Electroosmotic Flow and DNA Electrophoretic Transport in Micro/Nano Channels

Transcription

1 Electroosmotic Flow and DNA Electrophoretic Transport in Micro/Nano Channels Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Lei Chen, M.S. Graduate Program in Mechanical Engineering The Ohio State University 2009 Dissertation Committee: A.T. Conlisk, Advisor Joseph Heremans Vishwanath V. Subramaniam Sandip Muzumder

2 c Copyright by Lei Chen 2009

3 ABSTRACT In Micro/nano fluidic systems, electrokinetic transport is a convenient method to move materials, such as water, ions and particles for fast, high-resolution and low-cost analysis and synthesis. It has wide applications to drug delivery and its control, DNA and biomolecular sensing, manipulation, the manufacture laboratories on a microchip (lab-ona-chip) and many other areas. In the present work, electrokinetically driven fluid flow and particle transport in micro/nanoscale channels/pores with heterogeneous surface potential or converging shape are investigated theoretically and numerically. A step change in wall potential is found to induce a recirculation region in the bulk electroosmotic flow and interesting flow structures can be achieved by manipulating the surface heterogeneous patterns. Most of the previous work on this problem is based on the Debye-Huckel approximation and the validity of Boltzmann distribution for ionic species. In the present work, ionic species distributions in the electric double layers are found to be different from the Boltzmann distribution and this deviation is more noticeable for higher applied electric field. A mathematical model is developed to simulate the electroosmotic flow (EOF) and the transport of embedded particles in micro/nano nozzles/diffusers. Results can be used to estimate the mass transport of charged/uncharged species in micro/nano nozzles/diffusers which has a potential application in transdermal drug delivery. The model is extended to investigate the DNA electrophoretic transport through a converging nanopore for the ii

4 purpose of DNA sequencing. The flow field, the resistive forces acting on the DNA, the DNA velocity and the ionic current through the nanopore are calculated numerically based on the Poisson-Boltzmann theory and the lubrication approximation. It is found that the electroosmotic flow inside the nanopore plays an important role in the DNA translocation process and the resulting viscous drag decreases the effective driving force acting on the DNA substantially. Entropic forces, used to be considered as the main resistive forces in previous works, are found to be small and negligible. Modeling and simulations are validated by the good agreement with the experimental data for the tethering force and the DNA velocity. iii

5 ACKNOWLEDGMENTS I am deeply grateful to Professor A. T. Conlisk, whose help, stimulating suggestions and encouragement helped me in all the time of research for and writing of this thesis. I would also like to thank Prof. Vishwanathan Subramaniam, Prof. Sandip Muzumder and Prof. Joseph Heremans for accepting to be in the my Docotoral Examination Committee and for their valuable comments and advises on my thesis work. I would also like to thank Subhra for his discussions and supporting on my research. This work is supported by NSEC, Center for Affordable Nanoengineering of Polymeric Biomedical Devices (NSF Grant No. EEC ) and the PI is Professor James Lee from Department of Chemical and Biomolecular Engineering at OSU. The author is grateful for their support for this work. Especially, I would like to give my special thanks to my parents and my husband Xiaoyin for their encouraging and support. iv

6 VITA April 9, Born - Beijing, China July B.S. Mechanical Engineering Department of Thermal Engineering Tsinghua University, Beijing, China August June Graduate Engineer HVAC Lab Tsinghua University Beijing, China September 2003-Jan M.S. Mechanical Engineering Department of Mechanical Engineering The Ohio State University Columbus, OH Feb present.....theohio State University Columbus, OH PUBLICATIONS Research Publications Lei Chen and A. T. Conlisk, Modeling of DNA translocation in nanopores, 47th AIAA Aerospace Sciences Meeting, AIAA paper , 2009 Lei Chen and A. T. Conlisk, Effect of nonuniform surface potential on electroosmotic flow at large applied electric field strength, Biomedical Microdevices, Vol. 11, , 2009 Lei Chen and A. T. Conlisk, Electroosmotic flwo and particle transport in micro/nano nozzles and diffusers, Biomedical Microdevices, Vol. 10, , 2008 v

7 Lei Chen, P. Gnanaprakasam and A. T. Conlisk, Electroosmotic Flow in Micro/nano Nozzles, 45th AIAA Aerospace Sciences Meeting and Exhibit, AIAA paper , 2007 Lei Chen and A. T. Conlisk, Generation of nanovortices in electroosmotic flow in nanochannels, 4th AIAA Theoretical Fluid Mechanics Meeting, AIAA paper , 2005 Lei Chen and A. T. Conlisk, Electroosmotic flow in annular diatoms, 43rd AIAA Aerospace Sciences Meeting and Exhibit, AIAA paper , 2005 FIELDS OF STUDY Major Field: Mechanical Engineering Studies in Fluid Mechanics, Mass Transer and Electrokinetics: Professor A.T. Conlisk vi

8 TABLE OF CONTENTS Page Abstract Acknowledgments ii iv Vita v List of Tables x List of Figures xii Chapters: 1. Introduction Background Electrokinetic Phenomena The Molecular Biology of DNA Applications Literature Survey Electroosmotic Flow DNA Transport Present Work Governing Equations and General Assumptions Introduction Assumptions Governing Equations Poisson Equation Nernst-Planck Equation vii

9 2.3.3 Navier-Stokes Equation Dimensional Analysis Boltzmann Distribution and the Debye-Hückel Approximation Ionic Current and Current Density Summary Electroosmotic Flow with Nonuniform Surface Potential Introduction Governing Equations and Boundary Conditions Analytical Solution for Potential Based on the Debye-Hückel Approximation Numerical Methods Numerical Results Ionic Transport in a Nanopore Membrane with Charged Patches Summary Electroosmotic Flow and Particle Transport in Micro/Nano Nozzles/Diffusers Introduction Governing Equations and Boundary Conditions Lubrication Solutions for EOF in the Debye-Hückel Limit Known Surface Potential Known Surface Charge Density Results for EOF Based on the Debye-Hückel Approximation Results for EOF: Known Surface Potential Results for EOF: Known Surface Charge Density Analysis Based on the Thin Double Layer Limit (ɛ 1) Numerical EOF Calculation Based on Poisson-Boltzmann Model Numerical Solution using COMSOL Particle Transport Particle Transport Using Two-Phase Flow Modeling Modeling the Fabrication of the Nanonozzles Summary The Forces that Affect DNA Translocation Introduction Governing Equations and Boundary Conditions Electroosmotic Flow in an Annulus Lubrication Solutions for the Flow Field in a Conical Nanopore: The Viscous Drag Force Acting on the DNA inside the Nanopore viii

10 5.5 Uncoiling/recoiling Forces Acting on the DNA Viscous Drag on the DNA Outside the Nanopore Numerical Validation of the Lubrication Model using Tethering Force Data Summary DNA Translocation Velocity Introduction Governing Equations and Boundary Conditions Solution for the Flow Field Asymptotic Solution for potential φ and the EOF velocity u eof Results for DNA Translocation Velocity DNA Entering and Leaving the Nanopore Modeling DNA Delivery into a Cell through the Nanonozzles Summary Summary Appendices Appendices: A. Discretising the Equations A.1 Introduction A.2 Discritization A.3 The SIMPLE Algorithm A.4 Residuals ix

11 LIST OF TABLES Table Page 2.1 The scales used to nondimensionalize the equations listed in the previous section The numerical accuracy check for h =20nm channel with over-potential φ e =0.2. The results of grid are compared with the results of grid and two-digit accuracy has been attained Values of parameters used for simulations. Note that the concentration of species shown here are reservoir concentrations Parameters used for the numerical calculations for known surface charge. Note that the concentration of species shown here are reservoir concentrations The numerical accuracy check for the numerical calculation for the electroosmotic flow in the nano-diffuser shown in Figure The results for grid are compared with the results of grid and two-digit accuracy has been attained. The numerical results are shown in Figure Note that here the pressure p is rescaled on E I,x The numerical accuracy check for the results calculated using COMSOL shown in Figure The mesh I has 62, 442 triangular elements and the mesh II is a finer mesh, having 183, 712 triangular elements. The results shown for the electric potential φ (mv ), the velocity u (mm/s) and pressure p (Pa) are in dimensional form The dependence of the particle motion on the wall charge and the particle charge. Here u F is the bulk motion of the fluid flow and u EM is the velocity of particle due to electrical migration. The particle velocity is the summation of these two terms x

12 5.1 Comparison between the magnitudes of the uncoiling and recoiling force (FU and FR) calculated based on different models and the electrical driving force (Fe ) Comparison between the magnitudes of the viscous drag force acting on the DNA blob-like configuration outside the nanopore and the electrical driving force (Fe ). The viscous drag due to the approaching of the DNA blob to the nanopore is defined as Fblob1 and the viscous drag force acting on the DNA blob in Reservoir I due to the flow discharged from the nanopore is defined as Fblob The numerical accuracy check for the numerical calculation based on the experimental parameters (Keyser et al., 2006) (results are shown in Figure 5.10). The results for grid (Mesh I) are compared with the results of (Mesh II) grid and two-digit accuracy has been attained. Note that the pressure shown here is rescaled on E I,x ) Comparison between the magnitudes of the forces that may affect DNA translocation, calculated for a 16.5kbps double stranded DNA through a nanopore with a radius of 5nm at the small end. The applied voltage drop is 120mV. Note that the viscous drag force acting on the DNA inside the nanopore (Fd ) listed here is calculated for a DNA immobilized in the nanopore (u DNA =0) Comparison for DNA velocity between the numerical results (V Num in m/s) and the experimental data (V Exp in m/s). Here σ w (in C/m2 )is the surface charge density of the nanopore; L DNA is the length of the DNA used in the experiments; C KCl is the concentration of the KCl solution xi

13 LIST OF FIGURES Figure Page 1.1 The required pressure drop and voltage drop for nanochannels with different channel height. The required flowrate is 1µl/min; the length of the channel is 3.5µm and the width of the channel is 2.3µm. The zeta potential of the channel wall is 11mV and the working fluid is 1M NaCl (Conlisk, 2005) Micro and nanonozzles: (a) The Polymethyl Methacrylate (PMMA) micronozzles used to characterize fluid flow and particle transport. (b)the Polymethyl Methacrylate (PMMA) nanonozzles fabricated using sacrificial template imprinting (STI) methods (Wang et al., 2005) An illustration of the ion distribution near a glass surface and the electric double layer (EDL). The dash line denotes the approximate edge of the electric double layer Electric double layer around a charge particle DNA double-helix structure showing base pairs as vertical lines. The diameter of the spiral of the helix is about 2nm Conformation of DNA confined in a pore. Here h is the half height of the channel of the radius of the pore; R g is the radius of gyration of the DNA and l p is the persistence length of DNA Images of microneedles used for transdermal drug delivery. (a) Solid microneedles (150µm tall) etched from a silicon wafer were used in the first study to demonstrate microneedles for transdermal delivery (Henry et al., 1998). (b) Microneedle arrays (up to 300µm tall) in standard silicon wafer using potassium hydroxide (KOH) wet etching (Wilke et al., 2005) xii

14 1.8 Schematic drawing of a transdermal drug delivery system. There is a power source in the back. The foam ring with the drug reservoir is placed under the cathode and a microprojection array is integrated into the patch(lin et al., 2001) Gene delivery using nanotip array (Fei, 2007) under an applied electric field Concept for sequencing DNA by using a single protein pore. A singlestranded DNA (or RNA) molecule moves through the pore in the transmembrane electric field. As it passes a contact site each base produces a characteristic modulation of the amplitude in the single channel current (Alper, 1999) Nanopores used in the experiments to analyze DNA: (a) natural α-hemolysin nanopore and (b) synthetic nanopore The geometry of the nanopore and the DNA Cartesian coordinates and cylindrical coordinates used for modeling The geometry of the channel with a patch with over-potential φ p Analytical results for potential calculated from equation (3.23). The zeta potential of the channel wall and the patch is 1.3 mv and 6.5 mv respectively. The electrolyte solution is (a) 0.1M NaCl and (b) 0.001M NaCl, and the EDLs are not overlapped in (a) and overlapped in (b). The height of the channel is (a) h =20nm and (b) h =50nm A co-located mesh used for the numerical calculation The flowchart of the numerical calculation using SIMPLE algorithm Numerical results for potential, mole fractions calculated from the Poisson- Boltzmann equations. The imposed electric field is 10 6 V/m. The height of the channel is 20nm and the wall surface potential is 12mV. The electrolyte is 0.1M NaCl solution far upstream and the over-potential φ p = 2.0. The corresponding dimensionless parameters are ɛ = 0.05, Λ=0.77, A =1and Pe = xiii

15 3.6 Numerical results for potential, mole fractions, electric field lines and streamlines. The imposed electric field is 10 6 V/m. The height of the channel is 20 nm and the wall surface potential is 12 mv. The electrolyte is 0.1 M NaCl solution far upstream and the over-potential φ p =2.0. The corresponding dimensionless parameters are ɛ =0.05, Λ=0.77, A =1and Pe = Numerical results for potential, mole fractions and electric field lines calculated from the Poisson-Nernst-Plank equations. The imposed electric field is 10 7 V/m. The height of the channel is 20nm and the dimensional surface potential of the walls is 12 mv. The electrolyte is 0.1 MNaClsolution far upstream and the dimensional patch potential is 40 mv (φ p =2.0). The corresponding dimensionless parameters are ɛ =0.05, Λ=7.7, A =1and Pe = Cation mole fraction contours and streamwise velocity contours calculated from the Poisson-Nernst-Plank equations for Pe = 3.6 and Pe = 0. The filled colors show the results for Pe =3.6and the lines represent the results for Pe =0. The imposed electric field is 10 7 V/m. The height of the channel is 20 nm and the dimensional surface potential of the walls is 12 mv. The electrolyte is 0.1 MNaClsolution far upstream and the dimensional patch potential is 40 mv (φ p =2.0). The corresponding dimensionless parameters are ɛ =0.05, Λ=7.7, A = The dimensionless distance of the vortex center away from the patch as a function of the dimensionless over-potential of the patch. The center of the vortex is the point where u =0,v =0in the bulk flow. The imposed electric field is 10 7 V/m. The height of the channel is 20 nm and the dimensional surface potential of the walls is 12 mv. The electrolyte is 0.1 MNaClsolution far upstream and the corresponding dimensionless parameters are ɛ =0.05, Λ=7.7, A = Streamlines for the EOF in a nanochannel with a patch having negative over-potential. The height of the channel is 20nm and the aspect ratio of the patch A =1. The over-potential φ e = 0.4 and the imposed electric field is 10 7 V/m. The solution far upstream is 0.1M NaClelectrolyte buffer Streamlines for EOF in nanochannels with a patch having over-potential in the form of φ p = φ a cosnπx. The zeta potential of the wall is 1mV and the imposed electric field is 10 6 V/m. The solution far upstream is 0.1M NaCl xiv

16 3.12 The geometry of the reservoir and nanopore system (a) and the mesh used in the simulations (b). The height of the nanopores is 8 nm and the length of the pores is 1 µm. The size of the feed and the permeate reservoirs is 10 µm by 10 µm. The length of the patch is 500 nm Results for the electric potential, streamlines and concentration of albumin: (a) electric potential (b) streamlines and concentration contours of charged species. The ionic strength of the electrolyte solution is 0.14 M and the zeta potential of the charged patch is 46 mv. The concentration of the charged species (albumin) is 0.6 mm. The valence of albumin is 17 and the diffusion coefficient is m2/s. The height of the nanopores is 8nm and the length of the pores is 1 µm. The size of the feed and the permeate reservoirs is 10 µm by 10 µm. The length of the patch is 500 nm Results for albumin concentration at the centerline of the channel (a) with charged patch (b) without charged patch. The ionic strength of the electrolyte solution is 0.14 M and the zeta potential of the charged patch is 46mV. The concentration of the charged species (albumin) is 0.6 mm. The valence of albumin is 17 and the diffusion coefficient is m2/s. The height of the nanopores is 8 nm and the length of the pores is 1 µm. The size of the feed and the permeate reservoirs is 10 µm by 10 µm. The length of the patch is 500 nm The geometry of the micro-diffuser used in the experiments. The particle donor reservoir is on the right and the particle receiver reservoir is on the left of the diffuser. The walls are negatively charged and the positive electrode is placed in the particle receiver (left). The flow is from left to right and thus it is a diffuser. The polystyrene beads are shown and their motion is from right to left in a direction opposite to the bulk flow The geometry of the reservoir-nozzle system (a) the system used in MD simulations (b) the cooresponding dimensions. Here only three stripe of water molecules are shown (in grey and red) and Cl (green) and Na + (blue) ions dispersed in water are shown The geometry of the nozzle/diffuser used for modeling. The walls are negatively charged and for the polarity shown here, the flow is from left to right in both the nozzle and diffuser xv

17 4.4 The surface charge density boundary condition for the potential equation is φ y = ±σ y, ±h(x), where σ y is the component of the dimensionless surface charge density in the y direction The dimensionless imposed electric field (E I,x ) for a nozzle (Figure 4.3 a) and a diffuser (Figure 4.3 b). The length of the nozzle is 65nm; the height at the inlet is 13nm and the height at outlet is 8nm. The length of the diffuser is 650 µm; the height at the inlet is 20 µm and the height at the outlet is 130 µm. The cross-section of the nozzle is rectangular, with the width in the direction into the paper much larger than the height. The potential drop over the length of the nozzle/diffuser corresponds to 80V/cm. The solid line shows the electric field for a channel (80V/cm) Results for electroosmotic flow in the converging nanonozzle. The height of the nozzle is 13 nm at the inlet and 8 nm at the outlet and the length of the nozzle is 65 nm; ɛ =0.19 and the EDLs are overlapped. The imposed electric field is assumed to be 8000 V/m and the ζ-potential of the walls is 5mV. The pressure is zero both at inlet and the outlet and so the pressure shown here is induced by the presence of the EDLs Results for electroosmotic flow in the experimental micro-diffuser. The height of the diffuser is 20 µm at the inlet and 130 µm at the outlet and the length of the diffuser is 650 µm; ɛ = and the EDLs are thin compared to the diffuser. The imposed electric field is 8000V/m and the ζ-potential is 15mV. The pressure is zero both at the inlet and the outlet and so the pressure shown here is induced by the presence of the EDLs Results for pressure-driven flow in the experimental micro-nozzle as shown in Figure 4.3 (a). The height of the small end is 20 µm and 130 µm at the large end and the length of the nozzle is 650 µm. For (a), (b), (c), the dimensionless pressure is 1 at the large end and zero at the small end. For (d), (e), (f), the dimensionless pressure is 0 at the large end and 1 at the small end. The electroosmotic flow component is zero Streamwise velocity and the streamlines for the electroosmotic flow in a micro-nozzle with adverse pressure gradient. The height of the small end is 65µm and 130µm at the large end and the length of the nozzle is 650µm. The applied voltage drop is 80V/cm and the zeta potential of the wall is 5mV. The dimensionless pressure is p i =0and p o = xvi

18 4.10 Results for electroosmotic flow in the nano-nozzle. The height of the nozzle is 14.6 nm at the inlet and 4.6 µm at the outlet and the length of the diffuser is 15 nm; ɛ = The imposed electric field is 0.5V/nm and the surface charge density is 0.01C/m 2. The pressure is zero both at the inlet and the outlet and so the pressure shown here is induced by the presence of the EDLs Results for electroosmotic flow in the nano-nozzle. The height of the nozzle is 4.6 nm at the inlet and 14.6 µm at the outlet and the length of the diffuser is 15 nm; ɛ = The imposed electric field is 0.5V/nm and the surface charge density is 0.01C/m 2. The pressure is zero both at the inlet and the outlet and so the pressure shown here is induced by the presence of the EDLs The surface potential as a function of x. The large end of the nozzle/diffuser is 14.6 nm height; the small end is 4.6 nm height and the length is 15 nm. For (a), the surface charge density is 0.07C/m 2 and the electrolyte solution is 0.1 MNaCl. For (b), the surface charge density is 0.01C/m 2 and the electrolyte solution is M The geometry of the slowly varying channel with wall defined as h(x) = 1+δsin(wx). The walls are negatively charged and for the polarity shown here, the flow is from left to right The streamlines for electroosmotic flow in the channel shown in Figure Mapping from the original (x, y) coordinates to the (ξ, η) coordinates, which have a rectangular mesh, for easier numerical discretization Results for electroosmotic flow in the nano-nozzle. The height of the nozzle is 14.6 nm at the inlet and 4.6 nm at the outlet and the length of the diffuser is 15 nm; ɛ = The imposed electric field is 0.5V/nm and the surface charge density is 0.07C/m 2. The pressure is zero both at the inlet and the outlet and so the pressure shown here is induced by the presence of the EDLs xvii

19 4.17 Results for electroosmotic flow in the nano-diffuser. The height of the diffuser is 4.6 nm at the inlet and 14.6 nm at the outlet and the length of the diffuser is 15 nm; ɛ = The imposed electric field is 0.5V/nm and the surface charge density is 0.07C/m 2. The pressure is zero both at the inlet and the outlet and so the pressure shown here is induced by the presence of the EDLs The comparison of DH (based on the analysis in Section 4.3.2) and nonlinear results for streamwise velocity component u eof and pressure p. The height of the nozzle is 4.6 nm at the inlet and 14.6 µm at the outlet and the length of the diffuser is 15 nm; ɛ = The imposed electric field is 0.5V/nm and the surface charge density is 0.07C/m 2. The pressure is zero both at the inlet and the outlet and so the pressure shown here is induced by the presence of the EDLs The comparison of DH and nonlinear results for streamwise velocity component u eof and pressure p. The height of the nozzle is 4.6 nm at the inlet and 14.6µm at the outlet and the length of the diffuser is 15nm; ɛ = The imposed electric field is 0.5V/nm and the surface charge density is 0.01C/m 2. The pressure is zero both at the inlet and the outlet and so the pressure shown here is induced by the presence of the EDLs The Mesh used for the simulations. Close to the walls, extra fine mesh is used and in the center region, the mesh is coarser Results for ionic concentration and flow field for the EOF in the nozzlereservoir system calculated using COMSOL. The parameters are listed in table The results for EOF in the nozzle-reservoir system based on Molecular Dynamics simulations(shin & Singer, 2009) and continuum theory. The height of the nozzle is 14.6 nm at the inlet and 4.6 µm at the outlet and the length of the diffuser is 15 nm; ɛ = The imposed electric field is 0.5V/nm and the surface charge density is 0.07C/m The reservoir-nozzle system used to analyze particle transport. The reservoir on the right is the particle donor reservoir and the one on the left is the particle receiver reservoir. Negatively charged particles always tend to move from left to right due to electromigration (u EM < 0) xviii

20 4.24 Comparison of the analytical results and the experimental data (Wang & Hu, 2007). The length of the micro-diffuser is 650 µm; the inlet height is 20 µm and the outlet height is 130 µm. The electric field is 80 V/cm. The ζ-potential of the PMMA walls are 15 mv (Kirby & Hasselbrink, 2004). Both the particles and the walls are negatively charged and so u F > 0,u EM < Process schematic of EOF based dynamic assembly of silica(wang et al., 2005) The schematic diagram for the simplified models. (a) one-dimensional model: A homogeneous chemical reaction is happening in the bulk and a heterogeneous chemical reaction is happening at the liquid-solid interface at a different rate. The interface is moving with the deposition of the reaction product C. (b) two-dimensional model: EOF in rectangular channels with chemical reactions Schematic diagram of DNA translocation in a negatively charged nanopore The geometry of the nanopore and the DNA The geometry of a DNA placed in a cylindrical nanopore. The computational domain is the annular region between the DNA and the nanopore The results for potential and velocity calculated from equations 5.8 and The radius of the nanopore is 5nm and the radius of the DNA is 1nm. The surface charge density is 0.015C/m 2 on the DNA surface and 0.006C/m 2 on the wall The dimensionless shear stress τ = u at r = a as a function of electrolyte r concentration. The radius of the nanopore is 5nm and the surface charge density is 0.06C/m 2 while the radius of the DNA is 1nm and the surface charge density is 0.15C/m 2. The electrolyte concentration varies from 0.01M (ɛ =0.6) to0.1m (ɛ =0.06) The schematic diagram of the DNA-nanopore system used to investigate the uncoiling-recoiling force due to the DNA conformational change during the DNA translocation process (shown in (a)) and the resulting uncoilingrecoiling force (shown in (b)) for a 16.5kbps double stranded DNA xix

21 5.7 Schematic drawings of a DNA with one end dragged into a cylindrical nanopore. The radius of the nanopore is h and the force acting on the DNA is f The schematic diagram for (a) the experimental setup(keyser et al., 2006) and (b) the force balance. The DNA is attached to a polystyrene bead and a tethering force F t is used to immobilize the DNA. The electrical driving force F e and the viscous drag force F d are also shown The geometry of the nanopore used in Keyser s experiments. The nanopore is a 20nm long cylindrical nanopore with 5nm in radius connected to a 20nm long converging nozzle on one end and a 20nm long diverging diffuser on the other end. The slope of the converging or diverging wall is about Numerical results for the potential, concentration of cations, streamwise velocity and the streamlines calculated based on the lubrication approximation. The computational domain is the region between the DNA surface (left boundary) and the nanopore wall (right boundary). The nanopore is a 20nm long cylindrical nanopore with 5nm in radius connected to a 20nm long converging nozzle on one end and a 20nm long diverging diffuser on the other end. The surface charge of the nanopore is 0.06C/m 2 and the applied voltage drop is 120mV. The electrolyte solution is 0.1M KCl Comparison of tethering forces between the experimental data, the numerical results based on the lubrication approximation and the numerical results using COMSOL. The electrolyte solution is 0.1M KCl. Experimental data are recorded for one molecule at three different distances from the nanopore (filled squares 2.1µm, filled circles 2.4µm, filled triangles 2.9µm(Keyser et al., 2006). As shown in Figure 5.8, L b is the distance between the bead and the nanopore The geometry of the nanopore, the mesh and the numerical results for the concentration of anions, streamwise velcity. The nanopore is a 20nm long cylindrical nanopore with 5nm in radius connected to a 20nm long converging nozzle on one end and a 20nm long diverging diffuser on the other end. The surface charge of the nanopore is 0.06C/m 2 and the applied voltage drop is 120mV. The electrolyte solution is 0.1M KCl Illustration of DNA characterization system based on the translocation of DNA through a nanopore xx

22 6.2 The geometry of the nanopore and the DNA The algorithm used to calculate DNA velocity The inner region and outer region used for the asymptotic analysis The geometry of the nanopore used for calculation. A 300 nm long conical silica nanopore is connected to a 50 nm long cylindrical pore. The radii of the large and small ends of the conical pore are 50nm and 5nm respectively (Storm et al., 2003) Velocity contours in m/s calculated based on the lubrication solutions. The length of the conical pore is 300nm and the length of the cylindrical pore is 40 nm. The inlet radius of the nanopore is 50nm and the outlet radius is 5nm. The surface charge density of the nanopore is 0.06C/m 2 and the electrolyte solution is 0.1 MKCL. The radius of DNA is 1 nm The viscous drag force due to the electroosmotic flow component F eof and the pressure driven component F eof as a function of KCl concentration. The length of the conical pore is 300nm and the length of the cylindrical pore is 40 nm. The inlet radius of the nanopore is 50nm and the outlet radius is 5nm. The surface charge density of the nanopore is 0.06C/m DNA velocity as a function of electrolyte concentration (a) and pore surface charge (b). The length of the conical pore is 300nm and the length of the cylindrical pore is 40 nm. The inlet radius of the nanopore is 50nm and the outlet radius is 5nm. The surface charge density of the nanopore is 0.1 C/m 2 for (a) and the electrolyte concentration is 1M for (b) The results for current through the nanopore as a function of time. For the numerical results, the baseline current I baseline is I baseline = 7085 pa and the amplitude is I = 160 pa. The corresponding experimental data is I = 7100pA and I = 140 pa(storm et al., 2005a) Schematic of the partial entry case. The motion of the DNA is from left to right against the bulk electroosmotic flow which is from right to left xxi

23 6.11 DNA velocity and ionic current through the nanopore as DNA enters the pore. The length of the conical pore is 300nm and the length of the cylindrical pore is 40 nm. The pore is shown at the top. The inlet radius of the nanopore is 50nm and the outlet radius is 5nm. The surface charge density of the nanopore is 0.06 C/m 2 and the electrolyte concentration is 1M DNA velocity, ionic current and pressure through the nanopore as DNA enters the pore. The length of the conical pore is 300nm and the length of the cylindrical pore is 40nm. The pore is shown at the top. The inlet radius of the nanopore is 50nm and the outlet radius is 5nm. The surface charge density of the nanopore is 0.06 C/m 2 and the electrolyte concentration is 1M Velocity and ionic current change as a function of time during the entire DNA transloction process. The length of the DNA is 16.5kbps. The length of the conical pore is 300nm and the length of the cylindrical pore is 40nm. The inlet radius of the nanopore is 50nm and the outlet radius is 5nm. The surface charge density of the nanopore is 0.06 C/m 2 and the electrolyte concentration is 1M DNA velocity and ionic current through the nanopore as DNA enters a diverging nanopore. The length of the diverging nanopore is 300nm and the length of the straight cylindrical pore is 40 nm. The pore is shown at the top. The inlet radius of the nanopore is 5nm and the outlet radius is 223nm. The surface charge density of the nanopore is 0.06 C/m 2 and the electrolyte concentration is 1M DNA velocity and ionic current through the nanopore as DNA leaves a diverging nanopore. The length of the diverging nanopore is 300nm and the length of the straight cylindrical pore is 40 nm. The pore is shown at the top. The inlet radius of the nanopore is 5nm and the outlet radius is 223nm. The surface charge density of the nanopore is 0.06 C/m 2 and the electrolyte concentration is 1M Velocity and ionic current change as a function of time during the entire DNA transloction process. The length of the diverging nanopore is 300nm and the length of the straight cylindrical pore is 40 nm. The pore is shown at the top. The inlet radius of the nanopore is 5nm and the outlet radius is 223nm. The surface charge density of the nanopore is 0.06 C/m 2 and the electrolyte concentration is 1M xxii

