroosmosis’’ at microelectrode arrays to various dielectric and conducting structures in weak DC or AC electric fields. The basic effect produces micro vortices to enhance mixing in microfluidic devices, while various broken symmetries— controlled potential, irregular shape, nonuniform surface properties, and field gradients—can be exploited to produce streaming flows. Although the qualitative picture of ICEO was emphasized, the mathematical theory (for thin double layers and weak fields) was also briefly described and applied to a metal cylinder with a dielectric coating in a suddenly applied DC field. Experimental and numerical investigations of ICEO on a planar electrode surface directly in contact with a high conductivity electrolytic solution was reported [20]. Symmetric rolls of ICEO flow was produced on the electrode by placing it in an AC electric field. The slip velocity was measured for a range of AC voltages and frequencies using micro particle image velocimetry (µPIV). The slip velocity was also calculated by finite element simulations based on a linear and a nonlinear model of electrical double layer, respectively. The µPIV measurements were found to be much lower (two and half orders of magnitude) than the velocities predicted by the linear model. The linear model is valid only under Debye Huckel approximation which does not hold true for practical situations. The nonlinear model, on the other hand, predicts velocities which are lower than the linear model and closer to the experimental values. The nonlinearity reduces discrepancy between experimental and numerical results by approximately an order of magnitude. The nonlinear model accounts for nonlinear capacitance of the double layer and lateral conduction of charge in the double layer. 7 Numerical and experimental studies were also used to investigate the increase in efficiency of microfluidic AC electroosmotic pumps by introducing nonplanar geometries with raised steps on the electrodes [21]. The effect of the step height on AC electroosmotic pump performance was analyzed . AC electroosmotic pumps with three-dimensional electroplated steps were fabricated on glass substrates and pumping velocities of low ionic strength electrolyte solutions were measured systematically using a custom microfluidic device. The response of a model micro electrochemical cell to a large AC voltage of frequency was investigated [22]. To bring out the basic physics, the simplest possible model of a symmetric binary electrolyte was considered which was confined between parallel-plate blocking electrodes, ignoring any transverse instability or fluid flow. The resulting one-dimensional problem was analyzed by matched asymptotic expansions in the limit of thin double layers and two features are considered in the strongly nonlinear regime—significant salt depletion in the electrolyte near the electrodes and, at very large voltage, the breakdown of the quasi equilibrium structure of the double layers. The former leads to the prediction of “AC capacitive desalination” since there was a time-averaged transfer of salt from the bulk to the double layers, via oscillating diffusion layers. The latter was associated with transient diffusion limitation, which drives the formation and collapse of space-charge layers, even in the absence of any net Faradaic current through the cell. It was also predicted that steric effects of finite ion sizes (going beyond dilute-solution theory) act to suppress the strongly nonlinear regime in the limit of concentrated electrolytes, ionic liquids, and molten salts. Beyond the model problem, the reduced equations for thin double layers, based on uniformly valid matched asymptotic expansions, provided a useful mathematical framework to describe additional nonlinear responses to large AC voltages, such as Faradaic reactions, electroosmotic instabilities, and induced-charge electrokinetic phenomena. Theoretical models were presented for the time-dependent voltage of an electrochemical cell in response to a current step, including effects of diffuse charge (or “space charge”) near the electrodes on Faradaic reaction kinetics [23]. The full model was based on the classical Poisson-Nernst-Planck equations with generalized Frumkin- Butler-Volmer boundary conditions to describe electron-transfer reactions across the Stern layer at the electrode surface. In practical situations, diffuse charge is confined to thin diffuse layers (DLs) which poses numerical difficulties for the full model but allows simplification by asymptotic analysis. For a thin quasi equilibrium DL, effective boundary conditions were derived on the quasi-neutral bulk electrolyte at the diffusion time scale, valid up to the transition time, where the bulk concentration vanishes due to diffusion limitation. The thin-DL problem was integrated analytically to obtain a set of algebraic equations, whose (numerical) solution compares favorably to the full model. In the Gouy-Chapman and Helmholtz limits, where the Stern layer is thin or thick compared to the DL, respectively, simple analytical formulas were derived for the cell voltage versus time. The full model also described the fast initial capacitive charging of the DLs and super limiting currents beyond the transition time, where the DL expands to a transient non-equilibrium structure. The well-known Sand equation was extended for the transition time to include all values of the super limiting current beyond the diffusion-limiting current. 8 Two mathematical models of the electrokinetic flow were presented based on: (i) momentum balance of the electrolyte, (b) continuity equation, (c) molar balances of the components of the uni-univalent electrolyte, and (d) Poisson equation of electrostatics [24]. The electroosmotic flow was induced by the interaction of a surface electric charge with a perpendicularly imposed electric field. Both the models were characterized by the formation of extremely thin surface layers with gradients of electric potential, pressure, concentrations and velocity. In order to solve such problems, an anisotropic mesh of rectangular finite elements was developed. The first model described a classical electroosmotic problem – the electrolyte dosing in a microfluidic channel with an axially imposed DC electric field . Stationary distributions of the model variables were computed for various sets of model parameters: applied voltage, density of the surface electric charge, microchannel diameter, electrolyte concentration etc. The interaction of a low-concentrated biological analyte with a receptor bound on the microchannel walls was studied by means of dynamical simulations. The second model dealt with AC electroosmosis that was based on the application of AC voltage on two electrodes of different size. The formed asymmetric electric field pattern caused the zigzag motion of electrolyte in microcapillary with one dominant direction. The dynamical simulations were aimed at the effects of voltage frequency, electrode size and electrolyte concentration on velocity of the electrolyte flow. A full dynamic description was used, instead of the linearized model to show the results of the mathematical modeling of AC electroosmotic micropump [25]. Skewed hybrid discretization meshes were employed in order to accurately capture the main features of the studied system. Also, zig-zag electrode arrangements were introduced for traveling-wave electroosmotic micropumps. The detailed analysis of the system behavior was presented by means of the examination of the model properties. AC electroosmotic micropumps were suggested to be powerful tools for electrolyte dosing in various micro and nanofluidic systems. Two modeling approaches were compared for studying the AC electroosmosis in the following micro and nanochannel systems: (i) a traveling- wave AC pump with a spatially continuous wave of electric potential applied on a planar boundary, (ii) a traveling- wave AC pump with a wave of electric potential applied on a set of discrete planar electrodes, and (iii) an AC pump with a set of non-planar electrodes [26]. The equilibrium approach was based on the use of capacitor–resistor boundary conditions for electric potential and the slip boundary conditions for velocity at electrode surfaces. The non-equilibrium approach used the mathematical model based on the Poisson equation and the non-slip boundary conditions. Discrepancies have been observed between the predictions given by the both models and then their possible reasons were identified. The comparison of the equilibrium and non-equilibrium results further showed three important actualities: (a) how the equilibrium model overestimates or underestimates the net velocity, (b) how the velocity maxima in the frequency characteristics can be shifted, if the equilibrium model assumptions are not satisfied, (c) the parametric region where the equilibrium model is applicable. The limitations of the equilibrium and non-equilibrium models were discussed and selected predictions with available experimental data were compared. 