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

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 c Copyright by

Lei Chen

2009 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-on- a-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 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 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 Poly- meric Biomedical Devices (NSF Grant No. EEC-0425656) 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 Xi- aoyin for their encouraging and support.

iv VITA

April 9, 1979 ...... Born - Beijing, China

July 2001 ...... B.S. Mechanical Engineering Department of Thermal Engineering Tsinghua University, Beijing, China August 2001 - June 2003 ...... Graduate Engineer HVAC Lab Tsinghua University Beijing, China September 2003-Jan 2006 ...... M.S. Mechanical Engineering Department of Mechanical Engineering The Ohio State University Columbus, OH Feb 2006 - 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 2009-1121, 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, 251-258, 2009

Lei Chen and A. T. Conlisk, “Electroosmotic ﬂwo and particle transport in micro/nano nozzles and diffusers”, Biomedical Microdevices, Vol. 10, 289-298, 2008

v Lei Chen, P. Gnanaprakasam and A. T. Conlisk, “Electroosmotic Flow in Micro/nano Nozzles”, 45th AIAA Aerospace Sciences Meeting and Exhibit, AIAA paper 2007-0931, 2007

Lei Chen and A. T. Conlisk, “Generation of nanovortices in electroosmotic ﬂow in nanochannels”, 4th AIAA Theoretical Fluid Mechanics Meeting, AIAA paper 2005-5057, 2005

Lei Chen and A. T. Conlisk, “Electroosmotic ﬂow in annular diatoms”, 43rd AIAA Aerospace Sciences Meeting and Exhibit, AIAA paper 2005-0674, 2005

FIELDS OF STUDY

Major Field: Mechanical Engineering

Studies in Fluid Mechanics, Mass Transer and Electrokinetics: Professor A.T. Conlisk

vi TABLE OF CONTENTS

Page

Abstract ...... ii

Acknowledgments ...... iv

Vita...... v

List of Tables ...... x

List of Figures ...... xii

Chapters:

1. Introduction ...... 1

1.1 Background ...... 1 1.2 Electrokinetic Phenomena ...... 4 1.3 The Molecular Biology of DNA ...... 7 1.4 Applications ...... 11 1.5 Literature Survey ...... 20 1.5.1 Electroosmotic Flow ...... 20 1.5.2 DNA Transport ...... 23 1.6 Present Work ...... 28

2. Governing Equations and General Assumptions ...... 30

2.1 Introduction ...... 30 2.2 Assumptions ...... 30 2.3 Governing Equations ...... 32 2.3.1 Poisson Equation ...... 32 2.3.2 Nernst-Planck Equation ...... 34

vii 2.3.3 Navier-Stokes Equation ...... 37 2.4 Dimensional Analysis ...... 38 2.5 Boltzmann Distribution and the Debye-H¬uckel Approximation ...... 43 2.6 Ionic Current and Current Density ...... 45 2.7 Summary ...... 46

3. Electroosmotic Flow with Nonuniform Surface Potential ...... 47

3.1 Introduction ...... 47 3.2 Governing Equations and Boundary Conditions ...... 48 3.3 Analytical Solution for Potential Based on the Debye-H¬uckel Approxi- mation ...... 51 3.4 Numerical Methods ...... 55 3.5 Numerical Results ...... 60 3.6 Ionic Transport in a Nanopore Membrane with Charged Patches ..... 67 3.7 Summary ...... 74

4. Electroosmotic Flow and Particle Transport in Micro/Nano Nozzles/Diffusers . 76

4.1 Introduction ...... 76 4.2 Governing Equations and Boundary Conditions ...... 80 4.3 Lubrication Solutions for EOF in the Debye-H¬uckel Limit ...... 82 4.3.1 Known Surface Potential ...... 82 4.3.2 Known Surface Charge Density ...... 86 4.4 Results for EOF Based on the Debye-H¬uckel Approximation ...... 88 4.4.1 Results for EOF: Known Surface Potential ...... 88 4.4.2 Results for EOF: Known Surface Charge Density ...... 97 4.5 Analysis Based on the Thin Double Layer Limit ( 1)...... 100 4.6 Numerical EOF Calculation Based on Poisson-Boltzmann Model . . . . 108 4.7 Numerical Solution using COMSOL ...... 115 4.8 Particle Transport ...... 120 4.9 Particle Transport Using Two-Phase Flow Modeling ...... 125 4.10 Modeling the Fabrication of the Nanonozzles ...... 129 4.11 Summary ...... 134

5. The Forces that Affect DNA Translocation ...... 137

5.1 Introduction ...... 137 5.2 Governing Equations and Boundary Conditions ...... 139 5.3 Electroosmotic Flow in an Annulus ...... 141 5.4 Lubrication Solutions for the Flow Field in a Conical Nanopore: The Viscous Drag Force Acting on the DNA inside the Nanopore ...... 146

viii 5.5 Uncoiling/recoiling Forces Acting on the DNA ...... 148 5.6 Viscous Drag on the DNA Outside the Nanopore ...... 154 5.7 Numerical Validation of the Lubrication Model using Tethering Force Data157 5.8 Summary ...... 166

6. DNA Translocation Velocity ...... 169

6.1 Introduction ...... 169 6.2 Governing Equations and Boundary Conditions ...... 171 6.3 Solution for the Flow Field ...... 172 6.4 Asymptotic Solution for potential φ and the EOF velocity ueof ...... 174 6.5 Results for DNA Translocation Velocity ...... 178 6.6 DNA Entering and Leaving the Nanopore ...... 185 6.7 Modeling DNA Delivery into a Cell through the Nanonozzles ...... 198 6.8 Summary ...... 201

7. Summary ...... 204

Appendices ...... 211

Appendices:

A. Discretising the Equations ...... 211

A.1 Introduction ...... 211 A.2 Discritization ...... 211 A.3 The SIMPLE Algorithm ...... 231 A.4 Residuals ...... 232

ix LIST OF TABLES

Table Page

2.1 The scales used to nondimensionalize the equations listed in the previous section...... 39

3.1 The numerical accuracy check for h =20nm channel with over-potential φe =0.2. The results of 400 × 80 grid are compared with the results of 800 × 160 grid and two-digit accuracy has been attained...... 58

4.1 Values of parameters used for simulations. Note that the concentration of species shown here are reservoir concentrations...... 90

4.2 Parameters used for the numerical calculations for known surface charge. Note that the concentration of species shown here are reservoir concentrations...... 97

4.3 The numerical accuracy check for the numerical calculation for the electroosmotic ﬂow in the nano-diffuser shown in Figure 4.16. The results for 81×161 grid are compared with the results of 161×321 grid and two-digit accuracy has been attained. The numerical results are shown in Figure 4.17. Note that here the pressure p is rescaled on EI,x...... 112

4.4 The numerical accuracy check for the results calculated using COMSOL shown in Figure 4.21. 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...... 118

4.5 The dependence of the particle motion on the wall charge and the particle charge. Here uF is the bulk motion of the ﬂuid ﬂow and uEM is the velocity of particle due to electrical migration. The particle velocity is the summation of these two terms...... 123

x 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 )...... 154

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...... 157

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 121×161 grid (Mesh I) are compared with the results of 241 × 321 (Mesh II) grid and two-digit accuracy has been attained. Note that the pressure shown here is rescaled on EI,x)...... 162

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 (Fd ) listed here is calculated for a DNA immobilized in the ∗ =0 nanopore (uDNA )...... 167

6.1 Comparison for DNA velocity between the numerical results (VNum in ∗ 2 m/s) and the experimental data (VExp in m/s). Here σw (in C/m )is the surface charge density of the nanopore; LDNA is the length of the DNA used in the experiments; CKCl is the concentration of the KCl solution. . . 182

xi 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)...... 2

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)...... 3

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...... 4

1.4 Electric double layer around a charge particle...... 6

1.5 DNA double-helix structure showing base pairs as vertical lines. The diameter of the spiral of the helix is about 2nm ...... 8

1.6 Conformation of DNA conﬁned in a pore. Here h∗ is the half height of the channel of the radius of the pore; Rg is the radius of gyration of the DNA and lp is the persistence length of DNA...... 10

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

xii 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)...... 14

1.9 Gene delivery using nanotip array (Fei, 2007) under an applied electric ﬁeld. 15

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)...... 18

1.11 Nanopores used in the experiments to analyze DNA: (a) natural α-hemolysin nanopore and (b) synthetic nanopore...... 19

1.12 The geometry of the nanopore and the DNA...... 26

2.1 Cartesian coordinates and cylindrical coordinates used for modeling. . . . 32

3.1 The geometry of the channel with a patch with over-potential φp...... 48

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...... 54

3.3 A co-located mesh used for the numerical calculation...... 56

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

3.5 Numerical results for potential, mole fractions calculated from the Poisson- Boltzmann equations. The imposed electric ﬁeld is 106V/m. The height of the channel is 20nm and the wall surface potential is −12mV . The electrolyte is 0.1MNaClsolution far upstream and the over-potential φp =2.0. The corresponding dimensionless parameters are =0.05, Λ=0.77, A =1and Pe=0.36...... 60

xiii 3.6 Numerical results for potential, mole fractions, electric ﬁeld lines and streamlines. The imposed electric ﬁeld is 106 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 φp =2.0. The corresponding dimensionless parameters are =0.05, Λ=0.77, A =1and Pe=0.36...... 61

3.7 Numerical results for potential, mole fractions and electric ﬁeld lines calculated from the Poisson-Nernst-Plank equations. The imposed electric ﬁeld is 107 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 (φp =2.0). The corresponding dimensionless parameters are =0.05, Λ=7.7, A =1and Pe=3.6...... 63

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 107 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 (φp =2.0). The corresponding dimensionless parameters are =0.05, Λ=7.7, A =1...... 65

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 107 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 =0.05, Λ=7.7, A =1...... 66

3.10 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 φe = −0.4 and the imposed electric ﬁeld is 107V/m. The solution far upstream is 0.1MNaClelectrolyte buffer. 67

3.11 Streamlines for EOF in nanochannels with a patch having over-potential in the form of φp = φacosnπx. The zeta potential of the wall is −1mV and the imposed electric ﬁeld is 106V/m. The solution far upstream is 0.1M NaCl...... 68

xiv 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

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 10−10 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...... 71

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 10−10 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

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...... 77

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

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...... 79

xv 4.4 The surface charge density boundary condition for the potential equation is ∂φ = ± ± ( ) ∂y σy, h x , where σy is the component of the dimensionless surface charge density in the y direction ...... 87

4.5 The dimensionless imposed electric ﬁeld (EI,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)...... 89

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; =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 −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...... 92

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; =5× 10−5 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

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

4.9 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 pi =0and po =10...... 96

xvi 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; =0.065. The imposed electric ﬁeld is 0.5V/nm and the surface charge density is −0.01C/m2. 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...... 98

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; =0.065. The imposed electric ﬁeld is 0.5V/nm and the surface charge density is −0.01C/m2. 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

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/m2 and the electrolyte solution is 0.1 MNaCl. For (b), the surface charge density is −0.01C/m2 and the electrolyte solution is 0.001 M...... 100

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...... 107

4.14 The streamlines for electroosmotic ﬂow in the channel shown in Figure 4.13...... 108

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

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; =0.065. The imposed electric ﬁeld is 0.5V/nm and the surface charge density is −0.07C/m2. 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

xvii 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; =0.065. The imposed electric ﬁeld is 0.5V/nm and the surface charge density is −0.07C/m2. 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

4.18 The comparison of DH (based on the analysis in Section 4.3.2) and nonlinear results for streamwise velocity component ueof 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; =0.065. The imposed electric ﬁeld is 0.5V/nm and the surface charge density is −0.07C/m2. 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

4.19 The comparison of DH and nonlinear results for streamwise velocity component ueof 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; =0.065. The imposed electric ﬁeld is 0.5V/nm and the surface charge density is −0.01C/m2. 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

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...... 118

4.21 Results for ionic concentration and ﬂow ﬁeld for the EOF in the nozzle- reservoir system calculated using COMSOL. The parameters are listed in table 4.2...... 119

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; =0.065. The imposed electric ﬁeld is 0.5V/nmand the surface charge density is −0.07C/m2...... 121

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 (uEM < 0)...... 121

xviii 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 uF > 0,uEM < 0...... 124

4.25 Process schematic of EOF based dynamic assembly of silica(Wang et al., 2005) ...... 131

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 ...... 132

5.1 Schematic diagram of DNA translocation in a negatively charged nanopore. 137

5.2 The geometry of the nanopore and the DNA...... 140

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. . . 142

5.4 The results for potential and velocity calculated from equations 5.8 and 5.14. The radius of the nanopore is 5nm and the radius of the DNA is 1nm. The surface charge density is −0.015C/m2 on the DNA surface and −0.006C/m2 on the wall...... 144

= ∂u = 5.5 The dimensionless shear stress τ ∂r at r a as a function of electrolyte concentration. The radius of the nanopore is 5nm and the surface charge density is −0.06C/m2 while the radius of the DNA is 1nm and the surface charge density is −0.15C/m2. The electrolyte concentration varies from 0.01M ( =0.6)to0.1M ( =0.06)...... 144

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

xix 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...... 152

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 Ft is used to immobilize the DNA. The electrical driving force Fe and the viscous drag force Fd are also shown...... 158

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◦...... 159

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/m2 and the applied voltage drop is 120mV . The electrolyte solution is 0.1MKCl. . . 161

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 0.1MKCl. 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, Lb is the distance between the bead and the nanopore...... 163

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/m2 and the applied voltage drop is 120mV . The electrolyte solution is 0.1MKCl...... 165

6.1 Illustration of DNA characterization system based on the translocation of DNA through a nanopore...... 169

xx 6.2 The geometry of the nanopore and the DNA...... 171

6.3 The algorithm used to calculate DNA velocity...... 174

6.4 The inner region and outer region used for the asymptotic analysis...... 175

6.5 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)...... 178

6.6 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/m2 and the electrolyte solution is 0.1 MKCL. The radius of DNA is 1 nm...... 180

∗ 6.7 The viscous drag force due to the electroosmotic ﬂow component Feof and ∗ the pressure driven component Feof 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/m2. 181

6.8 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/m2 for (a) and the electrolyte concentration is 1M for (b)...... 182

6.9 The results for current through the nanopore as a function of time. For the numerical results, the baseline current Ibaseline is Ibaseline = 7085 pA and the amplitude is ∆I = 160 pA. The corresponding experimental data is I = 7100pA and ∆I = 140 pA(Storm et al., 2005a)...... 185

6.10 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. . . . . 186

xxi 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/m2 and the electrolyte concentration is 1M. . 191

6.12 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/m2 and the electrolyte concentration is 1M...... 192

6.13 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/m2 and the electrolyte concentration is 1M...... 193

6.14 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/m2 and the electrolyte concentration is 1M...... 195

6.15 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/m2 and the electrolyte concentration is 1M...... 196

6.16 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/m2 and the electrolyte concentration is 1M...... 197

xxii 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/m2 and the electrolyte concentration is 1M...... 197

6.18 The schematic for the gene delivery device...... 199

A.1 The upwind difference method and the central difference method. Here qe is the velocity of east face using upwind difference method and qe is the face velocity approximated by the central difference...... 214

A.2 An inner node on the mesh...... 215

A.3 A boundary node at the inlet...... 228

A.4 A boundary node at the outlet...... 229

A.5 A boundary node at the lower wall...... 230

A.6 The residuals of the numerical calculation...... 233

xxiii 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 multidis- ciplinary 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 rel- atively low imposed electric field can generate significant volume flowrates which can not

1 3 Q=1µ l/min,Pressure−driven flow Q=1µ l/min, EOF 2.5

1.5

0.5

Pressure Drop(Atm) and Applied Potential(Volts) 0 10 20 30 40 50 60 70 80 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 re-

quired pressure drop for a ﬂowrate of Q =1µl/min increases from 0.006 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 ∝ h3∆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 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 Methacry- late (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 En- gineering 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 Methacry- late (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 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 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 λd, 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 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 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 (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 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 0.34 × 16.5=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 = −log10[H ], where

[H+] is the concentration of hydrogen ions in the solution) because the phosphate groups

+ in the backbone tends to donate their protons (H3O ) to the solution to reach their acid- base equilibrium. Generally DNA has a line charge of −2e per base pair which is equivalent to a surface charge density of −0.15C/m2. 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 (lp), 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 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; Rg is the radius of gyration of the DNA and lp 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 (Rg) or the DNA persistence length (lp) (Muthukumar, 2007) as shown in Figure 1.6. Here the radius of gyration (Rg) 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 >Rg), 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 ∗ the radius of gyration of the DNA but larger than the persistence length (lp

(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

∗ restricted and in the strong conﬁnement limit (h lp), 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 6.

1.4 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 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 heteroge-

neous 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 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 1991.

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. Micronee- dles 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. Mi- croneedles 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 in- jury. 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 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 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 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 90nm 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 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 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. Biologi- cal 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) (Bay- ley & 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 (b) Sythetic nanopore (Storm et al., 2003) (a) α−hemolysin nanopore (Bayley & Cremer, 2001)

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 1.5 Literature Survey

1.5.1 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-H¬uckel approximation and Levine et al. (1975) examined electroosmotic ﬂow in a slit channel for both thin and overlapping EDLs. Con- lisk 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 to oppositely charged walls. The Debye-H¬uckel 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 mix-

tures 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 that work, the flow is modeled under the Debye-H¬uckel 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-H¬uckel 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 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.

1.5.2 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 0.1 ∼ 10A/µsû .