24 6.17 DNA velocity, ionic current and pressure through the nanopore as DNA enters a diverging pore. The length of the conical nanopore is 300nm and the length of the straight cylindrical pore is 40 nm. The inlet radius of the nanopore is 223nm and the outlet radius is 5nm. The surface charge density of the nanopore is 0.2 C/m 2 and the electrolyte concentration is 1M The schematic for the gene delivery device A.1 The upwind difference method and the central difference method. Here q e is the velocity of east face using upwind difference method and q e is the face velocity approximated by the central difference A.2 An inner node on the mesh A.3 A boundary node at the inlet A.4 A boundary node at the outlet A.5 A boundary node at the lower wall A.6 The residuals of the numerical calculation xxiii

25 CHAPTER 1 Introduction 1.1 Background The subject of micro- and nanofluidics deals with controlling and manipulating of fluid flows having length scale on the order of microns and nanometers and it is a multidisciplinary field comprised of physics, chemistry, engineering, and biotechnology. Micro and nanofluidics has applications to drug delivery and its control, DNA and biomolecular sensing, manipulation, the manufacture of laboratories on a microchip (lab-on-a-chip) and many other areas. The new ideas developed in these areas demand a better understanding of micro/nano scale fluid flow and particle transport phenomena. As dimensions shrink, the effective driving and dominating forces change radically. Conventional forces resulting from pressure, inertia, viscosity or gravity that usually plays the dominant role in macroscopic flows may not be practical in micro/nanofluidic systems while forces at interfaces such as surface tension become more important due to the increasing ratio of interfacial area and volume (Gad-el Hak, 1999). Electrokinetic transport is a convenient method to move materials, such as water, ions and particles, in miniature systems for fast, high-resolution and low-cost analysis and synthesis. For example, a relatively low imposed electric field can generate significant volume flowrates which can not 1

26 Pressure Drop(Atm) and Applied Potential(Volts) Q=1µ l/min,pressure driven flow Q=1µ l/min, EOF Channel Height (nm) Figure 1.1: The required pressure drop and voltage drop for nanochannels with different channel height. The required flowrate is 1µl/min; the length of the channel is 3.5 µm and the width of the channel is 2.3µm. The zeta potential of the channel wall is 11 mv and the working fluid is 1M NaCl (Conlisk, 2005). be achieved using the conventional pressure-driven flow. As shown in Figure 1.1, the required pressure drop for a flowrate of Q =1µl/min increases from atm to 3 atm as the height of the channel shrinks to 10 nm while the corresponding voltage drop changes from 0.05 Voltsto 0.33 Volts. Here the volume flowrate Q h 3 p (Conlisk et al., 2002) for the pressure-driven flow in a channel with rectangular cross section and Q h V for the corresponding electroosmotic flow (EOF), where Q is the flowrate; p and V are the required pressure and voltage drop to move the fluid. It is shown that the electroosmotic flow is more practical than the pressure-driven flow, and this makes electroosmotic flow widely used in drug delivery devices for lower requirements in cost and power. However, fluid flow and particle transport in microfluidic devices can be difficult to control due to many complex factors such as surface composition and buffer characteristics. Understanding of fundamental fluid transport and particle transport is valuable for device design and optimization. Due to the difficulties of measuring fluid motion in such a 2

27 small scale without disturbing the flow field, theoretical analysis and numerical simulation become essential tools. (a) PMMA micronozzle (b) PMMA nanonozzle Figure 1.2: Micro and nanonozzles: (a) The Polymethyl Methacrylate (PMMA) micronozzles used to characterize fluid flow and particle transport. (b)the Polymethyl Methacrylate (PMMA) nanonozzles fabricated using sacrificial template imprinting (STI) methods (Wang et al., 2005). This work is supported by the National Science Foundation Nanoscale Science and Engineering Center (NSEC) for Affordable Nanoengineering of Polymer Biomedical Devices (CANPBD). The primary goal of NSEC is to develop polymer-based, low-cost nanoengineering technology and then use it to produce nanofluidic devices. Polymethyl Methacrylate (PMMA) micro/nanonozzles (shown in Figure 1.2) have been fabricated using polymeric materials due to their biocompatibility and/or biodegradability, low cost, and recyclability. Efforts are being made to integrate these nanonozzles and into nanofluidic devices for both biomedical and non biomedical applications. The theory behind transport of species in these nanonozzles and nanopores is essential to design and implement any nano/micro fluidic device. In the present work, the fluid flow and transport of ionic species 3

28 and charged particles in nanochannels is addressed based on theoretical analysis and numerical simulations. In the following section, the basic concept of electrokinetic transport is discussed. 1.2 Electrokinetic Phenomena Figure 1.3: An illustration of the ion distribution near a glass surface and the electric double layer (EDL). The dash line denotes the approximate edge of the electric double layer. Electrokinetic phenomena refer to electroosmosis, electrophoresis, streaming potential and sedimentation potential, which are phenomena due to the interaction of the diffuse double layer and an applied electric field generally observed in porous medium or colloidal systems. Electroosmosis is the motion of fluid flow produced by the action of an applied electric field on a fluid having a net charge. Electrophoresis is defined as the relative motion of charged particles under an electric field. Streaming potential is the potential induced by the movement of the fluid. Sedimentation potential is the potential induced by the movement of charged particles (Williams et al., 1995). In this thesis, fluid flow (electroosmosis) and particle motion (electrophoresis) will be investigated. 4

29 When an electrolyte solution characterized by charged ions comes in contact with a charged wall, a predominantly charged layer near the wall will develop. For example, as shown in Figure 1.3, when a glass surface is immersed in aqueous solution, it undergoes chemical reactions resulting in negative surface charges. These negative surface charges, and thus negative surface potentials, attract the positive ions in the solution to the wall, while the negative charges in the solution are repelled away from the wall. In immediate contact with the surface, there is a layer of cations strongly bound to the wall, called the Stern layer. Outside the Stern layer, there is another layer where the cations are mobile called the diffuse layer. In these two layers, the surface potential significantly influences on the ion distribution and these two layers are usually termed the electric double layers (EDL). Away from the wall, the bulk of the solution remains electrically neutral. The thickness of EDL is characterized by the Debye length λ d, which is typically in the order of nanometers. Zeta potential (ζ) is widely used for quantification of the magnitude of the electrical charge at the double layer. It is defined as the electrical potential difference between the edge of the Stern layer and a location far away from the wall where the concentration of ionic species is the so called bulk concentration. If the finite size effect of the ions is neglected, the ζ potential becomes the potential difference between the wall and the far field. When electrodes are placed at the ends of a channel the cations are attracted to the cathode and the anions are attracted to the anode. In the bulk flow away from the walls, the concentration of cations and anions are the same and so the electric body force will balance. But in the EDLs, they are not the same and so there is a body force acting on the flow in the EDLs. The flow near the wall will move under such a body force and thus generate a 5

30 Figure 1.4: Electric double layer around a charge particle. bulk fluid movement through the channel. This flow is usually called electroosmotic flow (EOF) or electroosmosis. Electrophoresis is a well-known electrokinetic phenomena and it is a complement to electroosmosis. In 1807, Reuss observed that clay particles dispersed in water migrate under influence of an applied electric field (Zwolak, 1809). Electrophoresis deals with the migration of charged particles in a liquid medium under an electric field and the particles are usually rigid and non-conducting. For a charged solid particle placed in an electrolyte solution, an electric double layer of opposite charge will develop around the particle to maintain the electric neutrality of the system as shown in Figure 1.4. Under the external applied electric field, the particle will move towards the anode or the cathode due to the body force qe where q is the net charge of the particle. The electric field also induces a body force on the ions in the electrical double layer which generate viscous drag force to retard the motion of the particle. There is another force associated with the deviation of the double layer from spherical symmetry and surface conductivity due to the excess ions 6

31 in the diffuse layer, which is usually called the electrophoretic relaxation force (Hubbard, 2002). The viscous drag force balances all these forces which determine the velocity of the particle. Electrophoresis has been widely used to characterize and separate colloidal particles and macromolecules since different particles move at different electrophoretic velocities due to different charge to radius ratio. In microfluidic systems, it is also used to propel functionalized particles such as DNA molecules and proteins. In the present work, the DNA electrophoretic transport through a solid state nanopore is investigated in chapter 5 and 6. Some of the basic molecular biology of DNA is discussed in the following section. 1.3 The Molecular Biology of DNA DNA, or deoxyribonucleic acid, is a nucleic acid that contains the genetic instructions used in the development and functioning of all known living organisms and some viruses. It carries the information that characterizes a cell and the instructions for making a new cell (Silverstein & Nunn, 1002). DNA is a long thread-like macromolecule made of chemical units called nucleotides, each of which includes a phosphate, a sugar and a nitrogen base as shown in Figure 1.5 (a). Two polynucleotide chains are held together by hydrogen bonds and entwine like vines, in the shape of a double helix as shown in Figure 1.5 (b) forming a DNA molecule. There are four types of nitrogen bases found in DNA: A(adenine), T(thymine), C(cytosine) and G(guanine). Each type of base on one strand can form a bond with just one type of base on the other strand, which is called complementary base pairing. The four bases can only form two kinds of base pairing, either A-T or C-G. For example, if the sequence of one of the double strands is known as ATTCGG, the sequence of the other strand will be TAAGCC. This characteristic is very significant and it has been applied in some of the 7

32 (a) DNA chemical structure (b) DNA double helix structure Figure 1.5: DNA double-helix structure showing base pairs as vertical lines. The diameter of the spiral of the helix is about 2nm 8

33 primary DNA sequencing methods (Chang, 2005). The sequence of these four bases along the DNA backbone encodes the genetic information and it is translated to determine the sequence of the amino acids within proteins. DNA chain is 22 to 26 angstroms wide (2.2 to 2.6 nanometers) and one nucleotide unit is 3.4 angstroms (0.34nm) long(m et al., 1981). Based on the number of base pairs, one can easily obtain the contour length of the DNA. For example, for a 16.5kbps λ DNA, the contour length is =5.6 µm. The DNA from a human cell has a contour length of 2m however it can be compacted like a ball of string in cells, the diameter of which is only about 2 µm (Calladine et al., 2004). This gives DNA the ability to not only record huge amount of information but also to be packed into a micro sized cell. DNA molecules are negatively charged at physiological ph values (ph = log 10 [H + ], where [H + ] is the concentration of hydrogen ions in the solution) because the phosphate groups in the backbone tends to donate their protons (H 3 O + ) to the solution to reach their acidbase equilibrium. Generally DNA has a line charge of 2e per base pair which is equivalent to a surface charge density of 0.15C/m 2. Electrophoresis has become a very important mechanism that biologists use to manipulate DNA. In liquid solution, an isolated DNA chain is subjected to Brownian motion induced by the surrounding fluid molecules and behaves like a three dimensional random walk whose nodes moves randomly while keeping the connectivity of the chain. Usually for a long polymer, the persistence length (l p ), which is the maximum length of the uninterrupted polymer chain persisting in a particular direction(calladine & Drew, 1992), determines how a DNA segment should be described. For a double-stranded DNA, the persistence length is about 50 nm and for a contour length much smaller than 50 nm, the DNA behaves like a flexible elastic rod, while for the contour length much longer than 50 nm, the 9

34 properties of the DNA, such as DNA diffusivity and electrophoretic mobility, can be only described statistically (Salamone, 1996). Figure 1.6: Conformation of DNA confined in a pore. Here h is the half height of the channel of the radius of the pore; R g is the radius of gyration of the DNA and l p is the persistence length of DNA. A DNA molecule in a nanochannel may extend along the channel axis to a substantial fraction of its full contour length. The conformations of DNA in a nanochannel can be broadly classified into three groups based on the ratio of height or radius of the channel (h ) to the DNA radius of gyration (R g ) or the DNA persistence length (l p ) (Muthukumar, 2007) as shown in Figure 1.6. Here the radius of gyration (R g ) is defined as the average square distance of the chain segments from the centre of the mass of the chain and it is a statistical measure of the volume in which the DNA molecule is contained. The specific conformations that DNA adopts play an important role as the DNA interacts with its environment. If the pore or channel size is larger than the DNA radius of gyration (h >R g ), the DNA is free to coil in the nanochannel and the confinement does not alter the statistical mechanical properties of DNA (shown in Figure 1.6 (a)). If the height of the channel is smaller than 10

35 the radius of gyration of the DNA but larger than the persistence length (l p <h <R g ), the DNA still has three-dimensional freedom to coil, forming three-dimensional blobs, but extended along the axis of the channel, which is generally called the de Gennes regime (Bonthuis et al., 2009) as shown in Figure 1.6 (b). If the height of the channel is smaller than the persistence length of the DNA (h <l p ), the orientation of the DNA becomes restricted and in the strong confinement limit (h l p ), the DNA can be considered as the link of many rigid rods deflected by the boundary of the channel as shown in Figure 1.6 (c), which is usually called the Odijk regime (Odijk, 1983). In the present work, the DNA transport through a nanopore problem falls in the Odijk regime. Besides the restriction due to the nanopore, DNA is subjected to the extension due to the applied electrical driving force and the drag force, so that it translocates like a rigid rod. The details will be discussed in chapters 5 and Applications In this thesis, electroosmotic flow and the electrokinetic transport of ionic species and particles are investigated in two kinds of channels: a slit pore and a tapering nozzle with rectangular or circular cross-section. The electroosmotic flow in a slit pore with heterogeneous surface potential is discussed in chapter 3. The electroosmotic flow and particle transport in a tapering micro/nano nozzle is discussed in chapter 4. DNA transport through a converging nanopore is discussed in chapters 5 and 6. The potential applications of these micro/nanofluidic channel/pores are discussed in this section. Heterogeneous Surface Charge 11

36 The heterogeneous surface charge on the microchannels has been proposed to enhance species mixing in Lab-on-a-chip devices (Erickson & Li, 2002a). The effect of heterogeneous surface potential can also be used to reduce the band broadening. It is known that an analyte band will become distorted, or skewed, as it travels around a turn due to differences in the path length and differences in the magnitude of the electric field between the inside and the outside of the turn. Johnson et al. (2001) have introduced surface heterogeneity to the side wall of a polymeric microchannel and demonstrated how this could reduce sample band broadening around turns. Micro/nano Nozzle/Diffuser Transdermal Drug Delivery (a) (b) Figure 1.7: Images of microneedles used for transdermal drug delivery. (a) Solid microneedles (150µm tall) etched from a silicon wafer were used in the first study to demonstrate microneedles for transdermal delivery (Henry et al., 1998). (b) Microneedle arrays (up to 300µm tall) in standard silicon wafer using potassium hydroxide (KOH) wet etching (Wilke et al., 2005). 12

37 Transdermal drug delivery is an approach used to deliver drugs across the skin and into systemic circulation. Compared to other drug delivery approaches, it delivers drug directly to the blood stream, allowing the active ingredients to bypass the acid digestive environment of the stomach and filtration by the liver. Moreover, it provides a large surface area and ease of accessibility for drug administration. Transdermal drug delivery systems have been in use for over 20 years and this method of delivery has become widely recognized since the introduction of nicotine patches for smoking cessation in Transdermal delivery is severely limited by the inability of the large majority of drugs to cross skin at therapeutic rates due to the great barrier imposed by skin s outer stratum corneum layer. To increase the permeability of skin, microneedles are used. Microneedles are micron-dimensioned needles that can pierce the skin in a minimally invasive manner without causing pain. Upon piercing skin they create micro-conduits across stratum corneum and provide a direct route for transport of drugs and vaccines into the skin. Microneedles can be fabricated to be long enough to penetrate the stratum corneum, but short enough not to puncture nerve endings. This reduces the chances of pain, infection, or injury. Usually these microneedles are made of metal or silicon and as shown in Figure 1.7, the structures of these microneedles are similar to the micronozzles shown in Figure 1.2. In a transdermal drug delivery system, drug is generally loaded in a drug reservoir integrated into the patch and transport can occur by diffusion or electromigration or electrophoresis called iontophoresis if an electric field is applied. In the presence of an electric field transport can be greatly improved due to the presence of electroosmosis and electromigration. A transdermal drug delivery patch is shown in Figure 1.8 to increase antisense oligodeoxynucleotide (ODN) across the skin for cancer treatment. There is a power source 13

38 Figure 1.8: Schematic drawing of a transdermal drug delivery system. There is a power source in the back. The foam ring with the drug reservoir is placed under the cathode and a microprojection array is integrated into the patch(lin et al., 2001). in the back and the drug reservoir is placed under the cathode. A microneedle array is integrated in the patch. Delivery of ODN is greatly improved by the microneedles and it can be controlled by duration of the delivery, donor drug concentration, current density, and active patch area(lin et al., 2001). The theory discussed in chapter 4 can be used to characterize the drug delivery through the microneddles in this kind of devices. 14

39 Figure 1.9: Gene delivery using nanotip array (Fei, 2007) under an applied electric field. Gene Delivery into Cells A gene is a portion of DNA that determines a particular characteristic in an organism. Gene delivery is the process of introducing foreign DNA into host cells and it is one of the biggest challenges in the field of gene therapy. Viruses and liposomes have been widely used as carriers in animal testing, i.e. in vivo, but safety issues, such as immune response and cytotoxicity, have limited their clinical applications. Nanonozzles can be used to directly transfer repaired DNA into cells without using any carrier and thus can avoid the risks associated with introducing a secondary agent. The converging channels as shown in Figure 1.2 can pre-stretch and accelerate DNA molecules before sending them to cells. Moreover, the aperture of nanotips can carry a specific dosage of genes and leave it inside cells. A short penetration of these tiny nanotips would not cause permanent damage to the cells, the idea of which is shown in Figure 1.9. DNA is usually coiled in solution and the super-coiled DNA cannot pass through channels with dimensions smaller than its gyration diameter (usually 0.1µm) in a short time. Currently the nanopores fabricated by Wang et al. (2005) (shown in Figure 1.2) have small ends of 80nm and large ends of 1µm in 15

40 diameter. In the proposed gene delivery device, DNA can easily enter the large end. Inside the nanonozzle, the DNA is stretched and becomes a long worm-like shape with a much smaller radial size, and will eventually migrate through the channel. The stretching of the DNA is caused by the electrical field gradient due to the tapering geometry and the shear stress acting on the DNA due to the fluid flow. The transport of DNA through the nanopores in this problem falls in the de Gennes regime as discussed in the previous section. Boimolecular Sensing and DNA Sequencing The basic idea behind biomolecular sensing using a nanopore membrane is simple and already rather old. In 1950 s, Wallace Coulter (Coulter, 1953) first introduced a method to count particles or cells in solution. The principle of this system is to place the aperture in an electrolyte solution, containing the particles only on one side, and then constantly measure the conductance of the aperture. As the particles move through the aperture either by diffusion, by an external applied electric field or by a pressure gradient, they will cause a change in conductance due to the difference in conductance between particles and the solution. Initially this system was limited to the detection of micron-sized particles due to difficulty of fabricating small apertures in membrane. In 1970 s, DeBlois & Bean (1970); DeBlois et al. (1977) extended this method to detect submicron sized particles by using a single pore of a few hundred nanometers in diameter. The nanopore used is produced by the track-etching technique and the particles to detect are as small as 90 nm in diameter. Based on the same concept, Kasianowicz et al. (1996) proposed to characterize linear polymers like DNA using the protein, α-hemolysin, as the nanopore. DNA sequencing is the process of determining the nucleotide order of a given DNA fragment. It is useful in fundamental research into why and how organisms live, as well as in applied subjects such as the treatment of genetic diseases and forensic testing. Because 16

41 of the key nature of DNA to living things, knowledge of a DNA sequence may come in useful in practically any biological research. For example, in medicine it can be used to identify, diagnose and potentially develop treatments for genetic diseases. Due to the development in DNA sequencing, genome-based medicine has come ever closer to reality (Zwolak, 2008). Current sequencing processes are based on the chain termination method developed by Sanger et al. (1977). It be divided into four overall steps (Chan, 2005): i) DNA isolation, ii) sample preparation, iii) sequence production, and iv) assembly and analysis. It relies on very complex sample preparation and post processing of the data and the process is very expensive and time consuming. Based on current techniques, sequencing a single human genome involves costs of about 10 million USD and several months time (Fredlake et al., 2006). Although there has been some efforts to reduce the cost of the process, it is still hard to bring the overall cost lower than $10, 000 (Bayley, 2006). The NIH has set a remarkable challenge called the $1000 Genome which is to sequence the complete genome of an individual human quickly and at an accessible price. Nanopore sequencing, as shown in Figure 1.10, is one possible solution to bring the cost of DNA sequencing down and it has attracted many attentions recently. The sequencing method involves a membrane containing a single nano-scale pore between two halves of an electrochemical cell filled with an electrolyte solution. An external electric field is applied and the resulting ion current flowing through the electrolyte filled nanopore is recorded versus time. As the analyte, usually biomolecules and biopolymers with dimensions comparable to the nanopore diameter, is driven through the channel, the change in ion current is observed. The concentration of the analyte can be determined from the frequency of these current-changing translocation events and the identity of the analyte is encoded in 17

42 Figure 1.10: Concept for sequencing DNA by using a single protein pore. A single-stranded DNA (or RNA) molecule moves through the pore in the transmembrane electric field. As it passes a contact site each base produces a characteristic modulation of the amplitude in the single channel current (Alper, 1999). the magnitude and duration of the current change (Sexton et al., 2007). Although this technique is still in its early stage, the approach can already reveal limited information about base composition of DNA or RNA (Bayley & Martin, 2000). Both biological and artificial nanopores have been used for molecular sensing. Biological nanopore resistive-pulse sensors consist of a single transmembrane protein embedded in a planar lipid bilayer support. The most commonly used protein nanopore is α-hemolysin, shown in Figure 1.11 (a) and it is usually used to detect single strand DNA (ssdna) (Bayley & Cremer, 2001). Synthetic nanopores (shown in Figure 1.11 (b)), usually embedded in silicon nitride and silicon oxide nanopore membranes, are used to detect double strand DNA (dsdna) and proteins (Sexton et al., 2007; Storm et al., 2005a). For the design of the nanopore sensor, the key mechanical and electrical interactions between the DNA and the nanopore need to be well characterized (Chang & Yang, 2004). Water Purification A non-biomedical application of nanopore is the water purification. The world s water consumption rate is doubling every 20 years, outpacing by two times the rate of population 18

43 (a) α hemolysin nanopore (Bayley & Cremer, 2001) (b) Sythetic nanopore (Storm et al., 2003) Figure 1.11: Nanopores used in the experiments to analyze DNA: (a) natural α-hemolysin nanopore and (b) synthetic nanopore. growth. The supply of fresh water is on the decrease but water demand for food, industry and people is on the rise. Because of the potentially unlimited availability of seawater, people have made great efforts to try to develop feasible and cheap desalting technologies for converting salty water to fresh water. The development of membrane technology indicated a filter could be fabricated capable of separating liquids from colloidal particles or even low molecular weight dissolved solutes (ions). Surface properties of the membrane are important to the performance and the energy cost of the purification process. In chapter 3, the electrostatic interaction between a large charged molecule and the nanopore membrane is investigated to develop an artificial kidney. Extension of this work will provide foundational knowledge to the optimization of nanofiltration and reverse osmosis technologies for the purpose of water purification. 19

44 1.5 Literature Survey Electroosmotic Flow When in contact with electrolytes, many solid substrates acquire a surface charge and attracts opposite ions, creating thin layers of charges next to it, called electrical double layers. As shown in Figure 1.3, under an external electric field, the fluid in the electrical double layers acquires a momentum and drags the fluid in the bulk by viscosity. The resulting fluid motion is called electroosmotic flow (Probstein, 2003). The earliest model of the electrical double layer is usually attributed to Helmholtz (1879), who treated the double layer based on a physical model in which a single layer of ions is adsorbed at the surface. Later Guoy (1910) and Chapman (1913) made significant improvements by introducing a diffuse model of the electrical double layer, in which the potential at a surface decreases exponentially due to adsorbed counter-ions from the solution. The current classical electrical double layer model was introduced by Stern (1924) by combining the Helmholtz single adsorbed layer with the Gouy-Chapman diffuse layer. Von Smoluchowski (Smoluchowski, 1903, 1912) made several contributions to our understanding of electrokinetically driven flows, especially for conditions where the electric double layer thickness is much smaller than the channel height. Later the electroosmotic flow for an infinite capillary was studied by Rice & Whitehead (1965). Solutions in a narrow slit were obtained by Burgeen & Nakache (1964) in the context of the Debye-Hückel approximation and Levine et al. (1975) examined electroosmotic flow in a slit channel for both thin and overlapping EDLs. Conlisk et al. (2002) solved the problem for arbitrary ionic mole fractions and the velocity and potential for strong electrolyte solutions and have considered the case where there is a potential difference in the direction normal to the channel walls corresponding in some cases 20

45 to oppositely charged walls. The Debye-Hückel approximation has also been investigated by Conlisk (2005). Grahame (1953) investigated the electric double layer at a plane interface for an asymmetric electrolyte and calculated electric potential and ionic concentration in the double layers. Friedl et al. (1995) investigated the electroosmotic mobility for multivalent mixtures for the application in capillary electrophoresis and developed an empirical formula of mobility for a number of different acidic mixtures. Zheng et al. (2003) investigated the effect of multivalent ions on electroosmotic flow for multiple electrolyte components. In micro- and nanochannels having fixed surface charges, they found that adding multivalent counter-ions into a background electrolyte solution, even in very small amounts, caused significant reduction in electroosmotic flowrate in comparison with monovalent ions, while the multivalent co-ions have little effect on the electroosmotic flowrate. Datta et al. (2009) developed a site binding model to investigate the relation between the zeta potential and the divalent cations Ca +2 and Mg +2 in a background electrolyte solution. They found that the adsorption behaviors of the two divalent cations Ca +2 and Mg +2 are different, which can be explained by the stronger Ca +2 association to the silica surface, resulting in different zeta potential when same amount of Ca +2 and Mg +2 are added in to a background electrolyte solution. The importance of the heterogeneous surface charge on electrokinetic phenomena has been recognized in theoretical studies. The electro-osmotic flow in inhomogeneously charged pores was investigated initially by Anderson & Idol (1985). They have developed an infinite-series solution for the periodically varying ζ-potential in the flow direction and found recirculation regions. The electroosmotic flow between two slabs with periodic zeta potential has been investigated later by Ajdari (1995, 1996) to generate complex flows. In 21

46 that work, the flow is modeled under the Debye-Hückel approximation and the solution under the thin Debye layer (TDL) limit is also discussed. Chang & Yang (2004) have calculated the case with rectangular blocks located within the microchannel with the primary focus of enhancing a suitably defined mixing efficiency. The reversed flow pattern was also observed by Stroock et al. (2000) experimentally, and they also develop a model for the flow based on the Debye-Hückel approximation. The potential and velocity distribution close to a step jump in potential has also been investigated analytically by Yariv (2004). It is noted that one limitation of these studies including Ajdari (1996); Erickson & Li (2002b); Chang & Yang (2004), is the assumption that the net charge density field conforms to an equilibrium Boltzmann distribution. The Boltzmann distribution is based on the condition that the electrochemical potential must be constant everywhere at equilibrium and there is no imposed electric field and thus no bulk flow. However, axial non-uniformity in the surface potential will cause a deviation from the equilibrium that will necessitate the complete solution of Poisson-Nernst-Planck model augmented with convection of the flow field instead of a Poisson-Boltzmann equation based approach. This can be compared to corresponding classical developments in calculating the electrophoretic mobility of particles under the charge polarization and relaxation effects. EOF in micro/nano channels with varying cross-sectional area has also been investigated for different applications. Cervera et al. (2005) have developed a model for electroosmotic flow in conical nanopores. Their model focuses on the ionic transport and the flow field is not considered. Park et al. (2006) have investigated electroosmotic flow through a suddenly constricted cylinder and they have found eddies both in the center of the channel and along the perimeter due to the induced pressure gradient. Ramirez & Conlisk (2006) have examined the effects of sudden changes in channel cross-section area on the 22

47 EOF numerically and observed the formation of vortices or recirculation regions near the step face. Ghosal (2002) has investigated a slowly varying wall charge and slowly varying cross-section based on lubrication theory using asymptotic methods. The problem studied in the present work is similar to Ghosal s work but he used the Helmholtz-Smoluchowski (HS) slip approximation based on the assumption of infinitely thin EDLs, which leads to an EOF with uniform streamwise flow profile. However in view of application to nano-scale nozzle/diffusers, finite thickness EDLs are considered in the present model, which leads to an EOF with a nonuniform transverse profile. It also resolves induced pressure gradients in the problem that can not be resolved by analysis based on the HS approximation DNA Transport One recent application of nanopores is to use them as detectors and even analyzers for bio-polymers (e.g. DNA). When driven through a voltage-biased nanopore, bio-polymers cause modulation of ionic current through the pore, revealing the diameter, length and conformation of the polymers. Recently, the translocation of bio-polymers attracts increasing attention since it offers a possible solution to rapid DNA sequencing (Bayley, 2006). Both natural and synthetic nanopores are used to study the translocation process experimentally. Kasianowicz et al. (1996); Bayley & Cremer (2001); Meller (2001); Butler et al. (2007) experimentally investigated the single-stranded DNA (ssdna) molecules through a natural α-hemolysin pore (shown in Figure 1.11) to characterize the DNA translocation properties by analyzing the ionic current signals. The ssdna translocation velocity is found to be independent of DNA length if the length of the DNA is much longer than the nanopore length, and the experimental DNA translocation rate is about Å/µs. 23