9 The coplanar asymmetric arrangements of the forcing electrodes were analyzed by means of the following two mathematical models: 1) the classical slip model, which is based on a capacitor–resistor representation of the spatial domain, and 2) the nonslip model, which is based on the Poisson–Navier–Stokes–Nernst–Planck approach to the entire domain, including the electric double layers [27] . Both the models predicted similar results in many lowamplitude regimes. However, the nonslip model gave a much better insight on the highamplitude (nonlinear) behavior of the micropumps. Most important findings obtained by the nonslip model were summarized as follows: 1) There are optimal values of the electrode and gap size ratios that are generally different from those obtained by the slip model; 2) the micropump performance is relatively insensitive with respect to the electrode size ratio; 3) there is an optimal vertical confinement that enables to attain high net velocities; 4) flow reversals on frequency, amplitude, and certain geometry characteristics are observed; 5) the energy efficiency of these pumps is very low; and 6) the Joule heating effect is negligible. The nonslip model characteristics were also discussed to explain the observed differences between predictions of the models. Convergence analysis dealing with the precision of numerical results obtained by the nonslip model was presented. AC electroosmotic pumping of fluids in a microchannel was studied with two thin coplanar electrodes, deposited on one dielectric wall and placed in an aqueous solution [28]. Two different numerical formulations were compared. In the first one, the transport equations (mass, molar, and momentum balances) for the ion concentration and the Poisson equation for the electric field were solved simultaneously in the time domain without any simplification of the computational domain except that the dielectric walls are neglected. In the second one, a time average model was assumed, and the Debye layer was represented by an approximate 1-D model. In both cases, the finite-element method implemented in the COMSOL commercial software was used as a numerical tool. The results for the electric field, space charge, and velocity distributions were compared with a qualitative agreement between both models. Micropumps are able to manipulate small volumes of liquid samples and involve no moving parts. The pumps find their use in a variety of bio applications or clinical diagnostics. The micropump was realized by an array of microelectrodes coated along the microchannel, on which an AC electric field was applied. The AC voltage was typically in the range of few volts. Theoretical models describing the AC electroosmotic transport usually consider the Coulombic polarization of the microelectrodes. However, Faradaic currents can occur in such systems as a result of electrochemical reactions. A mathematical model of an AC electroosmotic system was presented with electrochemical reactions [29]. The model was based on the balances of mass and electric charge and the kinetic equations for the electrode reaction. The theory of the electrolyte dynamics at polarized surfaces for larger applied voltages taking into account the steric effects (formation of condensed ionic layers) has been also published by other authors. 10 The analytical validation of Poisson-Boltzmann (PB) equation was computed with Comsol Multiphysics, in the case of a polarized surface in contact with the electrolyte [30]. Comsol Multiphysics algorithms easily handled the highly nonlinear aspect of the PB equation. The limitations of the PB model, that considers ions as point like charges, were outlined. To account for the steric effects of the ion crowding at the charged surface, the Modified Poisson-Boltzmann model was analyzed for symmetric electrolytes. The MPB equation was then coupled to the complex AC electrokinetic and the Navier-Stokes equations to simulate the AC electroosmosis flow observed inside an interdigitated electrodes microsystem. 11 3. THEORETICAL DESCRIPTION OF THE MODEL 3.1 Electric Double Layer The electroosmotic transport can be induced by an external electric field in fluidic systems where the electric polarization (charging) of solid surfaces occurs. Some organic (polystyrene, plexiglass) or inorganic polymers (glass) gain a surface electric charge if immersed in an electrolyte. The fixed charge can arise, e.g., from dissociation of surface chemical groups of either the polymer substrate itself or of adsorbed additives. Counter ions present in the electrolyte are attracted to the charged solid phase via the coulombic force and the electric double layer(EDL) is formed. In the immediate proximity, the counter ions are tightly bound to the charged surface and thus become immobile. This thin part of EDL is called the Stern layer. OHP(Outer Helmholtz plane) Stern layer Diffuse Layer Distribution of Electric field potential Bulk solution Counter ion distribution Co ion distribution Electric double layer 0 Vs Figure 3.1: Structure of the electric double layer Further away from the solid phase in the diffusive part of EDL, the attraction columbic force is relatively weak and the counter ions remain mobile. The diffusive layer is usually much wider than the Stern layer. The imaginary surface between the diffusive and the Stern layers is called the outer Helmholtz plane – OHP, as shown in Fig. 3.1. The electric potential 12 localized on this surface and related to the reference potential value in the bulk solution ( 0 ) is an important characteristic of EDL, so-called zeta-potential ( ). Let us note that the EDL structure, depicted in Fig. 3.1 represents a typical steady state that is established when: (i) the dielectric or the electrode surface is in a contact with the electrolyte for a sufficient time, (ii) any temperature changes and bulk concentration changes do not occur in space and time, and (iii) the electrolyte does not move. These conditions are often not satisfied in real applications and thus the EDL structure can be more complex. The EDL width is approximately equal to the Debye length λ D. For a symmetric uni-univalent electrolyte, the Debye length can be estimated as 3.1 D RT /(2c0 F 2 ) where the symbols R, T, c0, and F denote the molar gas constant, temperature, the electrolyte concentration in the bulk, and Faraday’s constant, respectively. The permittivity of the environment ε is equal to the product of the vacuum permittivity ε0 and the dielectric constant (relative permittivity) of the environment r. The fixed electric charge on the solid phase is usually negative (depends on the substrate and the electrolyte pH value). Then, the electric potential decreases from the electrolyte bulk to the solid surface. In the same direction, the anion concentration decreases and the cation concentration increases with respect to the bulk concentration. Hence, the electrolyte in the EDL region does not satisfy the electro neutrality condition, i.e., a nonzero concentration of a mobile electric charge exists in EDL. 3.2 Electroosmosis Channel Wall EDL r a x EDL Channel Wall Electric Field, Ex Fig 3.2: Geometry of a microchannel used for modeling electroosmotic flow 13 Electrokinetic techniques provide some of the most popular small-scale non mechanical strategies for manipulating particles and fluids. A surface with charge density q in an aqueous solutions attracts a screening cloud of oppositely charged counter-ions to form the electrochemical ‘double layer’, which is effectively a surface capacitor. The excess diffuse ionic charge exponentially screens the electric field set up by the surface charge (Fig 3.2). An externally applied electric field exerts a body force on the electrically charged fluid in this screening cloud, driving the ions and the fluid into motion. The resulting electroosmotic fluid flow appears to ‘slip’ just outside the screening layer of width λD. This basic electrokinetic phenomenon gives rise to electroosmosis which find wide application in analytical chemistry, microfluidics, colloidal self-assembly, and other emerging technologies. Electroosmotic flow occurs when an electric field is applied along a channel with charged walls, wherein the electroosmotic slip at the channel walls gives rise to plug flow in the channel. Because the electroosmotic flow velocity is independent of channel size, (in contrast to pressure-driven flow, which depends strongly upon channel size), electroosmotic pumping presents a natural and popular technique for fluid manipulation in small channels. 