23 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 inde-

pendent 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 sim-

ulations (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 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-H¬uckel approximation. Ghosal (2006) developed a hydrodynamic model to in-

vestigate 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 cylin-

drical pores with constant cross-section are considered and the theoretical analysis is based

on the Debye-H¬uckel approximation.

For the application of nanopore sequencing using a synthetic nanopore, the DNA translo-

cation 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 elec-

trophoretically 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 (2005b) and Fyta et al. (2006) claim that the force due to the viscous drag on the blob- like 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 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 ac-

quire a conical region, connected to a small pore with constant radius, as shown in Figure

1.12. 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 consid-

ered the part with constant radius for simplicity. On the other hand, previous work on this

problem is generally based on the Debye-H¬uckel approximation (valid for absolute value

RT ∼ 26 = 300 −2 of φ F mV for T K). It is known that DNA has a charge of e per base pair where e =1.6 × 10−19C, equivalent to a surface charge of −0.15C/m2. 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-H¬uckel 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 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 ex-

clusion 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. Numer- ical 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 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 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 • 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 concen-

trations 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 axis- symmetric, only the coordinates x∗ (the axial coordinate) and r∗ (the radial coordinate) are considered. The applied electric ﬁeld is in x∗ direction.

31 (a) Cartesian coordinates

(a) Cylindrical coordinates

Figure 2.1: Cartesian coordinates and cylindrical coordinates used for modeling.

2.3 Governing Equations

2.3.1 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

2 ∗ e∇ ϕ = −ρe (2.2)

32 Here ρe is the charge density per unit volume and 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

∇2ψ∗ =0 (2.3)

The perturbation potential comes from the volume charge in the bulk and so

2 ∗ e∇ φ = −ρe (2.4)

For the applied electric ﬁeld, ∇• ∗ =0 Ei (2.5)

∗ = −∇ ∗ ∗ Since Ei ψ , in the integral form, the applied electric ﬁeld in x direction satisﬁes ∗ ∗ = Ei,xA constant.

The volume charge density ρe in equation (2.2) is deﬁned by

ρe = F zici = Fc ziXi (2.6) i i

= ci where ci, zi and Xi c are the molar concentration, the valence and the mole fraction of = + 0 species i and c cwater i ci is the total molar concentration; cwater is the mole con- 55 6 0 centration of water and it is . M; ci denotes the species concentration in the reservoirs where electrical neutrality is maintained and the corresponding mole fraction in the reser-

0 0 = ci voirs is given by Xi c ; F is the Faraday’s constant. Substituting ρe into equation (2.6), the potential equation becomes

2 ∗ e∇ φ = −Fc ziXi (2.7) i

33 Consider Cartesian coordinates shown in Figure 2.1 (a), the dimensional potential equation is 2 ∗ 2 ∗ 2 ∗ ∂ φ + ∂ φ + ∂ φ = −Fc ∗2 ∗2 ∗2 ziXi (2.8) ∂x ∂y ∂z e i In cylindrical coordinates, the dimensional equation for potential is 1 ∗ 2 ∗ ∂ ∗ ∂φ + ∂ φ = −Fc ∗ ∗ r ∗ ∗2 ziXi (2.9) r ∂r ∂r ∂x e i

2.3.2 Nernst-Planck Equation

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

∗ = − ∇ + ∗ + ∗ nA cDA XA mAzAcF XAE cXAu (2.10)

Here DA is the diffusion coefﬁcient of ionic species A in the mixture, XA is the mole fraction of species A, which can be either the anion or the cation, uA is the mobility, zA is the valence. The ionic species mobility mA is given by the Stokes-Einstein relation as

D m = A (2.11) A RT 34 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 0.6×10−9m2/s to 2 × 10−9m2/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 1MNaClsolution, 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 vectorsi, j and k is

∗ = ∗ + ∗ + ∗ = ∇ ( ∗ + ∗) E Exi Ey j Ez 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 = −∇ · n∗ + R (2.14) ∂t A A

35 where RA 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 ∂2X ∂2X ∂2X ∂X ∂X ∂X ∂X D A + A + A = A + u∗ A + v∗ A + w∗ A A ∂y∗2 ∂x∗2 ∂z∗2 ∂t ∂x∗ ∂y∗ ∂z∗ ∗ ∗ ∗ D F ∂X E ∂XAE ∂X E + A z A x + z y + z A z (2.15) RT A ∂x∗ A ∂y∗ A ∂z∗ ∗ = − ∂ψ∗ − ∂φ∗ ∗ = − ∂ψ∗ − ∂φ∗ ∗ = − ∂ψ∗ − ∂φ∗ Since Ex ∂x∗ ∂x∗ , Ey ∂y∗ ∂y∗ , Ez ∂z∗ ∂z∗ , equation (2.15) can be also written as ∂2X ∂2X ∂2X ∂X ∂X ∂X ∂X D A + A + A = A + u∗ A + v∗ A + w∗ A A ∂y∗2 ∂x∗2 ∂z∗2 ∂t ∂x∗ ∂y∗ ∂z∗ 2 ∗ 2 ∗ 2 ∗ ∗ ∗ −zAXADAF ∂ φ + ∂ φ + ∂ φ − zADAF ∂XA ∂φ + ∂ψ − RT ∂x∗2 ∂y∗2 ∂z∗2 RT ∂x∗ ∂x∗ ∂x∗ z D F ∂X ∂φ∗ ∂ψ∗ z D F ∂X ∂φ∗ ∂ψ∗ A A A + − A A A + (2.16) RT ∂y∗ ∂y∗ ∂y∗ RT ∂z∗ ∂z∗ ∂z∗ For cylindrical coordinates, the mass transfer equation is D ∂ ∂X ∂2X ∂X ∂X ∂X A A + D A = A + u∗ A + v∗ A r∗ ∂r∗ ∂r∗ A ∂x∗2 ∂t ∂x∗ ∂r∗ D Fz ∂X E∗ D Fz ∂ ∂X E∗ + A A A x + A A r∗ A r (2.17) RT ∂x∗ r∗RT ∂r∗ ∂r∗ Here the electric ﬁeld vector is given by

∂(ψ∗ + φ∗) ∂(ψ∗ + φ∗) E∗ = E∗i + E∗i = ∇ (ψ∗ + φ∗)= i + i x x r r ∂x∗ x ∂r∗ r

where x and r are the cylindrical unit vectors and the velocity vector is given by

∗ ∗ ∗ u = u ix + v ir

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 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 − H2O or KCl − H2O.

Note that most of the theory present in this work can be also applied to any combination of ionic constituents of arbitrary valences.

2.3.3 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

∂u∗ ∂v∗ ∂w∗ + + =0 (2.18) ∂x∗ ∂y∗ ∂z∗

In cylindrical coordinates, the continuity equation is

∂u∗ 1 ∂ (r∗v∗) + =0 (2.19) ∂x∗ r∗ ∂r∗

In vector form, the conservation of momentum is given by ∂u∗ ρ +(u∗ •∇)u∗ = −∇p∗ + µ∇2u∗ + ρ E ∗ (2.20) ∂t∗ e 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∗ ∂p∗ ρ + u∗ + v∗ + w∗ = − + ∂t∗ ∂x∗ ∂y∗ ∂z∗ ∂x∗

37 ∂2u∗ ∂2u∗ ∂2u∗ ∂(ψ∗ + φ∗) µ + + − Fc z X (2.21) ∂x∗2 ∂y∗2 ∂z∗2 i i ∂x∗

∂v∗ ∂v∗ ∂v∗ ∂v∗ ∂p∗ ρ + u∗ + v∗ + w∗ = − + ∂t∗ ∂x∗ ∂y∗ ∂z∗ ∂y∗ ∂2v∗ ∂2v∗ ∂2v∗ ∂(ψ∗ + φ∗) µ + + − Fc z X (2.22) ∂x∗2 ∂y∗2 ∂z∗2 i i ∂y∗

∂w∗ ∂w∗ ∂w∗ ∂w∗ ∂p∗ ρ + u∗ + v∗ + w∗ = − + ∂t∗ ∂x∗ ∂y∗ ∂z∗ ∂z∗ ∂2w∗ ∂2w∗ ∂2w∗ ∂(ψ∗ + φ∗) µ + + − Fc z X (2.23) ∂x∗2 ∂y∗2 ∂z∗2 i i ∂z∗ The corresponding Navier Stokes equation in cylindrical coordinates is ∂u∗ ∂u∗ ∂u∗ ∂p∗ ρ + u∗ + v∗ = − + ∂t∗ ∂x∗ ∂r∗ ∂x∗ ∂2u∗ 1 ∂ ∂u∗ ∂(ϕ + φ) µ + r∗ − Fc z X (2.24) ∂x∗2 r∗ ∂r∗ ∂r∗ i i ∂x∗

∂v∗ ∂v∗ ∂v∗ ∂p∗ ρ + u∗ + v∗ = − + ∂t∗ ∂x∗ ∂r∗ ∂r∗ ∂2v∗ 1 ∂ ∂v∗ ∂(ϕ + φ) µ + r∗ − Fc z X (2.25) ∂x∗2 r∗ ∂r∗ ∂r∗ i i ∂r∗

2.4 Dimensional Analysis

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 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∗ x L x L ∗ y∗ 0 = y h y h0 ∗ = z∗ z W z W ∗ r∗ r h0 r = ∗ hi ∗ RT φ∗ φ φ0 = ∼ 26mV φ = F φ0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Ex Ey Ez 0 = ( =0) = = = Ei,x,Ei,y,Ei,z,Ei,r E Ex,i x Ex 0 ,Ey 0 ,Ez 0 E ∗ ∗E ∗ E ∗ ∗ ∗ eRT E0 u v w u ,v ,w U0 = u = ,v = ,w = µF U0 U0 U0 ∗ ∗ µU0 p p p0 = p = h0 p0 2 ∗ ∗ h0 t t t0 = t = D0 t0

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 non-

linear and coupled equations and they are rarely solved in that form. Some simpliﬁca-

tions 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 two-

dimensional 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 ∂2φ ∂2φ β 2 + = − z X (2.26) 1 ∂x2 ∂y2 2 i i

39 and in the cylindrical coordinates, it is 1 2 ∂ ∂φ + 2 ∂ φ = − β r 1 2 2 ziXi (2.27) r ∂r ∂r ∂x i = h = c + 0 where the aspect ratio 1 L ; β I is the ratio of total concentration c=cwater i ci 2 0 λ eRT and the ionic strength I = z c ; = where λ = 2 is the Debye length. i i i hi IF For mole fraction of ionic species i, the dimensionless equation in Cartesian coordinates is 2 2 1 ∂ XA + 2 ∂ XA = ∂Xi + Pe ∂XA + ∂XA 2 1 2 1u v ∂y ∂x αi ∂t αi ∂x ∂y ∂XAEx ∂XAEy + 1z + z (2.28) A ∂x A ∂y

Pe = U0h0 Here D0 is usually called mass transfer Peclet number. Since the diffusion coef- ﬁcients for different species are not the same, the highest diffusion coefﬁcient is taken as

Di 0 0 = D and the ratio of other species diffusion coefﬁcients to this D is given by αi D0 .For computational ease, a single time scale is speciﬁed for the whole problem and it is taken to

2 h0 0 = ( ) be t D0 . Ex,Ey are the nondimensional electric ﬁeld in x, y directions are written as

h0E0 ∂φ Ex = Ei,x − 1 φ0 ∂x

h0E0 ∂φ Ey = Ei,y − φ0 ∂y

Λ= h0E0 The nondimensional imposed electric field is defined as φ0 , the dimensionless applied electric field is defined as Ei,x, Ei,y in the x and y directions respectively.

In cylindrical coordinates the equation is 1 2 1 ∂ ∂XA + 2 ∂ XA = ∂Xi + Pe ∂XA + ∂XA r 1 2 1u v r ∂r ∂r ∂x αi ∂t αi ∂x ∂r ∂XAEx ∂ ∂XAEr +1z + z r (2.29) A ∂x A ∂r ∂r

40 and the dimensionless electric ﬁeld components in r and x directions are

h0E0 ∂φ Ex = Ei,x − 1 φ0 ∂x

h0E0 ∂φ Ey = Ei,r − φ0 ∂r

The continuity equation is ∂u ∂v 1 + =0 (2.30) ∂x ∂y

Note that the x direction is the primary direction of ﬂow and the velocity scale is chosen as

eRT E0 0 = U µF . Using the same velocity scale, in cylindrical coordinates, the dimensionless continuity equation becomes ∂u ∂v 1 + =0 (2.31) ∂x ∂y and in cylindrical coordinates it is

∂u 1 ∂ (rv) 1 + =0 (2.32) ∂x r ∂r

The dimensionless Navier Stokes equation in Cartesian coordinates is 1 ∂u ∂u ∂u ∂p β 2 + Re 1u + v = −1 + z X E + ∇ u (2.33) Sc ∂t ∂x ∂y ∂x 2 i i x

1 ∂v ∂v ∂v ∂p β 2 + Re 1u + v = − + z X E + ∇ v (2.34) Sc ∂t ∂x ∂y ∂y 2 i i y

= ρU0h where Re µ is the Reynolds number and

∂2 ∂2 ∇2 = + 2 ∂y2 1 ∂x2

Here the same time scale is used as in the species mass transfer equation. The Schmidt

= µ = ρU0h0 number is given by Sc ρD0 and the Reynolds number is given by Re µ .For −9 2 the typical species diffusion coefﬁcient D0 ∼ 10 m /s, the Schmidt number Sc has a

41 value of 1000. Therefore, the acceleration terms in the momentum equation are of the ( 1 ) 1 order O Sc , which means that the velocity field responds much faster than the time 2 h0 t0 = scale chosen to non-dimensionalize the equation D0 . For the mass transfer equation, the transient term is of the order of O( 1 ) ∼ 1, indicating that the species respond on αi 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

−9 −6 neglected. The Reynolds number Re is usually very small since h0 ∼ O(10 − 10 ) 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 ∂p β ∇ u = 1 − z X E (2.35) ∂x 2 i i x

∂p β ∇2v = − z X E (2.36) ∂y 2 i i y ∇2 = 2 ∂2 + ∂2 ∇2 = For Cartesian coordinates, 1 ∂x2 ∂y2 and for cylindrical coordinates, 2 ∂2 + ∂2 + 1 ∂ 4+ 1 ∂x2 ∂r2 r ∂r. To summarize, for a electrolyte with n species, we have a series of n 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 2.5 Boltzmann Distribution and the Debye-Huckel¬ 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:

∗ ∇µi = −zie∇φ (2.37)

−19 where µi is the chemical potential and e = −1.602×10 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 dφ∗ = −z e (2.38) dy i dy and using the deﬁnition of the chemical potential in the form

= 0 + ln µi µi RT ni

where ni is the number of ions of species i per unit volume, we have ln 1 ∗ d ni = dni = − zie dφ dy ni dy RT dy ∗ =0 = 0 Integrating this equation from a point in the bulk solution where φ and ni ni leads to the Boltzmann equation: −z eφ∗ n = n0 exp( i ) (2.39) i i RT Boltzmann gives the local concentration of each species in the double layer region. Sub- stituting equation (2.39) into the equation of potential gives us the complete Poisson-

Boltzmann equation:

1 − ∗ ∇2 ∗ = − 0 exp( zieφ ) φ ni zie (2.40) eDA i RT 43 If φ∗ is small everywhere in the double layer, i.e. eφ∗ RT , we can expand the exponential using the relation

ex ≈ 1 − x

Then we get Debye-H¬uckel approximation of the problem 1 2 2 0 ∗ ∇2 ∗ = − 0 − zi e ni φ φ zieni (2.41) e i i RT

0 It is noted that i zieni must be zero to preserve electro neutrality in the bulk electrolyte and so the equation becomes 2 ∇2 ∗ = e 2 0 ∗ = 2 ∗ φ zi ni φ κ φ (2.42) eRT i where 1 e2 z2n0 2 κ = i i i eRT ( 0) The parameter κ depends mainly on the electrolyte concentration ni and it is referred to as the Debye-H¬uckel parameter.

The Boltzmann distribution and the Debye-H¬uckel 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-H¬uckel approximation applies a linear distribution

44 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 zini (2.43) i where ni is the dimensional ﬂux of ionic species i, which is given by

D n = −cD ∇X + c i z FX E∗ + cX u∗ (2.44) i i i RT i i i

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 ∂Xi ∂φ Pe Jx = zi + ziEi,xXi − zi + Xiu (2.45) i ∂x ∂x αi here the ionic current density is scaled on cU0 as

∗ Jx Jx = cF U0

The current through the channel (in x∗ direction) is the integral of the current density across the entire cross-section of the channel: ∗ = ∗ ∗ Ix JxdA (2.46) A 45 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-H¬uckel 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 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 elec-

trochemical 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 Figure 3.1: The geometry of the channel with a patch with over-potential φp.