48 Compared to the natural nanopores, synthetic solid-state nanopores are more stable, flexible in pore shape and surface properties, and they are easier to be integrated into micro/nanofluidic devices. To understand the relation between the DNA translocation rate and various experimental parameters such as the applied voltage bias and the concentration of the working solution, Chang et al. (2004); Fan et al. (2005); Li et al. (2003) and Storm et al. (2005a) investigated the translocation of double stranded DNA (dsdna) molecules through a synthetic nanopore experimentally. Li et al. (2003) demonstrated the capability of observing individual molecules of dsdna translocation, and the experimental translocation rate is 0.01m/s for 3kbps and 10kbps dsdnas. Chen et al. (2004) showed that the DNA translocation velocity increases linearly with the applied voltage drop, and it is independent of the DNA length based on the experimental data for 3kbps, 10kbps and 48.5kbps dsdnas. On the other hand, the experimental data of Storm et al. (2005a) on11.5kbps and 48.5kbps dsdna showed longer DNA transports slower through the nanopore, in conflict with Li et al. (2003). Modeling and simulations based on DNA dynamics have been conducted to understand the DNA translocation process, including Brownian dynamics simulations (Muthukumar, 2007; Murphy & Muthukumara, 2007; Forrey & Muthukumara, 2007), Monte Carlo simulations (Muthukumar, 1999; Kantor & Kardar, 2004; Kim et al., 2004; Chen et al., 2007) and Molecular Dynamic Simulations (O Keeffe et al., 2003; Fyta et al., 2006). These simulations generally focus on the DNA conformational change during the translocation process, ignoring the hydrodynamic interaction between the DNA and the nanopore. The DNA translocation is considered to be controlled by the entropic barriers related to the DNA conformation, and the predicted translocational velocity decreases nonlinearly with increasing DNA length. 24

49 Other hydrodynamic models are developed to investigate the hydrodynamic interactions locally inside the nanopore (Ashoke et al., 1997; Ghosal, 2006, 2007a,b). These models focus on the flow field and forces locally inside the nanopore, and the predicted DNA translocation velocity is independent of DNA length, which is consistent with Li et al. (2003) and Chen et al. (2004) s experimental findings. Ashoke et al. (1997) developed an analytical model for the translocation of a slender rod in a cylindrical channel based on the Debye-Hückel approximation. Ghosal (2006) developed a hydrodynamic model to investigate the electrically driven DNA across a converging nanopore, and showed that the proximity of the pore walls plays an important role on the magnitude of the viscous drag on DNA. In his model, the nanopore was assumed to be neutral and the Smoluchowski slip velocity was used to approximate the electroosmotic flow (EOF). Ghosal (2007a) also investigated the dependence of DNA transport on salt concentration and the predicted the force acting on a DNA inside a nanopore (Ghosal, 2007b). In these two papers, the cylindrical pores with constant cross-section are considered and the theoretical analysis is based on the Debye-Hückel approximation. For the application of nanopore sequencing using a synthetic nanopore, the DNA translocation rates reported in the experiments ( 1cm/s) are generally too high to measure the ionic current modulation resulting from the difference in shape and charge of the nitrogen bases on the DNA. To slow down the DNA transport, it is very important to understand the forces that control the translocation process. It is well accepted that the DNA is electrophoretically driven into the nanopore but the nature of the force resisting its transport is controversial in the literature. Some of the previous work including Muthukumar (1999, 2007); Kantor & Kardar (2004); Forrey & Muthukumara (2007) claim that the uncoiling force due to the DNA conformational change is the main resisting force while Storm et al. 25

50 (2005b) and Fyta et al. (2006) claim that the force due to the viscous drag on the bloblike DNA configuration outside the nanopore determines the DNA translocation rate. In the present work, all these forces are investigated and compared with the electrical driving force as well as the viscous drag force acting on the DNA inside the nanopore due to the electroosmotic flow. The results have shown that the uncoiling forces are usually negligible and the drag force acting on the blob-like DNA outside the nanopore is small compared to the electrical driving force. The main resisting force comes from the viscous drag due to the electroosmotic flow inside the nanopore. Future experimental design can be focused on the optimization of the parameters that can induce larger viscous drag on the DNA for the purpose of slowing down the DNA translocation, such as the surface charge of the nanopore and the geometry of the nanopore, Figure 1.12: The geometry of the nanopore and the DNA. An ideal model for DNA translocation would involve atomic-scale simulation of polymer, pore and ion dynamics in the presence of an applied electric field. However such a full MD approach is computationally very expensive, particularly when the simulation must be 26

51 conducted over microsecond timescales to capture blockade current events (O Keeffe et al., 2003). In this thesis, a hydrodynamic model is developed to investigate the fluid flow and ionic transport locally inside the nanopore. The nanopores considered here generally acquire a conical region, connected to a small pore with constant radius, as shown in Figure This configuration is widely employed in experimental nanopores since it is easier to fabricate and produces better current signals compared to cylindrical track-etched pores (Schiedt, 2007). Most of the previous work neglects the conical region and only considered the part with constant radius for simplicity. On the other hand, previous work on this problem is generally based on the Debye-Hückel approximation (valid for absolute value of φ RT F 26mV for T = 300K). It is known that DNA has a charge of 2e per base pair where e = C, equivalent to a surface charge of 0.15C/m 2. For such a high surface charge density, the zeta potential is as high as 109mV for a 0.1M KCl solution and 55mV for a 1M KCl solution. Obviously the Debye-Hückel approximation is not valid in this regime, and so Boltzmann distribution is used in this work to accurately describe the ionic distribution and fluid flow in the nanopore. It is well accepted that DNA navigates through the nanopore in a linear fashion, and previous experimental data also provided hard evidence by analyzing the change in ionic current due to the DNA blockage (Chen et al., 2004). Since the nanopore and the DNA both are negatively charged, the centroid of the DNA molecule tends to stay in the center of the channel due to the repulsion of the wall. Therefore, the DNA is modeled as rigid rod (1nm in radius) placed at the center of the nanopore in this work. Note that the smallest nanopore radius considered here is 5 nm and it may be argued that continuum theory may fail. Previously Zhu et al. (2005) have shown that the results 27

52 of Molecular Dynamic simulations for electroosmotic flow in a 6nm channel agree with the continuum results by easily adjusting the effective channel height h based on ion exclusion effects near the wall. Moreover, slip is not seen at the liquid-solid interface in their Molecular Dynamic Simulations. 1.6 Present Work In this thesis, the electrokinetically driven fluid flow and particle transport in converging/diverging channels are investigated. Electroosmotic flow and species transport modeling have been developed and results have been found using numerical methods. In chapter 2 the general governing equations for fluid flow and species transport are presented and the equations are nondimensionalized using appropriate length, time and velocity scales. The approximations made to simplify the equations are discussed. In chapter 3, electroosmotic flow in nanochannels with non-uniform wall potential is investigated. The nonlinear distributions of potential, velocity and mole fractions are calculated numerically based on the Poisson-Nernst-Planck (PNP) model including convective effects. The ionic transport of large charged molecules under a pressure-driven flow through a nanopore membrane is also studied to investigate the effect of the pore charge on the sieving of the molecule. In chapter 4, an analytical model has been developed to determine the electroosmotic flow in a converging or diverging channel based on the lubrication approximation. Numerical results for the full nonlinear case are also obtained and compared with results based on Molecular Dynamics (MD) simulations. In chapter 5, the flow field and forces acting on the DNA when it translocates through a nanopore is investigated. The forces acting on the DNA and the ionic current through the 28

53 nanopore are also calculated. The numerical results have been compared with the experimental data. In chapter 6, the DNA translocation velocity of DNA is analyzed based on the force calculations done in chapter 5. The DNA velocity during the processes of DNA entering and leaving the nanopore is also investigated. Forces resulting from the viscous drag and the random coiling of DNA have been considered. These forces have been included into the calculation of DNA velocity to improve the model. In chapter 7, the results are summarized and the prospects for future extension of this work have been discussed. 29

54 CHAPTER 2 Governing Equations and General Assumptions 2.1 Introduction In this chapter the general assumptions and the governing equations for electroosmotic flow is discussed. The Poisson equation, the Nernst-Plank equation and the Navier-Stokes equation are discussed for both Cartesian coordinates and cylindrical coordinates. The equations are non-dimensionalized using appropriate scales. Note that only the general equations are presented in this chapter and for each specific problem, the governing equations and the boundary conditions will be further discussed in the following chapters. 2.2 Assumptions The theoretical treatment of electroosmotic flow begin from the fundamental equations describing (i) the electrostatic potential (ii) the ionic current flow which are generated by the relative motion of the phases (iii) and the fluid flow. To simplify the problem we have made several assumptions including the ions are point charges 30

55 the solvent is continuous and is characterized by a constant permittivity which is not affected by the overall field strength or by the local field in the neighborhood of an ion the aqueous electrolyte is a dilute solution of a mixture of water or other neutral solvent and a salt compound such as sodium chloride; the charged species concentrations are so dilute that they do not interact with each other there is no chemical reaction happening in the solution all the solutes are entirely dissociated into cations and anions the fluid is incompressible and the no-slip boundary condition applies at the walls the Diffusion coefficient are not hindered by the size of the channel the fluid flow and species transport have reached their steady state A channel with rectangular cross-section is considered for Cartesian coordinates as shown in Figure 2.1 (a). The height of the channel is h. In Chapter 3, the height of the channel is a constant. In Chapter 4, the channel wall h (x) is converging or diverging along x axis and the height of the channel is a linear function of x. The width of the channel is W and the length of the channel is L. Usually W and L is much longer than h. There is an imposed potential drop along x direction and the primary flow is in x direction. A pore with circular cross-section is considered for the cylindrical coordinates as shown in Figure 2.1 (b). Since the geometry of the problems studied in this work is usually axissymmetric, only the coordinates x (the axial coordinate) and r (the radial coordinate) are considered. The applied electric field is in x direction. 31

56 (a) Cartesian coordinates (a) Cylindrical coordinates Figure 2.1: Cartesian coordinates and cylindrical coordinates used for modeling. 2.3 Governing Equations Poisson Equation The electric potential ϕ is written in the form of ϕ = ψ + φ (2.1) where φ is the perturbation potential due to the presence of the electrical double layers and ψ is the electric potential related to the applied electric field. The superscript indicates a dimensional quantity. The electrical potential satisfies the Gauss s Law as ɛ e 2 ϕ = ρ e (2.2) 32

57 Here ρ e is the charge density per unit volume and ɛ e is the permittivity of the mixture. Usually, the potential due to the applied electric field and the perturbation potential are considered separately. It is assumed that the volume charge does not distort the applied electric field and so 2 ψ =0 (2.3) The perturbation potential comes from the volume charge in the bulk and so ɛ e 2 φ = ρ e (2.4) For the applied electric field, E i =0 (2.5) Since E i = ψ, in the integral form, the applied electric field in x direction satisfies E i,x A = constant. The volume charge density ρ e in equation (2.2) is defined by ρ e = F i z i c i = Fc i z i X i (2.6) where c i, z i and X i = c i c are the molar concentration, the valence and the mole fraction of species i and c = c water + i c 0 i is the total molar concentration; c water is the mole concentration of water and it is 55.6 M; c 0 i denotes the species concentration in the reservoirs where electrical neutrality is maintained and the corresponding mole fraction in the reservoirs is given by X 0 i = c0 i c ; F is the Faraday s constant. Substituting ρ e into equation (2.6), the potential equation becomes ɛ e 2 φ = Fc i z i X i (2.7) 33

58 Consider Cartesian coordinates shown in Figure 2.1 (a), the dimensional potential equation is 2 φ x + 2 φ 2 y + 2 φ = Fc z 2 z 2 i X i (2.8) ɛ e i In cylindrical coordinates, the dimensional equation for potential is 1 r ( ) r φ r r Nernst-Planck Equation + 2 φ = Fc x 2 ɛ e z i X i (2.9) i Consider the mass transport in a liquid mixture of N components, for example water and an electrolytes with n species. The charged species will respond to three sets of stimuli: The electrical force. It is noted that the electric field, in general, is comprised of two components: an externally imposed electric field and a local electric field present near the solid surfaces of the channel corresponding to the presence of an electric double layer. Diffusion, tending to smooth out concentration variations. The bulk movement of charge carried along by the flow of the liquid (convective transport) Therefore the molar flux of species A for a dilute mixture is a vector and is given by n A = cd A X A + m A z A cf X A E + cx A u (2.10) Here D A is the diffusion coefficient of ionic species A in the mixture, X A is the mole fraction of species A, which can be either the anion or the cation, u A is the mobility, z A is the valence. The ionic species mobility m A is given by the Stokes-Einstein relation as m A = D A RT 34 (2.11)

59 Here R is the gas constant and T is the temperature which is taken to be 300K in the calculation. Different species have different diffusion coefficient but in dilute aqueous solutions, the diffusion coefficients of most ions are similar, in the range of m 2 /s to m 2 /s at room temperature. It is known that that in a system with multiple species, the diffusion of a species is governed not only by its own concentration gradient, but also by the concentration gradient of the other species in the system, which is generally referred to as multi-component diffusion. In such a multi-component system, use of Fick s law may results in violation of overall mass conservation (Kumar & Mazumder, 2007). However, the concentration of the ionic species considered in this work is very low and the mass fraction of the solution is higher than 90% (94.5% for 1M NaClsolution, 99.4% for 0.1M NaCl solution). In such a dilute solution, the diffusion of species are assumed to be independent of other species and the Fick s law is assumed to be valid (Bird et al., 2001) in such a system. The two vectors E and u can be resolved into their components along x, y and z axis. The electric field vector in terms of Cartesian unit vectors i, j and k is E = E x i + E y j + E z j = (ψ + φ ) (2.12) and the velocity vector is given by u = u i + v j + w k (2.13) The mass transfer equation is given by c A t = n A + R A (2.14) 35

60 where R A is the reaction rate of species A and it is assumed to be zero in the present work. The corresponding differential equation for ionic species A in Cartesian coordinates is D A ( 2 X A y 2 Since E x = ψ x also written as D A ( 2 X A y 2 z AX A D A F RT z A D A F RT + 2 X A x 2 + D AF RT ) + 2 X A = X ( A + u X A z 2 t x + X v A y + X ) w A z ( z A X A E x x φ, E x y = ψ y + 2 X A x 2 + z A X A E y y φ, E y z = ψ z + z A X A E z z ) (2.15) φ z, equation (2.15) can be ) + 2 X A = X ( A + u X A z 2 t x + X v A y + X ) w A z ( 2 φ x + 2 φ 2 y + 2 φ ) 2 z 2 ( ) φ X A y y + ψ y z AD A F RT z AD A F RT X A z ( ) X A φ x x + ψ x ( ) φ z + ψ z (2.16) For cylindrical coordinates, the mass transfer equation is ( ) D A XA 2 X A + D r r r A x = X ( A + u X A 2 t x + D AFz A RT X A E x x + D AFz A r RT ( r X AE ) r r r + X ) v A r (2.17) Here the electric field vector is given by E = Ex i x + Er i r = (ψ + φ )= (ψ + φ ) i x x + (ψ + φ ) i r r where x and r are the cylindrical unit vectors and the velocity vector is given by u = u i x + v i r. For each ionic species i in the electrolyte solution, there is a equation governing its transport and distribution. There are n mass transfer equations for n ionic species. In this 36

61 thesis, electrolyte solutions with a pair of monovalent ionic species of equal and opposite valence (n =2) is addressed, for example the mixture of NaCl H 2 O or KCl H 2 O. Note that most of the theory present in this work can be also applied to any combination of ionic constituents of arbitrary valences Navier-Stokes Equation The Navier-Stokes equations, arising from applying Newton s second law to fluid motion, describe the motion of fluid flow. It is one of the most useful equations and it has been used to simulate all kinds of fluid flow from atmosphere for weather prediction to blood flow in vessels for medical treatment. The Navier-Stokes equation expresses conservation of linear momentum, as well as the conservation of mass. In vector form, conservation of mass requires u =0 and in artesian coordinates, the dimensional equation is given by In cylindrical coordinates, the continuity equation is u x + v y + w =0 (2.18) z u x + 1 (r v ) =0 (2.19) r r In vector form, the conservation of momentum is given by ρ ( u t +( u ) u ) = p + µ 2 u + ρ e E (2.20) where, ρ is the density of the electrolyte, and µ is the fluid viscosity. The pressure field is p and the dimensional time is t. In Cartesian coordinates, the Navier-Stokes equation is ( ) u u u u ρ + u + v + w = p t x y z x + 37

62 µ ( 2 u x + 2 u 2 y + 2 u ) Fc (ψ + φ ) z 2 z 2 i X i (2.21) x µ ( ) v v v v ρ + u + v + w t x y z ( 2 v x + 2 v 2 y + 2 v ) 2 z 2 = p y + Fc z i X i (ψ + φ ) y (2.22) ( ) w w w w ρ + t u + v + w x y z ( 2 w x + 2 w 2 y + 2 w ) 2 z 2 µ = p z + Fc z i X i (ψ + φ ) z (2.23) The corresponding Navier Stokes equation in cylindrical coordinates is µ ( 2 u ( ) u u u ρ + u + v t x r x r = p x + ( )) r u Fc (ϕ + φ) z r r i X i (2.24) x 2.4 Dimensional Analysis µ ( ) v v v ρ + u + v t x r ( 2 v x + 1 ( )) r v 2 r r r = p r + Fc z i X i (ϕ + φ) r (2.25) In the previous section, the basic equations for electric potential, ionic species transport and fluid flow have been shown in the dimensional form both for Cartesian and cylindrical coordinates. To simplify the equations and prepare for the application of numerical simulation techniques, the equations are nondimensionalized based on proper scales. The scales chosen to nondimensionalize the equations are listed in table 2.1. Note that the length scale in y direction usually is the height of the channel or the nozzle/diffuser (as shown in Figure 38

63 1.2) at the inlet. Similarly, the applied field is scaled on the x component of the electric field at the inlet. The time scale shown in table 2.1 is called viscous time scale. dimensional variable scale dimensionless variable x L x = x L y h 0 y = y h 0 z W z = z W r h 0 r = r h i φ φ 0 = RT F 26mV φ = φ φ 0 E i,x,e i,y,e i,z,e i,r E 0 = E x,i(x =0) E x = E x E 0,E y = E y E 0,E z = E z E 0 u,v,w U 0 = ɛert E 0 µf u = u U 0,v = v U 0,w = w U 0 p p 0 = µu 0 h 0 t t 0 = h2 0 D 0 p = p p 0 t = t t 0 Table 2.1: The scales used to nondimensionalize the equations listed in the previous section. The equations shown in the previous section are fully three-dimensional, highly nonlinear and coupled equations and they are rarely solved in that form. Some simplifications can be made based on the characteristics of the problems considered in the present work. The geometries of the channel or the nozzle/diffusers can be simplified to be twodimensional since in z direction, there is no variation in surface properties and geometries and h W. Therefore only x and y axis are considered in Cartesian coordinates and for cylindrical coordinates, x and r are considered. Moreover, most of the problems are steady state and so the transient terms can be neglected in the equations. The two-dimensional steady-state dimensionless equation for potential in Cartesian coordinates is ɛ 2 2 φ 1 x + 2 φ 2 y = β zi X 2 ɛ 2 i (2.26) 39

64 and in the cylindrical coordinates, it is 1 r ( r r φ r ) + ɛ 2 2 φ 1 x = β z 2 ɛ 2 i X i (2.27) i where the aspect ratio ɛ 1 = h L ; β = c I is the ratio of total concentration c =c water + i c 0 i and the ionic strength I = i z 2 i c0 i ; ɛ = λ h i where λ = ɛert IF 2 is the Debye length. For mole fraction of ionic species i, the dimensionless equation in Cartesian coordinates is 2 X A + ɛ 2 2 X A y 2 1 x 2 + ( = 1 X i α i t + Pe ( ɛ 1 u X A α i x + v X A y ) ɛ 1 z A X A E x x + z A X A E y y ) (2.28) Here Pe = U 0h 0 D 0 is usually called mass transfer Peclet number. Since the diffusion coefficients for different species are not the same, the highest diffusion coefficient is taken as D 0 and the ratio of other species diffusion coefficients to this D 0 is given by α i = D i D 0.For computational ease, a single time scale is specified for the whole problem and it is taken to be t 0 = h2 0 D 0. E x,e y are the nondimensional electric field in (x, y) directions are written as E x = h 0E 0 φ E i,x ɛ 1 φ 0 x E y = h 0E 0 E i,y φ φ 0 y The nondimensional imposed electric field is defined as Λ= h 0E 0 φ 0, the dimensionless applied electric field is defined as E i,x, E i,y in the x and y directions respectively. In cylindrical coordinates the equation is 1 r ( r r X A r ) + ɛ 2 2 X A 1 x 2 +ɛ 1 z A X A E x x = 1 X i α i t + Pe ( ɛ 1 u X A α i x + v X ) A r ( + z A r X ) AE r r r (2.29) 40

65 and the dimensionless electric field components in r and x directions are E x = h 0E 0 φ E i,x ɛ 1 φ 0 x E y = h 0E 0 E i,r φ φ 0 r The continuity equation is ɛ 1 u x + v y =0 (2.30) Note that the x direction is the primary direction of flow and the velocity scale is chosen as U 0 = ɛert E 0 µf. Using the same velocity scale, in cylindrical coordinates, the dimensionless continuity equation becomes ɛ 1 u x + v y =0 (2.31) and in cylindrical coordinates it is u ɛ 1 x + 1 (rv) =0 (2.32) r r The dimensionless Navier Stokes equation in Cartesian coordinates is ( 1 u Sc t + Re ɛ 1 u u x + v u ) p = ɛ 1 y x + β ( ) zi X ɛ 2 i Ex + 2 u (2.33) ( 1 v Sc t + Re ɛ 1 u v x + v v ) = p y y + β ( ) zi X ɛ 2 i Ey + 2 v (2.34) where Re = ρu 0h µ is the Reynolds number and 2 2 = 2 y + 2 ɛ2 1 x 2 Here the same time scale is used as in the species mass transfer equation. The Schmidt number is given by Sc = µ ρd 0 and the Reynolds number is given by Re = ρu 0h 0 µ. For the typical species diffusion coefficient D m 2 /s, the Schmidt number Sc has a 41

66 value of Therefore, the acceleration terms in the momentum equation are of the order O( 1 ) 1, which means that the velocity field responds much faster than the time Sc scale chosen to non-dimensionalize the equation t 0 = h2 0 D 0. For the mass transfer equation, the transient term is of the order of O( 1 α i ) 1, indicating that the species respond on a longer time scale than the fluid flow. In the present work, most of the problems have assumed that the system has reached its steady state and so the transient terms are usually neglected. The Reynolds number Re is usually very small since h 0 O( ) for the problems considered in the present work and so the convective terms on the left hand side of the Navier Stokes equation can be neglected. Without the convective terms and the acceleration terms, the equation is usually called Stokes equation, which is easier to solve due to its linearity. For Re 1, the Stokes equation is given by 2 u = ɛ 1 p x β ɛ 2 ( zi X i ) Ex (2.35) For Cartesian coordinates, 2 = ɛ x 2 2 v = p y β ɛ 2 ( zi X i ) Ey (2.36) + 2 y 2 and for cylindrical coordinates, 2 = ɛ To summarize, for a electrolyte with n species, we have a series of 4+n x 2 r 2 r r equations in 4+n unknowns to solve for the two-dimensional velocity field, the pressure, the potential and the mole fractions for n species. Although simplifications have been made to the equations, these equations are still nonlinear and coupled with each other. Further simplifications can be made under certain conditions and the most common approximations for the electric potential and species distribution are discussed in the following section. 42

67 2.5 Boltzmann Distribution and the Debye-Hückel Approximation The Boltzmann distribution is usually used to predict the distribution of ions. It is based on the condition that the electrochemical potential of the ions must be constant everywhere at equilibrium which implies that the electrical force and diffusion of the ion must balance out: µ i = z i e φ (2.37) where µ i is the chemical potential and e = C is the charge on a single electron or proton. For a flat double layer the electrostatic potential and the chemical potential are constant in planes parallel to the wall so that equation (2.37) can be written as dµ dy = z ie dφ dy (2.38) and using the definition of the chemical potential in the form µ i = µ 0 i + RT ln n i where n i is the number of ions of species i per unit volume, we have d ln n i dy = 1 dn i n i dy = z ie dφ RT dy Integrating this equation from a point in the bulk solution where φ =0and n i = n 0 i leads to the Boltzmann equation: n i = n 0 i exp( z ieφ RT ) (2.39) Boltzmann gives the local concentration of each species in the double layer region. Substituting equation (2.39) into the equation of potential gives us the complete Poisson- Boltzmann equation: 2 φ = 1 n 0 i ɛ e D z ie exp( z ieφ A RT ) (2.40) i 43

68 If φ is small everywhere in the double layer, i.e. eφ RT, we can expand the exponential using the relation e x 1 x Then we get Debye-Hückel approximation of the problem 2 φ = 1 ɛ e ( i z i en 0 i i zi 2e2 n 0 ) i φ RT (2.41) It is noted that i z i en 0 i must be zero to preserve electro neutrality in the bulk electrolyte and so the equation becomes 2 φ = e2 ɛ e RT ( ) zi 2 n0 i φ = κ 2 φ (2.42) i where κ = ( e 2 i z 2 i n0 i ɛ e RT ) 1 2 The parameter κ depends mainly on the electrolyte concentration (n 0 i ) and it is referred to as the Debye-Hückel parameter. The Boltzmann distribution and the Debye-Hückel approximations are widely used in the present work to simplify the problem and derive analytical solutions. It is noted that the Boltzmann distribution assume slow and steady flow and neglects the convective term in the molar flux because of the small velocity. The nonlinearity of the problem has been removed due to the lack of inertial terms and the analysis and calculation becomes much easier. However, the Boltzmann distribution may fail for a very high electric field which will be discussed in the next chapter. What is more, the equilibrium condition indicates that the flow is steady and fully developed everywhere. Thus Boltzmann distribution may not hold any more for the unsteady electroosmotic flow and flow with velocity varying in flow direction as in our problem. The Debye-Hückel approximation applies a linear distribution 44

69 of the ions in the double layer and made the problem even simpler. Note that it is only valid for small potential φ 26mV. 2.6 Ionic Current and Current Density Ionic current is the electric current with ions as charge carriers. In micro/nanofluidic systems, ionic current is much easier to measure than other physics properties and it is often used for detection. In the problem discussed in Chapter 5 and 6, transient changes of ionic current indicate DNA translocation events and reveal important information on the translocation processes. In vector form, the dimensional equation for current density is J = F i z i n i (2.43) where n i is the dimensional flux of ionic species i, which is given by n i = cd i X i + c D i RT z ifx i E + cx i u (2.44) Using the same nondimensionalizing techniques as discussed in the previous section, in non-dimensional form, for example in the x direction the dimensionless current density is given by J x = ( Xi z i i x + z φ ie i,x X i z i x + Pe ) X i u α i here the ionic current density is scaled on cu 0 as (2.45) J x = J x cf U 0 The current through the channel (in x direction) is the integral of the current density across the entire cross-section of the channel: I x = A 45 J x da (2.46)

70 As shown in equation (2.44), the total current in the channel is comprised of three parts: the diffusion gradient, molar flux due to the migration of ions under the potential gradient, and the flux due to the bulk velocity of the electrolyte. Usually the migration term is dominant. 2.7 Summary In this chapter, the assumptions and the general governing equations for species transport, bulk velocity and electrical potential are presented in both Cartesian and cylindrical coordinates. These partial differential equations are nondimensionalized based on proper scales as shown in Table 2.1 for the numerical calculations. The Boltzmann distribution, which describes the ionic distribution at equilibrium, and the Debye-Hückel approximation, which linearizes the potential equation, are widely used to characterize the electric potential and ionic distributions in micro/nanofluidic systems. They are also used in the present work to simplify the equations and derive analytical solutions for certain problems. Note that the equations discussed in this chapter are in general form and they can be further simplified for specific problems in the following chapters. The boundary conditions for these equations are discussed for each problem considered. 46

71 CHAPTER 3 Electroosmotic Flow with Nonuniform Surface Potential 3.1 Introduction In this chapter, the electroosmotic flow in rectangular channels with heterogeneous surface potential is discussed. A single patch with different ζ potentials embedded in the wall of the channel is investigated based on the Poisson-Nernst-Planck equation. The surface modification can be effected by an initial reaction with a coupling reagent that will modify the ζ potential on the surface (Olesik, 2006). There are thus two objectives of this chapter: 1) to investigate the deviation of the species and potential distribution from the Boltzmann distribution; and 2) to investigate the structure of the vortical region over the potential patch. Since single molecule interrogation is usually done on the nanometer scale we consider channels from 20nm or up. Previous work done on the electroosmotic flow with heterogeneous surface potential, including Ajdari (1996); Erickson & Li (2002b); Chang & Yang (2004), is the assumption that the net charge density field conforms to the equilibrium Boltzmann distribution. As discussed in Chapter 2, the Boltzmann distribution is based on the condition that the electrochemical potential must be constant everywhere at equilibrium and there is no imposed electric field and thus no bulk flow. However, axial non-uniformity in the surface potential will cause a deviation from the equilibrium that will necessitate the complete solution of 47

72 Figure 3.1: The geometry of the channel with a patch with over-potential φ p. Poisson-Nernst-Planck model augmented with convection under the flow field instead of a Poisson-Boltzmann equation based approach. This can be compared to corresponding classical developments in calculating the electrophoretic mobility of particles under the charge polarization and relaxation effects. 3.2 Governing Equations and Boundary Conditions The geometry of the channel is shown in Figure 3.1. The spanwise width of the channel is much larger than the height of the channel and so the problem can be considered twodimensional. A patch having different ζ potentials is embedded in the lower wall of the channel and the length of patch is assumed to be on the order of the height of the channel. Note that the origin of the coordinate system (point (0, 0)) is at the center of the patch as shown in Figure 3.1 Electrodes are placed at the inlet and the outlet of the channel to generate a imposed electric field that is about V/min strength. As discussed in Chapter 2, it is assumed that the volume charge does not distort the applied electric field and so the applied electric 48