3.3 AC Electroosmosis Alternating electric fields can generate a net steady motion of aqueous saline solutions over microelectrode structures. The term AC electroosmosis (ACEO) refers to the fluid motion generated on top of electrodes by the interaction between an AC electric field and the electrical charge that this field induces at the electrode/electrolyte interface— i.e., the charge induced in the electrical double layer. E E Ey Ey Ex Ex Δ Fig 3.3: The principle of the AC electroosmotic flow above co-planar electrodes. ACEO flows are receiving growing interest in microfluidic applications because the integration of microelectrodes into microchannels makes possible the local actuation of electrolytes by means of electric fields. 14 In the most frequent co-planar arrangement of the electrodes, the AC electric field has the tangential and the normal components, as shown in Fig. 3.3. The normal component induces electrode polarization via coulombic force (capacitive charging). Then, the tangential component of the electric field forces the accumulated electric charge to move along the electrodes. The highest tangential coulombic force was predicted and observed at the electrode edges. f Δ Fig 3.4: Electrostatic volume force induces the eddy formation (solid line) and the net flow (dashed line) As the electric charge is formed by ions of finite size, the moving ionic particles pull the surrounding liquid via viscous forces, as shown in Fig. 3.4. The combination of coulombic, pressure and viscous forces in the liquid result in the formation of eddies above the electrodes. The velocity of the net flow strongly depends on several parameters: the frequency and the amplitude of the imposed electric field, the concentration of the used electrolyte and the geometric properties of the microfluidic system. The induced charge in the double layer lags behind the applied signal due to the finite charging time of the double layer. The ions are, therefore, subjected to a tangential electrical force in the direction of the traveling wave that, by viscous friction, is transmitted to the fluid. The characteristic charging time of the double layer is given by the product of the typical resistance of the bulk L /σ S and the typical capacitance of the double layer S / λD, tc = ( /σ) (L/ λD ). Here, and σ are the permittivity and conductivity of the liquid, respectively, L and S denote the typical length and area of the system, and λD is the Debye length. This charging time typically lies in the range of milliseconds. The frequency dependence is given by characteristic rates of two transport processes: (i) the rate of the electrode polarization (the formation of the EDL) and (ii) the rate/frequency of an 15 AC electric field. There are three qualitatively different regimes that can arise above the electrodes. If the period of the applied signal, T, is very large compared to tc , counterions have time to accumulate in the double layer. The electrical force is, then, mainly normal to the surface and negligible flow occurs. If the period of the applied signal T is much shorter than tc , the accumulation of induced charge in the double layer becomes negligible and, again, no lateral motion is generated. Maximum lateral force and fluid flow occur for T~ tc, which typically corresponds to an applied frequency in the range of kilohertz. A significant feature of ACEO flow is its dependence on the square of the electric field amplitude. This has important consequences for AC fields: if the direction of the electric field in the above picture (Fig 3.3) is reversed, so are the signs of the induced surface charge and screening cloud. The resultant ACEO flow, however, remains unchanged: the net flow generically occurs away from the poles, and towards the equator. 3.4 Model Geometry A 1:1 aqueous solution X+Y− e.g., H+Cl− placed on top of an array of electrodes subjected to an AC signal Fig. 3.5. The behavior of the system close to the plane of the electrodes is the area of interest. Above this plane, three distinct layers are distinguished: compact, diffuse, and diffusion. AC electroosmosis is usually explained in terms of the phenomena that happen in the combination of compact and diffuse layers known as electric double layer, abbreviated as EDL. In this paper, the effect of the charges situated in the diffusion layer is also examined. The outer Helmholtz plane (OHP) is localized between the compact and diffuse layers. . Bulk Diffusion Diffuse OHP 1 s Compact Vs Fig 3.5: Structure of the electric double layer (EDL) on an array of electrodes subjected to an AC potential. 16 The coplanar arrangement of AC electroosmotic pumps is considered (Fig.3.6). The studied microfluidic pump can be represented by a single segment of a long microfluidic channel. We assume that a microchannel consists of an infinitely large series of segments with periodic boundary conditions. L H y x G2/2 E1 G1 E2 G2/2 Fig 3.6: Scheme of one segment of the AC electroosmotic micropump. The dasheddotted lines indicate the periodic boundary conditions. The microchannel width is much larger than the other characteristic dimensions of the segments, so the AC electroosmotic pump can be described as a 2-D object with length L and height H. The electrodes have lengths E1 and E2, and they are separated by gap G1 . Both electrodes are G2/2 gap away from each periodic boundary. A symmetric mono-monovalent water electrolyte (e.g., Hydrogen chloride) is considered as the fluid in microchannels. No changes of density, viscosity, and temperature are expected. Faradic reactions are assumed on the electrodes. The formation of the condensed EDL is not considered in this paper. 3.5 Governing Equations (Equilibrium Model) At the electrode and electrolyte interfaces, double layers are formed with a typical thickness given by the Debye length. The Debye length is negligibly small compared to the typical size the system. Since the electrodes are long, both compared to gap width and the size of the periodic segment observed in the fluid, the problem can be considered to be two dimensional. When an AC voltage is applied to the electrodes, an electrical current is established in the solution. The following analysis assumes that electrolysis does not occur at the electrode surfaces, i.e., the electrodes are considered to be perfectly polarizable, and the double layer behaves in a linear manner. 3.5.1 The electrical field and ionic transport When the double layer is in quasi equilibrium, the bulk electrolyte behaves in a resistive manner and the double layer in a capacitive manner. 17 As a result, the potential in the bulk electrolyte satisfies Laplace’s equation 3.5.1(a) 2 0 with the boundary condition just outside the double layer on the electrode surface given by q EDL y t 3.5.1(b) where q EDL is the charge per unit area in the double layer. In this equation we assume that lateral currents along the double layer are negligible. The relationship between the charge and the potential drop across the double layer depends on the model used. If the voltage drop kT 0.025V there is a linear across the diffuse double layer is sufficiently small d e relationship between the charge and the voltage, i.e., q EDL C EDL ( V ) , and the equation can be written with complex amplitudes as = iq DL iC EDL ( V j ) y 3.5.1(c) where C EDL is the capacitance per unit of area of the total double layer (diffuse plus compact layers), and Vj is the potential applied to the electrode j. This capacitance C EDL is The compact layer capacitance is approximately independent of the electrolyte concentration. The properties of the diffuse part of the double layer depend on electrolyte concentration as given by the Gouy-Chapman theory [31]. Although experimentally the potential drop across the diffuse double layer can exceed 0.025 V, nevertheless the linear analysis can give useful information into the mechanism governing fluid flow. Since the conductivity of the solid surface is negligible, the boundary condition at the solid interface in the fluid is 0 3.5.1(d) y In deriving these boundary conditions, the presence of lateral currents along the double layer has been neglected. In this theoretical analysis, the applied and natural surface potentials are assumed to be small. In this case, the intrinsic mean ion density of the diffuse double layer is small and the lateral currents are negligible. For high ion concentrations in the double layer, surface currents might be comparable to the currents in the normal direction. Owing to electrode polarization, the electric field in the bulk electrolyte is frequency dependent. When the frequency is low, most of the applied voltage is dropped across the double layer but when the frequency is high most of the applied voltage is dropped across the bulk electrolyte. 