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 two- dimensional. 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 105-107V/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 ﬁeld satisﬁes ∗ ∗ = EI,xA constant

∗ ∗ Since the height of the channel h (x )=h0 is a constant, the imposed electric ﬁeld is

∗ a constant in x direction (in dimensionless form EI,x =1) due to the constant cross ∗ ∗ ∗ =0 section area A . The y component of the applied electric ﬁeld is given by EI,y since ∗ ∗ ∂EI,x + ∂EI,y =0 ∂x∗ ∂y∗ (equation (2.5)). The surface potential of the channel walls and the patch ∗ ∗ − are ζw and ζp respectively. For simplicity, a NaCl H2O 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

2 = h0 = length of the channel L discussed in Chapter and so the aspect ratio is 1 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

Λ= h0E0 parameter is deﬁned as φ0 , which is the ratio of the applied electric ﬁeld strength

φ0 E0 and the perturbation electric ﬁeld strength h0 . The simpliﬁed governing equations for potential, velocity and ionic species are

∂2φ ∂2φ β A2 + = − z X (3.1) ∂x2 ∂y2 2 i i 2 2 ∂ Xi + 2 ∂ Xi = Pe ∂Xi + ∂Xi 2 A 2 Au v ∂y ∂x αi ∂x ∂y ∂X ∂φ ∂X ∂φ ∂2φ ∂2φ +Az i Λ − A − z i − z X A2 + i =1...N (3.2) i ∂x ∂x i ∂y ∂x i i ∂x2 ∂y2 ∂u ∂u ∂p β ∂φ Re Au + v = −A + Λ − A z X + ∇2u (3.3) ∂x ∂y ∂x 2 ∂x i i ∂v ∂v ∂p β ∂φ Re Au + v = − − + ∇2v (3.4) ∂x ∂y ∂y 2 ∂y 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 = −Lx

49 is the upstream boundary and x = Lx is the downstream boundary, where Lx is usually chosen to be 10. Note that Lx is scaled on the length of the patch and so the dimensionless length of the computational domain is 2Lx =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 ∗ =0 ζw and in dimensionless form, it is assumed that φ at the walls for simplicity. Thus ∗− ∗ = φ ζw the dimensionless potential is related to the dimensional potential by φ φ0 . The =( ∗ − ∗ ) dimensionless patch potential is then φp ζp ζw /φ0. 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:

= − : = in = in = in =0 x Lx φ φ ,Xi Xi ,u u ,v

∂φ ∂X ∂u ∂v x = L : =0, i =0, =0, =0 x ∂x ∂x ∂x ∂x ∂X ∂φ =0 | | 1 =0: =0 =0 i + =0 φ if x > 2 y u ,v , ziXi = | |≤ 1 ∂y ∂y φ φp if x 2 ∂X ∂φ y =1:φ =0,u=0,v =0, i + z X =0 ∂y i i ∂y

in in in where φ , Xi and u are the fully developed proﬁles of φ, Xi and u. Note that the equation for ionic species can be written in the form of ∂ ∂Xi ∂φ ∂ ∂Xi ∂φ ∂Xi ∂XA ∂XA + z X +1 1 + z X 1 = 1z Λ +Pe 1u + v ∂y ∂y i i ∂y ∂x ∂x i i ∂x i ∂x ∂x ∂y (3.5)

The Boltzmann distribution for species

= 0 (− ) Xi Xi exp ziφ

50 is a solution for this equation if Λ=0and Pe=0under the equilibrium conditions, where

0 Xi 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 ∝ E0 and Λ ∝ E0, 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-Huckel¬ Ap- proximation

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-H¬uckel approximation. Note that the Debye-H¬uckel approximation is based on the Poisson-Boltzmann model and it is only valid for φ ∗ 26mV

= φ∗F 1 (φ RT ). The distribution of mole fraction of species will satisfy the Boltzmann distribution if the imposed electric ﬁeld term is neglected (Λ=0)

= 0 −ziφ Xi Xi e (3.6)

When the potential φ is small, the exponential can be expanded as

= 0(1 − ) Xi Xi ziφ (3.7)

The potential equation (2.26) becomes

∂2φ ∂2φ φ + A2 = (3.8) ∂y2 ∂x2 2 51 Equation (3.8) is a linear equation and the electric potential is then decomposed into

two components: φ = φedl + φ , where φedl 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 φedl satisﬁes ∂2φ ∂2φ φ edl + A2 edl = edl (3.9) ∂y2 ∂x2 2 and φ satisﬁes ∂2φ ∂2φ φ + A2 = (3.10) ∂y2 ∂x2 2

The boundary conditions for these two equations are

∂φ x = −L : edl =0,φ =0 x ∂x ∂φ x = L : edl =0,φ =0 x ∂x =0 | | 1 =0: = φ if x > 2 y φedl ζw, = − = | |≤ 1 φ ζp ζw φp if x 2

y =1:φedl = ζw,φ =0

= ∗ = ∗ where ζw ζw/φ0 and ζp ζp /φ0 are the dimensionless zeta potential of the wall and

2 2 ∂ φedl =0 the patch. Note that in equation (3.9), the term A ∂x2 since there is no streamwise variation in φedl. The equation becomes

∂2φ φ edl = edl (3.11) ∂y2 2 with boundary conditions φedl = ζw,y =0, 1 and the analytical solution is

sinh( 1 ) − sinh( y−1 ) = φedl ζw sinh( 1 )

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 differential equation becomes

X(x) Y (y) 1 + = (3.12) X(x) Y (y) 2 and it can be written as X(x) Y (y) 1 = −λ2 = − + (3.13) X(x) Y (y) 2 Now the equation becomes two separated ordinary differential equations

X(x) + λ2X(x)=0 (3.14)

1 Y (y) − (λ2 + )Y (y)=0 (3.15) 2

Since φ =0at x = ±Lx, the boundary condition for equation (3.14) is

x = ±Lx,X(x)=0

and the solution is given by nπ X(x)=An sin (x + Lx) = An sin(λn(x + Lx)) (3.16) 2Lx

= nπ where λn 2Lx and An 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 ⎛ ⎞ 1 Y (y)=B sinh ⎝ λ2 + (1 − y)⎠ (3.17) n 2 and the full solution for φ is now 1 φ = X(x)Y (y)=C sin(λ (x + L )) sinh( λ2 + (1 − y)) (3.18) n n x 2

where Cn = AnBn. On the lower wall y =0, φ is given by 1 φ = C sin(λ (x + L )) sinh( λ2 + )=ϕ sin(λ (x + L )) (3.19) n n x 2 n n x 53 (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.

= sinh( 2 + 1 ) where ϕn Cn λ 2 is a new coefﬁcient to be determined. The boundary =0 =0 | | 1 = | |≤ 1 condition for φ at y is given by φ if x > 2 and φ φp,if x 2 .

To apply this boundary condition, Fourier series is used as 1 Lx 1 1/2 ϕn = ϕ(t)sin(λn(t + Lx))dt = ϕ(t)sin(λn(t + Lx))dt (3.20) Lx −Lx Lx −1/2 and the integration gives 2a λ ϕn = sin(Lxλn)sin( ) (3.21) Lxλn 2

The coefﬁcient Cn is then given by

ϕn Cn = (3.22) sinh( 2 + 1 ) λ 2 Therefore, the solution for φ is given by ∞ 2 sin nπ sin λn sin( ( + )) sinh (1 − ) 1 + 2 φp 2 2 λn x L y 2 λn φ = 1 2 1 sinh + λnL 2 λn and the full solution for potential φ is

sinh( 1 ) − sinh( x−1 ) = φ ζw sinh( 1 ) 54 ∞ 2 sin nπ sin λn sin( ( + )) sinh (1 − ) 1 + 2 φp 2 2 λn x L y 2 λn + (3.23) 1 2 1 sinh + λnL 2 λn 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-H¬uckel 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 400 × 80.

55 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

∇•ρudV = ρu • ndS (3.24) V S 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 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 1980. 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 pk. The

velocity ﬁeld are denoted by uk and vk.

2. Use the values of pk to solve for velocities from the momentum equations. The

updated velocities are denoted as uˆ and vˆ.

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 pk.

The pressure correction equation has been discussed in the previous section. The

corrected pressure is

pk+1 = pk + p

and the corresponding velocity corrections u, v can be obtained from p such that

uk+1 = uk + u

vk+1 = vk + v

57 (x,y) 400 × 80 grid 800 × 160 grid φ =0.0440, u =0.0409 φ =0.0445, u =0.0412 (-5,0.5) g =0.002636, f =0.002636 g =0.002635, f =0.002635 v =0, p =0.0594 v =0, p =0.0601 φ =0.1160, u =0 φ =0.1171, u =0 (0,0.25) g =0.002638, f =0.002637 g =0.002637, f =0.002636 v =0, p = −0.7908 v =0, p = −0.7877 φ =0.09580, u =0 φ =0.09638, u =0 (-0.5,0.1) g =0.002703, f =0.002727 g =0.002705, f =0.002729 v =0.0189, p =0.8746 v =0.0191, p =0.8793

Table 3.1: The numerical accuracy check for h =20nm channel with over-potential φe = 0.2. The results of 400 × 80 grid are compared with the results of 800 × 160 grid and two-digit accuracy has been attained.

4. Designate the new value pk+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 400 × 80 grid to perform the numerical calculation and obtain two digit accuracy using that grid compared to a 800 × 160 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 Figure 3.4: The ﬂowchart of the numerical calculation using SIMPLE algorithm.

59 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 106V/m. The height of the channel is 20nm and the wall surface potential is −12mV . The electrolyte is 0.1MNaClsolution far upstream and the over-potential φp =2.0. The corresponding dimensionless parameters are =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 = −Lx = −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 (a) potential contours (b) mole fraction of cations contours

Figure 3.6: Numerical results for potential, mole fractions, electric ﬁeld lines and streamlines. The imposed electric ﬁeld is 106 V/m. The height of the channel is 20 nm and the wall surface potential is −12mV . The electrolyte is 0.1MNaClsolution far upstream and the over-potential φp =2.0. The corresponding dimensionless parameters are =0.05, Λ=0.77, A =1and Pe=0.36.

61 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 φp =2.0 is located

at the lower wall of the channel and the length of the patch is 20 nm (A =1). The imposed

electric field is 106 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

∇·(∇Xi + XiE)=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 (107V/m).

β Λ There are four dimensionless parameters that affect the results: 2 , , 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

0.1MNaClsolution under 106 V/mimposed electric ﬁeld will be equivalent to the results

for 200 nm channel with 0.001 MNaClsolution under 105 V/m imposed electric ﬁeld.

62 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

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 107 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 (φp =2.0). The corresponding dimensionless parameters are =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 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 106 V/m to 107 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 dis- ∂Xi + ∂Xi tribution is proportional to the convective term (Pe 1u ∂x v ∂y ) and the term related

Λ ∂Xi to the axial non-uniformity 1zi ∂x ). Increasing the applied electric ﬁeld increases these 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

∂Xi 1 Λ from the axial non-uniformity in the surface potential ( zi ∂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. Simultane- ously 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 (b) dimensionless streamwise velocity (a) cation mole fraction contours 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 107 V/m. The height of the channel is 20 nm 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 (φp =2.0). The corresponding dimensionless parameters are =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 3.10. Surplus cations are attracted to the patch, increasing

65 0.26

0.24

0.22

0.2

0.18

0.16

0.14

0.12

0.1

normal distance of the vortex center from patch 1 2 3 4 5 6 7 8 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 107 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 =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

φp = φacosnπx is discussed here. On Figure 3.11 shows the streamlines for the bulk ﬂow with the variable patch potential with n =2and φa =0.2, 1.0, 2.0.Forφa =0.2, two small vortex are found above the patch. Both of the vortices are recirculating clockwise. For φa =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 Figure 3.10: 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 φe = −0.4 and the imposed electric ﬁeld is 107V/m. The solution far upstream is 0.1MNaClelectrolyte buffer.

away from the patch. For a higher φa =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 10−10 m2/s. Note that

67 (a) φp =0.2cos2πx

(b) φp =1.0cos2πx

Figure 3.11: Streamlines for EOF in nanochannels with a patch having over-potential in the form of φp = φacosnπx. The zeta potential of the wall is −1mV and the imposed electric ﬁeld is 106V/m. The solution far upstream is 0.1MNaCl. 68 (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 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

P0 =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 × P0. 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 Chemi- cal 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 ele-

ments;

70 (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 10−10 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 • Solving the problem: Choose solver according to your problem, steady state or tran-

sient, 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 (cpermeate) and the feed (cfeed): c s = permeate cfeed 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 3.13. 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 3.14

72 (a) with charged patch

−7 3 x 10 Concentration of Albumin [mol/m ] 7 ] 3 6

5 feed nanopore permeate

1 Concentration of Albumin [mol/m

0 −1 −0.5 0 0.5 1 1.5 2 x −6 x 10

(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 10−10 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 (A)), the concentration in the permeate reservoir is 0.077 mM and the resulting sieving coefficient is 0.128. The rescaled concentration (c/cfeed) 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 concen-

tration in the permeate decrease to 3 M and the resulting sieving coefﬁcient is 0.01. 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 over- potential 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 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 elec-

tric 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 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 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/nm2 (e =1.6 × 10−19C is the elementary charge) which is equivalent to σ = −0.07C/m2 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 (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 y*

h* (x *)

2h i flow direction x * 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 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∗ − h∗ h∗(x)=h + o i x∗ i L

∗ ∗ where hi ,ho are the half heights of inlet and outlet of the channel and in dimensionless form h∗ − h∗ h(x)=1+ o i x hi

h∗ x∗ where h = ∗ and x = . The walls of the channel are deﬁned by y = ±h(x). hi 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

∇· ∗ =0 ∗ ( ) ∗( )= ∗ (0) ∗(0) ∗ in the form E so that EI,x x A x EI,x A , where A is the dimensional ∗ (0) cross-section area of the channel. If the imposed electric field at the inlet EI,x is taken as the scale of the imposed electric field, the dimensionless imposed electric field is thus = 1 EI,x h(x) and the corresponding y component is given by h = −1h = −1yh EI,y 2 dy 2 (4.1) −h h h

Usually in experiments potential drop is given and the value of this imposed electric ﬁeld = ∗ ( =0) ∗ ( ) scale E0 EI,x x can be found from the axial integration of EI,x x L L ∗ ∗ ∗ ∗ ∗ A (0) ∆φ = φ (L) − φ (0) = − E (x)dx = −E0 dx i i i 0 I,x 0 A∗(x)

80 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

2 approximation (1Re 1), the equations can be simpliﬁed as

2 ∂ φ = − β 2 2 ziXi ∂y i

2 ∂ u = − β + ∂p 2 EI,x 2 ziXi ∂y i ∂x Note that since the pressure is assumed to be not dependent on y, the pressure is scaled on

µU0 1 L and so the 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

φ = ζ,y = ±h(x)

∗ ζ = ζ ζ where φ0 is the dimensionless wall -potential. The other is based on the surface charge density σ∗ as ∂φ ∂φ = −σ ,y = −h; = σ ,y = h ∂y y ∂y y where σy is the dimensionless surface charge density in the y direction as shown in Figure

∗ ∗ φ0 σ hi 4.4. Note that σ is scaled on ∗ as σ = . The boundary condition for the species ehi eφ0 equation is that the walls are impermeable to the species, thus give rise to the no ﬂux boundary conditions

ni · nˆw =0 where n = ∂Xi + z X E + PeX u i + ∂Xi + z X E + PeX v j is the ﬂux of the i ∂x i i x αi i ∂y i i y αi i species i and nˆw is the unit vector normal to the wall. The boundary condition for the

81 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 = pi; x =1,p= po

where pi and po 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-Huckel¬ 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-H¬uckel 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.

4.3.1 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)

∂2φ 1 = φ (4.2) ∂y2 2 and the solution for the potential is

cosh y = φ ζ h (4.3) cosh

82 The velocity can be decomposed into two components: the ﬂow due to the electrical body force ueof and the ﬂow due to the pressure gradient up. The governing equation for the electroosmotic component of the velocity becomes 2 1 ∂ ueof = − β − 1 ∂φ 2 2 ziXi Λ (4.4) ∂y i h ∂x

1 1 ∂φ here h is the imposed electric ﬁeld and Λ ∂x is the electric ﬁeld due to the EDLs. Since the

∂φ potential gradient in x direction ∂x is small, this term is neglected. This equation becomes

∂2u 1 ∂2φ eof = (4.5) ∂y2 h ∂y2 and the solution for ueof is given by φ − ζ ζ cosh y = = − 1 ueof h (4.6) h h cosh

∂p Based on the lubrication approximation, the axial pressure gradient ( ∂x) is nearly independent of y and the pressure-driven component is given by

1 dp u = y2 − h2 (4.7) p 2 dx

The total streamwise velocity is then ζ cosh y 1 dp u = u + u = − 1 + y2 − h2 eof p h 2 h cosh dx

and the axial velocity gradient 2 y sinh h cosh y ∂u 1 d p dp ζh cosh ζh = y2 − h2 − hh − − 1 + ∂x 2 dx2 dx h2 cosh h cosh2 h h

The v velocity is obtained from the continuity equation as

y ∂u v = − 1 dy (4.8) −h(x) ∂x

83 Substituting the solution of the streamwise velocity (u = up +ueof ), the transverse velocity v is given by ⎡ ⎤ y y 1 d2p y3 dp ζh sinh h sinh = ⎣− − 2 + + ⎦ v 1 2 h y hh y 2 dx 3 dx h cosh2 h −h ⎡ ⎤ y y ζh sinh ζyh + ⎣ − ⎦ 2 h2 cosh h h −h so that h y 2 3 1 sinh sinh 1 d p y 2 2 3 dp ζh v = − − h y − h + 1 hh (y + h)+ + 2 dx2 3 3 dx cosh2 h h sinh y 2 h h ζh 1 1ζyh ζh 1tanh ζ1h tanh ζ1h − + + − (4.9) 2 cosh h h2 h h2 h h On the upper boundary y = h(x), the transverse velocity v =0due to the no-slip boundary condition and so 1 d2p dp ζh h ζh h ζh h3 + hh2 = − tanh2 − tanh + (4.10) 3 dx2 dx h h2 h

The pressure can be determined from this equation. Equation (4.10) can also be written as 1 d dp ζh h ζh h ζh h3 = − tanh2 − tanh + (4.11) 3 dx dx h h2 h

Integrating twice shows ζ h 6ζ p = − dh + C1x + C2 (4.12) 2 3 2 4 2h h h hi h h e +1 where C1 and C2 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 (2 1), the electroosmotic component can be simpliﬁed as (Conlisk, 2005) ζ −( h−y ) −( h+y ) u = e + e − 1 (4.13) eof h 84 Superimposing the velocities from equations (4.7) and (4.13), the total velocity is given by 1 dp 2 2 ζ −( h−y ) −( h+y ) u = y − h + e + e − 1 (4.14) 2 dx h and the axial gradient of u is

2 ∂u 1 d p 2 2 dp ζh −( h−y ) −( h+y ) ζh −( y−h ) −( h+y ) = y − h −hh − e + e − 1 − e + e ∂x 2 dx2 dx h2 h

The v-velocity can be obtained as y ∂u v = − 1 dy −h(x) ∂x 2 3 y 1 d p y 2 = − − h2y − h3 2 2 3 3 dx − h y dp 1ζh −( y−h ) −( h+y ) 1ζh −( y−h ) −( h+y ) + ( + )+ + − − − + 1 hh y h 2 e e y e e dx h h −h and the transverse velocity is given by 2 3 1 d p y 2 2 3 dp v = − − h y − h + 1 hh (y + h)− 2 dx2 3 3 dx 1ζh ζh 1 ζh 1 − h−y − h+y − 2h (y + h)+ + e − e − e +1 (4.15) h2 h2 h Applying the boundary condition at the upper boundary shows that

h3 d2p dp ζh + h + =0 3 dx2 dx h2 and the pressure solution is 2( 2 − 2)+ 2( 2 − 2) ( 2 + + 2) = poho h hi pihi ho h − ζ − ζ hi hiho ho − ζ p 2( 2 − 2) 3 2 ( + ) ( + ) h ho hi h h h h hiho hi ho h hiho hi ho (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().