73 field satisfies E I,x A = constant Since the height of the channel h (x )=h 0 is a constant, the imposed electric field is a constant in x direction (in dimensionless form E I,x = 1) due to the constant cross section area A. The y component of the applied electric field is given by E I,y =0since E I,x x + E I,y y =0(equation (2.5)). The surface potential of the channel walls and the patch are ζ w and ζ p respectively. For simplicity, a NaCl H 2O mixture is considered here. For multi-species, multivalent solutions, the results are pretty similar. The length scale in x direction is chosen to be the length of the patch instead of the length of the channel L discussed in Chapter 2 and so the aspect ratio is ɛ 1 = h 0 2b = A. For the two-dimensional problem shown in Figure 3.1, the governing equations are equations (2.26), (2.28), (2.31), (2.33) and (2.34) in Cartesian coordinates. A new dimensionless parameter is defined as Λ= h 0E 0 φ 0, which is the ratio of the applied electric field strength E 0 and the perturbation electric field strength φ 0 h 0. The simplified governing equations for potential, velocity and ionic species are ( X i +Az i x ( Re Λ A φ x Au u x + v u A 2 2 φ x φ y 2 = β ɛ 2 zi X i (3.1) 2 X i y + 2 X i 2 A2 x = Pe ( Au X i 2 α i x + v X ) i y ) ( ) X i φ z i y x z ix i A 2 2 φ x + 2 φ 2 y 2 i =1...N (3.2) ) = A p y x + β ( Λ A φ ) ( ) zi X ɛ 2 i + 2 u x (3.3) ( Re Au v x + v v ) = p y y β φ ɛ 2 y + 2 v (3.4) Since the channel is assumed to be sufficient long upstream and downstream, the flow is fully developed far upstream and downstream. In the numerical calculations, x = L x 49

74 is the upstream boundary and x = L x is the downstream boundary, where L x is usually chosen to be 10. Note that L x is scaled on the length of the patch and so the dimensionless length of the computational domain is 2L x = 20. At the walls, the no-slip boundary condition is used for the velocities and so u =0,v =0at y =0and y =1. Note that the potential is set to be zero at the walls so that away from the patch, the profile of the potential and the streamwise velocity are the same. The surface potential of the walls are ζ w and in dimensionless form, it is assumed that φ =0at the walls for simplicity. Thus the dimensionless potential is related to the dimensional potential by φ = φ ζw φ 0. The dimensionless patch potential is then φ p =(ζ p ζ w )/φ 0. The boundary condition for ionic species i is that the walls are impermeable to ions and so the flux into the wall is zero. To summarize, the boundary conditions in dimensionless form are: x = L x : φ = φ in,x i = X in i,u= u in,v =0 x = L x : φ x =0, X i x =0, u x y =0:u =0,v =0, X i y + z ix i φ y =0 v =0, x =0 { φ =0 if x > 1 2 φ = φ p if x 1 2 y =1:φ =0,u=0,v =0, X i y + z ix i φ y =0 where φ in, X in i and u in are the fully developed profiles of φ, X i and u. Note that the equation for ionic species can be written in the form of ( ) ( ) Xi y y + z φ X i ix i +ɛ 1 ɛ 1 y x x + z φ ix i ɛ 1 = ɛ 1 z i Λ X ( i x x +Pe ɛ 1 u X A x + v X ) A y The Boltzmann distribution for species (3.5) X i = X 0 i exp ( z iφ) 50

75 is a solution for this equation if Λ=0and Pe =0under the equilibrium conditions, where X 0 i is the mole fraction of ionic species in the upstream reservoir. The two terms on the right hand side are the source of non-equilibrium in the concentration distribution of the species and they are ignored in the Boltzmann distribution. If these two terms are large, the ionic concentration is different from the Boltzmann distribution, which will be shown in the numerical results. It is shown from equation (3.5) that this difference comes from the convection of ionic species and the non-uniform ionic distribution in x direction. Since Pe E 0 and Λ E 0, the right hand side of the equation is directly related to the applied electric field, which is also seen in the numerical results. 3.3 Analytical Solution for Potential Based on the Debye-Hückel Approximation The governing equations derived in the previous section are highly nonlinear and it is difficult to find the analytical solution. However, the equation of potential and mass flux can be linearized using the Debye-Hückel approximation. Note that the Debye-Hückel approximation is based on the Poisson-Boltzmann model and it is only valid for φ 26mV (φ = φ F RT 1). The distribution of mole fraction of species will satisfy the Boltzmann distribution if the imposed electric field term is neglected (Λ =0) X i = X 0 i e z iφ (3.6) When the potential φ is small, the exponential can be expanded as X i = X 0 i (1 z iφ) (3.7) The potential equation (2.26) becomes 2 φ y + 2 φ 2 A2 x = φ (3.8) 2 ɛ 2 51

76 Equation (3.8) is a linear equation and the electric potential is then decomposed into two components: φ = φ edl + φ, where φ edl is the potential due to the presence of the electrical double layers away from the patch and φ is the potential due to the patch. The component φ edl satisfies 2 φ edl y 2 + A 2 2 φ edl x 2 = φ edl ɛ 2 (3.9) and φ satisfies 2 φ y 2 + A2 2 φ x 2 = φ ɛ 2 (3.10) The boundary conditions for these two equations are x = L x : φ edl x =0,φ =0 x = L x : φ edl x =0,φ =0 y =0:φ edl = ζ w, { φ =0 if x > 1 2 φ = ζ p ζ w = φ p if x 1 2 y =1:φ edl = ζ w,φ =0 where ζ w = ζ w /φ 0 and ζ p = ζ p /φ 0 are the dimensionless zeta potential of the wall and the patch. Note that in equation (3.9), the term A 2 2 φ edl x 2 =0since there is no streamwise variation in φ edl. The equation becomes 2 φ edl y 2 = φ edl ɛ 2 (3.11) with boundary conditions φ edl = ζ w,y =0, 1 and the analytical solution is sinh( 1 y 1 ) sinh( ) ɛ ɛ φ edl = ζ w sinh( 1) ɛ The perturbed potential can be solved using the method of separation of variables, which assumes φ (x, y) =X(x)Y (y) Substitute this equation into equation (3.10), the partial 52

77 differential equation becomes X(x) X(x) + Y (y) Y (y) = 1 ɛ 2 (3.12) and it can be written as X(x) X(x) = λ2 = Y (y) Y (y) + 1 ɛ 2 (3.13) Now the equation becomes two separated ordinary differential equations X(x) + λ 2 X(x) =0 (3.14) Y (y) (λ )Y (y) =0 (3.15) ɛ2 Since φ =0at x = ±L x, the boundary condition for equation (3.14) is x = ±L x,x(x) =0 and the solution is given by ( ) nπ X(x) =A n sin (x + L x ) = A n sin(λ n (x + L x )) (3.16) 2L x where λ n = nπ 2L x and A n is the undetermined coefficient for the solution. This is the solution for a periodic array of patches. Similarly, φ =0at y =1and so the boundary condition for equation (3.15) is given by y =1,Y(y) =0. The solution for equation (3.15) is given by Y (y) =B n sinh λ 2 + 1ɛ2 (1 y) (3.17) and the full solution for φ is now φ = X(x)Y (y) =C n sin(λ n (x + L x )) sinh( λ (1 y)) (3.18) ɛ2 where C n = A n B n. On the lower wall y =0, φ is given by φ = C n sin(λ n (x + L x )) sinh( λ ɛ )=ϕ 2 n sin(λ n (x + L x )) (3.19) 53

78 (a) h =20nm (b) h =50nm Figure 3.2: Analytical results for potential calculated from equation (3.23). The zeta potential of the channel wall and the patch is 1.3 mv and 6.5 mv respectively. The electrolyte solution is (a) 0.1M NaCl and (b) 0.001M NaCl, and the EDLs are not overlapped in (a) and overlapped in (b). The height of the channel is (a) h =20nm and (b) h =50nm. where ϕ n = C n sinh( λ ɛ 2 ) is a new coefficient to be determined. The boundary condition for φ at y =0is given by φ =0if x > 1 and 2 φ = φ p,if x 1. 2 To apply this boundary condition, Fourier series is used as ϕ n = 1 Lx ϕ(t)sin(λ n (t + L x ))dt = 1 L x L x L x 1/2 1/2 ϕ(t)sin(λ n (t + L x ))dt (3.20) and the integration gives ϕ n = 2a L x λ n sin(l x λ n )sin( λ 2 ) (3.21) The coefficient C n is then given by C n = ϕ n sinh( λ ɛ 2 ) (3.22) Therefore, the solution for φ is given by 2φ φ p sin ( ) ( ) nπ 2 sin λn2 sin(λn (x + L)) sinh ( (1 y) ) 1 + λ ɛ 2 = 2 n 1 λ n L sinh 1 + λ ɛ 2 2 n and the full solution for potential φ is sinh( 1 x 1 ) sinh( ) ɛ ɛ φ = ζ w sinh( 1) ɛ 54

79 + 1 2φ p sin ( ) ( ) nπ 2 sin λn2 sin(λn (x + L)) sinh ( (1 y) ) 1 + λ ɛ 2 2 n λ n L sinh 1 ɛ 2 + λ 2 n (3.23) Note that this solution is for the channel with periodic patches since it is based on the Fourier series and the solution for an isolated patch in a channel will be derived in future. The distribution of mole fractions can be obtained using equation (3.7). It should be noted that since a pressure gradient is generated over the patch, the streamwise and transverse velocities are difficult to obtain analytically and thus numerical calculation is required. Figure 3.2 shows the analytical results for (a) thin double layer case and (b) overlapped double layer case calculated from equation (3.23). The zeta potential of the channel wall and the patch are 1.3mV and 6.5mV respectively. The electrolyte solution is 0.1M NaCl for (a) and the corresponding Debye length is 1nm. The NaCl concentration is 0.001M for (b) (10nm Debye length) and the height of the channel is h =50nm. In this case, the height of the channel is larger than (a) but the double layers are overlapped due to the large double layer thickness. It is seen that the patch causes a sharp rise in potential with a corresponding sharp change in the local concentration of the ionic species. 3.4 Numerical Methods In general, the Debye-Hückel approximation places a stringent limit in the magnitude of the potential and in most cases numerical solutions must be obtained. In Chapter 2, the general governing equations have been presented. There are 4+n equations in 4+n unknowns to solve for the two-dimensional velocity field, the pressure, the potential and the mole fractions of n species. These equations are nonlinear and coupled. A uniform co-located mesh in Cartesian coordinates as shown in Figure 3.3 has been used to perform the numerical calculation. Note that the mesh shown here is extremely coarse and much finer meshes are used in the actual calculation for example a grid size of

80 Figure 3.3: A co-located mesh used for the numerical calculation. A finite volume method is used to approximate the partial differential equations due to its advantage in solving conservation equations. The distinctive characteristic of the finite volume approach is that a balance of some physical quantity is made on the control volume and the conservation of the quantity is automatically satisfied. In the finite volume method the conservation statement (usually invoked in integral form) is applied to the entire partial difference equation and the divergence theorem is often utilized to obtain the appropriate form. The divergence theorem is given by V ρ udv = S ρ u nds (3.24) where V is the volume and S is the area of the control volume. The density ρ could be any of other scalars and u is the velocity vector here. The normal vector of the control volume surface is n. Using the divergence theorem, the integral of the equation over the control volume is related to the summation of some quantities at the surfaces. 56

81 The incompressible Navier-Stokes equations exhibit a mixed elliptic-parabolic behavior and it is found that the standard relaxation technique is not appropriate to solve the equations (Anderson, 1995). The pressure correction technique has been successfully used to solve a number of incompressible flow problem and the technique is developed by Patankar & Spalding (1972) in It is embodied in an algorithm called Semi-Implicit Method for Pressure-Linked Equations (SIMPLE). The SIMPLE algorithm was originally employed on staggered grids and Rhie & Chow (1983) have developed the SIMPLE algorithm on a co-located mesh based on the pressure weighted interpolation method (PWIM) as we have discussed in the previous section. The philosophy of the pressure correction method is as follows: 1. Start the iterative process by guessing the pressure field which is denoted by p k. The velocity field are denoted by u k and v k. 2. Use the values of p k to solve for velocities from the momentum equations. The updated velocities are denoted as û and ˆv. 3. Construct a pressure correction p using the continuity equation which will bring the velocity field more into agreement with the continuity equation when added to p k. The pressure correction equation has been discussed in the previous section. The corrected pressure is p k+1 = p k + p and the corresponding velocity corrections u, v can be obtained from p such that u k+1 = u k + u v k+1 = v k + v 57

82 (x,y) grid grid φ =0.0440, u = φ =0.0445, u = (-5,0.5) g = , f = g = , f = v =0, p = v =0, p = φ =0.1160, u =0 φ =0.1171, u =0 (0,0.25) g = , f = g = , f = v =0, p = v =0, p = φ = , u =0 φ = , u =0 (-0.5,0.1) g = , f = g = , f = v =0.0189, p = v =0.0191, p = Table 3.1: The numerical accuracy check for h =20nm channel with over-potential φ e = 0.2. The results of grid are compared with the results of grid and two-digit accuracy has been attained. 4. Designate the new value p k+1 as the new value of pressure field. Return to step 2 and repeat the process until a velocity field is found that does satisfy the continuity equation. This is the procedure of solving for pressure and velocity coupling. The discretized equations and the link coefficients are discussed in detail in Appendix A. The process of the whole numerical calculation is shown in Figure 3.4. The detailed numerical calculation process is also discussed in the appendix. It is known that the more grids in the mesh, the accurate the calculation is. However, a large grid will bring high compute cost. We have used a grid to perform the numerical calculation and obtain two digit accuracy using that grid compared to a grid as shown in Table 3.1. Three points are chosen to check the numerical accuracy: ( 5, 0.5) the upstream point, ( 0.5, 0.1) the point near the discontinuity of surface potential and (0, 0.25) the point above the patch. The use of first order differencing will cause dissipation of the results and so the numerical accuracy is needed. Higher order upwind 58

83 Figure 3.4: The flowchart of the numerical calculation using SIMPLE algorithm. 59

84 differencing may be more accurate but it is difficult to apply and will cause the numerical calculations become unstable. 3.5 Numerical Results (a) potential contours (b) cation mole fraction contours Figure 3.5: Numerical results for potential, mole fractions calculated from the Poisson- Boltzmann equations. The imposed electric field is 10 6 V/m. The height of the channel is 20nm and the wall surface potential is 12mV. The electrolyte is 0.1M NaCl solution far upstream and the over-potential φ p =2.0. The corresponding dimensionless parameters are ɛ =0.05, Λ=0.77, A =1and Pe =0.36. Figure 3.5 shows the potential and the cation mole fraction contours calculated from the Poisson-Boltzmann distribution. The height of the channel is 20 nm and the electrolyte is 0.1 MNaClsolution far upstream where x = L x = 10. The zeta potential is 12 mv for the channel and 40 mv for the patch. The Debye length is 0.8 nm and it is shown in the potential contours that there are thin Debye layers close to the walls. Since the potential of the patch is positive, there is a surplus of anions near the patch as shown in the anion 60

85 (a) potential contours (b) mole fraction of cations contours (c) electric field lines (e) streamlines Figure 3.6: Numerical results for potential, mole fractions, electric field lines and streamlines. The imposed electric field is 10 6 V/m. The height of the channel is 20 nm and the wall surface potential is 12mV. The electrolyte is 0.1M NaCl solution far upstream and the over-potential φ p =2.0. The corresponding dimensionless parameters are ɛ =0.05, Λ=0.77, A =1and Pe =

86 mole fraction contours. The mole fraction of the cations is lower above the patch and the distributions of both cations and anions are symmetric about the line of x =0. On Figure 3.6 are the results for the same parameters calculated based on the Poisson- Nernst-Planck equations. A patch having dimensionless over-potential φ p =2.0is located at the lower wall of the channel and the length of the patch is 20 nm (A =1). The imposed electric field is 10 6 V/mand Pe =0.36. The flow is reversed above the patch and a vortical motion is accompanied by a favorable pressure gradient above the patch. The electric field lines are shown in (c) and there are two components of this electric field including the applied electric field and the induced field by the patch. The direction of the applied field is in x direction, from left to right. The direction of the induced field is from the patch to the walls. According to the equation for the ionic species, neglecting the convective terms ( X i + X i E)=0 (3.25) and so the distribution of ionic species is affected only by the electric field, both from the perturbed electric field due to the EDLs and the applied field. Above the patch, the asymmetry of these electric field lines causes the asymmetry of the concentration distribution of charged ions as shown in (b). The ratio of the applied field to the local field strength is given by the dimensionless parameter Λ, which is usually small for the field strength used in most experiments Λ 1 and Λ O(1) for the high applied field strength (10 7 V/m). There are four dimensionless parameters that affect the results: β ɛ 2, Λ, A and Pein the equations. Dimensional analysis can be used to predict the results for micro-channels based on the results calculated for nanochannels. For example, the results for 20nm channel with 0.1M NaClsolution under 10 6 V/mimposed electric field will be equivalent to the results for 200 nm channel with MNaClsolution under 10 5 V/m imposed electric field. 62

87 Thus the results for a small channel with a dense solution are equivalent with the results of large channel with a dilute solution. (a) potential contours (b) cation mole fraction contours (c) mole fraction of anions contours (d) streamlines Figure 3.7: Numerical results for potential, mole fractions and electric field lines calculated from the Poisson-Nernst-Plank equations. The imposed electric field is 10 7 V/m. The height of the channel is 20nm and the dimensional surface potential of the walls is 12mV. The electrolyte is 0.1 MNaClsolution far upstream and the dimensional patch potential is 40 mv (φ p =2.0). The corresponding dimensionless parameters are ɛ =0.05, Λ=7.7, A =1and Pe =3.6. It is found that the difference of the species distribution from the Boltzmann distribution is more noticeable for higher electric fields. As shown in Figure 3.7, the species 63

88 distribution above the patch is considerably different compared to that shown in Figure 3.5. The flow above the patch is less reversed and the recirculation region is smaller than that in Figure 3.6. Note that in this case only the applied electric field is increased from 10 6 V/m to 10 7 V/m. As a result, both the Peclet number and the non-dimensional applied field Λ increase (Pe = 3.6 and Λ = 7.7). Other parameters are same as that of Figure 3.6. As shown in equation (3.5) the deviation of the species distribution (from the Boltzmann distribution is proportional to the convective term (Pe ( ɛ 1 u X i + v ) X i x y ) and the term related to the axial non-uniformity ɛ 1 z i Λ X i ). Increasing the applied electric field increases these x two terms simultaneously. The effect of the convective term is investigated by comparing the results calculated from Pe =3.6 and Pe =0. As shown in figure 3.8, the cation distribution and the velocity field is slightly different for Pe =0and Pe =3.6, which indicates that the convection of the ions is not the main cause for the deviation of the species distribution from the Boltzmann distribution. It is also shown that even for Pe =0the cation distribution is still quite different from the Boltzmann distribution. Therefore, this deviation comes mostly from the axial non-uniformity in the surface potential (ɛ 1 z i Λ X i x term in equation (3.5)). Higher over-potential creates higher reversed flow and thus a larger vortical region in the bulk. However, there is a limit of the size of the vortex because of the balance of the reversed body force and the favorable pressure gradient. A larger over-potential will attract more anions to the region near the patch and thus generate more reversed flow. Simultaneously the pressure gradient will increase and drive flow fun faster above the vortex to satisfy the mass continuity. As shown in Figure 3.9, the vortex center (where u =0,v =0) moves away from the patch as the over-potential increases, which indicates a larger recirculating region in the bulk. But this effect tends to saturate for higher over-potential. It is note that 64

89 (a) cation mole fraction contours (b) dimensionless streamwise velocity contours Figure 3.8: Cation mole fraction contours and streamwise velocity contours calculated from the Poisson-Nernst-Plank equations for Pe = 3.6 and Pe = 0. The filled colors show the results for Pe =3.6 and the lines represent the results for Pe =0. The imposed electric field is 10 7 V/m. The height of the channel is 20 nm and the dimensional surface potential of the walls is 12 mv. The electrolyte is 0.1 MNaClsolution far upstream and the dimensional patch potential is 40 mv (φ p =2.0). The corresponding dimensionless parameters are ɛ =0.05, Λ=7.7, A =1. further increasing the over-potential may cause the calculation to diverge due to the abrupt change of potential and ionic species concentration close to the patch. An extra-fine mesh can be used to resolve this problem but requires long-term calculation. The local streamlines in the vicinity of the step change in the surface potential are similar to the results given by Yariv (2004). In Yariv s calculation, the streamlines are calculated within the length scale of the electrical double layers (λ). Since we are interested in the bulk flow structure in the channel (length scale h, h λ), the mesh used in the calculation is not fine enough to resolve Yariv s problem. However, our calculations show that the local streamlines are qualitatively similar. If the patch over-potential is negative, there is a vortex close to the upper wall of the channel as shown in Figure Surplus cations are attracted to the patch, increasing 65

90 normal distance of the vortex center from the patch over potential ζ p Figure 3.9: The dimensionless distance of the vortex center away from the patch as a function of the dimensionless over-potential of the patch. The center of the vortex is the point where u =0,v =0in the bulk flow. The imposed electric field is 10 7 V/m. The height of the channel is 20nm and the dimensional surface potential of the walls is 12mV. The electrolyte is 0.1 MNaClsolution far upstream and the corresponding dimensionless parameters are ɛ =0.05, Λ=7.7, A =1. the body force acting on the fluid flow near the patch. The flow is accelerated above the patch and a vortex is found away from the patch. The flow in the vortex is recirculating counter-clockwise. Interesting flow structure can be obtained by manipulating the patch over-potential pattern. Here a special case where the zeta potential is in the form of φ p = φ a cosnπx is discussed here. On Figure 3.11 shows the streamlines for the bulk flow with the variable patch potential with n =2and φ a =0.2, 1.0, 2.0. Forφ a =0.2, two small vortex are found above the patch. Both of the vortices are recirculating clockwise. For φ a =1.0, a positive vortex appear between the two negative vortices due to the negative zeta potential in that region. Previous results have shown that negative zeta potential can induce positive vortex 66

91 Figure 3.10: Streamlines for the EOF in a nanochannel with a patch having negative overpotential. The height of the channel is 20nm and the aspect ratio of the patch A =1. The over-potential φ e = 0.4 and the imposed electric field is 10 7 V/m. The solution far upstream is 0.1M NaClelectrolyte buffer. away from the patch. For a higher φ a =2.0, the vortices become larger as shown in Figure 3.11 (c) due to the larger potential variation on the patch. 3.6 Ionic Transport in a Nanopore Membrane with Charged Patches In this section, a simple reservoir and nanopore membrane system is studied using COMSOL to investigate the interaction of charged species and a nanopore membrane with a charged patch located at the mouth of the pores. It is a part of a project to develop an artificial kidney using nanopore membranes. The geometry of the reservoirs and the nanopore membrane is shown in Figure 3.12 (a). The height of the nanopores is 8 nm and the length of the pores is 1 µm. The size of the feed and the permeate reservoirs is 10 µm by 10 µm. There are 8 nanopores calculated in this model. The ionic strength of the electrolyte solution is 0.14 M and the zeta potential of the charged patch is 46 mv. The concentration of the charged species (albumin) is 0.6 mm. The valence of albumin is 17 and the diffusion coefficient is m2/s. Note that 67

92 (a) φ p =0.2cos2πx (b) φ p =1.0cos2πx (c) φ p =2.0cos2πx Figure 3.11: Streamlines for EOF in nanochannels with a patch having over-potential in the form of φ p = φ a cosnπx. The zeta potential of the wall is 1mV and the imposed electric field is 10 6 V/m. The solution far upstream is 0.1M NaCl. 68

93 (a) geometry (b) mesh Figure 3.12: The geometry of the reservoir and nanopore system (a) and the mesh used in the simulations (b). The height of the nanopores is 8 nm and the length of the pores is 1 µm. The size of the feed and the permeate reservoirs is 10 µm by 10 µm. The length of the patch is 500 nm. 69

94 albumin is treated as a ionic species and so only the electrostatic interactions between the albumin and the membrane is investigated. In the feed reservoir, there is forced fluid flow moving perpendicular to the nanopores and the pressure at the feed inlet is assumed to be P 0 =2psi. Since the pressure drop on the feed side is negligible, the pressure at the outlet of the feed side is assumed to be 0.99 P 0. On the permeate side, the upper and lower boundary are both assumed to be open boundary with pressure equal to zero. Fluid flow will be forced into the nanopores due to the pressure drop across the membrane. The calculation is based on the Poisson-Nernst-Planck equations and the simulations are performed using COMSOL Multiphysics. COMSOL is a finite element analysis and solver software for solving scientific and engineering problems based on partial differential equations (PDEs). The advantage of COMSOL Multiphysics is its ability to solve physics-coupling problems. It has several application-specific modules including Chemical Engineering Module, MEMS Module, AC/DC Module and many more. It can import CAD data using CAD Import Module and also provides an interface to Matlab. Modeling and simulation using COMSOL is very straight forward and the procedure is Choose the modules according to your problem; Geometry modeling: Setting up geometry in the user interface; Modeling Physics and Equations: Define physical properties and specify boundary conditions; Creating Meshes: For 2D geometry, choose from triangular or quadrilateral mesh elements; For 3D geometry, choose from tetrahedral, hexahedral, or prism mesh elements; 70

95 (a) electric potential (b) streamlines and concentration of albumin Figure 3.13: Results for the electric potential, streamlines and concentration of albumin: (a) electric potential (b) streamlines and concentration contours of charged species. The ionic strength of the electrolyte solution is 0.14 M and the zeta potential of the charged patch is 46 mv. The concentration of the charged species (albumin) is 0.6 mm. The valence of albumin is 17 and the diffusion coefficient is m2/s. The height of the nanopores is 8nm and the length of the pores is 1µm. The size of the feed and the permeate reservoirs is 10 µm by 10 µm. The length of the patch is 500 nm. 71

96 Solving the problem: Choose solver according to your problem, steady state or transient, linear or nonlinear. Post processing. The goal of the calculations is to determine the sieving coefficient s of the membrane. The sieving coefficient is a measure of selectivity of the membrane toward a given solute; a low sieving coefficient implies a low degree of solute passage through the membrane. It can be defined as the ratio of the solute concentration in the permeate (c permeate ) and the feed (c feed ): s = c permeate c feed The mesh used for numerical calculations is shown in Figure 3.12 (b) and it is based on triangular elements. Inside the nanochannels, extra fine mesh grids are used to resolve the electric double layer structure near the channel walls. Outside in the feed and permeate reservoirs, coarse meshes are used. The results for the flow field and the electric potential are shown in Figure As shown in Figure 3.13 (a), at the entrance of the nanochannels, there is a small developing region where the surface potential changes from 0 to 46 mv. The potential is fully developed down the channel and the electric double layer structure is clearly shown in the figure. Flow is driven into the nanochannels by the pressure difference and the total flowrate across the membrane is calculated to be 0.06% of the flowrate within the feed side. The concentration contours of the charged species are also shown in Figure 3.13 (b). The corresponding concentration at the centerline of the nanopore is shown in Figure 3.14 and it decreases dramatically at the entrance of the nanopore and linearly on the charged patch. For the nanopore membrane with charged patch (shown in Figure

97 (a) with charged patch 7 x 10 7 Concentration of Albumin [mol/m 3 ] Concentration of Albumin [mol/m 3 ] feed nanopore permeate x x 10 6 (b) without charged patch Figure 3.14: Results for albumin concentration at the centerline of the channel (a) with charged patch (b) without charged patch. The ionic strength of the electrolyte solution is 0.14 M and the zeta potential of the charged patch is 46 mv. The concentration of the charged species (albumin) is 0.6 mm. The valence of albumin is 17 and the diffusion coefficient is m2/s. The height of the nanopores is 8 nm and the length of the pores is 1 µm. The size of the feed and the permeate reservoirs is 10 µm by 10 µm. The length of the patch is 500 nm. 73

98 (A)), the concentration in the permeate reservoir is mm and the resulting sieving coefficient is The rescaled concentration (c/c feed ) at the centerline of the channel for the nanopore membrane without the charged patches is shown in Figure 3.14 (b). Since the partitioning of the albumin molecules due to the size effect is not considered here, the concentration is constant all though the channel. Charged species is carried by the convection of the fluid flow and at steady state, the concentration of the species in the permeated and the feed will be equal. It is shown that the electrostatic interactions between the species and the membrane can help nanopore membrane sieve with same charge. Moreover, increasing the charge of the patch can further decrease the concentration of the charged species in the permeate reservoir significantly. For a patch having 100mV zeta potential, the concentration in the permeate decrease to 3 M and the resulting sieving coefficient is Note that this sieving coefficient will be smaller when the size of the albumin is considered. 3.7 Summary In this chapter, the flow and mass transport has been investigated for a micro/nanochannel having a discontinuity in wall zeta potential and thus a discontinuity in surface charge density, which could come from surface defects or designed surface modification. If the overpotential is positive a surplus of negative ions are attracted by the positive charged patch while a surplus of positive ions are attracted to the negatively charged wall. Due to the imposed electric field, the flow near the patch will have a direction opposite to the bulk flow which will generate a recirculation region. We have considered the case of negatively charged walls and have calculated the perturbation to the wall potential (i.e. the ζ potential). The potential, mole fractions and the flow field have been calculated based on the Poisson-Nernst-Plank equations. The distributions of the species and potential are found to 74