18 3.5.2 Fluid Flow Once the potential is solved, the electroosmotic velocity just outside the double layer at the surface of the electrodes can be calculated from this solution. a. The solution in the diffuse layer The fluid motion is caused by electrical stresses that are nonzero only in the diffuse double layer, since the charge density in the bulk is zero. These stresses result in a rapidly varying velocity profile in the diffuse double layer, changing from zero at the wall to a finite value just outside the double layer. This velocity can be imagined as slip velocity at electrode surface to calculate the bulk motion. In the thin double layer approximation, for diffuse layers in quasi equilibrium and on a perfectly polarizable metal surface, the slip electroosmotic velocity is given by the Helmholtz-Smoluchowski formula u d d Ex X 3.5.2(a) In this expression, Ex is the tangential field just outside the diffuse layer, is the fluid viscosity, and d = ( ) represents the difference between the potential on the outer side of the diffuse layer and the potential on the inner side of this layer, at the nonslip plane. This equation gives the tangential velocity on the electrodes; the normal velocity is zero. b. The solution in the bulk To obtain the velocity in the bulk, the Navier-Stokes equations must be solved. Since, for microelectrodes the Reynolds number is usually very small, muL / 102 , we neglect the inertial terms in the Navier-Stokes equations. In the absence of externally applied body forces, these equations reduce to 2 u p 0 and u 0 3.5.2(b) The boundary conditions are (i) the tangential AC electroosmotic velocity on the electrodes, Eqn.3.5.2(a); (ii) zero tangential velocity on the glass; and (iii) zero normal velocity on every boundary. 3.6 Necessity of Non-Equilibrium Model In Equilibrium model of AC electroosmosis, the computational domain is usually divided into the capacitor domains (vicinities of the polarized surfaces, where electric double layers (EDL) are formed) and the resistor domain (the electrolyte bulk). It is assumed that electric potential in an arbitrary point of the system can be represented by a product of a complex time-independent function and a time-dependent function (e.g., harmonic). Equilibrium 19 model is valid only for systems in the thermodynamic equilibrium. The AC electroosmotic systems are rarely close to the equilibrium because of the intensive convection transport at the electrode surfaces. Because of the mentioned limitation, more complex Poisson–Boltzmann models and non equilibrium models have been developed. The non-equilibrium models describe the electric potential distribution with the use of the Poisson equation and zero velocity is applied on solid surfaces. The models based on the Poisson equation should satisfactorily describe the behavior of the AC electroosmotic systems with non-equilibrated EDLs. The electric potential field satisfies the Poisson equation 2 q, 3.6(a) q F (c c ) In order to evaluate the field of electric charge density, two molar balances for the anion (-) and the cation (+) have to be used c J t 3.6(b) The total flux density of ions is given by the sum of the convective and electro migrationdiffusion contributions (the Nernst–Planck equation) J uc D(c c F ) RT 3.6(c) The velocity and pressure fields in the electrolyte are described by the Navier–Stokes equation and the continuity equation for incompressible Newtonian fluids ( u u u) 2 u p q , t .u 0 3.6(d) The non-slip boundary conditions are used on the electrolyte-solid interfaces. The Dirichlet boundary conditions are used for electric potential on the discrete electrodes. No Faradaic current through the solid-electrolyte interfaces is considered. The insulating boundary conditions are used for electric potential on the non-electrode solid boundaries. Periodical boundary conditions applied in all other boundaries. 3.7 Contribution of Faradaic Current The simplest form of AC electroosmosis theory is a linear theory that assumes that only forces in the diffuse layer are present and that the electrodes are perfectly polarizable, i.e., there are no Faradaic currents. At certain range of frequencies the direction of net fluid flow changes. The possible reason for this flow reversal is said to be Faradaic current. Faradaic 20 currents depolarize the electrode-electrolyte interface leading to lower electroosmotic velocities. Oscillations of total ionic concentration close to the electrodes is originated by Faradaic currents at low frequencies. In our model we only include two charged species and consider the pair of ions to be asymmetric in the sense that their mobilities are different. For simplicity, we assume that the cations react reversibly at the electrodes to produce neutral molecules according to a simple one-step one-electron redox process of the form X++e−↔X and that the electrodes are blocking for the anions. The sign of the reacting species, however, is not essential to the model. The introduction of the chemical reaction requires the inclusion of one neutral species, X, in the mathematical analysis. In our system, there is an AC applied voltage on the electrodes and this signal induces charges and motion in the liquid. The behavior of our system can be modeled in terms of four functions: the electric potential, and the concentrations of positive, negative, and neutral species, c+, c−, and c0, respectively. Our model describes a two-dimensional system, which approximates the behavior of a solution on top of an array of electrodes in the form of long strips. We denote as x the coordinate tangential to the plane of the electrodes and as y the coordinate normal to this plane. The different quantities verify the Poisson-Nernst-Planck (PNP) equations in the domain. The electric potential obeys Poisson’s equation, 2 q, q F (c c ) 3.7.1 The concentration of ions satisfy Nernst-Planck equations, c J , t J uc D (c c F ) RT 3.7.2(a) c J , t J uc D (c c F ) RT 3.7.2(b) c 0 J 0 , t J 0 uc 0 D 0 c 0 The diffusion coefficients D 9.312e 9 m2 s-1 (Diffusion Coefficient of H+) D 2.032e 9 m2 s-1 (Diffusion Coefficient of Cl-) D 0 2.2e 9 m2 s-1 (Diffusion Coefficient of Water)[31] To these equations we must add the Navier-Stokes equation for the liquid motion, 21 3.7.2(c) 3.7.3(a) u 0 ( u u u) 2 u p q t TABLE I DIMENSIONLESS PARAMETERS Symbol Parameter ~ xxL ~ yy L Dimensionless x-coordinate Dimensionless y-coordinate Dimensionless delta operator Dimensionless Debye length ~ L ~ D D L ~ h H L ~ re E1 E 2 ~ rg G 2 G1 ~ le E1 E 2 L ~ t t t 0 , t 0 D L D ~ f ft0 ~ c c c eq ~ c c c eq ~ c 0 c 0 ceq ~ RT , 0 0 F ~ A A 0 ~ u u D L p ~ p 0 2c RT 2 RT Ra D F Sc D ( / D ) ~ s ( s / s ) Height to length ratio Electrode length ratio Gap size ratio Equivalent length Dimensionless time Dimensionless frequency Dimensionless concentration of cation Dimensionless concentration of anion Dimensionless concentration of water Dimensionless potential Dimensionless amplitude Dimensionless velocity Dimensionless pressure Rayleigh number (Ra= 0.372) Schmidt number (Sc=348) Compact layer relative thickness 22 3.7.3(b) 3.8 Non-Dimensionalization of the Equations The model equations are transformed into a dimensionless form. The meaning of all the used symbols is summarized in Table I. The spatial coordinates and the segment dimensions are scaled by the factor L. Combination of Eqs. 3.7.1-3.7.3(b) and Table I gives the dimensionless form of the non-equilibrium mathematical model. The electric potential field satisfies the Poisson equation 3.8.1 1 ~ ~ 2 ~2 q~, D ~ ~ q 0.5(c ~ c) The concentrations of ions satisfy Nernst-Planck equations, c~ ~ ~ ~~ ~ ~ ~ ~ ~ D uc c c 0 ~t 3.8.2(a) c~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ D u c c c 0 t 3.8.2(b) c~ 0 ~ ~ ~~ 0 ~ ~ 0 ~ D u c c 0 t 3.8.2(c) The velocity and pressure fields in the electrolyte are described by the Navier-Stokes Equation and the continuity equation for an incompressible Newtonian fluid 1 ~ u ~ ~ ~ ~ ~ ~ 2 ~ Ra ~ ~ ~ ~ ~ u D u ~ p q ~ D u Sc t D ~ ~ u 0 3.8.3(a) 3.8.3(b) where Sc is the Schmidt number and Ra is the Rayleigh number defined by Ra RT D F 2 Sc D Hence in this section, the physical meaning of the dimensionless criteria is shortly discussed. Because the Schmidt number is high (particularly Sc=500>>1; the concentration and electric charge boundary layers will be much thinner than the momentum boundary layers. It means that the velocity field must be affected on a much higher distance from the electrodes than the concentration fields. Different behavior, not typical for microfluidic applications, can be expected for electrolytes with extremely low viscosities and with high diffusivities of ions. 23 From the definition, the Rayleigh number expresses a ratio between the forces that ~ destabilizes and stabilizes a static fluid. In our modification, the dimensionless ratio Ra / D2 is a relative ratio between the destabilizing electric body force and the stabilizing viscous force. This ratio is very high in the presented parametrical studies as well as in all AC microfluidic pumps. A low value of the ratio necessarily leads to zero net velocity. 