85 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)ix ± sin(α)iy at the lower and upper wall) is the surface normal vector

and it reduces to ∂φ = −σ at y = −h(x) ∂y y ∂φ = σ at y = h(x) ∂y y where σy 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 = σ ∂n and σy = σ cos(α) where α is the angle of the nozzle wall and the x-axis and α is small so

∂ ∂ 1 that ∂x ∂y. The solution for electric potential becomes

−σ cosh(y/) φ(y)= y (4.17) sinh(h/) and the EOF component of the streamwise velocity is σ cosh(y/) u = y coth(h/) − eof h sinh(h/)

The total velocity is 1 dp σ cosh(y/) u = u + u = y2 − h2 + y coth(h/) − eof p 2 dx h sinh(h/)

86 Figure 4.4: The surface charge density boundary condition for the potential equation is ∂φ = ± ± ( ) ∂y σy, h x , where σy is the component of the dimensionless surface charge density in the y direction .

and the axial gradient is 2 2 ∂u 1 d p dp σ h 1 − coth (h/) cosh(y/)cosh(h/) = y2 − h2 − hh − y + ∂x 2 dx2 dx h2 sinh2(h/) 1 − coth2( ) cosh( )cosh( ) +σh h/ + h/ y/ h sinh2(h/) Similar to the previous section, the transverse velocity v can be obtained from the inte-

gration the continuity equation analytically and the solution is 2 3 1 d p y 2 2 3 dp v = − − h y − h + 1 hh (y + h)− 2 dx2 3 3 dx 1h h sinh(y/)+ sinh(h/) (y + h)coth − − h2 sinh(h/) σh 1 − coth2(h/) cosh(h/)(sinh(y/)+sinh(h/)) (y + h)+ (4.18) h sinh2(h/) Applying the boundary condition for v at y = h, the pressure equation becomes

1 d2p dp 3σ2 h3 + hh2 = +3σ 1 − coth2(h/) (4.19) 3 dx2 dx h2

87 Rearranging the left hand side of the equation, equation (4.19) becomes d dp σ2 σ h3 =3h − dx dx h2 sinh2(h/) and integrating once shows dp 3σ2 3σ cosh(h/) = − + + C dx h4 h3 sinh(h/) where C is a integration constant. At this point further integration of this equation is difﬁ- cult and the solution for pressure has to be found numerically.

4.4 Results for EOF Based on the Debye-Huckel¬ Approximation

4.4.1 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 cross- section. 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 bound-

ary conditions. Second order ﬁnite difference methods are used to discretize the pressure

equation as h3 − 2 + − ζh ζh ζh j pj+1 pj pj−1 + 2 pj+1 pj−1 = − j 2 hj − j hj + j 3 ∆ 2 hjhj 2∆ tanh 2 tanh x x hj hj hj for equation (4.10) and h3 − 2 + − 3 2 j pj+1 pj pj−1 + 2 pj+1 pj−1 = σ +3 1 − coth2( ) 3 ∆ 2 hjhj 2∆ 2 σ hj/ x x hj 88 Figure 4.5: The dimensionless imposed electric ﬁeld (EI,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 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 E0 80 V/cm 80 V/cm 0 0 01 0 1 CNa+ . M . M 0 0 01 0 1 CCl− . M . M ζ −5 mV −15 mV 2hi 13 nm 130 µm 2ho 8 nm 20 µm L 65 nm 650 µm 0.19 5 × 10−5 Re 2 × 10−6 0.0035 pi,po 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 nanonoz-

zle at the inlet is 13nm and the length of the nozzle is 65nm; the height at the outlet is 8nm

2 and so the parameter 1 =0.04 1. The Debye length is 2.5 nm ( =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 1 and is small compared to the streamwise velocity (u). The nozzle is assumed

90 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 (up ∼ O(0.1)) due to this induced pressure is small compared to the electroosmotic ﬂow velocity (ueof ∼ 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 2 =0.04 1. 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 ( =5×10−5) 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 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 (a) potential (b) streamwise velocity

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; =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 (a) potential (b) streamwise velocity

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; =5× 10−5 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 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 1 1 x=0 x=0 x=0.2 x=0.2 x=0.4 x=0.4 0.5 x=0.6 0.5 x=0.6 x=0.8 x=0.8 x=1.0 x=1.0 y y 0 0

−0.5 −0.5

−1 −1 0 0.005 0.01 0.015 0.02 0.025 0.03 −2 −1 0 1 2 −3 U V x 10 (a) streamwise velocity (b) transverse velocity

1 1 x=0 0.5 x=0.2 x=0.4 0.5 x=0.6 0 x=0.8 x=1.0 −0.5

y 0 −1

−1.5 −0.5

−2 p dpdx −2.5 −1 0 1 2 3 4 5 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 x U (c) pressure and pressure gradient (d) streamwise velocity

1 2.5 x=0 p x=0.2 dpdx x=0.4 2 0.5 x=0.6 x=0.8 x=1.0 1.5

y 0 1

−0.5 0.5

−1 −2 −1 0 1 2 0 −3 0 1 2 3 4 5 V x 10 x (e) transverse velocity (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 1 1 x=0 x=0.2 x=0.4 0.5 x=0.6 0.5 x=0.8 x=1.0

y 0

Y 0

−0.5 -0.5

−1 −0.4 −0.2 0 0.2 0.4 -1 u 012345 X (a) streamwise velocity (b) streamlines

Figure 4.9: Streamwise velocity and the streamlines for the electroosmotic ﬂow in a micro- nozzle 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 pi =0and po =10.

96 4.4.2 Results for EOF: Known Surface Charge Density

Results for the EOF based on the Debye- H¬uckel approximation and the lubrication approximation for the known surface charge density boundary condition are shown in Fig- ures 4.10 and 4.11. 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/m2. 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 E0 0.5 V/nm 0.5 V/nm 0 0 1 0 1 CNa+ . M . M 0 0 1 0 1 CCl− . M . M σ −0.01 C/m2 −0.01 C/m2 2hi 14.6.3 nm 4.6 nm 2ho 4.6 nm 14.6 nm L 15 nm 15 nm 0.065 0.21 Re 0.16 0.05 pi,po 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 1 1 x=0 x=0.2 x=0.4 0.5 0.5 x=0.6 x=0 x=0.8 x=0.2 x=1.0 x=0.4 y 0 y 0 x=0.6 x=0.8 x=1.0 −0.5 −0.5

−1 −1 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0 0.5 1 1.5 φ u (a) potential (b) streamwise velocity

1 1 x=0 x=0.2 0 x=0.4 0.5 x=0.6 −1 x=0.8 x=1.0 −2

y 0 −3

−0.5 −4

−5 p dpdx −1 −6 −0.2 −0.1 0 0.1 0.2 0 0.2 0.4 0.6 0.8 1 v x (c) transverse velocity (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; =0.065. The imposed electric ﬁeld is 0.5V/nm and the surface charge density is −0.01C/m2. 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), h ζ = −σ coth y

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 4 3 x=0 3 x=0.2 2 x=0.4 2 x=0.6 x=0.8 x=0 1 1 x=0.2 x=1.0 x=0.4 y 0 y 0 x=0.6 −1 x=0.8 x=1.0 −1 −2 −2 −3 −3 −4 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 φ u (a) potential (b) streamwise velocity

0.05 3

2 0

1 −0.05

y 0 x=0 −0.1 −1 x=0.2 x=0.4 −2 x=0.6 −0.15 x=0.8 p −3 x=1.0 dpdx −0.2 −0.1 −0.05 0 0.05 0.1 0 0.2 0.4 0.6 0.8 1 v x (c) transverse velocity (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; =0.065. The imposed electric ﬁeld is 0.5V/nm and the surface charge density is −0.01C/m2. 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 −0.4 −0.515

−0.5

−0.6

−0.7 w φ w

φ −0.52 −0.8

−0.9

−1 diffuser nozzle nozzle diffuser −1.1 −0.525 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x (b) surface potential for a nozzle in 0.001M (a) surface potential for a nozzle in 1M NaCl 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/m2 and the electrolyte solution is 0.1 MNaCl. For (b), the surface charge density is −0.01C/m2 and the electrolyte solution is 0.001 M.

The surface potential ζ ∝ coth(h) and the curve of cotangent function shows that coth(x)

→ 0 h = h∗ has a larger change when x . Since λ , this change in surface potential is related 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 ( 1)

The electrical double layers (EDLS) are very thin compared to the height of the nozzle/diffuser (λD h) in micro-nozzles or micro-diffusers, making the numerical calcula-

tions hard to resolve the abrupt change of potential and mole fractions close to the wall.

100 An asymptotic analysis can be used to solve this problem. Compared to the solution given

for thin double layer limit under the Debye- H¬uckel 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

h − y η = so that as y = h, η =0; y =0, η →∞. Substituting η in to equation (4.2), the potential equation becomes 2 d φ = − 2 β ziXi (4.20) dη i The potential equation for a binary monovalent electrolyte is given by

d2φ =sinhφ dη2 with boundary conditions

η =0,φ= ζ ∂φ y →∞, =0 ∂η The equation can be written as 2 d dφ d = (2 cosh φ) (4.21) dη dη dη

Integrating once, the equation becomes

dφ = − 2coshφ + C1 dη

where C1 is a integration constant. In the bulk (η →∞), the concentration of cations and anions are the same and so →∞ + = − η ,Xi Xi

101 Based on the Boltzmann distribution, φ =0at η →∞and so C1 = −2. The equation for

potential in the inner region becomes dφ φ = − 2coshφ − 2=2sinh (4.22) dη 2

Integrating equation (4.22) the inner solution for potential is given by

ζ 2 − tanh η i e 2 φ =2ln ζ (4.23) 1 − 2 tanh η e 2 where the superscript i indicates that it is the inner solution for φ. Similarly, near the lower wall −h(x), η is deﬁned as y + h η = 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 ziXi =0) in the outer region, leading to the potential equation in the outer region: ∂2φ =0 (4.24) ∂y2 The boundary condition for this equation is given by the matching between the inner and outer solution at both walls

lim φo = lim φi y→±h η→∞ And the outer solution for potential is

φo =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 ueof is governed by equation d2u d2φ eof = E (4.25) dη2 dη2 I,x 102 with boundary conditions

du η =0,u =0; η = ∞, =0 eof dη

The inner solution is given by

i = ( − ) ueof EI,x φ ζ (4.26)

In the outer region, the equation for the electroosmotic component is given by

2 ∂ ueof =0 ∂y2 with the boundary condition

lim o = lim i ueof ueof y→±h η→∞

from the matching between the inner and outer solution

∂uo eof =0,y =0 ∂y

due to the symmetry of the problem. The outer solution for the electroosmotic component

is, using equation (4.26)

o = − ueof EI,xζ

For the transverse velocity in the inner region, the equation is

d2v d2φ eof = E dη2 dη2 I,y

and similarly, the inner solution is given by

i = ( − ) veof EI,y φ ζ

In the outer region, the equation for veof is given by

2 ∂ veof =0 ∂y2 103 with the boundary condition

lim o = lim i veof veof y→±h η→∞

from the matching between the inner and outer solution. The outer solution for the elec-

troosmotic component is

o = − veof EI,yζ

It is noted that unlike EI,x, which is independent of y, EI,y is a function of y, given by equation (4.1).

In the inner region, the pressure-driven velocity component is given by

∂2u p =0 (4.27) ∂η2 along with the boundary condition

du η =0,u =0; η = ∞, =0 p dη

i =0 and the solution is up . Similarly, the inner solution for the pressure-driven transverse i = velocity component is vp o. Outside the electrical double layer, the governing equation for the pressure-driven component is given by 2 ∂ up dp = 1 (4.28) ∂y2 dx and the solution is

1 dp uo = (y2 − h2) p 2 dx

In the outer region, the total outer velocity is then given by

1 ∂p uo = uo + uo = (y2 − h2) − E ζ eof p 2 ∂x I,x

104 = 1 Note that the non-dimensional applied electric ﬁeld EI,x h and so the outer solution for velocity is

1 dp ζ uo = (y2 − h2) − 2 dx h

The v velocity can be directly obtained by integrating from the continuity equation

2 y ∂u y 1 d p dp ζh o = − + = − 2 − 2 − + + v 1 dy C1 2 y h hh 2 dy C1 (4.29) −h ∂x −h 2 dx dx h so that 2 3 2 o 1 d p y 2 3 dp ζh v = − − h y − h + 1 hh (y + h)++ (y + h)+C1 (4.30) 2 dx2 3 3 dx h2 where C1 is the integration constant. The outer solution for the transverse velocity close

= ± o = o + o = − to the walls (y h) is given by v veof vp EI,yζ. On the upper boundary,

EI,y = EI,xh and on the lower boundary EI,y = −EI,xh . Substituting the boundary condition for transverse velocity on the lower boundary in to equation (4.30), the integration constant is given by ζh C1 = − h

Applying the boundary condition on the upper wall, the pressure equation is given by

3 2 ζh 2h d p dp 2 ζh = +21 h h + (4.31) h 3 dx2 dx h and so the pressure equation in the outer region is given by 1 ∂ 3 ∂p h 1 =0 (4.32) 3 ∂x ∂x

With the pressure boundary conditions pi = po =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 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- H¬uckel approximation, a pressure gradient is ∼ = λ → 0 induced by the EOF, which is usually p O h . If we take the limit ( ), 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- H¬uckel approximation discussed in Section 4.3.1.

Since there is no induced pressure gradient by the EOF, the velocity equation can then be re-written as ∂2u =0 (4.33) ∂y2 ∂2v =0 (4.34) ∂y2

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

∗ ∗ = eζ thin EDLs, the electroosmotic flow mobility can be defined as µe µ , the electroosmotic ∗ = ∗ ∗ velocity is then given by u µeEI,x. The flowrate is then given by

∗ ∗ ∗ = ∗ ∗( )= = u dA µeEI,xA x µeE0A0 constant A 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 ) = h0 + A0sin(w x ) as shown in Figure 4.13. The dimensionless form for the wall is h(x)=1+δsin(wx) where

106 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.

A0 ∗ ∗ ∗ = = ( ) ( )= 0 0 δ h0 and w w L. The applied electric ﬁeld is given by E i, x A x E A and ( )= 1 the dimensionless applied ﬁeld is E x h similar to the nozzle cases discussed before. The outer solution for the streamwise velocity is given by

ζ ζ u = − = − h 1+δsin(wx)

and v can be easily found from the integration shown in equation (4.29)

y y ( ) ( ) = h + (− )= − ζwδcos wx = ζwδcos wx v 2 dy v h 2 dy −h h −h (1 + δsin(wx)) 1+δsin(wx)

The streamlines are shown in Figure 4.14.

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 Figure 4.14: The streamlines for electroosmotic ﬂow in the channel shown in Figure 4.13.

4.6 Numerical EOF Calculation Based on Poisson-Boltzmann Model

In Section 4.4, the results are calculated based on the Debye- H¬uckel 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

∂2φ β = − z X (4.35) ∂y2 2 i i

108 = 0 −ziφ where Xi Xi e with boundary condition ∂φ y = ±h : = ±σ ∂x y

The equation is discretized using the second-order accurate central difference approxima-

tions for all the derivative terms. The ﬁnite difference equations are obtained in the form:

2 β 0 −ziφj φ +1 − 2φ + φ +1 = −dy z X e (4.36) j j j 2 i i 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 − φ New Old <δ (4.37) φNew where δ is a small number and it is usually taken as 10−4 as the convergence criteria.

Step 2 Solving ueof numerically For the streamwise velocity component ueof , the solution is calculated based on the equation

∂2u β eof = − z X (4.38) ∂y2 h2 i i and boundary conditions y = ±h : ueof =0. Central difference approximation is also used to discretize the equation and Thomas algorithm is used to solve the equation.