99 be different from the Boltzmann distribution and this deviation is more noticeable for higher applied electric field. It is shown that this deviation comes from the axial non-uniformity in the surface potential and the convective effects are usually small. Note that the Boltzmann distribution for species is independent of the applied electric field and for low applied electric fields (Λ << 1), the species distribution calculated from PNP equations is almost the same as the Boltzmann distribution. For higher applied electric field, the flow is less reversed above the patch due to this effect. Moreover, if the electric field is high enough, the vortex will disappear and this is also observed in a recent molecular dynamics simulation (Hassanali & Singer, 2006). Higher over-potential creates larger vortex in the bulk but this effect tends to saturate because of the balance of the reversed body force and the favorable pressure gradient. Ionic transport through a nanopore membrane is also investigated for the purpose of developing a novel renal replacement therapy. Results for pressure-driven flow past a nanopore membrane have shown that the charge of the nanopore has a significant effect on the ionic transport through the membrane. The combination of steric effect and electrostatic interactions between the particle and the nanopore membrane has the potential to significantly reduce the sieving coefficient of the membrane, which is the ultimate goal of this project. Extension of this work will provide foundational knowledge to the optimization of nanofiltration and reverse osmosis technologies for the purpose of water purification. 75

100 CHAPTER 4 Electroosmotic Flow and Particle Transport in Micro/Nano Nozzles/Diffusers 4.1 Introduction In this chapter, a mathematical model is developed to simulate EOF in these micro/nano nozzle and diffusers; particle transport in nozzle/diffusers is investigated as well as the various regimes of particle and bulk motion based on dimensional analysis. As shown on Figure 4.1 (Wang & Hu, 2007), a Polymethyl Methacrylate (PMMA) nozzle is fabricated by photolithography, followed by replica molding and wet etching respectively. Negatively charged and fluorescence-labeled polystyrene (PS) beads of size from 3 nm to 40 nm were used to examine the transport of particles in micro-nozzles. The length of the nozzle is 650 µm and the left end and right end heights are 20 µm and 130 µm, respectively; the nozzle is 40 µm deep and the walls are negatively charged. The particles are immersed in a 0.1 M NaCl solution and an 80 V/cm DC electric field was applied across the tapered channel with the positively charged electrode placed in the particle receiver region (left). Since the walls are negatively charged, cations will always be more populous than anions and will move to the negative electrode, dragging the bulk fluid flow to move from left to right. Thus in terms of the bulk velocity the device is a diffuser. The negatively charged polystyrene beads move from right to left in a direction opposite to the bulk flow due to 76

101 Figure 4.1: The geometry of the micro-diffuser used in the experiments. The particle donor reservoir is on the right and the particle receiver reservoir is on the left of the diffuser. The walls are negatively charged and the positive electrode is placed in the particle receiver (left). The flow is from left to right and thus it is a diffuser. The polystyrene beads are shown and their motion is from right to left in a direction opposite to the bulk flow. electromigration, so for these particles the device is a nozzle. The purpose of this chapter is to develop an accurate model to simulate the electroosmotic flow (EOF) and particle transport in these micro/nano nozzles/diffusers. To be consistent, the nozzle and diffuser mentioned in this chapter is in terms of the bulk flow. A similar nozzle-reservoir system was set up later (Shin & Singer, 2009) to investigate fluid flow and ion transport at nano-scale using Molecular Dynamics (MD) simulations. The nozzle-reservoir system for the molecular dynamics is shown in Figure 4.2. The length of the nanonozzle is 150 angstroms and the height of the large and small ends are 146 and 46 angstroms respectively. The wall charge is 0.43e/nm 2 (e = C is the elementary charge) which is equivalent to σ = 0.07C/m 2 and the electrolyte solution is 0.1M. The applied electric field is 0.5V/nm. At this length scale, the ionic transport and fluid flow might be described as the upper limit of what can be treated with microscopic atomistic theory, and the lower size limit where continuum theory is expected to be valid. It 77

102 (a) The geometry of the nozzle-reservoir system (b) The dimensions of the nozzle-reservoir system Figure 4.2: The geometry of the reservoir-nozzle system (a) the system used in MD simulations (b) the cooresponding dimensions. Here only three stripe of water molecules are shown (in grey and red) and Cl (green) and Na + (blue) ions dispersed in water are shown. 78

103 y * h* (x *) 2h i x * flow direction 2h o L (a) a nozzle (b) a diffuser Figure 4.3: The geometry of the nozzle/diffuser used for modeling. The walls are negatively charged and for the polarity shown here, the flow is from left to right in both the nozzle and diffuser. is important to note that the MD simulations will yield valuable molecular-scale information which can be input to the continuum approach used to design the device at the system level. Previous comparisons of EOF in a slit pore have shown that the continuum theory is capable of quantitative predictions when the appropriate boundary conditions, to which continuum results are quite sensitive, are imported from molecular calculations (Zhu et al., 2005). The applied electric field is very high in this problem and the Boltzmann distribution may not hold in this case (see Chapter 3). The Poisson-Nernst-Planck equations are used to investigate the fluid flow and ion distributions in the system and numerical simulations are conducted using COMSOL Multiphysics. The results based on the continuum theory are compared with the molecular dynamic simulation results in the following section. 79

104 4.2 Governing Equations and Boundary Conditions Consider a slowly varying channel whose height is nano-constrained as shown in Figure 4.3 (a) and (b), the primary direction of fluid flow is in the x-direction. For simplicity, the spanwise width of the channel is assumed to be much larger than the height of the channel so the problem can be considered two-dimensional. The length of the channel is L and for the purposes of this paper the channel height is assumed to be linear with h (x) =h i + h o h i L x where h i,h o are the half heights of inlet and outlet of the channel and in dimensionless form where h = h h i h(x) =1+ h o h i h i and x = x. The walls of the channel are defined by y = ±h(x). L Electrodes are placed at the inlet and the outlet of the channel and so there is an imposed electric field in the x-direction. The imposed electric field E satisfies Maxwell s equations in the form E =0so that E I,x (x)a (x) =E I,x (0)A (0), where A is the dimensional cross-section area of the channel. If the imposed electric field at the inlet EI,x (0) is taken as the scale of the imposed electric field, the dimensionless imposed electric field is thus E I,x = 1 and the corresponding y component is given by h(x) E I,y = h h x ɛ 1h h 2 dy = ɛ 1yh h 2 (4.1) Usually in experiments potential drop is given and the value of this imposed electric field scale E 0 = E I,x (x =0)can be found from the axial integration of E I,x (x) L φ i = φ i (L) φ i (0) = EI,x (x)dx = E L 0 A (0) A (x) dx

105 The dimensionless equations for potential, species i and flow field are the same as equations (2.26), (2.28), (2.31), (2.35), (2.36) shown in Chapter 3. Based on the lubrication approximation (ɛ 2 1Re 1), the equations can be simplified as 2 φ y = β z 2 ɛ 2 i X i 2 u y = E β 2 I,x z ɛ 2 i X i + p x Note that since the pressure is assumed to be not dependent on y, the pressure is scaled on i i µu 0 L and so the ɛ 1 in equation (2.35) is dropped. In most cases discussed in this chapter, the Boltzmann distribution is used to describe the ionic distribution in this chapter and so the governing equations and boundary conditions are not listed here. Two kinds of boundary conditions are used for the potential equation. One is that the zeta potential ζ is known where ζ = ζ φ 0 φ = ζ,y = ±h(x) is the dimensionless wall ζ-potential. The other is based on the surface charge density σ as φ y = σ y,y = h; φ y = σ y,y = h where σ y is the dimensionless surface charge density in the y direction as shown in Figure 4.4. Note that σ is scaled on φ 0 ɛ eh as σ = σ h i i ɛ eφ 0. The boundary condition for the species equation is that the walls are impermeable to the species, thus give rise to the no flux boundary conditions where n i = ( X i x n i ˆn w =0 + z ix i E x + Pe α i X i u ) i + ( X i + z y ix i E y + Pe α i X i v ) j is the flux of the species i and ˆn w is the unit vector normal to the wall. The boundary condition for the 81

106 velocity field is the no slip boundary condition u =0,v =0,y = ±h(x) and the boundary condition for pressure is x =0:p = p i ; x =1,p= p o where p i and p o are the pressure at the inlet and outlet of the nozzle/diffuser and usually they are assumed to be zero. 4.3 Lubrication Solutions for EOF in the Debye-Hückel Limit As discussed in the previous section, the potential, mole fraction and velocity equations are highly nonlinear and coupled, and so numerical calculations are necessary to find the solutions. However, analytical solutions can be derived based on the Debye-Hückel approximation and lubrication theory. In this section, the analytical solutions are discussed for two different kinds of potential boundary conditions: known zeta potential and known surface charge density Known Surface Potential Consider the case where the surface potential of the nozzle/diffuser walls are known. For small potential (φ 26mV ), the potential equation becomes (Conlisk, 2005) and the solution for the potential is 2 φ y 2 = 1 ɛ 2 φ (4.2) φ = ζ cosh y ɛ cosh h ɛ (4.3) 82

107 The velocity can be decomposed into two components: the flow due to the electrical body force u eof and the flow due to the pressure gradient u p. The governing equation for the electroosmotic component of the velocity becomes 2 u eof = β ( )( 1 z y 2 ɛ 2 i X i i h ɛ ) 1 φ Λ x (4.4) here 1 is the imposed electric field and ɛ 1 φ h Λ is the electric field due to the EDLs. Since the x potential gradient in x direction φ is small, this term is neglected. This equation becomes x 2 u eof y 2 = 1 2 φ (4.5) h y 2 and the solution for u eof is given by u eof = φ ζ h = ζ h ( cosh y ɛ cosh h ɛ 1 ) (4.6) Based on the lubrication approximation, the axial pressure gradient ( p ) is nearly indepen- x dent of y and the pressure-driven component is given by The total streamwise velocity is then u = u eof + u p = ζ h and the axial velocity gradient u p = 1 dp ( y 2 h 2) (4.7) 2 dx ( cosh y ɛ cosh h ɛ u x = ɛ 1 d 2 p ( y 2 h 2) hh dp ( 2 dx 2 dx ζh cosh y ɛ h 2 cosh h ɛ ) dp ( y 2 h 2) 2 dx The v velocity is obtained from the continuity equation as y v = h(x) 83 ) 1 + ζh sinh ( ) ( ) h ɛ cosh y ɛ ɛh cosh ( ) 2 h ɛ u ɛ 1 dy (4.8) x

108 Substituting the solution of the streamwise velocity (u = u p +u eof ), the transverse velocity v is given by v = ɛ 1 1 d 2 ( ) p y 3 2 dx 2 3 h2 y + dp + ɛζh sinh ( ) y ɛ h 2 cosh ( ) ζyh h h 2 ɛ so that v = ɛ 1 2 d 2 ( p y 3 dx 2 3 h2 y 2 ) dp 3 h3 + ɛ 1 ( dx hh y + ζh sinh h ɛ y h cosh 2 ( h ɛ h ) ( ) sinh y ɛ ) y h dx hh (y + h)+ ɛ ( ) ( ) 1ζh sinh h ɛ sinh y ɛ h cosh ( ) + 2 h ɛζh ɛ 1 sinh ( ) ( ) y ɛ h 2 cosh ( ) ɛ 1ζyh + ζh ɛ 1 tanh 2 h ɛ + ζɛ 1ɛh tanh ( ) h ɛ ζɛ 1h (4.9) h h 2 h h 2 h ɛ On the upper boundary y = h(x), the transverse velocity v =0due to the no-slip boundary ɛ condition and so 1 d2 p 3 h3 dx + 2 h h 2 dp dx = ζh h tanh2 ( ) h ɛ ζh ɛ h tanh 2 ( ) h + ζh ɛ h (4.10) The pressure can be determined from this equation. Equation (4.10) can also be written as ( 1 d h 3 dp ) ( ) ( ) = ζh h 3 dx dx h tanh2 ζh ɛ h ɛ h tanh + ζh (4.11) 2 ɛ h Integrating twice shows p = ζɛ h h 2 h 6ζɛ 3 ( h i h 2 h 4 2h e ɛ +1 ) dh + C 1 x + C 2 (4.12) where C 1 and C 2 are constants of integration and they can be determined by applying the boundary condition at the inlet and the outlet. At this point further integration of this equation is difficult and the solution for pressure has to be found numerically. For thin EDLs in the micro-nozzle/diffuser (ɛ 2 1), the electroosmotic component can be simplified as (Conlisk, 2005) u eof = ζ h ( e ( h y ɛ ) h+y ( + e ɛ ) 1 ) (4.13) 84

109 Superimposing the velocities from equations (4.7) and (4.13), the total velocity is given by and the axial gradient of u is u = 1 dp ( y 2 h 2) + ζ 2 dx h u x = ɛ 1 d 2 p ( y 2 h 2) hh dp 2 dx 2 dx ζh ( h y e ( h 2 ɛ ) + e The v-velocity can be obtained as + [ = [ ɛ 1 2 ɛ 1 dp dx hh (y + h)+ ɛ 1ɛζh h 2 ( e ( and the transverse velocity is given by v = ɛ 1 2 ɛ 1 ζh (y + h)+ h 2 d 2 ( p y 3 dx 2 ( e ( h y ɛ ) h+y ( + e ɛ ) 1 ) (4.14) h+y ( ɛ y u v = ɛ 1 h(x) x dy d 2 ( p y 3 dx 2 3 h2 y 2 3 h3 ) 1 ) ζh hɛ )] y h y h ɛ ) h+y ( + e ɛ ) y ) ɛ 1ζh h 3 h2 y 2 3 h3 ) ( ɛζh ) ɛ 1 + ζh ɛ 1 (e h y h 2 ɛ h ( e ( y h ɛ ) h+y ( + e ) ) ɛ ( e ( y h ɛ ) + e + ɛ 1 dp dx hh (y + h) Applying the boundary condition at the upper boundary shows that and the pressure solution is h 3 3 p = p oh 2 o (h2 h 2 i )+p ih 2 i (h2 o h2 ) h 2 (h 2 o h2 i ) d 2 p dx 2 + h dp dx + ζh ɛ h 2 =0 ɛ h+y ( ɛ ) )] y h e h+y ɛ e 2h ɛ +1 ) (4.15) [ ζ h h ζ (h2 i + h ih o + h 2 o ) ] 3 h h 2 h i h o (h i + h o ) ζ h h i h o (h i + h o ) (4.16) It is seen that the pressure distribution is induced by the inlet and outlet pressure difference (first term in equation (4.16)) as well as the presence of the EDLs (second term in equation (4.16)). If the inlet and outlet pressure are zero, the pressure in the nozzle/diffuser p = O(ɛ). 85

110 4.3.2 Known Surface Charge Density If the surface charge density is known, the potential equation is the same as equation (4.2) but the boundary condition is given by n φ = σ where n =cos(α) i x ± sin(α) i y at the lower and upper wall) is the surface normal vector and it reduces to φ y = σ y φ y = σ y at y = h(x) at y = h(x) where σ y is the projection of the dimensionless surface charge density in y direction as shown in Figure 4.4. Note that the ordinary boundary condition for potential based on the surface charge density is defined on the surface normal direction ( n) φ n = σ and σ y = σ cos(α) where α is the angle of the nozzle wall and the x-axis and α is small so that ɛ 1. The solution for electric potential becomes x y φ(y) = σ yɛ cosh(y/ɛ) sinh(h/ɛ) (4.17) and the EOF component of the streamwise velocity is The total velocity is u eof = σ yɛ h u = u eof + u p = 1 dp 2 dx ( coth(h/ɛ) cosh(y/ɛ) ) sinh(h/ɛ) ( y 2 h 2) + σ yɛ h 86 ( coth(h/ɛ) cosh(y/ɛ) ) sinh(h/ɛ)

111 Figure 4.4: The surface charge density boundary condition for the potential equation is φ y = ±σ y, ±h(x), where σ y is the component of the dimensionless surface charge density in the y direction. and the axial gradient is u x = ɛ 1 d 2 p ( y 2 h 2) hh dp 2 dx 2 dx σ yɛh ( 1 coth 2 (h/ɛ) + cosh(y/ɛ)cosh(h/ɛ) ) h 2 ɛ sinh 2 (h/ɛ) + σɛh h [ 1 coth 2 (h/ɛ) ɛ + cosh(h/ɛ)cosh(y/ɛ) ] sinh 2 (h/ɛ) Similar to the previous section, the transverse velocity v can be obtained from the integration the continuity equation analytically and the solution is v = ɛ 1 2 ɛ 1 ɛh [ (y + h)coth h 2 d 2 ( p y 3 dx 2 3 h2 y 2 ) dp 3 h3 + ɛ 1 dx hh (y + h) ( ) h ɛ ] ɛ sinh(y/ɛ)+ɛsinh(h/ɛ) sinh(h/ɛ) σɛh [ 1 coth 2 (h/ɛ) (y + h)+ cosh(h/ɛ)(sinh(y/ɛ)+sinh(h/ɛ)) h ɛ sinh 2 (h/ɛ) ] (4.18) Applying the boundary condition for v at y = h, the pressure equation becomes 1 3 h3 d2 p dx 2 + h h 2 dp dx = 3σɛ2 h 2 +3σ ( 1 coth 2 (h/ɛ) ) (4.19) 87

112 Rearranging the left hand side of the equation, equation (4.19) becomes ( d h 3 dp ) ( ) σɛ =3h 2 dx dx h σ 2 sinh 2 (h/ɛ) and integrating once shows dp dx = 3σɛ cosh(h/ɛ) 3σɛ2 + h 4 h 3 sinh(h/ɛ) + C where C is a integration constant. At this point further integration of this equation is difficult and the solution for pressure has to be found numerically. 4.4 Results for EOF Based on the Debye-Hückel Approximation Results for EOF: Known Surface Potential The dimensionless imposed electric field is shown in Figure 4.5 for a nozzle (Figure 4.3 a) and a diffuser (experimental setup, as shown in Figure 4.3 b) with rectangular crosssection. Unlike the applied electric field in a channel, there is a electric field gradient inside the nozzle/diffuser due to the converging/diverging geometry. The potential drop over the length of the nozzle/diffuser corresponds to the value used in the experiments shown in Figure 4.1, which is 80V/cm. It is seen that the magnitude of the electric field is increasing in a nozzle and decreasing in a diffuser in x-direction. As noted in the previous sections, numerical calculation is necessary to determine the pressure field based on equation (4.10) or equation (4.19) depending on the potential boundary conditions. Second order finite difference methods are used to discretize the pressure equation as h 3 j 3 p j+1 2p j + p j 1 x 2 + h jh 2 j p j+1 p j 1 2 x = ζh j h j tanh 2 ( ) hj ɛ ζh jɛ tanh h 2 j ( ) hj + ζh j ɛ h j for equation (4.10) and h 3 j 3 p j+1 2p j + p j 1 x 2 + h jh 2 j p j+1 p j 1 2 x = 3σɛ2 h 2 j +3σ ( 1 coth 2 (h j /ɛ) ) 88

113 Figure 4.5: The dimensionless imposed electric field (E I,x ) for a nozzle (Figure 4.3 a) and a diffuser (Figure 4.3 b). The length of the nozzle is 65 nm; the height at the inlet is 13 nm and the height at outlet is 8 nm. The length of the diffuser is 650 µm; the height at the inlet is 20 µm and the height at the outlet is 130 µm. The cross-section of the nozzle is rectangular, with the width in the direction into the paper much larger than the height. The potential drop over the length of the nozzle/diffuser corresponds to 80V/cm. The solid line shows the electric field for a channel (80V/cm). 89

114 for equation (4.19) at point j. The tridiagonal matrix algorithm (TDMA), also known as the Thomas algorithm is used to solve the discretized equations. For thin double layer cases, the pressure equation can be solved analytically from equation (4.16). Parameter Nano-nozzle Micro-diffuser E 0 80 V/cm 80 V/cm CNa M 0.1 M CCl M 0.1 M ζ 5 mv 15mV 2h i 13 nm 130 µm 2h o 8 nm 20 µm L 65 nm 650 µm ɛ Re p i,p o 0 0 Table 4.1: Values of parameters used for simulations. Note that the concentration of species shown here are reservoir concentrations. Figure 4.6 shows the electroosmotic flow in a nano-nozzle. The height of the nanonozzle at the inlet is 13nm and the length of the nozzle is 65nm; the height at the outlet is 8nm and so the parameter ɛ 2 1 = The Debye length is 2.5 nm (ɛ =0.19) and the EDLs are overlapped as shown in the parabolic potential and velocity profiles in Figure 4.6. The electrolyte is NaCl solution and the concentration in the reservoir is 0.1 M. The imposed electric field is 8000 V/m. The ζ-potential of the walls is 5 mv which is small enough to apply the solution derived in the previous section based on the Debye-Huckel approximation. The parameters are listed in Table 4.1. Since the walls are negatively charged, the fluid flow moves from left to right (u >0) and in terms of the bulk flow, the device is a nano-constrained nozzle. As shown in Figure 4.6 the transverse velocity (v) isinthe order of ɛ 1 and is small compared to the streamwise velocity (u). The nozzle is assumed 90

115 to be connected to reservoirs and so the pressure is zero both at the inlet and the outlet. It is noted that pressure shown here is only induced by the presence of the EDLs and the pressure-driven velocity (u p O(0.1)) due to this induced pressure is small compared to the electroosmotic flow velocity (u eof O(1)). Results for the EOF in the experimental microdiffuser configuration (shown in Figure 4.1) are shown in Figure 4.7. The height of the diffuser is 20 µm at the inlet and 130 µm at the outlet and the length of the diffuser is 650 µm; and the parameter ɛ 2 = The electrolyte concentration is 0.1M in the reservoir and the ζ-potential of the PMMA walls is 15mV (Kirby & Hasselbrink, 2004). The Debye length is only 0.7nm (ɛ = ) and the EDLs are very thin compared to the height of the diffuser which can be seen from the plug-like profile of potential and streamwise velocity; the velocity is a maximum at x =0 and a minimum at x =1on Figure 4.3. In this case, the pressure solution is obtained from equation (4.16) analytically. The pressure is assumed to be zero both at the inlet and the outlet since the channel is connected to large reservoirs; the pressure shown in Figure 4.7 (d) is also induced only by the presence of EDLs. The magnitude of this induced pressure is small (O(10 5 )) due to the small ɛ value. If there is no applied electric field or the wall charge is negligible, the electroosmotic flow component is zero. In this case, the flow field becomes the classic pressure-driven flow in a nozzle. The streamwise velocity, transverse velocity and the pressure are shown in Figure 4.8 for the pressure-driven flow under a favorable and adverse pressure gradient in the micronozzle shown in Figure 4.3 (a). In Figure 4.8, (a), (b) and (c) are for the case where higher pressure is on the side of the large end while (d), (e) and (f) are for the opposite case. The height of the nozzle is 130 nm at the large end and 20 nm at the small 91

116 (a) potential (b) streamwise velocity (c) transverse velocity (d) induced pressure and pressure gradient Figure 4.6: Results for electroosmotic flow in the converging nanonozzle. The height of the nozzle is 13 nm at the inlet and 8 nm at the outlet and the length of the nozzle is 65 nm; ɛ =0.19 and the EDLs are overlapped. The imposed electric field is assumed to be 8000 V/mand the ζ-potential of the walls is 5 mv. The pressure is zero both at inlet and the outlet and so the pressure shown here is induced by the presence of the EDLs. 92

117 (a) potential (b) streamwise velocity (c) transverse velocity (d) induced pressure and pressure gradient Figure 4.7: Results for electroosmotic flow in the experimental micro-diffuser. The height of the diffuser is 20 µm at the inlet and 130 µm at the outlet and the length of the diffuser is 650 µm; ɛ = and the EDLs are thin compared to the diffuser. The imposed electric field is 8000V/m and the ζ-potential is 15mV. The pressure is zero both at the inlet and the outlet and so the pressure shown here is induced by the presence of the EDLs. 93

118 end. The length is 650 nm. The typical parabolic velocity profile can be seen from (a) and (d) and the velocity is highest at the small end. Combining the electroosmotic flow and pressure driven flow, different flow structures can be obtained inside the nozzle. For example as shown in Figure 4.9 (b), the fluid flow in the center of the nozzle moves in the opposite direction to the fluid flow near the wall. The electroosmotic flow component is dominant near the walls due to the EOF body force in the electrical double layers while at the center, the adverse pressure gradient drives the flow to move in the opposite direction. Two shear layers are developed where the two parallel streams of fluids moving in the opposite directions meet. As shown in Figure 4.9 (b) The streamlines in these layers start from the inlet of the nozzle and return to the inlet. 94

119 1 0.5 x=0 x=0.2 x=0.4 x=0.6 x=0.8 x= x=0 x=0.2 x=0.4 x=0.6 x=0.8 x=1.0 y 0 y U (a) streamwise velocity V x 10 3 (b) transverse velocity y x=0 x=0.2 x=0.4 x=0.6 x=0.8 x= p dpdx x (c) pressure and pressure gradient U (d) streamwise velocity x=0 x=0.2 x=0.4 x=0.6 x=0.8 x= p dpdx y V x 10 3 (e) transverse velocity x (f) pressure and pressure gradient Figure 4.8: Results for pressure-driven flow in the experimental micro-nozzle as shown in Figure 4.3 (a). The height of the small end is 20 µm and 130 µm at the large end and the length of the nozzle is 650 µm. For (a), (b), (c), the dimensionless pressure is 1 at the large end and zero at the small end. For (d), (e), (f), the dimensionless pressure is 0 at the large end and 1 at the small end. The electroosmotic flow component is zero. 95

120 1 0.5 x=0 x=0.2 x=0.4 x=0.6 x=0.8 x= y 0 Y u (a) streamwise velocity X (b) streamlines Figure 4.9: Streamwise velocity and the streamlines for the electroosmotic flow in a micronozzle with adverse pressure gradient. The height of the small end is 65 µm and 130 µm at the large end and the length of the nozzle is 650 µm. The applied voltage drop is 80V/cm and the zeta potential of the wall is 5mV. The dimensionless pressure is p i =0and p o =10. 96

121 4.4.2 Results for EOF: Known Surface Charge Density Results for the EOF based on the Debye- Hückel approximation and the lubrication approximation for the known surface charge density boundary condition are shown in Figures 4.10 and The nozzle has a large end of 14.6nm and a small end of 4.6nm in height. The applied electric field is 0.5V/nm and the electrolyte concentration is 0.1M in the upstream reservoir. The corresponding Debye length is 1nm and the surface charge density of the nozzle/diffuser is 0.01C/m 2. The parameters are listed in Table 4.2, and the results for the electric potential and the velocity field are similar to the results based on the known zeta potential boundary condition shown in the previous section. It is shown in Figure 4.10 that at the entrance of the nozzle, the electric double layer are not overlapped which can be seen from the potential and streamwise velocity profile. Along x axis, the electric double layers at the upper and lower walls come closer due to the tapering shape of the nozzle. The parabolic potential and streamwise velocity profiles at x =1indicate overlapped EDLs at the outlet of the nozzle. Parameter Nano-nozzle Nano-diffuser E V/nm 0.5 V/nm CNa M 0.1 M CCl M 0.1 M σ 0.01 C/m C/m 2 2h i nm 4.6 nm 2h o 4.6 nm 14.6 nm L 15 nm 15 nm ɛ Re p i,p o 0 0 Table 4.2: Parameters used for the numerical calculations for known surface charge. Note that the concentration of species shown here are reservoir concentrations. 97

122 y x=0 x=0.2 x=0.4 x=0.6 x=0.8 x=1.0 y x=0 x=0.2 x=0.4 x=0.6 x=0.8 x= φ (a) potential u (b) streamwise velocity y x=0 x=0.2 x=0.4 x=0.6 x=0.8 x= v (c) transverse velocity 5 p dpdx x (d) induced pressure and pressure gradient Figure 4.10: Results for electroosmotic flow in the nano-nozzle. The height of the nozzle is 14.6 nm at the inlet and 4.6 µm at the outlet and the length of the diffuser is 15 nm; ɛ = The imposed electric field is 0.5V/nm and the surface charge density is 0.01C/m 2. The pressure is zero both at the inlet and the outlet and so the pressure shown here is induced by the presence of the EDLs. The surface potential can be derived from equation (4.17) by taking y = ±h(x), ζ = σ y ɛ coth ( ) h ɛ and there is a slight change in surface potential along the wall as shown in Figure 4.12 (a). Note that the change in surface potential increases near the sharp end of the nozzle/diffuser. 98

123 y y x=0 x=0.2 x=0.4 x=0.6 x=0.8 x= φ (a) potential v (c) transverse velocity x=0 x=0.2 x=0.4 x=0.6 x=0.8 x=1.0 y x=0 x=0.2 x=0.4 x=0.6 x=0.8 x= u 0 (b) streamwise velocity 0.15 p dpdx x (d) induced pressure and pressure gradient Figure 4.11: Results for electroosmotic flow in the nano-nozzle. The height of the nozzle is 4.6 nm at the inlet and 14.6 µm at the outlet and the length of the diffuser is 15 nm; ɛ = The imposed electric field is 0.5V/nm and the surface charge density is 0.01C/m 2. The pressure is zero both at the inlet and the outlet and so the pressure shown here is induced by the presence of the EDLs. 99

124 φ w 0.52 φ w nozzle diffuser x (a) surface potential for a nozzle in 1M NaCl 1 diffuser nozzle x (b) surface potential for a nozzle in 0.001M NaCl Figure 4.12: The surface potential as a function of x. The large end of the nozzle/diffuser is 14.6 nm height; the small end is 4.6 nm height and the length is 15 nm. For (a), the surface charge density is 0.07C/m 2 and the electrolyte solution is 0.1 MNaCl. For (b), the surface charge density is 0.01C/m 2 and the electrolyte solution is M. The surface potential ζ coth(hɛ) and the curve of cotangent function shows that coth(x) has a larger change when x 0. Since h = h, this change in surface potential is related ɛ λ to the ratio of the height of the channel and the Debye length. For the region where EDL are overlapped, the surface potential changes more than the region where EDLs are thin compared to the height of the nozzle. As shown in Figure 4.12 (b), the change in surface potential is larger for a more dilute solution (0.001 M NaCl). 4.5 Analysis Based on the Thin Double Layer Limit (ɛ 1) The electrical double layers (EDLS) are very thin compared to the height of the nozzle/diffuser (λ D h) in micro-nozzles or micro-diffusers, making the numerical calculations hard to resolve the abrupt change of potential and mole fractions close to the wall. 100