3.9 Boundary Conditions The boundary conditions at the OHP (y=0) for these functions are the continuity of the displacement vector, assuming a linear compact layer of width λs and permittivity s 3.9(a) s s s y where s denotes the voltage drop across the compact layer (Fig 3.5) s Vs (t ) ( y 0), Vs ,1 A sin 2ft, Vs , 2 0 3.9(b) We assume that the electrodes are blocking to the anions, D c F c D 0 RT y y 3.9(c) whereas the fluxes of positive and neutral species are related through Faradaic currents, Dc F c J F D RT y y e D0 J c 0 F y e 3.9(d) 3.9(e) To model this current we use the Butler-Volmer equation JF F s (1 ) F s K 0 c 0 exp K c exp e RT RT with as the transfer coefficient, where the reaction constants K0 = 1e-10 m3mol-1s-1 (Reaction rate constant for water) K+ = 1e-7 m3mol-1s-1 (Reaction rate constant for H+) 24 3.9(f) The boundary condition for the liquid velocity at the OHP is the no-slip condition u0 3.9(g) The equations can be made dimensionless using adequate scales. Equation 3.9(a) becomes ~ ~ ~ ~ s s D ~ y ~ ~ ~ s Vs (t ) ( y 0), 3.9(h) ~ ~ ~ Vs ,1 A sin 2f ~t , Vs , 2 0 3.9(i) ~ ( / D ) where s which is the compact layer thickness. It gives the ratio between the ( s / s ) capacitance of the diffuse layer to the compact layer. This can be interpreted as a ratio between effective thicknesses: ~ s ( s / s ) / D Equation 3.9(d)-3.9(g) becomes ~ ~ c ~ c ~ ~ 0 y y 3.9(j) ~ JFL ~ c ~ c ~ ~ y y eD c eq 3.9(k) J L c~ 0 ~ F0 y eD c eq 3.9(l) and the Butler-Volmer equation becomes JFL L ~ ~ [K0~ c 0 exp s K ~ c exp (1 ) s ] eDc eq D 3.9(m) Equation 3.9(h) becomes ~ u0 3.9(n) 25 4. NUMERICAL SOLUTION METHODOLOGY 4.1 Finite Element Method Finite element method (FEM) is employed for solving the governing equations of AC electroosmosis. The method is well described in literature [32] and is widely used for solving differential equations in many areas of engineering and science. In finite element method, a geometrically complex domain of the problem is discussed with simple elements. The elements are connected to each other at nodal points. The responses of the dependent variables are assumed a priori. State variables are approximated by basis functions, which are formulated from polynomials. Approximating piecewise function for state variables are known as test functions. Replacing the shape function in the discretized weak equations by the shape functions associated to each computational node gives a single algebraic equation which satisfies the discretized form of the governing equations. In this way a system of algebraic equations are formed. These equations describe the relationship between the coefficients of the test functions. Solving these algebraic equations for these coefficients the behavior of the dependent variable is obtained. 4.2 Computer Implementation of the Model Poisson, Nernst Planck, Navier-Stokes and continuity equations are solved in a time dependent domain. In present analysis a two dimensional geometry is discretized by triangular mesh. Higher number of mesh elements is used on the electrode surface. This facilitates capturing the higher electric field and/or ion concentration gradient near the electrode surface . A general feature of all types of the AC electrokinetic systems is that the systems do not work in steady condition. After the initial transient, they can attain a stable periodic regime. This aspect of the AC systems substantially increases the demands on numerical analysis of model equations. From technical point of view, the most interesting velocity characteristic of the flow is the net velocity which is obtained as the tangential velocity averaged over one period of the stable periodic regime. Intel Core i3 processor and 4GB RAM are employed to obtain the solutions presented here. 26 4.3 Mesh Sensitivity Analysis The mesh sensitivity analysis is performed by observing the improvement in the calculated values of time averaged net velocity with increasing number of mesh elements. A slit ~ ~ ~ H microchannel having h 0.333, ~ re 1, ~ rg 9 , le 0.333 is considered. Here, h , is L E the dimensionless height of the channel, ~ re 1 , is the ratio of the electrode dimensions, E2 G ~ E E2 ~ rg 2 , is the ratio of the gap sizes, le 1 , is the relative size of the electrode G1 L domains. Tests are performed from nearly 2000 elements to 6000 elements. The problem is time dependent, memory requirement for computation is higher than the stationary problem. Figure 4.1: Mesh sensitivity analysis for time averaged net velocity Moreover, Poisson and Nernst-Planck equations are coupled with the Navier-Stokes and continuity equations, which increases the number of degrees of freedom for each mesh node. This means, at each node dependent variables c+, c-, c0 and V are solved with the dependent variables of Navier-Stokes equation (u, v, p). For this reason, number of mesh elements is taken adjustable to minimize the memory and time requirement. 27 Time averaged net velocity for different number of mesh elements is shown in Fig. 4.1. It is well noticeable that, the value cannot be improved significantly by taking higher number of elements above 6000. 4.4 Flowchart of the Overall Solution Methodology Geometry modeling (Finite channel length approximation) Defining governing equations Non-dimensionalization of the governing equations Defining boundary conditions Non-dimensionalization of boundary conditions Discretizing the geometry into finite number of mesh elements Applying boundary conditions Solving Poisson, Nernst Planck, Navier-Stokes and continuity equations simultaneously with time dependent solver Calculating horizontal component of velocity for one cycle when the system goes to a steady state from transient phase Calculating Time averaged net velocity for one complete cycle 28 5. RESULTS AND DISCUSSION This chapter shows the dependence of time averaged net velocity and volumetric flow rate on different parameters. The developed model utilized in present analysis is validated by comparing the calculated values of time averaged net velocity with those available in the literature. Later, time averaged net velocity is presented for different values of frequency, Debye length, compact layer thickness, microchannel height and electrodes position. 5.1 Validation of the Model Electroosmotic flow in a straight tube is presented having a low surface potential [31]. The flow is considered as fully developed. A microchannel of radius a with a charged surface bearing a surface potential of is considered as illustrated in Fig 3.2. The coordinate system used is cylindrical with r representing the radial direction and x representing the axial direction. A symmetric ( z : z) electrolyte is considered flowing in the channel. The electroosmotic velocity in x-direction is given by u x (r ) u x a 2 px 4 r 2 1 a I 0 ( r / D ) 1 I (a / ) E x 0 D 5.1 dp , is electrolyte viscosity, is electrolyte permittivity, D is Debye dx length and I 0 is the zeroth-order modified Bessel function of the first kind. where, p x 29 Fig 5.1: Model validation with analytical result A finite element solution is obtained for =1, = 1e-3 Pa s, =8.8541878e-12*78.5 C2 N-1 m-2 and D =10-7m. From Fig 5.1, it is seen that the result obtained from present model is in good agreement with the analytical results. Values of time averaged net velocity are calculated utilizing the developed non-equilibrium model. L H y x G2 /2 E1 G1 E2 G2/2 Fig 5.2: Schematic diagram of the AC electroosmotic mircropump. The dashed dotted line indicates the periodic boundary conditions. 30 These calculated values are compared with the published numerically computed results [27]. Fig 5.2 indicates the Cervenka model of an AC electroosmotic micropump. The coplanar arrangement of AC electroosmotic pump is considered here. A periodic segment of a long microfluidic channel is represented by this model. Length of this segment is L and height is H. The length of the electrodes are E1 and E2, and the gap between them are G1 and gap from ~ each periodic boundary is G /2. For this model, the values are h 0.333, ~ r 1.667, 2 e ~ ~ H E ~ rg 10, le 0.2667 where h , is the dimensionless height of the channel, ~ re 1 , L E2 G ~ E E2 is the ratio of the electrode dimensions, ~ , rg 2 , is the ratio of the gap sizes, le 1 L G1 is the relative size of the electrode domains. That means, E1 is 1.667 times larger than E2 i.e. the electrodes are arranged in an asymmetric way. Fig 5.3: Comparison of time averaged net velocity between Cervenka model and present model Time averaged net velocity for different dimensionless frequency is shown in Fig 5.3 for Cervenka model and present model. Only magnitude of the time averaged net velocity is considered here. Dimensionless frequency are taken in the range of 0.1 to 50. The velocity of any point varies with the varying electric field caused by AC cycle. The net velocity is defined as the average of the velocity over a cross section of the microchannel. For a certain 31 dimensionless frequency, the net velocity goes to a steady state from transient states. For example, it takes 100 cycles for the system to go in a steady state when the dimensionless ~ frequency is, f = 0.4. The average value of the velocities at 100th cycle is taken as the time averaged net velocity. In Fig. 5.3, the net velocities are taken at the middle point of the two electrodes. v when compared to the The percentage of error for the value of calculated net velocity, ~ ~ ~ values of [27] for dimensionless frequency, f 1 are within 0.949 % . Above f 1 , the maximum error is 5.3%. So, values of dimensionless frequency is taken in the range of 0.01 to 1 in the present study. Parametric dependencies computed by present model are plotted by empty markers. The Cervenka results are represented by filled markers (Fig 5.3). 