3 ∂ueof Step Calculating ∂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 = − 1 dy −h ∂x 109 (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 and at the upper boundary, h h ∂u ∂ueof + up v = − 1 dy = − 1 dy =0 −h ∂x −h ∂x

Thus the EOF component ueof can be related to the pressure-driven up component by h ∂u h ∂u p dy = − eof dy −h ∂x −h ∂x

Substituting the analytical solution of the pressure driven component shown in equation

(4.7), the pressure equation is 2 3 2 h h ∂ p +2 2 ∂p = ∂ueof 2 h h dy (4.39) 3 ∂x ∂x −h ∂x

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.

∂u To ﬁnd ∂x numerically, one has to map x, y to a rectangular mesh in ξ,η coordinates, which 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

= = ∂u ∂u ∂u ξ x and η y/h. The term ∂x can be written in terms of ∂ξ and ∂η as

∂u = ∂u ∂ξ + ∂u ∂η ∂x ∂ξ ∂x ∂η ∂x

= = ∂ξ =1 ∂η = −yh Since x ξ and η y/h, ∂x and ∂x h2 . Therefore

∂u = ∂u − ηh ∂u ∂x ∂ξ h ∂η and for point i, j, − − ∂u = ui+1 ui−1 − ηh uj+1 uj−1 ∂x 2∆ξ hi 2∆η ∆ = ∆ = dy where ξ dx and η h is a constant. The transverse velocity v can be calculated y − ∂u from −h ∂xdy using numerical integration based on the trapezoidal rule. 111 x =0.25,y = −1.2541 x =0.75,y = −1.9708 81 × 161 φ = −1.2414, ueof =1.0072,p = −0.0187 φ = −1.1026, ueof =1.0185,p = −0.1365 161 × 321 φ = −1.2419, ueof =1.0083,p = −0.0182 φ = −1.1013, ueof =1.0242,p = −0.1327

x =0.6,r =0.08 x =0.8,r =0.4175 81 × 161 φ = −0.0314, ueof =1.3229,p = −0.7735 φ = −0.0975, ueof =1.0246,p = −0.1600 161 × 321 φ = −0.0313, ueof =1.3248,p = −0.7749 φ = −0.1013, ueof =1.0342,p = −0.1632

Table 4.3: The numerical accuracy check for the numerical calculation for the electroosmotic ﬂow in the nano-diffuser shown in Figure 4.16. The results for 81 × 161 grid are compared with the results of 161 × 321 grid and two-digit accuracy has been attained. The numerical results are shown in Figure 4.17. Note that here the pressure p is rescaled on EI,x.

Step 4 Solving the pressure equation numerically The pressure equation (4.39) can

be solved using Thomas algorithm. The pressure-driven component up 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 4.11. A 81 × 161 mesh is used to perform the numerical calculation and two digit accuracy is obtained by comparing the results calculated on the 81 × 161 mesh and 161 × 321 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/m2, higher than that listed in the table. For such a high surface charge

(σ = −0.07C/m2) the Debye-H¬uckel solution calculated based on the analysis in Sec- tion 4.3.2 is about 30% off due to the high electric potential as shown in Figure 4.18. In this case, the Debye- H¬uckel 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- H¬uckel solution is also about 30% higher than the nonlinear result as shown in Figure 4.18 (b). For a smaller surface charge, the Debye-H¬uckel results will be more accurate. For example as shown

112 1 1 x=0 x=0.2 x=0.4 0.5 0.5 x=0.6 x=0 x=0.8 x=0.2 x=1.0 x=0.4 y 0 y 0 x=0.6 x=0.8 x=1.0 −0.5 −0.5

−1 −1 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0 2 4 6 8 10 φ u (a) potential (b) streamwise velocity

1 10 x=0 x=0.2 x=0.4 0.5 x=0.6 0 x=0.8 x=1.0

y 0 −10

−0.5 −20 p dpdx −1 −30 −1.5 −1 −0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 v x (c) transverse velocity (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; =0.065. The imposed electric ﬁeld is 0.5V/nm and the surface charge density is −0.07C/m2. 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 4 3 x=0 3 x=0.2 2 x=0.4 2 x=0.6 x=0.8 x=0 1 1 x=0.2 x=1.0 x=0.4 y 0 y 0 x=0.6 −1 x=0.8 x=1.0 −1 −2 −2 −3 −3 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0 0.5 1 1.5 2 2.5 3 φ u (a) potential (b) streamwise velocity

0.2 3

2 0

1 −0.2

y 0 x=0 −0.4 −1 x=0.2 x=0.4 −2 x=0.6 −0.6 x=0.8 p −3 x=1.0 dpdx −0.8 −0.5 0 0.5 0 0.2 0.4 0.6 0.8 1 v x (c) transverse velocity (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; =0.065. The imposed electric ﬁeld is 0.5V/nmand the surface charge density is −0.07C/m2. 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 in Figure 4.19, the surface charge density is −0.01C/m2 and the corresponding surface potential is about ζ =0.5. The discrepancy is much less between the nonlinear numerical calculations and the Debye- H¬uckel 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 E0 =0.5 V/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 10

4 eof u 2 x=0,DH x=0.5,DH 0 x=1,DH x=0,nonlinear −2 x=0.5,nonlinear x=1,nonlinear −4 −1 −0.5 0 0.5 1 y/h

(a) ueof

5 DH Nonlinear 4

3 p

0 0 0.2 0.4 0.6 0.8 1 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 ueof 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; =0.065. The imposed electric ﬁeld is 0.5V/nmand the surface charge density is −0.07C/m2. 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 1.6

1.4

1.2

0.8 eof u 0.6

0.4 x=0,DH x=0.5,DH 0.2 x=1,DH x=0,nonlinear 0 x=0.5,nonlinear x=1,nonlinear −0.2 −1 −0.5 0 0.5 1 y/h

(a) ueof

5 DH Nonlinear 4

3 p

0 0 0.2 0.4 0.6 0.8 1 x

(b) p

Figure 4.19: The comparison of DH and nonlinear results for streamwise velocity component ueof 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; =0.065. The imposed electric ﬁeld is 0.5V/nm and the surface charge density is −0.01C/m2. 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 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∗ = −0.0061, u∗ =0.0121,p∗ = 23265.09 φ∗ =0.00085, u∗ =0.0114,p∗ = 8896.86 Mesh II φ∗ = −0.0060, u∗ =0.0118, p∗ = 23494.47 φ∗ = −0.00081, u∗ =0.0112,p∗ =8809.56

x∗ =20nm, y∗ = −1nm x∗ =40nm, y∗ =5nm Mesh I φ∗ = −0.0034, u∗ =0.0124,p∗ = 15721.80 φ∗ =0.0062, u∗ =0.0096,p∗ = 10156.73 Mesh II φ∗ = −0.0030, u∗ =0.0121,p∗ = 15074.45 φ∗ = −0.0059, u∗ =0.0095,p∗ = 10418.42

Table 4.4: The numerical accuracy check for the results calculated using COMSOL shown in Figure 4.21. 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 4.4.

118 (a) Cl− in the nozzle (b) u in the nozzle

Figure 4.21: Results for ionic concentration and ﬂow ﬁeld for the EOF in the nozzle- reservoir system calculated using COMSOL. The parameters are listed in table 4.2.

119 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

∗ = − ∇ + ∗ + ∗ nA DA cA uAzAFcAE cAu

and the velocity in the x− direction is

∗ = −DA ∂cA + ∗ + ∗ uA uAzAFEx u (4.40) cA ∂x

120 (a) velocity contours (MD) (b) velocity contours (continuum)

Figure 4.22: The results for EOF in the nozzle-reservoir system based on Molecular Dy- namics 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; =0.065. The imposed electric ﬁeld is 0.5V/nm and the surface charge density is −0.07C/m2.

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 (uEM < 0).

121 = 650 × 10−6 =10−9 m2 Based on the experimental data (L m and DA sec ) the magnitude of the ﬁrst term, the diffusion term is

D ∂c m2 ∆c ∆c m A A ∼ 10−9 /(650 × 10−6m) A ∼ 10−6 A cA ∂x sec cA cA sec

Similarly the second term, the electrical migration term is

mole m2 Coul m u Fz E∗ ∼ 4 × 10−14 × 96500 × 80V/(10−2m)z ∼ 3.9 × 10−4z A A x J sec mole A A sec

The last term, the convective term is of the order

E∗RT m u∗ ∼ e x ∼ 1.6 × 10−4 Fµ 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

∗ = ∗ + ∗ = + uA uAzAFEx u uEM uF

where uEM is the velocity component due to the electrical migration and uF 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 negatively charged wall positively charged wall uncharged particles uF > 0,uEM =0 uF < 0,uEM =0 positively charged particles uF > 0,uEM > 0 uF < 0,uEM > 0 negatively charged particles uF > 0,uEM < 0 uF < 0,uEM < 0

Table 4.5: The dependence of the particle motion on the wall charge and the particle charge. Here uF is the bulk motion of the ﬂuid ﬂow and uEM 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 (uEM =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 (uEM < 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 (uEM < 0,uF < 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 (uEM < 0,uF > 0)and the particle motion depends on the parameter RCE,

123 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 uF > 0,uEM < 0.

which is ratio of the convective term and the electrical migration term:

∗ zADAE RCE = φ0U0

In the experiments (Figure 4.1), the walls and the particles are both negatively charged

(uF > 0,uEM < 0) and so the particle motion depends on the parameter RCE. For the experimental parameters (Wang & Hu, 2007), RCE =0.02zA. The calculated particle velocity at the nozzle centerline (y =0) is compared with the experimental data in Fig.

4.24. 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 =

zAe =16 × 10−19 πd2 where e . C and d is the diameter of the particles. The calculated results 124 compare well with the experimental data for σ = −1µC/cm2 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 4.23.

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 manipulation and combination of those balance equations into modiﬁed forms, by intro-

ducting 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

σA ασ · 4πa2 3ασ ρ = α = = e,p V 4πa3/3 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

2 e∇ φ = −ρe, the dimensionless equation for potential becomes

2 2 2 ∂ φ 2 ∂ φ 2 ∂ φ β + + = − z X − π1α (4.41) ∂y2 1 ∂x2 2 ∂z2 2 i i

3σ 1 = where π aF c is the dimensionless volume charge density of the particle phase. Here the volume charge from the particle phase and the ﬂuid phase are both included.

126 For ionic species A in the ﬂuid phase, the dimensionless equation is given by 2 2 2 ∂ XA 2 ∂ XA 2 ∂ XA ∂XA ∂XA ∂XA + + = Pe 1u + v + 2w ∂y2 1 ∂x2 2 ∂z2 ∂x ∂y ∂z ∂XAEx ∂XAEy ∂XAEz + 1z + z + 2z (4.42) A ∂x A ∂y A ∂z

For the ﬂuid phase, the conservation of momentum is given by the Navier Stokes equation as, ∂u∗ ρ +(u∗ •∇)u∗ = −∇p∗ + µ∇2u∗ + ρ E ∗ − F − ρg (4.43) ∂t∗ e D

here FD is the Stokes Drag body force. For a single spherical particle, the Stokes Drag

force is

FD,p =6πµa(u − up) and for the particle phase, the Stokes Drag force per unit volume is given by

F 9αµ(u − u ) F = α D,P = p = ηα(u − u ) D V 2a2 p

= 9µ where η 2a2 . The three momentum equations for an incompressible, steady ﬂow are, in dimensionless form, ∂u ∂u ∂u ∂u ∂p β 1 ∂φ 2 +Re 1u + v + 2w = −1 + z X 1 − +∇ u−ηα(u−u ) ∂t ∂x ∂y ∂z ∂x 2 i i Λ ∂x p (4.44)

∂v ∂v ∂v ∂v ∂p β ∂φ 2 Re +Re 1u + v + 2w = − − z X +∇ v−ηα(v−v )− ∂t ∂x ∂y ∂z ∂y Λ2 ∂y i i p Fr2 (4.45)

∂w ∂w ∂w ∂w ∂p β ∂φ 2 + Re 1u + v + 2w = −2 − z X + ∇ w (4.46) ∂t ∂x ∂y ∂z ∂z Λ2 ∂z i i

127 2 U0 ρh here the Froude number is given by Fr = √ and the time scale is t0 = . These gh µ 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 + ∇·(αu∗)=0 ∂t∗ p and the steady state dimensionless continuity equation is given by

∂αup ∂αvp ∂αwp 1 + + 2 =0 ∂x ∂y ∂z

∗ u = up here the particle velocity is also scaled on p U0 . The velocity equation is given by, ∂u ∗ ρ α p +(u ∗ •∇)u ∗ = ρ E ∗ + F − αρ g + αρg (4.47) p ∂t∗ p p e,p D p

∗ here ρp is the density of the particle phase, up is the dimensional velocity vector of the particle phase. In dimensionless form, this equation can be written as ∂up ∂αup ∂αup ∂αup β 1 ∂φ α + γRe 1u + v + 2w = π1α 1 − + ηα(u − u ) ∂t p ∂x p ∂y p ∂z 2 Λ ∂x p (4.48)

∂vp ∂αvp ∂αvp ∂αvp βπ1 ∂φ αRe α +γRe 1u + v + 2w = − α +ηα(v−v )− (γ − 1) ∂t p ∂x p ∂y p ∂z Λ2 ∂y p Fr2 (4.49)

∂wp ∂αwp ∂αwp ∂αwp βπ1 ∂φ α +γRe 1u + v + 2w = − α +ηα(w −w ) (4.50) ∂t p ∂x p ∂y p ∂z Λ2 ∂z p

= ρp where γ ρ is the ratio of the particle phase and ﬂuid phase densities. 128 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,,

γ, π1 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/m2 (Ohsawa et al., 1986) surface charge density in 0.1MNaClsolution. The relative dielectric constant for Polystyrene is

2.5 and the density is 1050kg/m2. The ratio of the electric body force to the viscous drag

body force is 3σE0 2 3 7 FElectric α a σa . = ∗ ∗ = ≈ 9µ(u −up) FDrag 3eφ0(u − up) u − up α 2a2

where u and up 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

3σE0 FElectric α a 3 = ≈ 6 × 10 E0 FGravity g(ρp − ρ) 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.

4.10 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 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 aque-

ous 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 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 (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(OC2H5)4 +2H2O SiO2 +4C2H5OH

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 + u ·∇c = D ∇2c + R (4.51) ∂t A AB A A 132 where cA is the concentration for species A and u is the velocity vector; DAB is the diffusion of species A; RA is the volumetric reaction rate of A in the solution. Note that for TEOS,

it is the consumption rate (RA < 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

∂c ∂2c A = D A + R (4.52) ∂t AB ∂y2 A with boundary conditions:

t =0,h=0,cA = c0

0 = = | t> ,y h, ny k2 cA y=s

where ξ is a parameter related to the reaction rate on the surface and ny is the ﬂux of species

A into the wall due to the reaction at the surface. Note that RA = −k1CA is also a function of the concentration of species A and the height of the interface h(t) is given by

t h = γ ny| = dt 0 y s 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 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 nozzle-

reservoir 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 in-

cluding chemical kinetics, surface growth, ﬂuid ﬂow and mass transport will be consid-

ered 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.

4.11 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- H¬uckel 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 in a nozzle and decreasing in a diffuser in the axial direction due to the change of cross-

sectional 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 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 (up) due to this induced pressure is usually small compared to the electroosmotic ﬂow velocity (ueof ).

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- H¬uckel approximation will not hold. For the cases of thin EDLs ( 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- H¬uckel approximation by taking the limit → 0. Comparison between the results calculated from the Debye- H¬uckel solution and the nonlinear numerical solution has shown that the Debye- H¬uckel solutions over-predict the streamwise velocity and the induced pressure by ∼ 30% for a surface charge density −0.07C/m2.

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 RCE. The results shown in this chapter

135 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 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 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 (uDNA). 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 (FE and

∗ FR) related to the conformational change of DNA in transition. As the blob-like conﬁg-

uration 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 (Fblob2)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 con-

∗ ∗ formational change (FE and FR), the viscous drag force acting on the linear DNA inside

∗ the nanopore (Fd ) and the viscous drag acting on the blob-like DNA outside the nanopore

∗ ∗ (Fblob1 and Fblob2).