125 An asymptotic analysis can be used to solve this problem. Compared to the solution given for thin double layer limit under the Debye- Hückel approximation as discussed in Section 4.3.1, the analysis is based on the Poisson-Boltzmann equations for a binary monovalent electrolyte. The solution for the full nonlinear problem is derived in the outer region in this section. For the region close to the upper wall (y = h(x)), a new variable is defined as η = h y ɛ so that as y = h, η =0; y =0, η. Substituting η in to equation (4.2), the potential equation becomes d 2 φ dη = β z 2 i X i (4.20) i The potential equation for a binary monovalent electrolyte is given by with boundary conditions d 2 φ dη 2 =sinhφ η =0, φ = ζ The equation can be written as y, φ η =0 d dη Integrating once, the equation becomes ( ) 2 dφ = d (2 cosh φ) (4.21) dη dη dφ dη = 2coshφ + C 1 where C 1 is a integration constant. In the bulk (η ), the concentration of cations and anions are the same and so η,x + i = X i 101

126 Based on the Boltzmann distribution, φ =0at η and so C 1 = 2. The equation for potential in the inner region becomes dφ dη = 2coshφ 2=2sinh φ 2 (4.22) Integrating equation (4.22) the inner solution for potential is given by φ i =2ln e ζ 2 tanh η 2 1 e ζ 2 tanh η 2 (4.23) where the superscript i indicates that it is the inner solution for φ. Similarly, near the lower wall h(x), η is defined as η = y + h ɛ and the inner solution is the same as equation (4.23). Outside the EDLs, the electrolyte solution is electrically neutral so that the volume charge density is zero (Fc z i X i =0) in the outer region, leading to the potential equation in the outer region: 2 φ =0 (4.24) y2 The boundary condition for this equation is given by the matching between the inner and outer solution at both walls And the outer solution for potential is lim y ±h φo = lim φ i η φ o =0 The inner solution for the streamwise velocity can be found based on the similarity between the potential equation and the velocity equation for EOF component. For the inner region near both walls, the inner solution for u eof is governed by equation d 2 u eof dη 2 = d2 φ dη 2 E I,x (4.25) 102

127 with boundary conditions The inner solution is given by η =0,u eof =0; η =, du dη =0 u i eof = E I,x (φ ζ) (4.26) In the outer region, the equation for the electroosmotic component is given by with the boundary condition 2 u eof y 2 =0 lim y ±h uo eof = lim η ui eof from the matching between the inner and outer solution u o eof y =0,y =0 due to the symmetry of the problem. The outer solution for the electroosmotic component is, using equation (4.26) u o eof = E I,x ζ For the transverse velocity in the inner region, the equation is d 2 v eof dη 2 = d2 φ dη 2 E I,y and similarly, the inner solution is given by v i eof = E I,y (φ ζ) In the outer region, the equation for v eof is given by 2 v eof y 2 =0 103

128 with the boundary condition lim y ±h vo eof = lim η vi eof from the matching between the inner and outer solution. The outer solution for the electroosmotic component is v o eof = E I,yζ It is noted that unlike E I,x, which is independent of y, E I,y is a function of y, given by equation (4.1). In the inner region, the pressure-driven velocity component is given by along with the boundary condition 2 u p =0 (4.27) η2 η =0,u p =0; η =, du dη =0 and the solution is u i p =0. Similarly, the inner solution for the pressure-driven transverse velocity component is v i p = o. Outside the electrical double layer, the governing equation for the pressure-driven component is given by and the solution is 2 u p y = ɛ dp 2 1 dx u o p = ɛ 1 2 dp dx (y2 h 2 ) (4.28) In the outer region, the total outer velocity is then given by u o = u o eof + uo p = ɛ 1 2 p x (y2 h 2 ) E I,x ζ 104

129 Note that the non-dimensional applied electric field E I,x = 1 and so the outer solution for h velocity is u o = ɛ 1 2 dp dx (y2 h 2 ) ζ h so that The v velocity can be directly obtained by integrating from the continuity equation y v o u y = ɛ 1 h x dy + C ɛ 1 1 = h 2 v o = ɛ 1 2 d 2 p dx 2 ( y 2 h 2) hh dp dx + ζh h 2 dy + C 1 (4.29) d 2 ( p y 3 dx 2 3 h2 y 2 ) dp 3 h3 + ɛ 1 dx hh (y + h)++ ζh h (y + h)+c 2 1 (4.30) where C 1 is the integration constant. The outer solution for the transverse velocity close to the walls (y = ±h) is given by v o = v o eof + vo p = E I,yζ. On the upper boundary, E I,y = E I,x h and on the lower boundary E I,y = E I,x h. Substituting the boundary condition for transverse velocity on the lower boundary in to equation (4.30), the integration constant is given by C 1 = ζh h Applying the boundary condition on the upper wall, the pressure equation is given by ζh h = ɛ 2h 3 d 2 p 3 dx +2ɛ dp 2 1 dx h2 h + ζh h (4.31) and so the pressure equation in the outer region is given by ɛ 1 3 ( ) h 3 p ɛ 1 =0 (4.32) x x With the pressure boundary conditions p i = p o =0, the pressure solution is p =0along x direction. Note that this pressure solution is solved in the outer region without considering the inner solutions, in the limit that the double layers are extremely thin compare to 105

130 the height of the nozzle so that the detailed distribution of potential and flow field in the electrical double layers can be neglected at least initially. In the previous calculation for EOF in nozzle/diffusers based on the Debye- Hückel approximation, a pressure gradient is induced by the EOF, which is usually p O ( ) ɛ = λ h. If we take the limit (ɛ 0), the induced pressure becomes zero and the solution (4.16) is consistent with the pressure outer solution discussed in this section. The inner solution derived for potential and velocity in this section is based on the Boltzmann distribution, not limited to low potential (φ 1) as the solution based on the Debye- Hückel approximation discussed in Section Since there is no induced pressure gradient by the EOF, the velocity equation can then be re-written as 2 u =0 y2 (4.33) 2 v =0 y2 (4.34) Note that the solution discussed here is not limited to nozzles/diffusers, it can be applied to any slowly varying channel with constant wall potential/charge. For these problems with thin EDLs, the electroosmotic flow mobility can be defined as µ e = ɛeζ, the electroosmotic µ velocity is then given by u = µ ee I,x. The flowrate is then given by A u da = µ e E I,xA (x) =µ e E 0 A 0 = constant at any x location. It is seen that the electroosmotic flow satisfies the continuity equation itself and that is the reason that pressure gradient in x direction is not induced. Knowing this can greatly simplify the calculations for EOF in slowly varying channels, which will be shown in the following example. For example, consider a slowly varying channel with h(x ) = h 0 + A 0 sin(w x ) as shown in Figure The dimensionless form for the wall is h(x) =1+δsin(wx) where 106

131 Figure 4.13: The geometry of the slowly varying channel with wall defined as h(x) = 1+δsin(wx). The walls are negatively charged and for the polarity shown here, the flow is from left to right. δ = A 0 h 0 and w = w L. The applied electric field is given by E (i, x)a (x) =E 0 A 0 and the dimensionless applied field is E(x) = 1 h similar to the nozzle cases discussed before. The outer solution for the streamwise velocity is given by u = ζ h = ζ 1+δsin(wx) and v can be easily found from the integration shown in equation (4.29) v = y h h h 2 dy + v( h) = y h The streamlines are shown in Figure ζwδcos(wx) ζwδcos(wx) dy = (1 + δsin(wx)) 2 1+δsin(wx) It is seen that the calculation process is greatly simplified since the induced pressure gradient is zero. Due to the similarity between the potential equation and the electroosmotic velocity component equation, the velocity field can be easily obtained. This analysis applies to the electroosmotic flow in any slowly varying channels with constant surface potential and thin double layers. 107

132 Figure 4.14: The streamlines for electroosmotic flow in the channel shown in Figure Numerical EOF Calculation Based on Poisson-Boltzmann Model In Section 4.4, the results are calculated based on the Debye- Hückel approximation, which is usually valid for low surface potential. Poisson-Boltzmann equation can give a more accurate predict to the potential and velocity field when the zeta potential is high. In this section, the model is extended based on the non-linear Poisson-Boltzmann equation and numerical differentiation and integration are used to solve the governing equations. The procedure of the numerical calculation is listed in the following. Step 1 Solving Poisson-Boltzmann equations numerically In the nonlinear calculations, the electrical potential due to the presence of electric double layers (EDLs) are calculated numerically based on the Poisson-Boltzmann equation in the lubrication limit. The equation for electric potential is 2 φ y 2 = β ɛ 2 zi X i (4.35) 108

133 where X i = X 0 i e z iφ with boundary condition y = ±h : φ x = ±σ y The equation is discretized using the second-order accurate central difference approximations for all the derivative terms. The finite difference equations are obtained in the form: φ j+1 2φ j + φ j+1 = dy 2 β ɛ 2 zi X 0 i e z iφ j (4.36) for point j. The difference equations for the mole fractions and potential are solved iteratively using a Gauss-Seidel scheme since they are coupled. The equation for potential (Equation (4.36)) is solved using Successive Over-relaxation (SOR) in 1D. The iterative procedure is said to converge if φ New φ Old <δ (4.37) φ New where δ is a small number and it is usually taken as 10 4 as the convergence criteria. Step 2 Solving u eof numerically For the streamwise velocity component u eof, the solution is calculated based on the equation 2 u eof y 2 = β hɛ 2 zi X i (4.38) and boundary conditions y = ±h : u eof =0. Central difference approximation is also used to discretize the equation and Thomas algorithm is used to solve the equation. Step 3 Calculating u eof x numerically To find the transverse velocity and pressure field, previous derivation based on the lubrication is still applicable but there is no analytical solution can be derived from the nonlinear equations and numerical differentiation and integration has to be used. Based on the continuity equation, the transverse velocity is given by y u v = ɛ 1 h x dy 109

134 (a) x, y coordinates (b) ξ,η coordinates Figure 4.15: Mapping from the original (x, y) coordinates to the (ξ,η) coordinates, which have a rectangular mesh, for easier numerical discretization. 110

135 and at the upper boundary, h u h v = ɛ 1 h x dy = h u eof + u p ɛ 1 dy =0 x Thus the EOF component u eof can be related to the pressure-driven u p component by h h u p x dy = h h u eof x dy Substituting the analytical solution of the pressure driven component shown in equation (4.7), the pressure equation is 2h p x 2 +2h2 h p x = h h u eof x dy (4.39) The right hand side of equation (4.39) must be determined first. Unlike the regular rectangular meshes we usually used, the mesh established here has a different shape as shown in Figure 4.15 (a). Ordinary numerical differentiation cannot be used for this case. To find u numerically, one has to map x, y to a rectangular mesh in ξ,η coordinates, which x transforms the nozzle geometry into a rectangular geometry for easier to discretization as shown in Figure 4.15 (b). To transform between x, y and η, ξ coordinates, ξ, η is defined as ξ = x and η = y/h. The term u u u can be written in terms of and as x ξ η Since x = ξ and η = y/h, ξ η =1and x and for point i, j, where ξ = dx and η = dy h from y h u x u x = u ξ ξ x + u η η x = yh x h 2. Therefore u x = u ξ ηh u h η u x = u i+1 u i 1 ηh u j+1 u j 1 2 ξ h i 2 η is a constant. The transverse velocity v can be calculated dy using numerical integration based on the trapezoidal rule. 111

136 x =0.25,y = x =0.75,y = φ = , u eof =1.0072,p = φ = , u eof =1.0185,p = φ = , u eof =1.0083,p = φ = , u eof =1.0242,p = x =0.6,r =0.08 x =0.8,r = φ = , u eof =1.3229,p = φ = , u eof =1.0246,p = φ = , u eof =1.3248,p = φ = , u eof =1.0342,p = Table 4.3: The numerical accuracy check for the numerical calculation for the electroosmotic flow in the nano-diffuser shown in Figure The results for grid are compared with the results of grid and two-digit accuracy has been attained. The numerical results are shown in Figure Note that here the pressure p is rescaled on E I,x. Step 4 Solving the pressure equation numerically The pressure equation (4.39) can be solved using Thomas algorithm. The pressure-driven component u p can be calculated based on equation (4.7) and the transverse velocity can be calculated from equation (4.29). Results for potential, velocity and pressure shown in Figures 4.16 and 4.17 are similar to the results shown in Figures 4.10 and A mesh is used to perform the numerical calculation and two digit accuracy is obtained by comparing the results calculated on the mesh and mesh as shown in Table 4.3. The parameters used for the calculation are listed in table 4.2, except that the surface charge density used here is 0.07C/m 2, higher than that listed in the table. For such a high surface charge (σ = 0.07C/m 2 ) the Debye-Hückel solution calculated based on the analysis in Section is about 30% off due to the high electric potential as shown in Figure In this case, the Debye- Hückel results over-predict the potential and streamwise velocity as well as the pressure. Since the pressure distribution is directly related to the EOF velocity component by equation (4.39), the pressure calculated from the Debye- Hückel solution is also about 30% higher than the nonlinear result as shown in Figure 4.18 (b). For a smaller surface charge, the Debye-Hückel results will be more accurate. For example as shown 112

137 y x=0 x=0.2 x=0.4 x=0.6 x=0.8 x=1.0 y x=0 x=0.2 x=0.4 x=0.6 x=0.8 x= φ (a) potential u (b) streamwise velocity y x=0 x=0.2 x=0.4 x=0.6 x=0.8 x= v (c) transverse velocity 20 p dpdx x (d) induced pressure and pressure gradient Figure 4.16: Results for electroosmotic flow in the nano-nozzle. The height of the nozzle is 14.6 nm at the inlet and 4.6 nm at the outlet and the length of the diffuser is 15 nm; ɛ = The imposed electric field is 0.5V/nm and the surface charge density is 0.07C/m 2. The pressure is zero both at the inlet and the outlet and so the pressure shown here is induced by the presence of the EDLs. 113

138 y y x=0 x=0.2 x=0.4 x=0.6 x=0.8 x= φ (a) potential v (c) transverse velocity x=0 x=0.2 x=0.4 x=0.6 x=0.8 x=1.0 y x=0 x=0.2 x=0.4 x=0.6 x=0.8 x= u 0 (b) streamwise velocity 0.6 p dpdx x (d) induced pressure and pressure gradient Figure 4.17: Results for electroosmotic flow in the nano-diffuser. The height of the diffuser is 4.6 nm at the inlet and 14.6 nm at the outlet and the length of the diffuser is 15 nm; ɛ = The imposed electric field is 0.5V/nm and the surface charge density is 0.07C/m 2. The pressure is zero both at the inlet and the outlet and so the pressure shown here is induced by the presence of the EDLs. 114

139 in Figure 4.19, the surface charge density is 0.01C/m 2 and the corresponding surface potential is about ζ =0.5. The discrepancy is much less between the nonlinear numerical calculations and the Debye- Hückel solution. 4.7 Numerical Solution using COMSOL A reservoir-nozzle system (Figure 4.2 (a)) has been investigated using both molecular and continuum techniques. Molecular dynamic simulation is a computer simulation technique calculates the time dependent behavior of a molecular system. Generally molecular dynamic systems consist of a vast number of particles and generate information of these particles at the microscopic level, including atomic positions and velocities. These microscopic information is converted to macroscopic observables such as pressure, energy, heat capacities, etc., using statistical mechanics (Rapaport, 1995). It is often used to study of biological molecules since it can provide detailed information on the fluctuations and conformational changes of proteins and nucleic acids. In the MD simulations, the nozzle walls are described using the amorphous silica model. Unlike the much simpler case of flow in a uniform pore (e.g. slit or cylindrical pore), the ion distribution and flow pattern depend on the applied field through the ratio of the time scale for ion diffusion across the channel to the time scale for flow past features of the nozzle (Shin & Singer, 2009). The applied electric field is very high E 0 =0.5V/nm and the Boltzmann distribution may not be able to describe the ionic distribution in across the nozzle because of the ion polarization and relaxation. The Poisson-Nernst-Planck equations are necessary to simulate this problem. In COMSOL Multiphysics, the Laplace equation, Poisson equation, Stokes equation and two Nernst-Planck equations for each ionic species are chosen for the simulations. The source terms and body force terms are set up 115

140 u eof x=0,dh x=0.5,dh 0 x=1,dh x=0,nonlinear 2 x=0.5,nonlinear x=1,nonlinear y/h (a) u eof 5 4 DH Nonlinear 3 p x (b) p Figure 4.18: The comparison of DH (based on the analysis in Section 4.3.2) and nonlinear results for streamwise velocity component u eof and pressure p. The height of the nozzle is 4.6 nm at the inlet and 14.6 µm at the outlet and the length of the diffuser is 15 nm; ɛ = The imposed electric field is 0.5V/nm and the surface charge density is 0.07C/m 2. The pressure is zero both at the inlet and the outlet and so the pressure shown here is induced by the presence of the EDLs. 116

141 u eof x=0,dh x=0.5,dh 0.2 x=1,dh x=0,nonlinear 0 x=0.5,nonlinear x=1,nonlinear y/h (a) u eof 5 4 DH Nonlinear 3 p x (b) p Figure 4.19: The comparison of DH and nonlinear results for streamwise velocity component u eof and pressure p. The height of the nozzle is 4.6 nm at the inlet and 14.6 µm at the outlet and the length of the diffuser is 15 nm; ɛ = The imposed electric field is 0.5V/nm and the surface charge density is 0.01C/m 2. The pressure is zero both at the inlet and the outlet and so the pressure shown here is induced by the presence of the EDLs. 117

142 Figure 4.20: The Mesh used for the simulations. Close to the walls, extra fine mesh is used and in the center region, the mesh is coarser. x =0nm, y = 5nm x =10nm, y =3nm Mesh I φ = , u =0.0121,p = φ = , u =0.0114,p = Mesh II φ = , u =0.0118, p = φ = , u =0.0112,p = x =20nm, y = 1nm x =40nm, y =5nm Mesh I φ = , u =0.0124,p = φ =0.0062, u =0.0096,p = Mesh II φ = , u =0.0121,p = φ = , u =0.0095,p = Table 4.4: The numerical accuracy check for the results calculated using COMSOL shown in Figure The mesh I has 62, 442 triangular elements and the mesh II is a finer mesh, having 183, 712 triangular elements. The results shown for the electric potential φ (mv ), the velocity u (mm/s) and pressure p (Pa) are in dimensional form. according to equations (2.3), (2.8), (2.15), (2.21) and (2.22). An extra fine mesh is used for the region close to the channel walls to resolve the drastic change of the ionic concentration due to the electric double layers. Away from the walls, coarser mesh is used as shown in Figure 4.20 and two digit accuracy is obtained for a mesh with 62, 442 triangular elements by comparing with the results obtained on a mesh with 183, 712 triangular elements as shown in Table

143 (a) Cl in the nozzle (b) u in the nozzle (c) Cl in the diffuser (d) u in the diffuser Figure 4.21: Results for ionic concentration and flow field for the EOF in the nozzlereservoir system calculated using COMSOL. The parameters are listed in table

144 Results calculated from Molecular Dynamic simulations for the electroosmotic flow in the nozzle-reservoir system are shown in Figure 4.22 (Shin & Singer, 2009) (a) and (c). The corresponding results for the flow field and Na + concentration calculated based on the continuum theory are shown in Figure 4.21 (b) and (d). It is shown that the results for both the streamwise velocity and ionic concentration are qualitatively similar to each other. In the reservoir on the left hand side, both the MD and continuum results show a depletion region at the center, indicating a smaller streamwise velocity than the region closed to the walls, which is due to the local adverse pressure gradient induced by the electroosmotic flow. It is seen that the streamwise velocity contours are asymmetric about the centerline of the system, which is due to the asymmetric setup in surface charge in the MD simulations. This can be improved in the future calculations, either by incorporating more detailed surface charge density distribution in the continuum model or by using more even surface charge setup in the MD simulations. 4.8 Particle Transport A simplified model is used to calculate the velocity of the polystyrene beads shown in Figure 4.1. The polystyrene particles are treated as another charged species A immersed in the electrolyte mixture. The flux equation for this species A is n A = D A c A + u A z A Fc A E + c A u and the velocity in the x direction is u A = D A c A c A x + u Az A FE x + u (4.40) 120

145 (a) velocity contours (MD) (b) velocity contours (continuum) (c) Na + concentration contours (MD) (d) Na + concentration contours (continuum) Figure 4.22: The results for EOF in the nozzle-reservoir system based on Molecular Dynamics simulations(shin & Singer, 2009) and continuum theory. The height of the nozzle is 14.6 nm at the inlet and 4.6 µm at the outlet and the length of the diffuser is 15 nm; ɛ = The imposed electric field is 0.5V/nm and the surface charge density is 0.07C/m 2. Figure 4.23: The reservoir-nozzle system used to analyze particle transport. The reservoir on the right is the particle donor reservoir and the one on the left is the particle receiver reservoir. Negatively charged particles always tend to move from left to right due to electromigration (u EM < 0). 121

146 Based on the experimental data (L = m and D A =10 the first term, the diffusion term is D A c A c A x 10 9 m2 sec /( m) c A c A Similarly the second term, the electrical migration term is u A Fz A E x mole J m 2 The last term, the convective term is of the order 9 m2 sec 10 6 c A c A ) the magnitude of m sec sec 96500Coul mole 80V/(10 2 m)z A m z A sec u ɛ ee x RT Fµ m sec Note that the diffusion term is two orders of magnitude less than the other two terms while the electrical migration term and the electroosmotic term are of the same order. Hence particles can be easily pumped against its concentration gradient. Neglecting the diffusion term, the particle velocity is u A = u A z A FE x + u = u EM + u F where u EM is the velocity component due to the electrical migration and u F is the velocity component due to the bulk fluid flow. The particle motion depends on the directions and the magnitudes of these two components. A reservoir-nozzle system similar to the experimental setup (shown in Figure 4.1) is shown in Figure 4.23, which includes a particle donor reservoir and a particle receiver reservoir and a nozzle/diffuser connecting the two reservoirs. Different combinations of wall charge and particle charge determine the direction of fluid flow motion and particle motion as shown in Table 4.5. Consider the transport of an uncharged bio-molecule such as glucose. The velocity within the nozzle responds instantaneously to the inception of the imposed electric field 122

147 negatively charged wall positively charged wall uncharged particles u F > 0,u EM =0 u F < 0,u EM =0 positively charged particles u F > 0,u EM > 0 u F < 0,u EM > 0 negatively charged particles u F > 0,u EM < 0 u F < 0,u EM < 0 Table 4.5: The dependence of the particle motion on the wall charge and the particle charge. Here u F is the bulk motion of the fluid flow and u EM is the velocity of particle due to electrical migration. The particle velocity is the summation of these two terms. and the glucose, being neutral, is only transported by the bulk fluid motion (u EM =0). The time scale of this transient period is on the order of millisecond or even smaller. Thus, the total amount of a given species that reaches the receiver is linear with time and this is characteristic of what is termed zeroth order release. If the nozzle wall is positively charged, the fluid flow is from particle donor to the particle receiver and the glucose in solution will be transported into the receiver. If the nozzle is negatively charged, the uncharged species will not be transported to the receiver. Suppose the analyte of interest is a negatively charged species such as albumin or polystyrene beads. The electrostatic force acting on the species will always drive the species to the positive electrode (u EM < 0). The flow direction depends on the nozzle wall charge. If the wall is positively charged, the electromigration term and electroosmotic term will have the same sign (u EM < 0,u F < 0) and the particles will readily move to the receiver and even adsorb on the walls of the nozzle. Because such a molecule can adsorb to the surface, not all of the particles which originated in the donor will reach the receiver. If the wall is negatively charged, the electromigration term and electroosmotic term have different sign (u EM < 0,u F > 0)and the particle motion depends on the parameter R CE, 123

148 Figure 4.24: Comparison of the analytical results and the experimental data (Wang & Hu, 2007). The length of the micro-diffuser is 650 µm; the inlet height is 20 µm and the outlet height is 130 µm. The electric field is 80 V/cm. The ζ-potential of the PMMA walls are 15 mv (Kirby & Hasselbrink, 2004). Both the particles and the walls are negatively charged and so u F > 0,u EM < 0. which is ratio of the convective term and the electrical migration term: R CE = z AD A E φ 0 U 0 In the experiments (Figure 4.1), the walls and the particles are both negatively charged (u F > 0,u EM < 0) and so the particle motion depends on the parameter R CE. For the experimental parameters (Wang & Hu, 2007), R CE = 0.02z A. The calculated particle velocity at the nozzle centerline (y =0) is compared with the experimental data in Fig The experimental data are obtained from visualizing the motion of 80nm polystyrene beads on the centerline of the nozzle. The dots correspond to the distribution of velocities of the nozzle measured in the experiments and the solid lines are the model velocity for different particle charge. The effective surface charge density can be calculated from σ = z A e πd 2 where e = C and d is the diameter of the particles. The calculated results 124

149 compare well with the experimental data for σ = 1µC/cm 2 and the calculated transit time is 1.65 seconds. It is also seen that the particles are accelerated near the outlet region due to the higher electric field, which indicates a potential application for drug and gene delivery. 4.9 Particle Transport Using Two-Phase Flow Modeling In the previous section, the particles are treated as point charges and the motion of these charged particles under the external applied electric field is modeled as electromigration. In this section, a two-phase flow model, that treats the dispersed solid particles as one phase and the liquid solution as another, is discussed to better resolve the transport of particles in the nozzle-reservoir system as shown in Figure A mixture of two medium of different phases is often encountered in flow systems, such as liquid-gas or liquid-vapor mixtures in condensers and evaporators, dust/ash particles in environmental flows, transport of micron-scale particles in viscous fluids, and liquid-liquid mixtures when dealing with emulsions. Two-phase flow is a difficult subject principally because of the complexity of the form in which the two fluids exist. Historically, interest in two-phase flows dates from the 1930 s due to the necessity to simulate composed and very complex processes of the solid fuel combustion in rockets nozzles. Later, it was observed that adding dust to a gas flowing through a pipe results in reduction of a pressure gradient required to maintain the flow at its original rate in experiments. Two-phase flows always involve relative motion of one phase with respect to the other, and so a two-phase-flow problem should be formulated in terms of two velocity fields. Historically, there have been two main approaches. One approach, called two-fluid model, is based on individual balance equations for each of the fluids, while the second involves 125

150 manipulation and combination of those balance equations into modified forms, by introducting ancillary functions such as the fractional flow function(binning & Celia, 1999). In this section, a dusty-gas model, which belongs to the first approach, is discussed describe the particle transport problem (Hibiki & Ishii, 2003) discussed in the previous section. This model treats each phase separately and formulates two sets of conservation equations governing the balance of mass, momentum and energy of each phase. For this kind of two-phase models, the main difficulties in modeling usually arise from the existence of interfaces between phases and discontinuities associated with them (Marble, 1970). The basic assumption in the dusty gas model is that each material can be described as a continuous phase, occupying the same region in space and they can interact with the other through terms in the governing equations that correspond to inter-phase drag forces. Therefore each phase has its own velocity, and in the flow region both velocities exist at every point. Consider spherical particles, the volume charge density is given by ρ e,p = α σa V ασ 4πa2 = 4πa 3 /3 = 3ασ a where the radius of the particle is a, σ is the surface charge density and α is the volume fraction of the particle phase. Note that the volume fraction is related to the concentration of the particles and it is also a function of position and time. Since the potential is given by ɛ e 2 φ = ρ e, the dimensionless equation for potential becomes 2 φ y + 2 φ 2 ɛ2 1 x + 2 φ 2 ɛ2 2 z = β ( zi X 2 ɛ 2 i π 1 α ) (4.41) where π 1 = 3σ is the dimensionless volume charge density of the particle phase. Here the af c volume charge from the particle phase and the fluid phase are both included. 126

151 For ionic species A in the fluid phase, the dimensionless equation is given by 2 X A + ɛ 2 2 X A y ɛ 2 2 X A x 2 2 z 2 + ( ɛ 1 z A X A E x x = Pe + z A X A E y y ( ɛ 1 u X A x + v X A y + ɛ 2w X ) A z ) X A E z + ɛ 2 z A z (4.42) For the fluid phase, the conservation of momentum is given by the Navier Stokes equation as, ( ) u ρ t +( u ) u = p + µ 2 u + ρ e E F D ρ g (4.43) here F D is the Stokes Drag body force. For a single spherical particle, the Stokes Drag force is F D,p =6πµa(u u p ) and for the particle phase, the Stokes Drag force per unit volume is given by F D = α F D,P V = 9αµ(u u p) 2a 2 = ηα(u u p ) where η = 9µ 2a 2. The three momentum equations for an incompressible, steady flow are, in dimensionless form, ( u t +Re ɛ 1 u u x + v u y + ɛ 2w u ) p = ɛ 1 z x + β ( ( ) zi X ɛ 2 i 1 ɛ ) 1 φ + 2 u ηα(u u p ) Λ x (4.44) ( v t +Re ɛ 1 u v x + v v y + ɛ 2w v ) = p z y β φ ( ) zi X Λɛ 2 i + 2 v ηα(v v p ) Re y Fr 2 (4.45) ( w t + Re ɛ 1 u w x + v w y + ɛ 2w w ) p = ɛ 2 z z β φ ( ) zi X Λɛ 2 i + 2 w (4.46) z 127