5.2 Inclusion of Faradaic Current The above models consider that the electrodes are perfectly polarizable i.e., there is no Faradaic current. For certain ranges of frequency, the direction of predicted flow may change while considering Faradaic current. Assuming a linear compact layer width s and Debye length D , a dimensionless parameter can be defined as Compact layer relative thickness, ( / D ) ~ , where s is the permittivity of electrolyte at the compact layer and is the s ( s / s ) permittivity of electrolyte at other regions. It gives the ratio between the capacitance of the diffuse layer to the compact layer. This can be interpreted as a ratio between effective thicknesses i.e., ratio between compact layer and diffuse (Debye) layer thickness. ~ s (s / s ) / D 32 Fig 5.4: Effect of Faradaic current on net velocity. ~ ~ ~ ~ ~ re 1.667, ~ rg 10 , le 0.2667 , A = 0.75, s =4 , D =3.3e-3 h 0.333, ~ The magnitude of time averaged net velocity is shown for asymmetric arrangements of electrodes taking Faradaic currents and no Faradaic currents in consideration. No Faradaic currents means considering the electrodes as perfectly polarizable. Faradaic currents depolarize the electrode-electrolyte interface i.e charge leaks from the double layer leading to lower electroosmotic velocity. This phenomenon is seen in Fig 5.4. Net velocity decreases by an order because of the effect of Faradaic currents. Net velocity is taken at logarithmic scale to show the difference of order in two cases. The shape of the net velocity profile with increasing dimensionless frequency is somewhat similar. Sharp changes are noticed for net velocity when considering Faradaic current. The slope of the curve is relatively small without Faradaic current. 33 Fig 5.5: Net velocity without Faradaic current. ~ ~ ~ ~ ~ ~ h 0.333, re 1.667, ~ rg 10 , le 0.2667 , A = 0.75, s =4 , D =3.3e-3 No flow reversal is seen for this system when taking no Faradaic currents. Spatial direction x is taken positive from left to right side. Negative sign of velocity indicated the net flow is in the opposite direction of positive x. In the range of dimensionless frequency mentioned above in Fig 5.5, velocities are negative. That means fluid flows in the same direction for ~ dimensionless frequency, f 0.01 1 while considering no Faradaic current. 34 Fig 5.6: Net velocity with Faradaic current. ~ ~ ~ ~ ~ re 1.667, ~ rg 10 , le 0.2667 , A = 0.75, s =4 , D =3.3e-3 h 0.333, ~ ~ Flow reversal occurs after dimensionless frequency, f ≈ 0.2 in this system while taking Faradaic current into consideration. Fluid velocity goes positive from negative direction. That means the direction of fluid flow changes in the above range of dimensionless ~ frequency, f 0.01 1 while considering Faradaic current. In both cases, the change in the magnitude of net velocity is low up to frequency 0.2. After this frequency, net velocity significantly increases. For no Faradaic current, net velocity dramatically increases and for Faradaic current inclusion, flow reversal occurs and net velocity also increases in that opposite direction. Again it is seen in more detail that the changes are sharp in the net velocity profile while considering Faradaic current. The changes in net velocity is very low up to frequency 0.2 for no Faradaic current. It remains almost the same. Faradaic current predicts (i) flow reversal and (ii) lower electroosmotic velocity. So, the calculation of time averaged net velocity and its dependences on selected parameters should be done taking Faradaic current into consideration. The electrode arrangement of above 35 models are asymmetric. In the next section, time averaged net velocity is explored for two different arrangements of electrodes i.e., symmetric and asymmetric. The following sections show the characteristics of velocity and electric field using Faradaic current at electrode surface. 5.3 Asymmetric vs Symmetric Electrodes ~ The spatial parameters for asymmetric electrodes are ~ re 1.667, ~ rg 10 and le 0.2667, which means one electrode is 1.667 times larger than the other one and both of them are at equal distance away from periodic boundaries. ~ Fig 5.7: Net velocity comparison for asymmetric h 0.333, ~ re 1.667, ~ rg 10 , ~ ~ ~ le 0.2667 , A = 0.75, s =4 , D =3.3e-3 and symmetric electrode arrangement ~ ~ ~ ~ r 9 , l 0.333 , A = 0.75, =4, =3.3e-3 h 0.333, ~ r 1, ~ e g e s 36 D ~ Symmetric electrode arrangement ~ re 1, ~ rg 9 , le 0.333 shows that two electrodes are equal in size and again they are equal distance away from periodic boundaries. Time averaged net velocity is calculated at the middle plane between two electrodes for the two ~ ~ cases. Dimensionless amplitude is taken as, A = 0.75, compact layer relative thickness is, s =4, which means compact layer is four times larger than diffuse (Debye) layer and ~ dimensionless Debye length is taken as, D =3.3e-3. Figure 5.7 shows the comparison of net velocity for asymmetric and symmetric electrode arrangements. Negative sign of velocity shows the direction of fluid flow as it is in the opposite direction of positive x-direction considered in the model geometry. Fluid usually flows in the opposite direction of produced electric field. In comparison of magnitude, it is seen that for lower frequency symmetric electrode arrangement exerts lower net velocity than asymmetric arrangements. For higher frequencies, symmetric electrodes exert higher net velocities than asymmetric electrodes. For symmetric electrodes, flow reversal occurs at dimensionless frequency ≈ 0.6 and for asymmetric electrodes, it happens after the value of dimensionless frequency ≈ 0.2. That means flow reversal occurs at a higher frequency when electrodes are symmetric. So, symmetric electrode arrangement is a better design considering flow reversal as for a certain range of frequency, desired net flow can be obtained. ~ In the range of dimensionless frequency, f 0.01 1 , the value of time averaged net velocity is higher for symmetric arrangement than for asymmetric arrangement. The further calculation of the velocity and Electric field characteristics, behavior of time averaged net velocity and flow rate within a range of dimensionless frequency are discussed below while considering Faradaic current in symmetric arrangement of electrodes. Effect of Debye length, microchannel height, compact layer relative thickness, the ratio of the electrode dimensions, the ratio of the gap sizes and the relative size of the electrode domains are calculated using Faradaic currents in the following sections. 5.4 Velocity and Electric Field Characteristics Electroosmosis depends on surface charges at the electrode/electrolyte interface. Most materials will acquire fixed surface charges when coming into contact with a fluid that contains ions (either an electrolyte or a dielectric liquid with ionic impurities or generated locally via reactions). The surface charges attract counterions from the solution and repel coions from the surface to maintain local charge neutrality. Consequently, an excess of charges is built up near the electrode surface, thus forming an electrical double layer. The ions in the electrical double layer are mobile and will migrate under the influence of an electric field tangential to the electrode surface. Due to fluid viscosity, fluid surrounding the ions will move along, inducing so-called electroosmosis. 