138 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 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∗ − h∗ h∗(x∗)=h + o i x∗ i L∗

∗ ∗ where hi ,ho are the radii of inlet and outlet of the pore and in dimensionless form

∗ − ∗ ( )=1+ho hi h x ∗ x hi

h∗ x∗ where h = ∗ and x = ∗ . hi L 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

∇· ∗ =0 ∗ ( ) ∗( )= ∗ (0) ∗(0) form E so that the imposed electric field satisfies EI,x x A x EI,x A . ∗ (0) If the imposed electric field at the inlet EI,x is taken as the scale of the imposed electric

140 1−a2 a∗ field, the dimensionless imposed electric field is thus EI,x = 2 2 where a = ∗ . h(x) −a hi 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 1 ∂ φ + ∂φ = − β 2 2 ziXi (5.1) ∂r r ∂r i 2 1 ∂ u + ∂u = − β + ∂p 2 EI,x 2 ziXi (5.2) ∂r r ∂r i ∂x 2 1 1 ∂ v + ∂v = − β + ∂p 2 EI,y 2 ziXi (5.3) ∂r r ∂r i 1 ∂r ∂u 1 ∂rv 1 + =0 (5.4) ∂x r ∂r Note that since the pressure is assumed to be independent of r, the pressure is rescaled

= µU0 on p0 L and so the 1 in equation (2.35) is dropped. The boundary conditions are ∂φ r = h(x): = σ ,u=0,v =0 ∂r w ∂φ r = a : = −σ ,u=0,v =0 ∂r d where σw and σd are the surface charge density of the nanopore and the DNA, and σ is

∗ ∗ φ0 σ hi scaled on ∗ as σ = . ehi eφ0

5.3 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 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 1 ∂ φ + ∂φ = − β 2 2 ziXi (5.5) ∂r r ∂r i 2 1 2 ∂ u + ∂u = − β 2 2 ziXi (5.6) ∂r r ∂r i

Note that in this case h =1, EI,x =1all through the pore and there is no pressure gradient

in the x direction. The boundary conditions are

∂φ = σ ,u=0,r=1 ∂r w ∂φ = −σ ,u=0,r = a ∂r d

These equations can be solved numerically and an analytical solution may be obtained using the Debye-H¬uckel approximation. Based on the Debye-H¬uckel approximation, the potential equation becomes 1 ∂ ∂φ φ r = (5.7) r ∂r ∂r 2

142 This is a modiﬁed Bessel equation and the solution is r a 1 r a 1 I0 σwK1 + σdK1 + K0 σwI1 + σdI1 φ (r)=− (5.8) a 1 1 a 1 1 − 1 1 I K I K At r = a, the DNA surface potential is given by equation 5.8 as a a 1 a a 1 I0 σwK1 + σdK1 + K0 σwI1 + σdI1 ζ = φ (r = a)=− d a 1 − 1 a I1 K1 I1 K1 (5.9) and at r =1, the nanopore surface potential is given by 1 a 1 1 a 1 I0 σwK1 + σdK1 + K0 σwI1 + σdI1 ζ = φ (r = a)=− w a 1 − 1 a I1 K1 I1 K1 (5.10)

The corresponding velocity solution can be derived from equation 1 ∂ ∂φ 1 ∂ ∂u r = r (5.11) r ∂r ∂r r ∂r ∂r and so

∂u ∂φ C1 = + (5.12) ∂r ∂r r

The solution is given by

u = φ +lnC1 + C2 (5.13) where C1 and C2 are integration constants. Applying the boundary conditions for u and φ

shows

ζw − ζd C1 = ln(a) and C2 = −ζw where ζd (equation (5.9)) and ζw (equation (5.10)) are the dimensionless zeta potential of the DNA and the wall. Substituting C1 and C2 into equation (5.13), the velocity is given by ζ − ζ u(r)=φ(r)+ w d ln(r) − ζ (5.14) ln(a) w

143 0 0.4 0.1M 0.35 1M −0.1 0.3 −0.2 0.25 φ −0.3 u 0.2

0.15 −0.4 0.1 −0.5 0.1M 0.05 1M 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 r r (a) potential (b) velocity

Figure 5.4: The results for potential and velocity calculated from equations 5.8 and 5.14. The radius of the nanopore is 5nm and the radius of the DNA is 1nm. The surface charge density is −0.015C/m2 on the DNA surface and −0.006C/m2 on the wall.

τ 4.9

4.8

4.7

4.6 dimensionless shear stress

4.5 0 0.2 0.4 0.6 0.8 1 Electrolyte concentration (M)

= ∂u = Figure 5.5: The dimensionless shear stress τ ∂r at r a as a function of electrolyte concentration. The radius of the nanopore is 5nm and the surface charge density is −0.06C/m2 while the radius of the DNA is 1nm and the surface charge density is −0.15C/m2. The electrolyte concentration varies from 0.01M ( =0.6)to0.1M ( =0.06).

144 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/m2 on the DNA surface and −0.006C/m2

on the wall. It is shown in Figure 5.8 (a) that the electric double layers are overlapped for

0.1MKClsolution (Debye length λ =0.96nm). The Debye length for 1MKClsolution 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 is dr du ζ − ζ = + w d σd ln( ) dr r=a a a ∗ τ µU0 τ = ,τ0 = h0 which is scaled as τ0 h0 (see Table 2.1 for the deﬁnition of ). The corresponding dimensionless viscous drag acting on the DNA surface is 1 ∂u ∂u Fd = dx ∝ τ = 0 ∂r r=a ∂r r=a

∗ ∗ ∗ ∗ ∗ Fd ∗ L ∗ ∂u ∗ 2πµU0a L = = 2 ∗ 0 = if this force is scaled as Fd F0 , where Fd 0 πµa ∂r dx and F h0 . 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 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 ∗ 5 3 nanopore (Fd ). In Section . , the analytical solution for EOF in an annulus is derived based on the Debye-H¬uckel 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 (ueof ) and the pressure driven ﬂow (up) as in Chapter 4. The velocity equation for each component is given by 1 ∂ ∂ueof = − β r 2 EI,x ziXi (5.15) r ∂r ∂r i

1 ∂ ∂u ∂p r p = − (5.16) r ∂r ∂r ∂x with boundary conditions

r = a : ueof =0,up =0

r = h : ueof =0,up =0

The pressure equation is based on mass conservation inside the pore Q ≡ Qeof + Qp = constant where Q is the total ﬂowrate and Qeof and Qp are the ﬂowrate due to the EOF and

146 pressure-driven flow components. For the electroosmotic component, the flowrate Qeof can = h be obtained by Qeof a ueof rdr where ueof 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 dp r2 a2ln(r/h) − h2ln(r/a) u = + (5.17) p dx 4 4ln(h/a)

The ﬂowrate Qp is then given by

h dp (a2 − h2)2 a4 − h4 Qp = uprdr = + (5.18) a dx 16ln(h/a) 16

Note that Qeof ,Qp are all varying along x axis but from the conservation of mass, the total = = ∂Q = ﬂowrate should be a constant (Q constant). Therefore the axial derivative Q ∂x + = ∂Qeof + ∂Qp =0 Qeof Qp ∂x ∂x ; substituting Qeof and Qp into the equation, the pressure equation becomes

d2p (a2 − h2)2 a4 − h4 dp (a2 − h2)2 a4 − h4 + + + + Q =0 (5.19) dx2 16ln(h/a) 16 dx 16ln(h/a) 16 eof

This is a second order differential equation and it can be solved numerically with the bound-

ary 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 L ∗ = 2 ∗ ∗ ∗ ∗ =2 ∗ ∗ ∗ ∗ ∗ =2 ∗ ∗∆ Fe πa σdEI,xdx πa σd EI,xdx πa σd V (5.20) 0 0 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 ∗ − 0 15 2 density of the DNA (σd . C/m ) is assumed to be a constant here. The typical applied

147 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 L ∗ ∗ = 2 ∗ ∂u ∗ Fd πµa dx (5.21) 0 ∂r∗

∗ F = Fd and in dimensionless form ( d F0 ) 1 ∂u Fd = dx (5.22) 0 ∂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 5.3.

5.5 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 (a) DNA in transition (Muthukumar, 1999)

0.015

0.01

0.005

−0.005 entropic force (pN)

−0.01

−0.015 0 0.2 0.4 0.6 0.8 1 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 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-to-

end 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 ex-

tensions, 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 (1 − x/L)−2 1 x p = − + (5.23) kT 4 4 L 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/lp, which is about 0.1pN for a double stranded

DNA (lp =50nm).

150 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 =0.5(ln(m)+ln(N − m)) kT 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/lp is the total number of segments and m is the number of segments in region I. The corresponding entropic uncoiling/recoiling force (FE)is

− 2 = dH = kT N m FE 2 (5.24) dx 2lp Nm− m 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 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 lp LDNA 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 = A0kT 1 3 (5.25) / 1/3 R// lp h where R// is the length of the pore occupied by the polymer and A0 =1.1036 is a constant.

The corresponding entropic force is

= dH = A0kT FE 1 3 (5.26) / 1/3 dR// lp h 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 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 k Tx ∆H = B A w−α−1 + Bwδ−1 + Cw−1 (5.27) 2h∗ where w = s(2h∗/a)−1+1/υ; α =(3υ − 1)−1; δ =(1− υ)−1; where υ =0.58765 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

dH k T F = = B A w−α−1 + Bwδ−1 + Cw−1 (5.28) E dx 2h

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 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 (FR) in the receiver reservoir is given by Prinsen et al. (2009)

kT FR = lp

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, lp =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.

Force Magnitude ∗ ∼ 0 01 FU (Muthukumar, 1999) . pN ∗ 0 67 FU (Odijk, 1983) . pN ∗ 1 9 FU (Klushin et al., 2008) . pN ∗ 0 083 FR (Prinsen et al., 2009) . pN ∗ 113 Fe pN

∗ Table 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 ).

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 simple model based on the balance of the electrical force and the viscous drag on the blob-

like 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

∗ =6 ∗ Fblob1 πµRgvblob (5.29)

∗ where vblob is the approaching velocity of the DNA blob-like configuration outside the pore and Rg 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 dR ξdLυ v∗ = − g = (5.30) blob dt dt where υ is the scaling index for a real chain and ξ is defined as an effective friction coef-

ﬁcient, 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 (uDNA) 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 = Llp Rg 6 (5.31)

where L is the DNA contour length and lp is the persistence length. The velocity of the

DNA blob is then given by dR 1 l dL v∗ = − = − p (5.32) blob dt 2 6L dt 155 Since the reduction of the contour length of the blob-like DNA outside the nanopore is due

∗ = − dL to the DNA translocation in the pore, uDNA dt and so the velocity of the blob-like conﬁguration is related to the DNA translocation velocity by 1 l v∗ = p u∗ (5.33) blob 2 6L DNA

∗ The viscous drag acting on the DNA outside the pore Fblob 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

∗ ∗ = πµlpuDNA Fblob1 2

Note that this force is independent of the DNA contour length. Using the experimental ∗ =001 DNA translocation velocity uDNA . m/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 Ll F ∗ =6πµR U ∗ =6πµU∗ p blob2 g i 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 of the nanopore is 340nm. The surface charge of the pore is −0.2C/m2 and the applied voltage drop is 120mV . The working solution is 1M KCL solution and the electroosmotic ∗ ∼ (10−5 ) ﬂow velocity at the nanopore entrance is given by Ui O m . 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.

Force Magnitude ∗ 7 86 Fblob1 (Storm et al., 2005b) . pN ∗ ∼ 0 1 Fblob2 . pN ∗ 113 Fe pN

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 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 Ft is used to immobilize the DNA. The electrical driving force Fe and the viscous drag force Fd 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

∗ = ∗ − ∗ Ft Fe Fd

158 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 Ft is the tethering force; Fe is the electrical driving force and Fd 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 fo- cusing the electron beam on a 60 nm thin SiO2/SiN/SiO2 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 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 numeri-

cally. 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:

∆ rj r 2 β 0 −ziφj φ +1 − 2φ + φ +1 + (φ +1 − φ −1)=−∆r z X e (5.34) j j j 2 j j 2 i i

0 for point j, where Xi 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 Suc-

cessive Over-relaxation (SOR) method at each iteration. The iterative procedure is said to

converge if φ − φ new old <δ (5.35) φnew where δ is a small number and it is usually taken as 10−4 as the convergence criteria.

Step 2: Solving ueof numerically

Equation (5.15) for the streamwise velocity component ueof is descritized using central

160 1.2 1.2

PHI 1 1 -0.2 Cl -0.4 -0.6 0.09 -0.8 0.08 0.8 -1 0.8 0.07 -1.2 0.06 -1.4 0.05 -1.6 0.04 Y 0.6 -1.8 Y 0.6 0.03 -2 0.02 -2.2 0.01 -2.4 0.4 -2.6 -2.8 0.4 -3 -3.2 -3.4 0.2 -3.6 0.2

0 0 0.2 0.4 0.6 0.8 1 1.2 0 X 0 0.2 0.4 0.6 0.8 1 1.2 X

(a) potential (b) Concentration of Cl−

1.2

0.8

Y 0.6

0.4

0.2

0 0.2 0.4 0.6 0.8 1 1.2 X (c) Streamwise Velocity (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/m2 and the applied voltage drop is 120mV . The electrolyte solution is 0.1MKCl.

161 Mesh x =0.2,r =0.1525 x =0.4,r =0.06 121 × 161 φ = −0.0014, ueof =2.9439, p =0.010086 φ = −0.5984, ueof =2.4184, p =0.157109 241 × 321 φ = −0.0014, ueof =2.9443,p =0.009751 φ = −0.5985, ueof =2.4196, p =0.153938

x =0.6,r =0.08 x =0.8,r =0.4175 121 × 161 φ = −0.9808, ueof =1.8381,p =4.803718 φ = −0.0028, ueof =1.8363, p =2.496149 241 × 321 φ = −0.9808, ueof =1.8395, p =4.808123 φ = −0.0028, ueof =1.8391, p =2.508760

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 121 × 161 grid (Mesh I) are compared with the results of 241 × 321 (Mesh II) grid and two-digit accuracy has been attained. Note that the pressure shown here is rescaled on EI,x).

differencing as

∆ ∆ 2 rj r r EI,xβ 0 −ziφj u +1 − 2u + u +1 + (u +1 − u −1)=− z X e (5.36) j j j 2 j j 2 i i for point j and solved using the Thomas algorithm.

Step 3: Integrating Qeof numerically

Based on the solution for ueof , Qeof 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 up is then calculated based on equation (4.7). The transverse velocity can be

calculated from equation (4.29) numerically using the trapezoidal rule. A 121 × 161 mesh

is used for the numerical calculations and two digit accuracy is obtained by comparing with

the numerical results calculated based on a 241 × 321 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/m2 for Keyser’s experiments by (Smeets et al., 2006); the voltage drop is 120mV

162 and the working ﬂuid is 0.1MKClsolution. 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.

35 Numerical Lubrication Experiment,L =2.1µ m 30 b Experiment,L =2.4µ m b 25 Experiment,L =2.9µ m b COMSOL 20

Tethered Force (pN) 10

0 0 20 40 60 80 100 120 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 0.1MKCl. 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, Lb 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 5.11. 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 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 ∼ 25% ∗ = ∗ + ∗ tethering force accounts for of the total electric driving force (Fe Fd Ft )) 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

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

2 ∂2 the axial derivative terms (1 ∂x2 ) 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 (∼ 106V/m in the central cylindrical nanopore), inducing concentration polarization,

or ion density gradients in the nanopores. In the central cylindrical nanopore, the elec-

trical double layers are overlapped and the electrostatic interaction between the ions and

164 (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/m2 and the applied voltage drop is 120mV . The electrolyte solution is 0.1M KCl.

165 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∗ =1× 10−8m in Figure 5.12 (c)) of higher concentration. As the Cl− ions migrate out of the central nanopore (near x∗ = −1 × 10−8m 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 Ft Fe Fd , 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 5 ∗ the pore is nm at the small end. It is shown here that the electrical driving force (Fe )is

166 100 ∗ ∗ about pN, and the uncoiling and recoiling forces (FU and FR) 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 (Fblob1

∗ and Fblob2) is about one or three orders of magnitude less. In some of the previous work,

∗ ∗ ∗ these forces, including FU , FR and Fblob1, 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 Magnitude ∗ ∼ 0 01 FU (Muthukumar, 1999) . pN ∗ 0 67 FU (Odijk, 1983) . pN ∗ 1 9 FU (Klushin et al., 2008) . pN ∗ 0 083 FR (Prinsen et al., 2009) . pN ∗ 7 86 Fblob1 (Storm et al., 2005b) . pN ∗ ∼ 0 1 Fblob2 . pN ∗ 113 Fe pN ∗ 83 5 Fd . pN

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 (Fd ) listed here is calculated for a DNA ∗ =0 immobilized in the nanopore (uDNA ).

As shown in the table, the only force comparable to the electrical driving force is the vis-

∗ cous drag force acting on the DNA inside the nanopore (Fd ). Note that in the calculations ∗ =0 for this drag force, the DNA is assumed to be immobilized in the nanopore (uDNA ) to match with the experimental data given by Keyser et al. (2006). When the DNA is re-

leased, this force is even larger due to the movement of the DNA, which will be discussed

in Chapter 6.

167 ∗ The force acting on the DNA inside the nanopore (Fd ) 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 (Ft Fe Fd ), 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 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 ∗ =0 (uDNA ) 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 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-H¬uckel 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 =1.6 × 10−19C which is equivalent to a surface charge of

−0.15C/m2 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 0.1MKCl

solution and ∼−55mV for a 1MKClsolution (thin EDL limit). Obviously the Debye-

H¬uckel 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 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 uDNA. 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

∂φ r = h(x): = σ ,u=0,v =0 ∂r w ∂φ r = a : = −σ ,u= u ,v =0 ∂r d d

171 Note that the only difference between the governing equations and boundary conditions in

∗ 5 6 u = u u = uDNA Chapter and is the moving DNA surface d, where d U0 is the dimensionless translocation velocity.

6.3 Solution for the Flow Field

Unlike Chapter 5, another velocity component, the Couette flow component (uct), 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 (ueof ), the pressure driven flow (up) and the Couette flow (uct). The velocity equation for each component is given by 1 ∂ ∂ueof = − β ( ) r 2 E x ziXi (6.1) r ∂r ∂r i

1 ∂ ∂u ∂p r p = − (6.2) r ∂r ∂r ∂x

1 ∂ ∂u r ct =0 (6.3) r ∂r ∂r with the boundary conditions

r = a, ueof =0,up =0,uct = ud

r = h, ueof =0,up =0,uct =0

The pressure equation is based on conservation mass inside the pore Q ≡ Qeof + Qp +

Qct = constant where Q is the total flowrate and Qeof , Qp and Qct are the flowrate due to the three components. For the electroosmotic component, the flowrate Qeof can be obtained = h by Qeof a ueof rdr where ueof is the numerical solution of equation (6.1). The solution

172 for the Couette component can be found analytically by solving equation (6.3) with its

boundary conditions: ln(r/h) u = u (6.4) ct d ln(a/h) and the ﬂowrate Qct is given by

h 2 2 2 ud(−h + a − 2a ln(a/h)) Qct = uctrdr = (6.5) a 4lna/h

The pressure driven ﬂow component solution is given by equation (5.17) and the ﬂowrate

Qp is given by equation (5.18). Note that Qeof ,Qct,Qp are all varying along x axis but from the conservation of mass, the total ﬂowrate should be a constant (Q = constant).