152 here the Froude number is given by Fr = U 0 and the time scale is t 0 gh = ρh2 µ. These equations are the classical Navier-Stokes equations for constant density and viscosity which govern fluid flow of Newtonian liquids in a continuum. The particle phase also satisfies the continuity equation: α t + (αu p )=0 and the steady state dimensionless continuity equation is given by ɛ 1 αu p x + αv p y + ɛ αw p 2 z =0 here the particle velocity is also scaled on u p = u p U 0. The velocity equation is given by, ρ p α ( up t +( u p ) u p ) = ρ e,p E + F D αρ p g + αρ g (4.47) here ρ p is the density of the particle phase, u p is the dimensional velocity vector of the particle phase. In dimensionless form, this equation can be written as α u ( ) p t + γre αu p ɛ 1 u p x + v αu p p y + ɛ αu p 2w p = β ( z ɛ π 1α 1 ɛ ) 1 φ + ηα(u u 2 p ) Λ x (4.48) α v ( ) p t +γre αv p ɛ 1 u p x + v αv p p y + ɛ αv p 2w p = βπ 1 z Λɛ α φ 2 y +ηα(v v p) αre (γ 1) Fr2 (4.49) α w ( ) p t +γre αw p ɛ 1 u p x + v αw p αw p p + ɛ 2 w p = βπ 1 y z Λɛ α φ 2 z +ηα(w w p) (4.50) where γ = ρp ρ is the ratio of the particle phase and fluid phase densities. 128

153 To summarize, for an electrolyte with n species, we have a series of 9+n equations in 9+n unknowns to solve for the three-dimensional velocity field of the two phases, the pressure, the potential, the volume fraction of particle phase and the mole fractions for the n species. There are seven dimensionless parameters that govern the problem: Re, Pe, Λ,ɛ, γ, π 1 and η. The body force terms are then need to be determined. The particle considered here is Polystyrene beads having 10nm in radius, with a 0.01C/m 2 (Ohsawa et al., 1986) surface charge density in 0.1M NaClsolution. The relative dielectric constant for Polystyrene is 2.5 and the density is 1050kg/m 2. The ratio of the electric body force to the viscous drag body force is F Electric = α 3σE0 a F Drag = α 9µ(u u p) 2a 2 2σa 3ɛ e φ 0 (u u p ) 3.7 u u p where u and u p are dimensionless velocities for fluid phase and particle phase respectively. The ratio of the electric body force term to the gravity force for the particle phase F Electric F Gravity = α 3σE0 a g(ρ p ρ) E 0 The electrical body force and the Stokes drag are in the same order while the gravity body force is much smaller. Based on specific problems, these equations can be further simplified and solutions calculated numerically. In the future, this system of equations will be used to investigate particulate problems Modeling the Fabrication of the Nanonozzles In biomedical applications, polymers are often the material of choice because of their biocompatibility and/or biodegradability, low cost, and recyclability. However, the typical methods of fabricating nanofluidic channels use top down approaches such as X- ray lithography, electron beam lithography (EBL) or ion beam lithography and most of 129

154 the nanochannels are made of silicon or other hard materials (Ehrfeld & Schmidt, 1999; Madou, 2002). A method to produce hybrid nanochannels is proposed (Zeng et al., 2005) by growing silica within a polymer template, from the precipitation reaction from the aqueous solution. It is called self-assembly and it is a combination of top down manufacturing and bottom up synthesis. The growing silica within the channel would lead to a reduction in channel size and it can provide much needed reinforcement to the polymer nanostructure. Moreover, using this technique, other chemically and biologically functional precursors can also be driven into the nanochannel array to add needed functionality onto the channel wall through the surface reaction/assembly. This would entail fabricating multi-functional nanochannels that are not usually achievable in a conventional way(zeng et al., 2005). In the self-assembly process, the fabricated device depends on the parameters including the applied electric field, the concentration of TEOS, the surface charge etc. If these parameters are well chosen, a uniform silica layer will be formed on the channel wall otherwise only nanoparticle aggregates will appear in the bulk. Experimental attempts to obtain the optimal combination of these parameters involve high cost and high experimental workload. Modeling and simulations can provide valuable support to the experimental design which can help interpret the experimental data and reduce study failure. Optimal use of modeling and simulation results can cut the development time and cost for this new technology substantially. On Figure 4.25 shows the schematic of the dynamic assembly process done (Zeng et al., 2005). They have utilized electrokinetic flow to guide the silicification reaction within micro/nanochannels to further reduce the size and improve mechanical properties of the channels. As shown in the figure, the inner wall of the nano-array was treated to anchor a cationic polymer, polyallyamine hydrochloride (PAH), to enhance the surface reaction rate 130

155 Figure 4.25: Process schematic of EOF based dynamic assembly of silica(wang et al., 2005). by catalyzing silica condensation. In the bulk, a nucleation process occurs homogeneously which produces randomly distributed silica nanoparticle aggregates. On the liquid-solid interface, the nucleation process happens at a higher reaction rate which could produce a dense and smooth inorganic film. The ultimate goal is to achieve substantial silica growth on the liquid-solid interface while suppressing the homogeneous nucleation in the bulk. On the other hand, to consistently supply silica source into the interior region of the channel, a electric field is applied to generate electroosmotic flow. To better understand the dynamic assembly process, the fluid flow, the mass transfer of reactants and the chemical reaction on the liquid-solid surface and in the bulk flow have to be considered. Note that these three components are coupled with each other. The electroosmotic flow is determined by the surface charge at the liquid-solid interface while the deposition of silica will modify the surface charge. On the other hand, the supply of the reactants is carried by the electroosmotic flow. Therefore, a comprehensive model for this problem has to include all of these three components. 131

156 (a) (a) Figure 4.26: The schematic diagram for the simplified models. (a) one-dimensional model: A homogeneous chemical reaction is happening in the bulk and a heterogeneous chemical reaction is happening at the liquid-solid interface at a different rate. The interface is moving with the deposition of the reaction product C. (b) two-dimensional model: EOF in rectangular channels with chemical reactions Development of realistic models of the growth of silica inside the nanonozzle will involve several steps. As the first step, a simplified one-dimensional mathematic model as shown in Figure 4.26 (a) will be set up to investigate the mass transfer of TEOS and the homogeneous and heterogeneous chemical reactions. TEOS has the remarkable property of easily converting into silicon dioxide and good conformality of coating. This precipitation reaction occurs upon the addition of water: Si(OC 2 H 5 ) 4 +2H 2 O SiO 2 +4C 2 H 5 OH The side product is ethanol and the rates of this conversion are sensitive to the presence of acids and bases, both of which serve as catalysts. In the experiments, the surface was treated to anchor a cationic polyelectrolyte, polyallyamine hydrochloride (PAH) to serve as catalysts. The mass transfer equation for species A is given by c A t + u c A = D AB 2 c A + R A (4.51) 132

157 where c A is the concentration for species A and u is the velocity vector; D AB is the diffusion of species A; R A is the volumetric reaction rate of A in the solution. Note that for TEOS, it is the consumption rate (R A < 0). For simplicity, a one-dimensional model (shown in Figure 4.26) will be used to study the relationship between the chemical reactions and the growth of the interface. The one-dimensional mass transfer equation is with boundary conditions: c A t = D 2 c A AB y + R 2 A (4.52) t =0,h=0,c A = c 0 t>0,y = h, n y = k 2 c A y=s where ξ is a parameter related to the reaction rate on the surface and n y is the flux of species A into the wall due to the reaction at the surface. Note that R A = k 1 C A is also a function of the concentration of species A and the height of the interface h(t) is given by h = γ t 0 n y y=s dt here γ is a parameter describing the relationship between the height of the interface and the total amount of silica deposited on the surface. The species considered in the modeling will be TEOS. This equation will be solved numerically with appropriate parameters. The relationship between the height of the interface h(t) and the concentration of TEOS in the bulk, the surface concentration of catalysts will be investigated. Based on the one-dimensional model, a two-dimensional rectangular channel will be studied as shown in Figure 4.26 (b). An external electric field will be applied and so there will be electroosmotic flow inside the channel. Since TEOS is carried by the fluid flow, unlike the one-dimensional model, the convective term in the mass transfer equation has to 133

158 be considered. As silica is deposited onto the channel surface, the surface charge density will be modified. Recirculation zones may be found in the bulk flow. At this stage, the two dimensional fluid flow and the mass transfer equations will be solved numerically and the basic physics of this surface modification process will be revealed. The next step is to investigate the experimental configuration, which is the nozzlereservoir system shown in Figure 4.1. It could be a rectangular channel or a cylindrical cone depending on the experimental configuration. This model would help interpret the experimental data and guide future experimental design. Based on the simulations, the effect of different parameters such as geometry, concentration, surface properties will be studied. In the future, a comprehensive model will be established to investigate the self-assembly of silica as a technique to fabricate nano-scale channels/nozzles. The coupling physics including chemical kinetics, surface growth, fluid flow and mass transport will be considered in this model. Predicted results from this model will help interpret the experimental data and obtain the optimal combination of the controlling parameters for the experimental setup Summary In this chapter, a model has been established for the electroosmotic flow in a micro/nano nozzle or diffuser with rectangular cross section. Analytical solutions have been derived based on the lubrication theory and the Debye- Hückel approximation for two kinds of boundary conditions on potential: known surface potential or known surface charge density. Unlike the applied electric field in a channel, the magnitude of the electric field is increasing 134

159 in a nozzle and decreasing in a diffuser in the axial direction due to the change of crosssectional area. Generated by the motion of ions under the applied electric field, the EOF usually moves faster in the region closed to the small end of the nozzle/diffuser and moves slower at the large end. The transverse velocity (v) is in the order of ɛ 1 and is usually small compared to the streamwise velocity (u). A pressure gradient is induced by the electroosmotic flow and the corresponding pressure-driven velocity (u p ) due to this induced pressure is usually small compared to the electroosmotic flow velocity (u eof ). Based on the lubrication approximation, a non-linear calculation based on the Poisson- Boltzmann equation is also discussed for the high potential cases where the Debye- Hückel approximation will not hold. For the cases of thin EDLs (ɛ 1), analytical solutions are derived based on the asymptotic analysis for a symmetric monovalent electrolyte solution for both inner and outer regions. Results have shown that there is no induced pressure gradient by the EDLs in the outer region (p =0) and this is consistent with the results derived based on the Debye- Hückel approximation by taking the limit ɛ 0. Comparison between the results calculated from the Debye- Hückel solution and the nonlinear numerical solution has shown that the Debye- Hückel solutions over-predict the streamwise velocity and the induced pressure by 30% for a surface charge density 0.07C/m 2. A full numerical solution is calculated using COMSOL based on the Poisson-Nernst- Planck equations to compare with the MD results, in which the Poisson-Boltzmann equation cannot be used due to the high applied electric field. The comparisons have shown some similarity in the results for velocity field and the ionic concentration. This can be used to validate the feasibility of continuum theories at such small length scale. Particle motion in microdiffusers is investigated and it is found to be dependent on the wall charge, the particle charge and the parameter R CE. The results shown in this chapter 135

160 can be used to estimate the mass transport of charged/uncharged species in micro/nano nozzles/diffusers which has a potential application in drug and gene delivery. A two-phase flow model is discussed to better resolve the particle motion in the nozzle-reservoir system. The general governing equations are derived using the dusty-gas model, which treats the solid particles as a continuous phase, occupying the same region in space as the fluid flow and interacting with the fluid phase by the inter-phase forces. Future work will be focused on the numerical calculations for the fluid flow and particle motion based on the two-phase model. Nanonozzles are fabricated using a dynamic assembly process to further reduce the height of the nozzle. A comprehensive model is proposed for future work to calculate the fluid flow, the mass transfer of reactants and the chemical reaction on the liquid-solid surface in such a dynamic assembly process. Predicted results from this model will help interpret the experimental data and obtain the optimal combination of the controlling parameters for the experimental setup. 136

161 CHAPTER 5 The Forces that Affect DNA Translocation 5.1 Introduction In this chapter, the forces that affect DNA translocation, including the electrical driving force, the uncoiling and recoiling forces due to DNA conformational change and the viscous drag force due to the hydrodynamic interactions between the DNA and the nanopore, are investigated to understand the mechanisms involved in the DNA translocation process. Figure 5.1: Schematic diagram of DNA translocation in a negatively charged nanopore. 137

162 Generally, DNA contour length, which is the total length of the DNA when it is stretched completely, is several micros, much longer than the length of a nanopore. Therefore, during the DNA translocation, only a small part of DNA is inside the nanopore while most of the DNA overhangs in the reservoirs, forming blob-like configurations as shown in Figure 5.1. The negatively charged DNA is subjected to an electrical driving force (Fe ) under the external applied electric field, moving to Reservoir II (the reservoir with the anode as shown in Figure 5.1) at a translocation velocity (u DNA). Locally inside the negatively charged nanopore, an electroosmotic flow is generated due to the motion of cations in the EDLs. The bulk flow in the negatively charged nanopore is in the opposite direction to the DNA motion, exerting a drag force (Fd ) on the DNA surface. As the DNA moves from Reservoir I (the reservoir with the cathode as shown in Figure 5.1) to Reservoir II, the DNA strand gradually uncoils from the blob-like configuration in Reservoir I, navigates through the nanopore and then recoils in Reservoir II, involving uncoiling and recoiling forces (F E and F R) related to the conformational change of DNA in transition. As the blob-like configuration of DNA in Reservoir I uncoils, the center of this DNA blob gradually approaches the nanopore, inducing a viscous drag force (Fblob1 ) on the DNA blob due to its motion (vblob ). On the other hand, the DNA blob stays close to the opening of the nanopore, facing the fluid flow discharged from the nanopore, introducing another viscous drag (F blob2 )on the blob in Reservoir I. To summarize, the DNA is subjected to four categories of forces: the electrical driving force (Fe ), the uncoiling and recoiling forces due to the DNA conformational change (F E and F R), the viscous drag force acting on the linear DNA inside the nanopore (F d ) and the viscous drag acting on the blob-like DNA outside the nanopore (F blob1 and F blob2 ). 138

163 It is well accepted that the DNA is electrophoretically driven into the nanopore but the nature of the resisting force is controversial in the literature. In some of the previous work (Muthukumar, 1999, 2007; Kantor & Kardar, 2004; Forrey & Muthukumara, 2007), the uncoiling and recoiling forces were considered to be the main resisting force balancing with the applied electrical driving force, determining the DNA translocation rate. On the other hand, the viscous drag force acting on the blob-like DNA configuration outside the nanopore is considered as the force resisting DNA passage through the nanopore by Storm et al. (2005b) and Fyta et al. (2006). In this chapter, the aforementioned four categories of forces are investigated and the magnitudes of these forces are compared. Results show that the viscous drag force acting on the DNA inside the nanopore (Fd ) is the main resisting force. To calculate this viscous drag (Fd ), a hydrodynamic model is established to calculate the fluid flow in a nanopore with a DNA placed at the center. This model is extended from the model for EOF in a converging nozzle discussed in Chapter 4 and the model is validated by comparing with the experimental data given by Keyser et al. (2006) for a tethered DNA for which the translocation velocity vanishes. Previous work on this problem(ghosal, 2007b) has assumed a cylindrical pore with constant cross-section, neglecting the converging/diverging geometry used in Keyser et al. (2006). In this chapter, the converging and diverging geometry similar to the experimental configurations is addressed. 5.2 Governing Equations and Boundary Conditions In this section, the model used to calculate the flow field in the nanopore and viscous drag force (Fd ) acting on the DNA inside the nanopore is discussed. A typical nanopore 139

164 Figure 5.2: The geometry of the nanopore and the DNA. geometry is depicted on Figure 5.2. The geometry of the nanopore is similar to the converging nozzle discussed in Chapter 4, and the only difference is that the cross-section of the nanopore is circular instead of rectangular. The analysis in this chapter is also based on lubrication theory, similar to Chapter 4. In the radial direction, the inner radius, which is the DNA surface, is defined by r = a and the outer radius r = h (x ), which is the wall of the nanopore, is a linear function of x as discussed previously, h (x )=h i + h o h i L where h i,h o are the radii of inlet and outlet of the pore and in dimensionless form x where h = h h i and x = x L. h(x) =1+ h o h i h i Electrodes are placed at the inlet and the outlet of the pore and so there is an imposed electric field in the x-direction. The electric field E satisfies Maxwell s equations in the form E =0so that the imposed electric field satisfies E I,x (x)a (x) =E I,x (0)A (0). If the imposed electric field at the inlet EI,x (0) is taken as the scale of the imposed electric x 140

165 field, the dimensionless imposed electric field is thus E I,x = 1 a2 h(x) 2 a 2 where a = a. h i Outside the nanopore (in the regions x<0 and x>1), the cross-sectional area is much larger than that inside the pore. Since the electric field is negligible outside the pore, the electrical driving force on DNA is only felt in the direct vicinity of the pore. The governing equations for electric potential, mole fractions of ionic species and the velocity field are given by equations (2.27), (2.32), (2.31), (2.35) and (2.36) discussed in Chapter 2. Similar to the analysis in Chapter 4, the governing equations can be simplified based on the lubrication approximation in cylindrical coordinates as 2 φ r + 1 φ 2 r r = β ɛ 2 2 u r + 1 u 2 r r = E β I,x i 2 v r + 1 v 2 r r = E β I,y ɛ 2 u ɛ 1 x + 1 rv r r i ɛ 2 i z i X i (5.1) z i X i + p x z i X i + 1 ɛ 1 p r (5.2) (5.3) =0 (5.4) Note that since the pressure is assumed to be independent of r, the pressure is rescaled on p 0 = µu 0 L and so the ɛ 1 in equation (2.35) is dropped. The boundary conditions are r = h(x) : φ r = σ w,u=0,v =0 r = a : φ r = σ d,u=0,v =0 where σ w and σ d are the surface charge density of the nanopore and the DNA, and σ is scaled on φ 0 ɛ eh i as σ = σ h i ɛ eφ Electroosmotic Flow in an Annulus Prior to considering the geometry of the experimental nanopores, DNA transport in a cylindrical nanopore of constant cross-section is first investigated. For the case where the 141

166 Figure 5.3: The geometry of a DNA placed in a cylindrical nanopore. The computational domain is the annular region between the DNA and the nanopore. slope of the wall is zero (h =0), the dimensionless governing equations for fully developed electroosmotic flow can be simplified as a one-dimensional problem with governing equations ( 2 φ r + 1 ) φ = β z 2 r r ɛ 2 i X i (5.5) i ( ɛ 2 2 u r + 1 ) u = β z 2 r r ɛ 2 i X i (5.6) Note that in this case h =1, E I,x =1all through the pore and there is no pressure gradient in the x direction. The boundary conditions are φ r = σ w,u=0,r=1 φ r = σ d,u=0,r = a These equations can be solved numerically and an analytical solution may be obtained using the Debye-Hückel approximation. Based on the Debye-Hückel approximation, the i potential equation becomes 1 r ( r r φ r ) = φ ɛ 2 (5.7) 142

167 This is a modified Bessel equation and the solution is φ (r) = ɛi ( )[ ( ) ( )] ( )[ ( ) ( )] r 0 σw K a ɛ 1 ɛ + σd K 1 1 ɛ + r ɛk0 σw I a ɛ 1 ɛ + σd I 1 1 ɛ ( ) ( ) ( ) ( ) I a 1 1 K1 ɛ ɛ 1 a (5.8) I1 K1 ɛ ɛ At r = a, the DNA surface potential is given by equation 5.8 as ζ d = φ (r = a) = ɛi ( )[ ( ) ( )] ( )[ ( ) ( )] a 0 σw K a ɛ 1 ɛ + σd K 1 1 ɛ + a ɛk0 σw I a ɛ 1 ɛ + σd I 1 1 ɛ ( ) ( ) ( ) ( ) I a 1 1 K1 ɛ ɛ 1 a I1 K1 ɛ ɛ (5.9) and at r =1, the nanopore surface potential is given by ζ w = φ (r = a) = ɛi ( )[ ( ) ( )] ( )[ ( ) ( )] 1 0 σw K a ɛ 1 ɛ + σd K 1 1 ɛ + 1 ɛk0 σw I a ɛ 1 ɛ + σd I 1 1 ɛ ( ) ( ) ( ) ( ) I a 1 1 K1 ɛ ɛ 1 a I1 K1 ɛ ɛ (5.10) The corresponding velocity solution can be derived from equation and so The solution is given by 1 r ( r r φ r ) = 1 r u r = φ r + C 1 r ( r u ) r r (5.11) (5.12) u = φ +lnc 1 + C 2 (5.13) where C 1 and C 2 are integration constants. Applying the boundary conditions for u and φ shows C 1 = ζ w ζ d ln(a) and C 2 = ζ w where ζ d (equation (5.9)) and ζ w (equation (5.10)) are the dimensionless zeta potential of the DNA and the wall. Substituting C 1 and C 2 into equation (5.13), the velocity is given by u(r) =φ(r)+ ζ w ζ d ln(a) ln(r) ζ w (5.14) 143

168 M 1M φ M 1M r (a) potential u r (b) velocity Figure 5.4: The results for potential and velocity calculated from equations 5.8 and The radius of the nanopore is 5nm and the radius of the DNA is 1nm. The surface charge density is 0.015C/m 2 on the DNA surface and 0.006C/m 2 on the wall. 5 dimensionless shear stress τ Electrolyte concentration (M) Figure 5.5: The dimensionless shear stress τ = u at r = a as a function of electrolyte concentration. The radius of the nanopore is 5nm and the surface charge density is 0.06C/m 2 r while the radius of the DNA is 1nm and the surface charge density is 0.15C/m 2. The electrolyte concentration varies from 0.01M (ɛ =0.6) to0.1m (ɛ =0.06). 144

169 The results for potential and streamwise velocity obtained from equations (5.8) and (5.14) are plotted in Figure 5.4 for two different electrolyte concentrations: 0.1M and 1M in the upstream reservoir. The radius of the nanopore is 5nm and the radius of the DNA is 1nm. The surface charge density is 0.015C/m 2 on the DNA surface and 0.006C/m 2 on the wall. It is shown in Figure 5.8 (a) that the electric double layers are overlapped for 0.1M KClsolution (Debye length λ =0.96nm). The Debye length for 1M KClsolution is 0.3nm and the electric double layers are thin compared to the gap between the DNA and the nanopore. Based on equation (5.14), the dimensionless shear stress acting on the DNA surface τ = du dr is du = σ dr d + ζ w ζ d r=a a ln(a) which is scaled as τ = τ τ 0,τ 0 = µu 0 h 0 (see Table 2.1 for the definition of h 0 ). The corresponding dimensionless viscous drag acting on the DNA surface is F d = 1 0 u dx τ = u r r=a r r=a if this force is scaled as F d = F d F 0, where F d = L 0 2πµa u r dx and F 0 = 2πµU 0a L h 0. As shown in Figure 5.5, the dimesionless shear stress τ at r = a on the DNA surface is smaller for higher electrolyte concentration and the viscous drag acting on the DNA is thus lower. It is evident from this plot that the concentration of the electrolyte in the upstream reservoir plays an important role in determining the forces acting on the DNA surface. It is noted that the viscous drag force acting on the DNA moving in a converging nanopore due to the bulk flow is larger for higher electrolyte concentration, in conflict with the results shown here. It is mainly due to the induced pressure gradient by EOF and DNA movement in the converging nanopore. In the nanopore with constant cross-section as discussed in this section, there is no induced pressure gradient. 145

170 5.4 Lubrication Solutions for the Flow Field in a Conical Nanopore: The Viscous Drag Force Acting on the DNA inside the Nanopore In Section 5.2, the governing equations and boundary conditions are discussed to calculate the flow field and the viscous drag force acting on the DNA immobilized in the nanopore (Fd ). In Section 5.3, the analytical solution for EOF in an annulus is derived based on the Debye-Hückel approximation to simulate the problem of a DNA placed at the center of a cylindrical nanopore with constant cross section. In this section, the solutions are generalized to investigate the flow field in a conical nanopore with a immobilized DNA. Due to the electrostatic repulsion of the negatively charged nanopore and the negatively charged DNA molecule, the DNA molecule tends to stay at center of the nanopore and it is modeled as a rigid rod placed at the center of the nanopore. Similar to Chapter 4, the analysis discussed here is based on the lubrication approximation. To simplify the problem, the streamwise velocity is decomposed into two components: the electroosmotic flow (u eof ) and the pressure driven flow (u p ) as in Chapter 4. The velocity equation for each component is given by 1 r ( r r u eof r ) = β ɛ E 2 I,x z i X i (5.15) i 1 r ( r r u p r ) = p x (5.16) with boundary conditions r = a : u eof =0,u p =0 r = h : u eof =0,u p =0 The pressure equation is based on mass conservation inside the pore Q Q eof + Q p = constant where Q is the total flowrate and Q eof and Q p are the flowrate due to the EOF and 146

171 pressure-driven flow components. For the electroosmotic component, the flowrate Q eof can be obtained by Q eof = h a u eofrdr where u eof is the numerical solution of equation (5.15). The pressure-driven flow component can be solved from equation (5.16) and the analytical solution is u p = dp ( r 2 dx 4 + a2 ln(r/h) h 2 ) ln(r/a) 4ln(h/a) (5.17) The flowrate Q p is then given by Q p = h a u p rdr = dp ( (a 2 h 2 ) 2 dx 16ln(h/a) + a4 h 4 ) 16 (5.18) Note that Q eof,q p are all varying along x axis but from the conservation of mass, the total flowrate should be a constant (Q = constant). Therefore the axial derivative Q = Q x = Q eof + Q p = Q eof x equation becomes + Qp x =0; substituting Q eof and Q p into the equation, the pressure d 2 ( p (a 2 h 2 ) 2 dx 2 16ln(h/a) + a4 h 4 ) + dp ( (a 2 h 2 ) 2 16 dx 16ln(h/a) + a4 h 4 ) + Q eof 16 =0 (5.19) This is a second order differential equation and it can be solved numerically with the boundary conditions p =0, x =0, 1 The flow field can be determined from numerical calculations based on equations (5.15), (5.16), (5.19) and the forces acting on the DNA can be calculated as: L Fe = 2πa σd E I,x dx =2πa σd 0 L 0 EI,x dx =2πa σd V (5.20) in dimensional form. (van der Heyden et al., 2005) have shown that varying the electrolyte concentration has little effects on the surface charge density and so the surface charge density of the DNA (σ d 0.15C/m2 ) is assumed to be a constant here. The typical applied 147

172 voltage drop used in experiments is 120mV ; a = 1nm is the radius of the DNA and the corresponding electrical driving force acting on the DNA is 113pN calculated from equation (5.20). The viscous drag force can be calculated from the shear stress on the DNA surface F d = L 0 2πµa u r dx (5.21) and in dimensionless form (F d = F d F 0 ) F d = 1 0 u dx (5.22) r r=a This force can only be calculated based on the numerical results for the flow field in the nanopore. The calculation of this force and its magnitude will be discussed in Section 5.6. Note that the solution derived in this section applies to a converging/diverging nanopore (h 0) as well as a pore with constant radius (h =0). It is a more generalized solution compared to the solution discussed in Section Uncoiling/recoiling Forces Acting on the DNA The previous section discussed the modeling of the fluid flow and viscous drag force acting on the DNA inside the nanopore. In this section, the uncoiling and recoiling force due to DNA conformational changes during the translocation process is investigated. For a DNA in transition as shown in Figure 5.6 (a), the DNA segments are gradually uncoiled in region I and recoiled in region II as the DNA transports through the nanopore. In this section, the free energy and forces related to this uncoiling-recoiling process are investigated to determine if they play important roles on determining the DNA velocity. As mentioned in Chapter 2, the DNA chain in the liquid solution continuously changes its conformation in response to random Brownian forces. One of the inherent properties 148

173 (a) DNA in transition (Muthukumar, 1999) entropic force (pn) fraction of DNA in region I (d) uncoiling-recoiling force Figure 5.6: The schematic diagram of the DNA-nanopore system used to investigate the uncoiling-recoiling force due to the DNA conformational change during the DNA translocation process (shown in (a)) and the resulting uncoiling-recoiling force (shown in (b)) for a 16.5kbps double stranded DNA. 149

174 of an isolated flexible polymer chain in solution is its ability to assume a large number of conformations. Different models have been used to investigate the deformation, stretching and relaxation of single polymer chains. Among these models, the simplest one is the ideal chain model, in which the DNA are modeled as randomly moving nodes while keeping the connectivity. To model a real chain, the hydrodynamic interactions, backbone rigidity and the excluded volume of the chain need to be considered, making it impossible to find the theoretical solution (Dutcher & Marangoni, 2004). Unlike a rigid rod, DNA coils randomly in solution, resulting in an average end-toend distance much shorter than its contour length. Pulling the DNA into a more extended chain is entropically unfavorable, as there are fewer possible conformations at longer extensions, and the resulting uncoiling force increases as a random coil is pulled from the two ends (Bustamante et al., 2003). A Worm-Like-Chain model (Dutcher & Marangoni, 2004), which considers a polymer as a line that bends smoothly under the influence of random thermal fluctuations, is often used to describe DNA entropic elasticity. Based on the Worm-Like-Chain model, the force is related to the extension of a DNA as (Bustamante et al., 2003) Fl p kt = (1 x/l) x L (5.23) where F is the force acting on the ends of the DNA, x is the extension of the DNA, k is the Boltzmann constant, T = 300K is the temperature of the environment and L is the contour length of DNA. Calculations based on this formula show that the required force to extend the DNA significantly is on the order of kt/l p, which is about 0.1pN for a double stranded DNA (l p =50nm). 150