37 Electrodes Fig 5.8: Velocity stream lines for ACEO in a slit microchannel ~ ~ ~ ~ ~ ~ re 1, ~ h 0.333, ~ rg 9 , le 0.333 D =3.3e-3 , A = 0.75, s =4, f =0.3 , Figure 5.8 shows the stream lines within slit microchannel. Two vortices are created at the electrode edges. One central flow is noticed in the middle of the electrodes. Electrodes Fig 5.9: Velocity contour for ACEO in a slit microchannel ~ ~ ~ ~ ~ ~ re 1, ~ rg 9 , le 0.333 D =3.3e-3 , A = 0.75, s =4, f =0.3 h 0.333, ~ , 38 Electrodes Fig 5.10: The local orientation and the magnitude of the velocity vector by arrows for ACEO in a slit microchannel ~ ~ ~ ~ ~ ~ ~ ~ h 0.333, re 1, rg 9 , le 0.333 D =3.3e-3 , A = 0.75, s =4, f =0.3 , The interaction of the electric field and the space charge in the EDL generates the electric force, which moves the fluid. The calculated flow velocity distribution is very non uniform (Fig. 5.9 ). At points closer to the channel center and ends, the velocity rapidly decreases, but at all points, the fluid uniformly flows in the horizontal direction (Fig 5.10). Electrodes Fig 5.11: Electric field and stream line for ACEO in a slit microchannel ~ ~ ~ ~ ~ ~ h 0.333, ~ re 1, ~ rg 9 , le 0.333 D =3.3e-3 , A = 0.75, s =4, f =0.3 , 39 As the electric force is a product of the space charge density and the electric field, the strongest force occurs close to the electrode edges ( Fig 5.11). ~ ~ ~~ When positive potential Vs A sin(2f t ) and 0 potential are applied on left and right electrodes respectively, electric field is created as shown in Fig 5.12. Fig 5.12: Plot of tangential field above electrodes. ~ ~ ~ ~ ~ ~ h 0.333, ~ re 1, ~ rg 9 , le 0.333, D =3.3e-3 , A = 0.75, s =4, f =0.3 Tangential component of electric field created above electrodes are shown in Fig 5.12. It is seen that x component of electric field is maximized at two edges of electrodes. 40 Fig 5.13: Horizontal component of the velocity vector along a horizontal line 0.001µm ~ ~ ~ ~ ~ above the electrodes. h 0.333, ~ r 1, ~ r 9 , l 0.333, =3.3e-3 , A = 0.75, =4, e e g D s ~ f =0.3 A detailed information about the flow characteristics is provided in Fig. 5.13. The local velocity vector along a horizontal line just above the electrodes (0.001 µm above electrodes) varies rather dramatically; the maximum value is more than 0.19 µm/s at a point between both electrodes. Flow direction changes at both edges of the electrodes. Velocity reaches 0.5 µm/s for outer edge of positive electrode but changes direction and goes to -0.13 µm/s near other end. At the inner edge of the negative electrode, the flow reaches maximum velocity and then reverses its direction and reaches a value of about 0.08 µm/s. 41 Fig 5.14: Horizontal component of the velocity vector along a vertical line at the channel outlet (far from the electrodes) ~ ~ ~ ~ ~ ~ ~ ~ h 0.333, re 1, rg 9 , le 0.333, D =3.3e-3 , A = 0.75, s =4, f =0.3 The net velocity along a vertical direction at the channel outlet is shown in Fig 5.14. At frequency 0.3, the system goes at steady state for 100 cycle. Net velocity is calculated for every 1/10th time of 100th cycle. 0t means the starting of the cycle and 1t means the end of the cycle. It is observed that the velocity changes direction in the middle of the cycle that is at the time 0.5t. The value of net velocity goes to a high from 0t to 0.1t but then decreases with time. At 0.5t, flow is reversed. After that, net velocity increases then decreases and finally at the end of the cycle the flow is again at the previous direction that is opposite to created tangential electric field. The magnitude is also very close for the starting and ending of the cycle. The net velocity at channel outlet is parabolic in nature. It goes to a maximum at the middle point of the microchannel and velocity is 0 at walls. At time 0.1t, the velocity reaches maximum at the middle point of channel radius, the value is -0.1 µm/s. After flow reversal, at time 0.7t, it goes to maximum at opposite direction, the value is 0.09 µm/s. 42 Fig 5.15: Time averaged net velocity along a vertical line at the channel outlet (far from the electrodes) ~ ~ ~ ~ ~ ~ ~ ~ h 0.333, re 1, rg 9 , le 1 D =3.3e-3 , A = 0.75, s =4, f =0.3 , Fig 5.15 shows the average values of Fig 5.14. In the vertical direction at the channel outlet, the flow profile is parabolic with the maximum value of about 0.0048 µm/s (see Fig. 5.15). 43 Fig 5.16: Horizontal component of the velocity vector along a vertical line at the midpoint between the electrodes ~ ~ ~ ~ ~ ~ ~ ~ h 0.333, re 1, rg 9 , le 0.333, D =3.3e-3 , A = 0.75, s =4, f =0.3 The net velocity along a vertical direction at the middle point between electrodes is shown in Fig 5.16. It is observed that the velocity changes direction in the middle of the cycle that is at the time 0.5t, at the same time as the velocity at discharge. At 0.14 µm , net velocity changes direction for every fraction of time of the cycle. The value of net velocity goes to a high from 0t to 0.1t but then decreases with time, just like above case. At 0.5t, flow is reversed. After that, net velocity increases then decreases and finally at the end of the cycle the flow is again at the previous direction that is opposite to created tangential electric field. The magnitude is also very close for the starting and ending of the cycle. The net velocity changes sharply up to 0.14 µm above electrodes. After that point, velocity is parabolic in nature. It goes to a maximum value of 0.05 µm/s at 0.24 µm from and velocity is 0 at walls. At time 0.1t, the velocity reaches maxim at 0.3 µm from electrodes, the value is -0.45 µm/s. 44 Fig 5.17: Time averaged net velocity vector along a vertical line at the midpoint between the electrodes ~ ~ ~ ~ ~ ~ ~ ~ h 0.333, re 1, rg 9 , le 0.333, D =3.3e-3 , A = 0.75, s =4, f =0.3 Fig 5.17, shows the time averaged net velocity in the middle plane between two electrodes. Near the electrodes the velocity increases with the increasing distance from electrode, goes to a maximum at 0.025 µm above electrodes. The maximum velocity is 0.02 µm/s for the above ~ case of dimensionless frequency f =0.3. From 0.025 µm above the electrodes, the velocity decreases and goes to zero at 0.149 µm above electrodes. Then the velocity changes direction i.e. flow reversal occurs. Dramatic change in net velocity is observed in the range of 0 µm to 0.025 µm range area above the electrodes. Rate of change of the magnitude of net velocity is comparatively lower at 0.025 µm to 0.1 µm region. After the flow reversal, the region of 0.2 to 0.3 µm above electrodes shows that the magnitude of velocity is nearly equal. The velocity is parabolic in shape from the point 0.0125 µm to the wall (0.33 µm). The velocity is 0 at both wall. Parabolic nature velocity is in the reversed direction. It reaches maximum of magnitude 0.025 µm/s at the region of 0.225 µm to 0.275 µm above electrodes. 45 5.5 Effect of AC Frequency on Time Averaged Net Velocity and Flow Rate Fig 5.18: Dependence of the time-averaged net velocity on the AC frequency. ~ ~ ~ ~ ~ h 0.333, ~ re 1, ~ rg 9 , le 1, D =3.3e-3 , A = 0.75, s =4. The dependences of the dimensionless time-averaged net velocity on the dimensionless frequency of the electric field are shown in Fig 5.18. Velocity decreases in the domain of lower frequency, increases in relatively higher frequency and reaches maximum. At higher frequency, flow reversal occurs and again it goes to another maximum in the opposite ~ direction of fluid flow. Velocity maxima (in the absolute value) is located around f ≈ 0.3. It should be noted that the precise location of the maximum depends on the particular choice of the characteristic length. In this paper, the length of the periodic domain is chosen to be the ~ characteristic length. Flow reversal occurs at f ≈ 0.6. The volumetric flow per width of microchannel shows the same nature as the net velocity while changing with frequency. It goes to a maximum at frequency 0.3. The maximum flow rate at channel outlet is 0.005 µm3/s. 46 Fig 5.19: Flow rate at channel outlet and midpoint between electrodes ~ ~ ~ ~ ~ re 1, ~ rg 9 , le 0.333, D =3.3e-3 , A = 0.75, s =4. h 0.333, ~ Comparative flow rate between channel outlet and middle point (Fig 5.19) shows that flow rate at channel outlet is more or less greater than that of middle point for every frequency. Both the curves show same nature. It goes to a maximum at frequency 0.3. The value of maximum flow rate is 0.008 µm3/s. 47 5.6 Dependence of Time Averaged Net Velocity on Debye length and Compact Layer Relative Thickness Fig 5.20: Dependence of the time-averaged net velocity on Debye length ~ ~ ~ ~ re 1, ~ rg 9 , le 0.333, f = 0.3 , A = 0.75 h 0.333, ~ Fig 5.20 shows that the effect of reducing the equilibrium concentration of neutral species ( HCl Solution) on net velocity. The velocity of the net flow dramatically increases with an increasing Debye length. As Debye length is proportional to ceq -0.5 according to Eq. (3.1), it can be said that net flow increases with decreasing electrolyte concentration of neutral species. Compact layer relative thickness can be interpreted as a ration between effective thickness of compact layer and diffuse (Debye) layer. Increasing Compact layer relative thickness means ~ decreasing length of Debye length. Here, s represents Compact layer relative thickness. ~ Velocity decreases with increasing s . Negative sign shows the direction of fluid flow. Flow ~ reversal is noticed for Dimensionless frequency 1 in case of s = 4. That is, Fluid flow ~ changes direction at this point. But no flow reversal occurs in the same frequency if s is = 48 ~ ~ 0.5 or 0.2 ( when s is less than 1). s value is greater than or equal to 1 means, compact ~ layer is of the same order or larger than the diffuse layer. Increasing value of s causes decreasing electroosmotic velocity. From equation 3.5.2(a) it can be seen that electroosmotic velocity is proportional to difference between the potential on the outer side of the diffuse layer and the potential on the inner side of this layer, at the nonslip plane. Fig: 5.21: Dependence on the compact layer thickness. ~ ~ ~ r 1, ~ r 9 , l 0.333 , A = 0.75 h 0.333, ~ e e g The presence of a thick compact layer reduces the voltage drop across the diffuse layer, lowering the electroosmotic velocity. In all the three conditions of relative compact layer frequency, the time averaged net velocity increases with increasing dimensionless AC frequency. ~ Increasing value of s means decreasing value of Debye length thickness. It can be said that ~ with the decreasing value of s , electroosmotic velocity increases. That means with the increasing Debye length thickness, velocity increases ( Fig 5.21). 49 Diffusion Diffuse Compact layer ~ ~ ~ ~ ~ Fig 5.22: Flow reversal. h 0.333, ~ re 1, ~ rg 9 , le 0.333, D =3.3e-3 , A = 0.75, s =4, ~ f =1 If the compact layer is very thin there is no reverse flow, but if it is of same order or larger than the diffuse layer, there could be flow reversal since the voltage drop in the diffuse (Debye) layer is greatly reduced and can be overcome by the opposite voltage drop across the diffusion layer. 50 5.7 Effect of Electrode and Channel Geometry on Time Averaged Net Velocity Fig 5.23: Effect of Microchannel height on time averaged net velocity. ~ ~ ~ ~ ~ ~ re 1, ~ rg 9 , le 0.333, D =3.3e-3 , A = 0.75, s =4, f =0.3 In this section, the effects of the geometric parameters of the system on the micropump performance is discussed. The dependences of the dimensionless time averaged net velocity on the microchannel height are shown in Fig 5.23. There is clear maxima of the time averaged net velocity in the computed dependences. It means that a certain confinement leads to a higher net velocities than those in the unconfined channels. Micropumps with a larger vertical dimension are not affected by the top solid boundary. E The relative size of electrodes is defined by ~ re 1 which means the ration of the lengths of E2 electrodes. ~ r =1 means the electrodes are of same size. Here we see the effect of increasing e ~ re on time averaged net velocity. Four values of ~ re are presented here. Value of ~ re is 51 Fig 5.24: Effect of relative size of electrodes ~ ~ ~ ~ ~ ~ rg 9 , le 0.333, D =3.3e-3 , A = 0.75, s =4, f =0.3 increasing from 1 to 3. ~ re =3 means left electrode is 3 times greater than right electrode. Here, left electrode is positive and right electrode is negative. It seems that with the increasing length of positive electrodes, time average net velocity increases. Maximum value is 0.0024 µm/s for the given system which is for ~ re =1 i.e. symmetric electrodes. For ~ re =2, flow reversal occurs and then velocity increases in that direction with increasing relative size of electrode lengths ( ~ re ). 52 6. CONCLUSION AND RECOMMENDATIONS 6.1 Concluding Remarks A fully nonlinear analysis is performed to understand the mechanism of AC electroosmosis with Faradaic currents. The mobilities of the species are taken different and the compact layer is assumed to be not very thin compared to the diffuse layer. ACEO with Faradaic current for symmetric electrodes is considered in this analysis. In this chapter, conclusions are drawn based on the analysis performed in the previous chapter. The analyses are performed first through investigating the effect of dimensionless AC frequency on the time averaged net velocity of fluid moving along a slit microchannel by varying the frequency from 0.01 to 1. Later the analyses are performed for different Debye length, compact layer relative thickness, microchannel height and electrodes position. From the results of time averaged net velocity and direction of flow of fluid flowing through silt micro channel for different parameters; the following inferences can be drawn: - At very low frequency, net velocity is higher when considering Faradaic current. After frequency 0.02, net velocity with Faradaic current is lower than net velocity without Faradaic current. Changes in velocity profile is sharper with Faradaic current than without Faradaic current. - In the range of frequency from 0.01 to 1, there is no flow reversal while taking no Faradaic current. But in the same region of frequency, flow reversal occurs after frequency ≈0.2 while taking Faradaic current into consideration. - Taking absolute value of net velocity, symmetric electrode arrangement exerts lower net velocity than asymmetric arrangements at lower frequency. And for higher frequencies, symmetric electrodes exert higher net velocities than asymmetric electrodes. For symmetric electrodes, flow reversal occurs at dimensionless frequency ≈ 0.6 and for asymmetric electrodes, it happens after the value of dimensionless frequency ≈ 0.2. - For the dimensionless amplitude of applied sinusoidal signal at the positive electrode being 0.75, time averaged net velocity decreases at lower frequency region and increases at higher frequency region. Maximum velocity attained at dimensionless frequency 0.3. Flow reversal occurs at dimensionless frequency 0.6 when relative compact layer thickness is 4. Volumetric flow rate is more or less same for the middle point between electrodes and at the channel outlet. The maximum flow rate at channel outlet is 0.005 µm3/s. - The velocity of the net flow dramatically increases with an decreasing electrolyte concentration and increasing Debye length up to frequency 0.02; then it increases 53 slowly at higher frequency region. Electroosmotic velocity increases with decreasing compact layer relative thickness. - Up to frequency 0.1, the time averaged net velocity drastically changes from going to maximum and then decreases sharply with the increase of height to length ration of slit microchannel. After that point, net velocity decreases slowly. Top solid boundary of microchannel is not affected when the vertical dimension is large. Maximum velocity is attained at electrode length ratio 1. When positive electrode is double than negative electrode, flow reversal occurs. Optimal set of model parameters for AC pump can be recommended based of the present numerical analysis of the dimensionless model. Real parameters for any particular case can be computed from all the dimensionless parameters. Geometrical parameters can be set to attain particular net velocity for different electrolyte concentration. These results can be a guideline for the selection of an experimental geometry. 6.2 Recommendations for Future Work Some possible directions of the future works are as follows: - Frequency values are taken from 0.01 to 1. Investigation can be carried out in a much broader parameter range to precisely examine the qualitative and quantitative behavior of AC electroosmosis in slit microchannel with symmetric electrode. - Zero potential is taken for the negative electrode in the present study. Negative AC signal can be incorporated at the bottom electrode. A DC offset can be added to the AC signal so that one electrode potential remains greater than zero and the other lower than zero. - Geometrically more complex electrodes should be taken in consideration. Since it is predicted that the counter rotating regions of fluid observed above the electrode arrays inhibit net flow, a three-dimensional electrode design should be incorporated where the inhibition should be taken care of. - A simple model for Faradaic currents is used here to investigate its effect on the electrokinetic flows generated on top of coplanar symmetric electrodes. The liquid is taken as symmetric uni-univalent electrolyte where only one ionic species react at the electrodes. More reacting ionic species should be included that is different metal/electrolyte systems should be investigated for a better understanding of the electrochemical process in a more realistic way. AC electroosmotic pumps can be a promising tool for various microfluidic applications. Further analysis can be carried out to predict the effect of Faradaic current on the pumping behavior of symmetric and asymmetric electrodes subjected to AC signal. 54 7. REFERENCES [1]. 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