= ∂Q = + + = ∂Qeof + ∂Qct + ∂Qp =0 Therefore the axial derivative Q ∂x Qeof Qct Qp ∂x ∂x ∂x .

Substituting Qeof , Qct and Qp into this equation, the pressure equation is then given by

d2p (a2 − h2)2 a4 − h4 dp (a2 − h2)2 a4 − h4 + + + + Q + Q =0 (6.6) dx2 16ln(h/a) 16 dx 16ln(h/a) 16 eof ct

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 Qct term that accounts for the ﬂow

ﬁeld due to the DNA motion. ∗ = ∗ + ∗ =0 A force balance ( F Fe Fd ) is used as an additional condition to ﬁnd DNA

∗ ∗ velocity (udna) where the dimensional electrical driving force Fe is given by equation (5.20)

∗ and Fd (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 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 5.

6.4 Asymptotic Solution for potential φ and the EOF velocity ueof

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 ( 1),

174 asymptotic analysis can be used to ﬁnd the solution for the potential and the electroosmotic

ﬂow component (ueof ).

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 h − r y =

λ so that as r = h, y =0and r =0, y →∞. Here = ∗ is the ratio of the Debye length to hi the radius of the nanopore at the inlet. Substituting y the potential equation becomes

2 d φ = − + ( ) 2 β ziXi O dy i For small , the second term on the right hand side is neglected and for binary monovalent species, the potential equation is given by

d2φ = sinhφ dy2 with boundary conditions dφ y =0, = σ dy w ∂φ y →∞, =0 ∂y 175 This equation is similar to equation (4.20) discussed in Chapter 4 and the inner solution is

given by equation (4.23) as ζw y e 2 − tanh i =2ln 2 φ ζw 1 − 2 tanh y e 2 where the superscript i indicates that it is the inner solution for φ.

= r−a For the region close to the DNA surface, a similar variable can be deﬁned as η , and the equation for potential becomes

d2φ 1 dφ + = −β z X 2 + a i i dη η dη i dφ = λ = (1) Note that the dη term cannot be neglected since the term a a∗ O . Unfortunately there is no simple analytical solution to the potential equation and the DNA surface potential ζd and the inner solution φi must be found numerically.

In the bulk region away from the walls, usually called the outer region, the equation for potential becomes 1 d dφ r =0 r dr dr The boundary condition can be obtained from the matching between the inner and outer solution as

lim φo = lim φi; lim φo = lim φi r→a η→∞ r→h y→∞

where the superscript o indicates the outer solution and thus

φo =0

The asymptotic solution for the streamwise velocity near r = h can be found based

on the similarity between the velocity equation and the potential equation. For the inner

= h−r region near the wall, the inner solution is governed by equation (y ) d2u d2φ eof = E dy2 dy2 I,x 176 with boundary conditions

du y =0,u =0; y = ∞, =0 eof dy and the inner solution is

i = ( − ) ueof Ei,x φ ζw

For the inner region near the DNA surface (r = a), the EOF component of the velocity

= r−a satisﬁes (η ) d du d dφ (η + a) eof = (η + a) E dη dη dη dη i,x

with the boundary conditions

du η =0,u =0; η = ∞, eof =0 eof dη

i = ( − ) and the inner solution is ueof EI,x φ ζd .

In the outer region, the EOF velocity equation is given by 1 ∂ ∂u r =0 r ∂r ∂r and the boundary conditions are obtained from matching conditions

lim uo = lim ui r→a eof η→∞ eof

lim o = lim i ueof ueof r→h y→∞

and the outer solution for the EOF velocity is (ζ − ζ )lnr ζ lnh − ζ lna uo = E w d + d w eof I,x ln(a/h) ln(a/h)

1 In the calculations, the asymptotic analysis is only used for the region where h(x) . The ∼ (1) numerical calculation is used for the region where h(x) O .

177 Figure 6.5: 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 50 nm and 5nm respectively (Storm et al., 2003).

6.5 Results for DNA Translocation Velocity

In most of the experiments (Storm et al., 2005a;Liet al., 2003), a conical nanopore is usually connected to a cylindrical nanopore. To model this conﬁguration, a similar geometry is used for the calculations as shown in Figure 6.5. A slowly varying 300 nm long conical nanopore is connected to a 50 nm long cylindrical. The inlet radius of the converging nanopore is 50nm and the outlet radius is 5nm. Since it is a slowly varying (1 =0.15)

2 conical pore and the lubrication approximation is applicable (1 1). The geometry of this nanopore is shown in Figure 6.5. The applied voltage drop over the nanopore is 120 mV and the surface charge density of the silica wall is −0.06 C/m2(Smeets et al., 2006). The

DNA and the nanopore are immersed in 0.1MKClsolution and the corresponding Debye length is ∼ 1 nm.

The contour plots of the electroosmotic ﬂow component ueof and the streamwise velocity u are shown in Figure 6.6. As shown in (a) the value of EOF velocity component ueof

178 is negative all through the nanopore due to the negative surface charge of the nanopore and

the DNA. The streamwise velocity u near the lower boundary (DNA surface) is positive

which indicates a positive DNA velocity. Driven by the electric force, DNA is moving

from left to right and dragging the ﬂuid ﬂow moving in the same direction due to viscosity.

However, the EOF velocity component is dominant in the region far away from the DNA

surface. Inside the 5 nm cylindrical nanopore, the abrupt change in velocity near the DNA

surface generates large shear stress and the resulting viscous drag balances most of the ap-

plied electric driving force (113 pN). The viscous drag force (71 pN) resulting from the

EOF component accounts for ∼ 63% of the total drag force and the remaining drag force

comes from the DNA motion uct (Couette component, 23 pN about 20% of the total drag force) and the induced pressure driven ﬂow component (up, 19 pN about 17% of the total drag force).

As shown in Figure 6.8 both the electrolyte concentration and wall charge of the nanopore play important roles on determining DNA velocity. The DNA velocity is smaller for higher electrolyte concentration as shown in Figure 6.8 (a). Note that DNA velocity is determined from the force balace between the electrical driving force and the viscous drag force ∗ ∝ ∗ = ∗ − ∗ − ∗ (ud Fct Fe Feof Fp ). As shown in Figure 6.7, the magnitude of the viscous drag

∗ from the electroosmotic component Feof is smaller for higher KCl concentration but the magnitude of the total viscous force due to the EOF component and the pressure-driven ∗ + ∗ component (Fp Feof )islarger for higher KCl concentration. In Chapter 5, the dimensionless shear stress acting on the DNA placed in a nanopore with constant radius is plot as a function of the electrolyte concentration in Figure 5.5. Since Fd ∝ τ, the viscous drag acting on the DNA in the straight pore is smaller for higher KCl concentration, leading

179 1 Ueof 1 U -0.005 0.025 -0.01 0.02 -0.015 0.015 0.8 -0.02 0.8 0.01 -0.025 0.005 -0.03 0 -0.035 -0.005 0.6 -0.04 0.6 -0.01 -0.045 -0.015 -0.05 -0.02 Y -0.055 Y -0.025 -0.06 -0.03 0.4 -0.065 0.4 -0.035 -0.07 -0.04 -0.075 -0.045 -0.05 0.2 0.2 -0.055

01234567 01234567 X X

(a) EOF velocity component ueof contours (b) total streamwise velocity u contours

Figure 6.6: 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/m2 and the electrolyte solution is 0.1 MKCL. The radius of DNA is 1 nm.

to the opposite conclusion for the dependency of the DNA velocity on the electrolyte con- ∗ ∝ ∗ = ∗ − ∗ centration if ud Fct Fe Feof is used to determine the DNA velocity. Therefore, the converging/diverging geometry has to be addressed in the model. Modeling a converging nanopore using the model derived for a straight pore, as performed in some of the previous modeling works, cannot account for the induced pressure gradient and the resulting pressure-driven ﬂow in the nanopore. This will lead to errors in the prediction of the DNA velocity, which may also give inappropriate conclusions in parameter studies.

The DNA velocity is almost a linear function of wall surface charge (shown in (b)) since the magnitude of the electroosmotic ﬂow is determined by the surface charge of the pore

2 and the DNA. At σw = −0.14C/m the DNA velocity is zero and for the nanopore surface charge lower than this value, DNA moves in the opposite direction to the bulk ﬂow. If the

180 −86.5

−87

−87.5

−88

−88.5 F* eof −89 F* +F* eof p −89.5 DNA velocity

−90

−90.5

−91

−91.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 KCl concentration

∗ Figure 6.7: The viscous drag force due to the electroosmotic ﬂow component Feof and the ∗ pressure driven component Feof 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/m2.

surface charge density is higher than this value, the viscous drag obtained from the EOF is larger than the applied electrical driving force acting on the DNA and thus the DNA is moving with the bulk flow. Since it is assumed that DNA is already placed inside the pore in our model, the DNA velocity is negative in these cases. In the experimental setup, if the pore wall charge falls in this region, DNA cannot translocate through the nanopore to the reservoir on the side with the anode (Figure 6.1) due to the intensive fluid flow through the nanopore in the opposite direction.

The geometry of the nanopore used in Storm’s experiments is similar to the pore shown in Figure 6.5 and the inlet radius of the nanopore is 223nm and the total length of the pore is

340nm; the 40nm long cylindrical pore with 5nm in radius is connected to the converging nanopore. In this case the radius of the nanopore at the inlet region is very large compared to the Debye length and the asymptotic solution is used to calculate the electroosmotic

≤ 0 01 ﬂow. In the calculation, the asymptotic solution is used for the region where h(x) .

181 (b) DNA velocity for different wall charge (a) DNA velocity for different concentration

Figure 6.8: 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/m2 for (a) and the electrolyte concentration is 1M for (b).

∗ ( ) ( ) Source σw LDNA µm CKCl M VExp VNum Storm et al. (2005a) −0.2 3.91 1 0.013 0.015 Smeets et al. (2006) −0.14 16.5 0.5 0.012 0.012 Li et al. (2003) −0.14 3.4 1 0.010 0.012

Table 6.1: Comparison for DNA velocity between the numerical results (VNum in m/s) ∗ 2 and the experimental data (VExp in m/s). Here σw (in C/m ) is the surface charge density of the nanopore; LDNA is the length of the DNA used in the experiments; CKCl is the concentration of the KCl solution.

182 0 01 and lubrication solution is numerically calculated for h(x) > . . The nanopore used in the Li et al. (2001)’s experiments is 50nm in inlet radius and 500nm in length; the

radius of the cylindrical pore with constant cross-section is 5nm and 10nm in length. The

results for the DNA velocity calculated numerically compare well with the experimental

data. The geometry of the nanopore used in Smeets et al. (2006)’s experiments is different

from the pores discussed before. It is a 20nm long cylindrical nanopore with 5nm in radius

connected to a 20nm long converging nozzle on one end and a 20nm diverging diffuser on

the other end. The slope of the converging or diverging wall is about 23◦. It is shown that the numerically calculated DNA velocities compare well with the experimental data. The results for DNA velocity are compared with experimental data in Table 6.1 and the good agreement between the experimental data and the numerical results on DNA translocation velocity has validate the model.

The electric current through the nanopore is given by ∗ = ∗ ∗ = 2 ∗ ∗ ∗ Ix JxdA πr Jx dr (6.7) A A

∗ where Jx is the current density, given by ∗ = − ∂Xi + Di ∗ + Jx ziF cDi ∗ c ziFXiEI,x cXiu (6.8) i ∂x RT Consider the ionic current density through a nanopore for a 1MKClsolution. The ﬁrst term in equation (6.8) is the diffusion of the ionic species due to the concentration gradient in the x direction. The magnitude of this term is, for the cations (K + in a KCl solution), ∂ ziXi −3 3 −9 2 ziXi C cD + ∼ 57.6×10 mol/m ×96500C/mol×10 m /s ∼ 16.1 K ∂x∗ 350 × 10−9m m2s

The second term is due to the electromigration and the magnitude of this term is

D 10−9m2/s cF i z2FX E∗ = (96500C/mol)2 × 57.6 × 10−3mol/m3 × RT i i I,x 8.314J/(Kmol˙ ) × 300K

183 × 2 × 106 ∼ 188 9 2 C zi Xi V/m . zi Xi 2 i m s and the last term is due to the convection

= 96500 ×57 6×10−3 3× ×0 0526 ∼ 256 9 C FcXiu C/mol . mol/m ziXi . m/s . ziXi 2 i i m s For a symmetric monovalent electrolyte solution, for example KCl, ziXi = XK+ −XCl−

2 = + + − + − and zi Xi XK XCl , where XK and XCl are of the same order. Usually the

2 magnitude of zi Xi is larger than ziXi but the ratio of the magnitude of these two terms depends on the local ionic distributions in the nanopore. As an example, in a 5nm h 2 = cylindrical nanopore with a DNA molecule placed at the center, the term a r zi Xidr 0 1321 h =00486 0 1 . and a r ziXidr . for . M upstream reservoir concentration. Note that the diffusion term is much smaller than the electromigration and convection terms, and the ionic current becomes to leading order, in dimensional form,

h∗ cDF 2 I∗ = z2X E∗ + z X cu∗r∗dr∗ (6.9) x ∗ i i x i i a i RT

In dimensionless form, the ionic current is given by

h 1 = 2 + Ix zi XiEI,x ziXiurdr (6.10) a i Pe

∗2 and Ix is scaled on cF U0hi . The results are compared with the experimental data in Figure 6.9. The current results compare very well with the experimental data from Storm et al. (2005a) for 11.5 kbps

DNA. The baseline current Ibaseline, which is the current through the nanopore without

DNA in it, is Ibaseline = 7085 pA and the amplitude is ∆I = 160 pA. The corresponding

experimental data is Ibaseline = 7100pA and ∆I = 140 pA. The relative current change

(∆I/I)is0.020 for experimental data and 0.022 from the calculation. Note that Figure

184 7200 Calculated Experimental(Storm2005) 7150

7100

7050 I(pA)

7000

6950

6900 0 0.5 1 1.5 2 t(ms)

Figure 6.9: The results for current through the nanopore as a function of time. For the numerical results, the baseline current Ibaseline is Ibaseline = 7085 pA and the amplitude is ∆I = 160 pA. The corresponding experimental data is I = 7100pA and ∆I = 140 pA(Storm et al., 2005a).

6.9 is just a illustration based on the two current levels: DNA ﬁlled the nanopore or no

DNA in the nanopore. DNA partially entry or exit is not considered and the corresponding current change is not shown. In the next section, DNA entering and leaving process will be modeled and the current change during the entire translocation process will be shown.

6.6 DNA Entering and Leaving the Nanopore

The process of DNA entering the nanopore is simulated as a quasi-steady-state process.

Since the Reynolds number is very low (Re O(10−5)), the convection terms are taken to be negligibly small. Moreover, during the entry and exit processes, the transient term in the momentum equation may be safely neglected since this term is only important during the time when the axial length of the DNA inside the tube ins on the order of the gap between the DNA molecule and the tube. During this period, there is a rapid development ( 1 ) of a pressure gradient on this short time scale (of O Sc ) which is much shorter than the 185 (a) DNA entry

(a) DNA leaving

Figure 6.10: 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.

186 total DNA translocation time. Upon neglecting the transient term, the process thus can be

considered to be quasi-steady, with the external variable, the velocity of the DNA in this

case, being the time-dependent parameter in the otherwise steady ﬂow (Smith, 1987).

As shown in Figure 6.10, the two cases where the particle partially enters the pore and

partially exits the pore are considered in addition to the case where the DNA ﬁlls the entire

length of the nanopore (discussed in the previous section). The computational region can

be divided into two parts from the DNA leading/trailing edge x = xDNA: region I with a

DNA placed at the centerline and region II without a DNA. The governing equations in

both regions are the same as equations (5.1), (5.2) and (5.4) shown in Chapter 5.

The boundary conditions for potential and each velocity component in region I are

given by ∂φ r = h(x): = σ ,u=0,v =0 ∂r w ∂φ r = a : = σ ,u= u ,v =0 ∂r d d and in region II, ∂φ h(x): = σ ∂r w ∂φ r =0: =0 ∂r

The streamwise velocity is decomposed into three components (ueof , up and uct) for region

I and two components (ueof , up) in region II. The corresponding boundary conditions for the velocity components are, in region I,

r = a, ueof =0,up =0,uct = ud

r = h, ueof =0,up =0,uct =0 and in region II, ∂u ∂u r =0: eof =0, p =0 ∂r ∂r 187 r = h : ueof =0,up =0

The boundary condition for pressure is the same as discussed in Chapter 5, given by

p =0;x =0, 1 (6.11)

Similar to the analysis in Chapter 5, the total ﬂowrate should be a constant along x axis and so in region I, the pressure equation is given by equation (5.19) and the pressure gradient is related to the ﬂowrates of each velocity components as

dp Qtotal − Qeof (x) − Qct(x) = 2 2 2 (6.12) dx (a −h ) + a4−h4 16ln(h/a) 16

Integrating equation (6.12), the pressure in region I is given by ⎡ ⎤ x − ( ) − ( ) = ⎣Qtotal Qeof x Qct x ⎦ + p1 ( 2− 2)2 4 4 dx C1 (6.13) 0 a h + a −h 16ln(h/a) 16 where C1 is the integration constant and p1 is the pressure distribution in region I.