175 For the configuration shown in Figure 5.6 (a)(muthukumar, 1999), the free energy for a DNA in transition is calculated based on the formula of a polymer anchored to a wall as H kt =0.5(ln(m)+ln(N m)) for a Gaussian chain (Muthukumar, 1999). Here H = U TS is the Helmholtz free energy, which measures the useful work obtainable from a closed thermodynamic system; U is the internal energy of the system and S is the entropy; N = L/l p is the total number of segments and m is the number of segments in region I. The corresponding entropic uncoiling/recoiling force (F E )is F E = dh dx = kt N 2m (5.24) 2l p Nm m 2 and it is plot in Figure 5.6 (b) for a 16.5kbps double stranded DNA (L =5.6µm). The persistence length of the DNA is 50nm and T = 300K. It is shown that this entropic force only depends on the number of the DNA segments in region I and II, and the magnitude of this force is very small (O(0.01pN)) compared to the electrical driving force (113pN) discussed in the previous section. This is a very simple model describing the force related to DNA uncoiling and recoiling in the DNA translocation process, which was considered to be the key mechanism controlling DNA translocation velocity in some of the previous work (Muthukumar, 1999, 2007; Kantor & Kardar, 2004; Forrey & Muthukumara, 2007). Note that the uncoiling/recoiling force given by equation (5.24) cannot account for the effects due to the confinement of the nanopore and the flexibility of the polymer and it is shown here that the resulting uncoiling/recoiling force is negligible compared to the electrical driving force (usually O(100pN)). Odijk (1983) has considered the free energy of a DNA partially confined in a cylindrical pore, with the effects of the polymer rigidity and the confinement due to the nanopore. The 151

176 Figure 5.7: Schematic drawings of a DNA with one end dragged into a cylindrical nanopore. The radius of the nanopore is h and the force acting on the DNA is f. pore diameter is in the regime of h l p L DNA where h is the radius of the pore, which is consistent with the case discussed in the present work. Odijk showed that the free energy, which is the work required to reversibly insert the polymer into the nanopore, is given by H R // = A 0kT l 1/3 p h 1/3 (5.25) where R // is the length of the pore occupied by the polymer and A 0 = is a constant. The corresponding entropic force is F E = dh dr // = A 0kT l 1/3 p h 1/3 (5.26) Note that this entropic force is related to the height of the nanopore and the persistence length of DNA, which is a measure of polymer flexibility. For a h =5nm cylindrical pore, the entropic uncoiling force is calculated to be 0.667pN to insert a double stranded DNA into the pore. Monte Carlo simulations by Klushin et al. (2008) investigate the process of dragging DNA into the nanotube as shown in Figure 5.7. The position of one end of the DNA (x) is progressively moved further inside the nanotube in a quasi-equilibrium process. Due to the thermodynamic interactions between the DNA and the surrounding molecules, the 152

177 tail outside the nanopore remains as an un-deformed coil while the part of the chain inside forms a linear string. This configuration is called flower structure (Klushin et al., 2008) since the coiled tail resembles a flower and the chain inside resembles a stem. Klushin et al. (2008) have shown that the free energy H associated with this flower structure is given by H = k BTx 2h A [ w α 1 + Bw δ 1 + Cw 1] (5.27) where w = s(2h /a) 1+1/υ ; α =(3υ 1) 1 ; δ =(1 υ) 1 ; where υ = is the scaling index for a three-dimensional self-avoiding chain; s = x /na; n is the instantaneous number of confined monomers in the stem and x is the length of the stem; a =1 is the lattice spacing used in the Monte Carlo simulations; A = 1.48, B = 0.67 and C =1.98 are the parameters obtained from matching with the numerical solutions based on the Monte Carlo simulations. The corresponding entropic uncoiling force is F E = dh dx = k BT 2h A [ w α 1 + Bw δ 1 + Cw 1] (5.28) Note that this uncoiling force is not a function of DNA contour length since only the part of the DNA entering the nanopore contributes to the free energy increase during the translocation process and it is proportional to 1/h which means that a higher force is needed to drag DNA into smaller pores. For a nanopore with h =5nm, the uncoiling force is calculated to be 1.9pN according to Klushin s formula, larger than the force calculated from Odijk s formula. According to Klushin et al. (2008), the stem of the partially confined conformation is stretched more strongly due to the additional stretching force coming from the deformation of DNA when it enters the pore, which was not considered in Odijk s work. Once the chain is confined completely, there is no tail outside the tube and the reaction force at the controlled chain 153

178 end disappears. The reverse process in which the chain leaves the confinement does not have this effect and it occurs smoothly without any jumps. The entropic force from DNA recoiling (F R ) in the receiver reservoir is given by Prinsen et al. (2009) F R = kt l p It is not a function of the length of the chain outside the nanopore and it is about 82.9fN for a double stranded DNA (T = 300K, l p =50nm). In this section, the uncoiling and recoiling forces due to DNA conformational changes during the translocation are investigated based on different models. To summarize, the magnitude of these forces are listed in Table 5.1 as well as the electrical driving force for a 16.5kbps double stranded DNA through a nanopore (h =5nm). It is shown here that these forces are very small compared to the electrical driving force and thus they cannot be the main resisting force determining the DNA translocation velocity. F U FU FR Force (Muthukumar, 1999) FU (Odijk, 1983) (Klushin et al., 2008) 1.9pN (Prinsen et al., 2009) 0.083pN F e Magnitude 0.01pN 0.67pN 113pN Table 5.1: Comparison between the magnitudes of the uncoiling and recoiling force (F U and F R ) calculated based on different models and the electrical driving force (F e ). 5.6 Viscous Drag on the DNA Outside the Nanopore In this section, viscous drag forces acting on the DNA blob-like configuration outside the nanopore (as shown in Figure 5.1) are investigated. Storm et al. (2005b) proposed a 154

179 simple model based on the balance of the electrical force and the viscous drag on the bloblike DNA configuration outside the nanopore to determine the translocation velocity. They used the classical formula for flow past a sphere to calculate this viscous drag acting on the gyration of the DNA outside the pore (shown in Figure 5.1) which is given by F blob1 =6πµR gv blob (5.29) where vblob is the approaching velocity of the DNA blob-like configuration outside the pore and R g is the radius of gyration, defined as the average square distance of the chain segments from the centre of the mass of the chain, which characterizes the size and shape of the polymer. The velocity of the blob-like DNA outside the pore moving toward to the pore is given by v blob = dr g dt = ξdlυ dt (5.30) where υ is the scaling index for a real chain and ξ is defined as an effective friction coefficient, the value of which is determined by experimental data fitting in his model. They focused on the deduction of the relation between the DNA translocation velocity (u DNA ) and the length of the DNA (L) using the scaling law and the magnitude of this force is not estimated. In their model, the viscous drag from the electroosmotic flow inside the nanopore is assumed to be much smaller than the viscous drag acting on the blob. If the radius of gyration for an ideal chain is given by R g = Llp 6 (5.31) where L is the DNA contour length and l p is the persistence length. The velocity of the DNA blob is then given by vblob = dr dt = 1 lp dl 2 6L dt 155 (5.32)

180 Since the reduction of the contour length of the blob-like DNA outside the nanopore is due to the DNA translocation in the pore, u DNA = dl dt and so the velocity of the blob-like configuration is related to the DNA translocation velocity by v blob = 1 2 lp 6L u DNA (5.33) The viscous drag acting on the DNA outside the pore F blob is based on the classical theory of flow past a sphere given by equation (5.29). Substituting equations (5.31) and (5.33) into equation (5.29), the viscous drag force acting on the blob-like DNA configuration is F blob1 = πµl pu DNA 2 Note that this force is independent of the DNA contour length. Using the experimental DNA translocation velocity u DNA =0.01m/s (Storm et al., 2003; Li et al., 2001) for a double stranded DNA through a synthetic nanopore, this viscous drag force is 7.85 pn. The viscosity of water is used in the calculation µ =0.001Pa s. As shown in Figure 5.1, dragged by the linear DNA moving inside the nanopore, the blob-like DNA tends to stay close to the opening of the nanopore, facing the fluid flow discharged from the nanopore in he opposite direction. The simplest way to characterize this force is to use the flow past a solid sphere model Llp F blob2 =6πµR gu i =6πµU i 6 where Ui is the velocity of the fluid flow discharged from the nanopore at the opening. Note that this force is related to the length of the DNA L outside the nanopore and it vanishes as the entire DNA moves to the other side of the nanopore. The experimental parameters of Storm et al. (2003) are used to estimate the magnitude of this force. As shown in Figure 5.1, the nanopore has a large end of 223nm and a small end of 5nm in radius. The total length 156

181 of the nanopore is 340nm. The surface charge of the pore is 0.2C/m 2 and the applied voltage drop is 120mV. The working solution is 1M KCL solution and the electroosmotic flow velocity at the nanopore entrance is given by U i O(10 5m). The viscous drag on the DNA (16.5kbps double stranded DNA) outside the nanopore is about 0.1pN, if µ =0.001Pa s is taken for the viscosity of the solution. F blob1 Force (Storm et al., 2005b) 7.86pN Fblob2 Fe Magnitude 0.1pN 113pN Table 5.2: Comparison between the magnitudes of the viscous drag force acting on the DNA blob-like configuration outside the nanopore and the electrical driving force (Fe ). The viscous drag due to the approaching of the DNA blob to the nanopore is defined as Fblob1 and the viscous drag force acting on the DNA blob in Reservoir I due to the flow discharged from the nanopore is defined as Fblob2. In this section, the viscous drag forces acting on the DNA blob-like configuration outside the nanopore are calculated. As shown in Table 5.2, these two viscous forces are small compared to the electric driving force. Therefore, they are not the most important forces controlling DNA transport. 5.7 Numerical Validation of the Lubrication Model using Tethering Force Data In Section 5.3, a generalized solution is derived to calculated the flow field and the viscous drag force acting on an immobilized DNA molecule inside a nanopore. In this section, numerical results are calculated based on based on the lubrication solutions derived 157

182 in Section 5.4. The calculated tethering force is compared with the experimental data to validate the model. (a) Experimental setup (b) Force balance Figure 5.8: The schematic diagram for (a) the experimental setup(keyser et al., 2006) and (b) the force balance. The DNA is attached to a polystyrene bead and a tethering force F t is used to immobilize the DNA. The electrical driving force F e and the viscous drag force F d are also shown. To understand the mechanism of DNA transport through a nanopore, Keyser et al. (2006) measured the force acting on the DNA experimentally. As shown in Figure 5.8, they attached a DNA to a polystyrene bead and placed the DNA in a nanopore. Under an applied electric field, the DNA tends to move to the anode and the bulk electroosmotic flow moves to the cathode in the negatively charged nanopore, exerting a drag force on the DNA surface. A tethering force (Ft ) is applied on the polystyrene bead to immobilize the DNA by an optical tweezer and the force balance on the DNA inside the pore is given by F t = F e F d 158

183 Figure 5.9: The geometry of the nanopore used in Keyser s experiments. The nanopore is a 20nm long cylindrical nanopore with 5nm in radius connected to a 20nm long converging nozzle on one end and a 20nm long diverging diffuser on the other end. The slope of the converging or diverging wall is about 23. where F t is the tethering force; F e is the electrical driving force and F d is the viscous drag force. Note that the DNA is immobilized in the nanopore for the force measurement. To simulate Keyser s experiments, the DNA velocity is assumed to be zero in this section. The geometry of the nanopore used in Keyser s experiments (Figure 5.8) is shown in Figure 5.9. The nanopores are drilled in a transmission electron microscope by tightly focusing the electron beam on a 60 nm thin SiO 2 /SiN/SiO 2 membrane. Focused electron beam irradiation leads to the formation of a small hole by sputtering atoms from the membrane(keyser et al., 2006). The resulting nanopore is a 20nm long cylindrical nanopore with 5nm in radius connected to a 20nm long converging nozzle on one end and a 20nm long diverging diffuser on the other end. The slope of the converging or diverging wall is about 23. As mentioned before, the solutions derived in Section 5.4 is a generalized solution applicable to converging/diverging nanopores as well as cylindrical nanopores with constant cross section. In this problem, the solution is valid for all the three regions of the 159

184 nanopore shown in Figure 5.9, while the solution discussed in Section 5.3 can only be used in the region where the radius of the nanopore is constant. Note that the computational domain and the coordinate system shown in Figure 5.2 is turned 90 degrees clockwise to match with Keyser s configuration shown in Figure 5.8. The computational domain is the region between the DNA surface (left boundary) and the nanopore wall (right boundary) as shown in Figure 5.8. As discussed in Section 5.3, the pressure-driven flow component is solved analytically but the electroosmotic flow component and the pressure field must be computed numerically. To find the flow field, the numerical calculations include 4 steps: Step 1: Solving the Poisson-Boltzmann equations numerically The equation is discretized using the second-order central difference approximations for all the derivative terms. The finite difference equations are obtained in the form: φ j+1 2φ j + φ j+1 + r j r 2 (φ j+1 φ j 1 )= r 2 β ɛ 2 zi X 0 i e z iφ j (5.34) for point j, where X 0 i is the mole fraction for species i in the upstream reservoir. The coupling equations for the mole fractions and potential are solved iteratively using a Gauss- Seidel scheme, while the equation for potential (Equation (5.34)) is solved using the Successive Over-relaxation (SOR) method at each iteration. The iterative procedure is said to converge if φ new φ old <δ (5.35) φ new where δ is a small number and it is usually taken as 10 4 as the convergence criteria. Step 2: Solving u eof numerically Equation (5.15) for the streamwise velocity component u eof is descritized using central 160

185 Y PHI Y Cl X (a) potential X (b) Concentration of Cl Y (c) Streamwise Velocity X (d) Streamlines Figure 5.10: Numerical results for the potential, concentration of cations, streamwise velocity and the streamlines calculated based on the lubrication approximation. The computational domain is the region between the DNA surface (left boundary) and the nanopore wall (right boundary). The nanopore is a 20nm long cylindrical nanopore with 5nm in radius connected to a 20nm long converging nozzle on one end and a 20nm long diverging diffuser on the other end. The surface charge of the nanopore is 0.06C/m 2 and the applied voltage drop is 120mV. The electrolyte solution is 0.1M KCl. 161

186 Mesh x =0.2,r = x =0.4,r = φ = , u eof =2.9439, p = φ = , u eof =2.4184, p = φ = , u eof =2.9443,p = φ = , u eof =2.4196, p = x =0.6,r =0.08 x =0.8,r = φ = , u eof =1.8381,p = φ = , u eof =1.8363, p = φ = , u eof =1.8395, p = φ = , u eof =1.8391, p = Table 5.3: The numerical accuracy check for the numerical calculation based on the experimental parameters (Keyser et al., 2006) (results are shown in Figure 5.10). The results for grid (Mesh I) are compared with the results of (Mesh II) grid and two-digit accuracy has been attained. Note that the pressure shown here is rescaled on E I,x ). differencing as u j+1 2u j + u j+1 + r j r 2 (u j+1 u j 1 )= r2 E I,x β ɛ 2 zi X 0 i e z iφ j (5.36) for point j and solved using the Thomas algorithm. Step 3: Integrating Q eof numerically Based on the solution for u eof, Q eof can be integrated numerically based on the trapezoidal rule. Step 4: Solving the pressure equation numerically The pressure equation (5.19) is solved using the Thomas algorithm, and the pressure-driven component u p is then calculated based on equation (4.7). The transverse velocity can be calculated from equation (4.29) numerically using the trapezoidal rule. A mesh is used for the numerical calculations and two digit accuracy is obtained by comparing with the numerical results calculated based on a mesh as shown in Table 5.3. Results for the potential, the concentration of ions and the flow field are shown in Figure 5.10 for Keyser s experimental parameters. The surface charge density of the nanopore is 0.06C/m 2 for Keyser s experiments by (Smeets et al., 2006); the voltage drop is 120mV 162

187 and the working fluid is 0.1M KClsolution. The double layers are overlapped in the central cylindrical pore as shown in the potential and Cl profiles. Since the applied electric field is inversely proportional to the cross-sectional area of the nanopore, the streamwise velocity u is much higher in the small cylindrical pore than the converging and diverging regions. The fluid flow moves in the direction from the bottom to the top of the nanopore as shown in Figure 5.10 (d), in the same direction as the applied electric field (not shown in the figure), while the negatively charged DNA wants to move in the opposite direction but cannot because of the tethering force. Tethered Force (pn) Numerical Lubrication Experiment,L b =2.1µ m Experiment,L b =2.4µ m Experiment,L b =2.9µ m COMSOL Applied Voltage Drop (mv) Figure 5.11: Comparison of tethering forces between the experimental data, the numerical results based on the lubrication approximation and the numerical results using COMSOL. The electrolyte solution is 0.1M KCl. Experimental data are recorded for one molecule at three different distances from the nanopore (filled squares 2.1µm, filled circles 2.4µm, filled triangles 2.9µm(Keyser et al., 2006). As shown in Figure 5.8, L b is the distance between the bead and the nanopore. The calculated tethering force based on the lubrication approximation is compared with the experimental data in Figure As shown in Figure 5.11, the tethering force is a linear function of the applied voltage drop. The total electrical driving force is 113pN for 163

188 an applied voltage drop 120mV and the viscous drag force due to the electroosmotic flow acting on the DNA inside the nanopore is 83.5pN and the tethering force is 28.5pN. The tethering force accounts for 25% of the total electric driving force (Fe = F d + F t )) and the remaining 75% comes from the viscous drag force due to the bulk flow for all the applied voltage drops shown in the Figure. Note that the nanopore has a large slope with ɛ 1 =0.83 = O(1) but the results still compare very well with the experimental data even though the lubrication approximation is not satisfied. To validate the results (shown in Figure 5.10) based on the lubrication approximation, a full numerical calculation is carried out for the same problem using COMSOL. In this model, Poisson-Nernst-Planck equations are used to calculate the potential and ionic concentrations and the Stokes equation is used to calculate the flow field, without neglecting the axial derivative terms (ɛ x 2 ) terms. Finite element methods (FEM) are used to numerically calculate the equations. As shown in the two-dimensional geometry of the computational domain (Figure 5.12 (a)), two big reservoirs are connected by the nanopore and the DNA is placed at the center of the nanopore (not shown). Extra fine meshes are used to resolve the double layers and coarser meshes are used in the reservoirs (shown in (b)). As shown in Figure 5.12 (c), the concentration distribution are quite different from previous results calculated based on the Poisson-Boltzmann equations and the lubrication approximation (shown in Figure 5.10). As discussed in Chapter 3, Boltzmann distribution is derived for a system under equilibrium, while in the problem considered here, the ions are migrating under the applied electric field ( 10 6 V/m in the central cylindrical nanopore), inducing concentration polarization, or ion density gradients in the nanopores. In the central cylindrical nanopore, the electrical double layers are overlapped and the electrostatic interaction between the ions and 164

189 (a) geometry (b) mesh (c) Cl concentration (d) streamwise velocity Figure 5.12: The geometry of the nanopore, the mesh and the numerical results for the concentration of anions, streamwise velcity. The nanopore is a 20nm long cylindrical nanopore with 5nm in radius connected to a 20nm long converging nozzle on one end and a 20nm long diverging diffuser on the other end. The surface charge of the nanopore is 0.06C/m 2 and the applied voltage drop is 120mV. The electrolyte solution is 0.1M KCl. 165

190 the negatively charged surfaces are stronger than in the converging and diverging regions. Negatively charged ions (Cl ) migrate in the applied electric field and feel a stronger repulsion when entering the central cylindrical pore, forming a region (near x = m in Figure 5.12 (c)) of higher concentration. As the Cl ions migrate out of the central nanopore (near x = m in Figure 5.12 (c)), the withdrawal of the ions is not compensated for by the intake from the nanopore, leading to a decrease in concentration. Although the results calculated from the Boltzmann distribution and the lubrication approximation are different from the results calculated using Poisson-Nernst-Planck model (COMSOL), the tethering force calculated based on these two sets of results is very close. As shown in Figure 5.11, COMSOL results fit the experimental data better but the difference between the full numerical solution (COMSOL) and the lubrication approximation results is not significant. In this section, numerical results have been calculated based on the lubrication solutions derived in Section 5.3 for the flow field and the viscous drag force acting on the DNA. The tethering force, which is given by F t = F e F d, is calculated and compared with the experimental data to validate the model. The good agreement between the numerical results and the experimental data has proved that the viscous drag force acting on the DNA inside the nanopore is the main resisting force for DNA passage. 5.8 Summary In this chapter, the forces that may affect DNA translocation are investigated, and the magnitude of these forces are listed in Tabel 5.4. These forces are calculated for a 16.5kbps double stranded DNA in a nanopore under a 120mV applied voltage drop. The radius of the pore is 5nm at the small end. It is shown here that the electrical driving force (Fe )is 166

191 about 100pN, and the uncoiling and recoiling forces (F U and F R) due to DNA conformational change are about two orders of magnitude less than the this driving force, while the viscous drag force acting on the DNA blob-like configuration outside the nanopore (F blob1 and Fblob2 ) is about one or three orders of magnitude less. In some of the previous work, these forces, including FU, F R and F blob1, are considered to be the main resisting forces balancing with the electric driving force and it is shown here that they are not the key forces controlling DNA translocation. Force FU (Muthukumar, 1999) FU (Odijk, 1983) 0.67pN FU (Klushin et al., 2008) 1.9pN FR (Prinsen et al., 2009) Fblob1 (Storm et al., 2005b) Fblob2 Fe Fd Magnitude 0.01pN 0.083pN 7.86pN 0.1pN 113pN 83.5pN Table 5.4: Comparison between the magnitudes of the forces that may affect DNA translocation, calculated for a 16.5kbps double stranded DNA through a nanopore with a radius of 5nm at the small end. The applied voltage drop is 120mV. Note that the viscous drag force acting on the DNA inside the nanopore (F d ) listed here is calculated for a DNA immobilized in the nanopore (u DNA =0). As shown in the table, the only force comparable to the electrical driving force is the viscous drag force acting on the DNA inside the nanopore (Fd ). Note that in the calculations for this drag force, the DNA is assumed to be immobilized in the nanopore (u DNA =0) to match with the experimental data given by Keyser et al. (2006). When the DNA is released, this force is even larger due to the movement of the DNA, which will be discussed in Chapter

192 The force acting on the DNA inside the nanopore (F d ) is calculated from the model based on the lubrication approximation, which is extended from the model developed for EOF in a converging nozzle in Chapter 4. Results have shown that the electroosmotic flow introduces a large viscous drag on the DNA surface and the magnitude of this drag force is usually about 60% 80% of the electric driving force for a immobilized DNA placed inside a nanopore. Therefore, the hydrodynamic interaction between the DNA and the nanopore plays an important role in determining the DNA translocation velocity. The numerical results (both lubrication and COMSOL results) are in good agreement with the experimental data given by Keyser et al. (2006) for the tethering force (F t = F e + F d ), validating the model for an immobilized DNA placed in a converging/diverging nanopore discussed in this chapter. This model is then extended to calculate the DNA translocation velocity in the next chapter. 168

193 CHAPTER 6 DNA Translocation Velocity 6.1 Introduction Figure 6.1: Illustration of DNA characterization system based on the translocation of DNA through a nanopore. In the last chapter, the forces affecting DNA translocation are investigated and the viscous drag force acting on the DNA inside the nanopore has been shown to be the main resisting force. Based on the balance between this force and the electrical driving force, a model is established to calculate the tethering force acting on a DNA immobilized (u DNA =0) in the nanopore, validated by the good agreement with the experimental data. In this chapter, this model is extended to account for the movement of DNA. 169

194 The electrical characterization of DNA using a nanopore has been of interest in the recent years. Compared to the conventional techniques of DNA sequencing, this technique can be more efficient and can have higher throughputs. The basic idea of the characterization is shown in Figure 6.1. The two reservoirs are filled with saline buffer solution (usually KCl in the experiments), and separated by a membrane having a nanopore; an anode and a cathode are set in each reservoir, and the pore provides the only path for ionic currents and the electrophoretic movement of DNA. When DNA electrophoretically moves from the cathode to the anode, the ionic current through the nanopore changes due to pore blockades, and the current fluctuations can determine the characteristics of the DNA. In this chapter, a hydrodynamic model is developed to investigate the fluid flow, ionic transport and DNA velocity for a moving DNA in a converging nanopore. Previous work on this problem is often based on the Debye-Hückel approximation (valid for absolute value of φ 26mV ) and have assumed a constant pore size. It is known that DNA has a charge of 2e per base pair where e = C which is equivalent to a surface charge of 0.15C/m 2 if DNA is treated as a rigid cylindrical rod having 1 nm in radius. For such a high surface charge density, the zeta potential is as high as 109mV for a 0.1M KCl solution and 55mV for a 1M KClsolution (thin EDL limit). Obviously the Debye- Hückel approximation is not valid in this regime. An asymptotic solution is developed to resolve the abrupt change of potential and velocity close to the nanopore and the DNA surface when the nanopore radius is much larger than the Debye length. The entry and exit problem is also considered. 170

195 Figure 6.2: The geometry of the nanopore and the DNA. 6.2 Governing Equations and Boundary Conditions A typical nanopore geometry is depicted on Figure 6.2. A converging nanopore is connected to a cylindrical nanopore at the small end, which is the general geometry of the solid-state nanopores used in experiments. The variables are defined similar to Chapter 5 and the only difference is that DNA is moving at a speed u DNA. The purpose of this chapter is to determine this translocation velocity. The governing equations for electric potential, mole fractions of ionic species and the velocity field is given by equations 2.27, 2.32, 2.31, 2.35 and 2.36 discussed in Chapter 2. As discussed in chapter 5, the governing equations can be simplified to equations 5.1, 5.2 and 5.4 based on the lubrication approximation. The boundary conditions are r = h(x) : φ r = σ w,u=0,v =0 r = a : φ r = σ d,u= u d,v =0 171

196 Note that the only difference between the governing equations and boundary conditions in Chapter 5 and 6 is the moving DNA surface u = u d, where u d = u DNA U 0 is the dimensionless translocation velocity. 6.3 Solution for the Flow Field Unlike Chapter 5, another velocity component, the Couette flow component (u ct ), is added into the flow field to account for the movement of the DNA, and the streamwise velocity is then decomposed into three components: the electroosmotic flow (u eof ), the pressure driven flow (u p ) and the Couette flow (u ct ). The velocity equation for each component is given by 1 r ( r r u eof r ) = β ɛ 2 E(x) i z i X i (6.1) 1 r ( r r u p r ) = p x (6.2) 1 r ( r r u ct r ) =0 (6.3) with the boundary conditions r = a, u eof =0,u p =0,u ct = u d r = h, u eof =0,u p =0,u ct =0 The pressure equation is based on conservation mass inside the pore Q Q eof + Q p + Q ct = constant where Q is the total flowrate and Q eof, Q p and Q ct are the flowrate due to the three components. For the electroosmotic component, the flowrate Q eof can be obtained by Q eof = h a u eofrdr where u eof is the numerical solution of equation (6.1). The solution 172

197 for the Couette component can be found analytically by solving equation (6.3) with its boundary conditions: and the flowrate Q ct is given by u ct = u d ln(r/h) ln(a/h) (6.4) Q ct = h a u ct rdr = u d( h 2 + a 2 2a 2 ln(a/h)) 4lna/h (6.5) The pressure driven flow component solution is given by equation (5.17) and the flowrate Q p is given by equation (5.18). Note that Q eof,q ct,q p are all varying along x axis but from the conservation of mass, the total flowrate should be a constant (Q = constant). Therefore the axial derivative Q = Q x = Q eof + Q ct + Q p = Q eof x + Qct x + Qp x =0. Substituting Q eof, Q ct and Q p into this equation, the pressure equation is then given by d 2 ( p (a 2 h 2 ) 2 dx 2 16ln(h/a) + a4 h 4 ) + dp ( (a 2 h 2 ) 2 16 dx 16ln(h/a) + a4 h 4 ) + Q eof 16 + Q ct =0 (6.6) This is a second order differential equation and it can be solved numerically with the boundary conditions p =0;x =0, 1 Note that the only difference between equation (6.6) with the pressure equation (5.19), derived in Chapter 5 is that equation (6.6) includes the Q ct term that accounts for the flow field due to the DNA motion. A force balance ( F = F e + F d =0) is used as an additional condition to find DNA velocity (u dna ) where the dimensional electrical driving force F e is given by equation (5.20) and F d (dimensional) is given by equation (5.21). The algorithm used to solve for the flow field and the DNA velocity is shown in Figure 6.3. Since the DNA velocity is unknown, an initial value for DNA velocity is guessed and the three components of the streamwise 173

198 Figure 6.3: The algorithm used to calculate DNA velocity. velocity can be then calculated. Next the drag force is calculated based on the flow field and the force balance is checked. If the force balance is not been achieved, a new DNA velocity is used to repeat the process until the force balance is satisfied. Second order finite difference methods are used to discretize the potential equation, EOF velocity equation and pressure equation and the Thomas algorithm is used to solve these equations. The details of the numerical methods are similar to those in Chapter Asymptotic Solution for potential φ and the EOF velocity u eof Note that in some cases (Storm et al., 2005a), the radius of the converging nanopore changes from O(100 nm) at the inlet to O(1 nm) at the outlet and the Debye length is O(1 nm). For the regions near the inlet, the radial space between the DNA surface and the wall is large comparing to the Debye length. Thus the numerical calculation cannot resolve the abrupt change of potential and mole fractions at the DNA surface and the wall. Similar to the analysis discussed in Section 4.5 for the EOF in a micronozzle/diffuser (ɛ 1), 174

199 asymptotic analysis can be used to find the solution for the potential and the electroosmotic flow component (u eof ). Figure 6.4: The inner region and outer region used for the asymptotic analysis. The equation for potential is given by equation (2.27) and for the region close to the wall of the nanopore where r = h, usually called the inner region (shown in Figure 6.4), a new variable is defined as y = h r ɛ so that as r = h, y =0and r =0, y. Here ɛ = λ h i is the ratio of the Debye length to the radius of the nanopore at the inlet. Substituting y the potential equation becomes d 2 φ dy 2 = β i z i X i + O (ɛ) For small ɛ, the second term on the right hand side is neglected and for binary monovalent species, the potential equation is given by d 2 φ dy 2 = sinhφ with boundary conditions y =0, y, dφ dy = σ wɛ φ y =0 175