In region II, the pressure driven component is given by the classical Poiseuille ﬂow in a pipe as 1 dp u = y2 − h2 (6.14) p 4 dx and the corresponding ﬂowrate is

h4 dp Q = − (6.15) p 16 dx

The mass conservation is then given by

h4 dp − + Q (x)=Q (6.16) 16 dx eof total for region II and so the pressure gradient in x direction is

dp 16 = − (Q − Q (x)) (6.17) dx h4 total eof 188 Similarly, integrating this equation gives the pressure in region II as

x 16 = − ( − ( )) + p2 4 Qtotal Qeof x dx C2 (6.18) xDNA h where C2 is the integration constant and p2 is the pressure distribution in region II.

Other than the pressure boundary condition given by equation (6.11), the pressure at the interface between region I and region II should match, which gives another condition that at x = xDNA, p1 = p2. Substituting the boundary conditions and the matching condition into equations (6.13) and (6.18), the integration constants can be determined as

C1 =0 ⎡ ⎤ xDNA − ( ) − ( ) = ⎣Qtotal Qeof x Qct x ⎦ C2 2 2 2 4 4 dx 0 (a −h ) + a −h 16ln(h/a) 16 and the total ﬂowrate is given by xDNA Qeof(x)+Qct(x) 1 16Qeof (x) 0 2 2 2 4 4 dx − 4 dx (a −h ) + a −h xDNA h = 16ln(h/a) 16 Qtotal xDNA 1 1 16 (6.19) 0 2 2 2 4 4 dx − 4 dx (a −h ) + a −h xDNA h 16ln(h/a) 16

Note that Qeof (x) is a function of x and it can be determined numerically by Qeof = h = h a ueof rdr in region II in region I or Qeof 0 ueof rdr in region II. Qct is given by h 2 2 2 ud(−h + a − 2a ln(a/h)) Qct(x)= uctrdr = (6.20) a 4lna/h

Substituting C1, C2 and Qtotal into equation (6.13) and (6.18), the total pressure ﬁeld can be determined.

The procedure of solving for the DNA velocity is similar to the solving process shown in Section 6.5. A DNA velocity is guessed and the corresponding Couette ﬂow component uct is calculated from equation (6.3). The pressure equation is then solved to ﬁnd the

pressure ﬁeld and pressure driven component up. Next the drag force is calculated and the

189 force balance is checked. If the force balance is not achieved, the DNA velocity is guessed again and the flow field is calculated again based on the new DNA velocity until the force balance is satisfied. The numerical methods are similar to that discussed in Section 6.5 based on the same finite difference numerical calculations. The ionic current through the nanopore is calculated in the straight cylindrical nanopore.

As shown in Figure 6.11, the DNA velocity increases as the leading edge moves progressively into the nanopore and it increases dramatically as it moves into the straight cylindrical pore due to the locally high electric ﬁeld in the pore. After it ﬁlls the pore, the velocity is a constant until the other end of the DNA enters the pore. The entering of the DNA into the straight nanopore causes a drop in ionic current as shown in Figure 6.11.

As DNA leaves the conical section of the nanopore, the DNA velocity increases as the trailing edge approaches the straight pore due to the smaller viscous drag force on the

DNA tail as shown in Figure 6.12. After the trailing edge moves into the straight nanopore, the DNA velocity starts to decrease since the electrical driving force becomes smaller and smaller as more DNA segments move out of the nanopore. The ionic current remains the same as the DNA trailing edge moves into the straight section of the nanopore.

The DNA velocity and the ionic current through the nanopore during the entire translocation process, including DNA entering the nanopore (Figure 6.11), ﬁlling the nanopore and leaving the nanopore (Figure 6.12), are shown in Figure 6.13 for a double standed

16.5kbps DNA. The problem for DNA filling the nanopore is discussed in Section 6.5 with the geometry shown in Figure 6.5 and the corresponding flow field shown in Figure

6.6. The time of DNA entry (0.22ms) is on the same order of the translocation time, in which DNA ﬁlls the entire nanopore (0.66ms). The time of DNA leaving is much shorter

(0.02ms) than for the other two processes. As the leading edge of the DNA enters the

190 Figure 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/m2 and the electrolyte concentration is 1M.

191 Figure 6.12: 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 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/m2 and the electrolyte concentration is 1M.

192 0.035

0.03 DNA Entering DNA fills the nanopore 0.025 DNA Leaving 0.02 DNA u 0.015

0.01

0.005

0 0 0.2 0.4 0.6 0.8 1 t −3 x 10 (a) DNA velocity

5140

5120

5100

5080

5060 |I|(pA)

5040

5020

5000

0 0.2 0.4 0.6 0.8 1 t(ms)

(a) ionic current

Figure 6.13: 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 40 nm. The inlet radius of the nanopore is 50nm and the outlet radius is 5nm. The surface charge density of the nanopore is −0.06 C/m2 and the electrolyte concentration is 1M.

193 nanopore, the DNA accelerates and then reaches a constant velocity when it ﬁlls the entire

nanopore system, including the conical section and the straight pore. As the trailing edge

of the DNA enters the nanopore, the DNA velocity increases again and leaves the nanopore

in a much shorter time scale. The entry of DNA into the straight nanopore causes a drop

in ionic current and as the trailing edge moves out of the straight nanopore, the current

changes back to the initial value.

The velocity and ionic current for a DNA entering and leaving a diverging nanopore

are shown in Figures 6.14 and 6.15. As the DNA enters the nanopore from the straight

pore side, the DNA velocity increases since it experiences a high electric driving force

in the straight nanopore. The DNA then reaches its maximum velocity when the leading

edge moves out of the straight nanopore. As the DNA moves into the diverging region,

the increase of the viscous drag acting on the DNA exceeds the increase of the driving

force, leading to a reduction in the DNA velocity. As the DNA trailing edge moves into the

straight nanopore, the DNA moves slower due to the smaller driving force on the DNA. The

exit of the trailing edge from the straight nanopore causes a rise in ionic current. Velocity

and ionic current change as a function of time during the entire DNA transloction process

is shown in Figure 6.16. In this case, the time of DNA entry is much shorter than the

other two processes, including DNA leaving the pore and DNA ﬁlling the pore. At t =0, the entry of the DNA into the straight pore causes a current drop and after the trailing edge of the DNA leaves the straight pore, the current changes back to the initial value.

Note that in experiments, DNA usually enters the nanopore from the converging region because the larger the pore opening is, the higher probability DNA can ﬁnd the nanopore and transport through the nanopore. DNA entering from the straight pore side is seldom seen in experiments.

194 Figure 6.14: 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/m2 and the electrolyte concentration is 1M.

195 Figure 6.15: 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/m2 and the electrolyte concentration is 1M.

196 0.04 DNA entering DNA fills the pore DNA leaving (m/s) 0.02 DNA u 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t(ms) 5150

5100

|I|(pA) 5050

5000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 t(ms)

Figure 6.16: 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/m2 and the electrolyte concentration is 1M.

0.06 Entry DNA fills the pore 0.04 (m/s)

DNA 0.02 u Exit 0 0 0.1 0.2 0.3 0.4 0.5 t(ms)

7100

7050

|I|(pA) 7000

6950 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 t(ms)

Figure 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/m2 and the electrolyte concentration is 1M.

197 The DNA velocity and ionic current during the entire translocation process for Figure

6.9 are shown in Figure 6.17. 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/m2 and the electrolyte concentration is 1M. The velocity proﬁle is similar to that of Figure 6.13 since the geometries of the nanopores are very similar. Note that in this ﬁgure, the time axis (t) starts from the moment that the leading edge of the DNA enters into the converging nanopore (t =0,xDNA =0). It is seen that the ionic current drops when the leading edge of the DNA moves all the way into the straight pore (t =0.125ms) and it rises when the

DNA trailing edge leaves the straight pore (t =0.52ms). In Figure 6.9, it is assumed that

there is no DNA in the nanopore for t =0− 0.5ms and the DNA ﬁlls the entire nanopore for t =0.5 −0.87ms. The transitions between these two scenarios are neglected. In Figure

6.17, these transition processes on DNA partially enters or leaves the pore are accounted

for.

6.7 Modeling DNA Delivery into a Cell through the Nanonozzles

A major challenge in gene therapy and drug delivery is to deliver genes and high molecular weight drugs into mammalian cells with high efﬁciency and minimal cell damage.

Viruses and liposomes have been widely used as in vivo carriers, but safety issues, such as immune response and cytotoxicity, have limited their clinical applications. Nanonozzles are proposed to use to directly transfer repaired DNA into cells without using any carrier and avoid the risks associated with introducing a secondary agent. 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.

198 (b) (a)

Figure 6.18: The schematic for the gene delivery device.

The nanonozzle used for gene delivery is shown in Figure 6.18. The nanonozzle is connected to the cell at one end and the reservoir on the other end. To better understand the delivery process, cell mechanics, DNA dynamics, electroosmosis and electrophoretic transport inside the nozzle need to be studied.

The cell is the smallest anatomical and physiological unit in the human body that can live and reproduce on its own. For the design of the gene delivery devices, parameters have to be carefully chosen to avoid cell death caused by the intervention. The working fluid must be similar to the intracellular fluid, usually called cytosol. Cytosol is composed of water, salts and organic molecules(Alberts & Raff, 2003) and it makes up about 70% of the cell volume. For the applied electric field, if the field strength is too high, the cytoskeleton of the cells becomes damaged which causes cell death. Other aspects of the design of the device need close collaboration with biologists. The size (∼ 10µm) and the proper shape

of the cell need to be considered in the model. The composition of the working ﬂuid and

concentration of the charged ions is important on determine the ﬂow rate in the nanonozzle.

199 With the biased electrical potential, the distribution of the electric ﬁeld strength in the

nozzle and the cell must be investigated to determine the rate of delivery.

DNA is usually coiled in solution and the super-coiled DNA cannot pass through chan-

nels with dimensions smaller than its gyration diameter (usually ∼ 0.1µm) in a short time.

In the proposed gene delivery device, DNA can easily enter the large end of the nozzle,

which is larger than its gyration diameter. Inside the nanonozzle, the DNA is stretched

and becomes a long worm-like shape with a much smaller radial size, and will eventu-

ally migrate through the channel. The stretching of the DNA is caused by the electrical

ﬁeld gradient due to the tapering geometry and the hydrodynamic interactions between the

DNA and ﬂuid ﬂow. Moreover, the electrophoretic mobility of coiled DNA also needs to

be addressed. The DNA mobility depends on a couple of parameters such as the buffered

solutions, DNA size and its conformation. When transported inside the nanonozzle, the

mobility of the coiled DNA will be a function of the location and its conformation. DNA

dynamic modeling will be necessary.

The ﬁrst step of the modeling is the computation of the electric ﬁeld. The different

conductivity and permittivity, which is the ability of a dielectric medium to transmit an

applied electric ﬁeld, need to be considered in the modeling to calculate the voltage drop

across the nanonozzle. On the other hand, the nanonozzle is negatively charged and the cell

membrane also has a negatively surface charge (∼−0.03C/m2) (Boal, 2002). The electric

ﬁeld across the cell membrane will be distorted. All of these factors need to be integrated into a model that describes the electric ﬁeld strength and direction in the system.

200 The second step of the modeling is to investigate the flow field of the system. The electroosmotic flow and the transport of ionic species will be studied based on the Poisson-

Nernst-Planck equations. Previous research on EOF in nanonozzles can be used with mod- iﬁed boundary conditions on the cell side.

Next step is to investigate the DNA mobility as a function of the applied ﬁeld and its conformation. Previous research has been done by Fei (2007) to measure the instantaneous

DNA electrophoretic mobility in the nanonozzles. An empirical solution from their experimental data can be used directly in the simulations. If such a solution is not available, dynamic modeling of DNA chain using Worm-like-Chain model (Dutcher & Marangoni,

2004) may be necessary.

The last step is to integrate the electric field, the fluid flow, the transport of DNA together. The effect of the applied field, the surface charge of the nanonozzle, the size of DNA and the concentration of the working fluid on the delivery process will be investigated.

In this section, modeling and simulations are proposed for the future work to investigate the delivery of DNA molecules into a living cell using a converging nanonozzle/nanopore.

Different aspects that should be considered to establish a practical model have been discussed. The model discussed in the present work can be further extended to simulate the

DNA transport in such a system.

6.8 Summary

In the previous chapter, results have shown that the viscous drag force acting on the

DNA inside the nanopore and the electrical driving force are the most important forces determining DNA translocation velocity. Based on the balance of these two forces, a model using the lubrication approximation is established in Chapter 5 for a immobilized DNA

201 placed in a nanopore and it is validated by the comparison with the experimental data for the

tethering force. In this chapter, this model is extended to calculate the DNA translocation

velocity.

As discussed in Chapter 5, the viscous drag force accounts for 60 ∼ 80% of the total

electrical driving force acting on the immobilized DNA inside the nanopore. In this chapter,

the viscous drag force acting on the DNA inside the nanopore is assumed to be balanced

with the entire electrical driving force, and the increased viscous drag (20% ∼ 40% of the electric driving force) is due to the DNA motion against the bulk ﬂow, which introduces a larger shear stress on the DNA surface and induces larger pressure gradient in the nanopore.

The electric field, the potential, the concentration of ionic species and the flow field are calculated based on the lubrication approximation. Results have shown that the bulk flow moves in the opposite direction to the DNA translocation while in the region close to the

DNA surface, the ﬂuid ﬂow, dragged by the DNA, moves in the same direction with the

DNA molecule. The viscous drag force resulting from the electroosmotic ﬂow component accounts for ∼ 63% of the total drag force and the remaining drag force comes from the induced pressure-driven ﬂow component (17%) and the DNA motion (20%). The DNA translocation velocity is compared to the experimental data to validate the model and they are in good agreement. This has validated the assumption of the force balance between the electrical driving force and the viscous drag force for a DNA in motion.

The DNA velocity is found to be dependent on the parameters that affect the flow field in the nanopore, for example the concentration of the electrolyte solution and the surface charge density of the nanopore. In this chapter, it is found that the converging geometry induces a pressure-driven flow component, which also introduces a viscous drag force on the DNA surface. The resulting total drag force on the DNA placed in the converging

202 nanopore is larger for higher electrolyte, in conﬂict with the results for the DNA placed in a nanopore with constant cross-section where no induced pressure gradient as discussed in Chapter 5. Therefore, modeling the nanopore based on the model derived for the pore with constant cross-section, as performed in some of the previous modeling works, cannot account for the induced pressure gradient and the resulting pressure-driven ﬂow in the nanopore. This will lead to errors in the prediction of the DNA velocity, which may draw inappropriate conclusions in parameter studies.

Results have shown that the direction of DNA translocation depends on the surface charge density of the nanopore. In the previous experimental works, DNA usually moves in the opposite direction to the bulk ﬂow. However, if the surface charge density of the negatively charge nanopore is higher than the DNA surface charge, DNA may move in the same direction to the bulk ﬂow.

DNA entering and leaving the nanopore is also investigated. It is shown that DNA accelerates as the leading edge moves into the nanopore. The velocity increase drastically as the DNA enters the cylindrical nanopore with constant cross-section due to the high electric ﬁeld inside the pore, causing a drop in ionic current. As the DNA leave the nanopore,

DNA velocity increases as the trailing edge moves into the cylindrical pore with constant cross-section due to the decrease of viscous drag.

Modeling of a gene delivery device is proposed for future work. Understanding the physics of DNA translocation will provide guidance for the proposed experimental setup.

203 CHAPTER 7

Summary

In this thesis, electroosmotic flow and DNA electrophoretic transport in micro/nano channels are investigated. Several distinct problems have been solved and these problems have a variety of application. Most of the present work focuses on the nanochannels or nanopores with tapering geometry and the applications of these nozzles/pores in particle delivery and DNA sequencing. The basic theories used to describe the fluid flow and particle transport in these problems are electroosmosis, electrophoresis and fluid dynamics.

Theoretical and numerical calculations are used to analyze and simulate ﬂuid ﬂow, particle transport and DNA translocation in these nozzles/pores.

Electroosmotic Flow with Non-uniform Surface Potential

Two-dimensional electroosmotic ﬂow in a rectangular channel having a step change in surface potential is investigated in Chapter 3. Since the ζ-potential and/or surface charge density of the channel wall is often non-uniform due to the adsorption of biomolecules or fabrication defects, the problem studied here will help us better understand the complex

fluid flow in micro/nano channels. Fluid flow and ionic transport are modeled based on the

Poisson-Nernst-Planck (PNP) equations and the Navier-Stokes equation. A ﬁnite volume method is used to solve the coupled non-linear equations numerically on a uniform co- located mesh using on SIMPLE algorithm.

204 Results have shown that if the over-potential 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 that looks similar to a classical Lamb vortex. It is noticed that the electric double layer structure near the patch is different from that calculated based on 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 this difference is directly related to the external applied electric

ﬁeld strength and under the ﬁeld strength in ordinary experimental conditions (smaller than

105V/m), the species distribution calculated from PNP equations is almost the same as the

Boltzmann distribution.

A simple reservoir and nanopore membrane system is also studied using COMSOL in

Chapter 3 to investigate the interactions between charged species and the nanopore membrane for the application of an artificial kidney. The pressure-driven flow and the ionic transport of large molecules are simulated based on the Poisson-Nernst-Planck equations using COMSOL and the results have shown that nanopore charge has a significant effect on the sieving of the charged molecules. Extension of this work will provide foundational knowledge to the optimization of nanofiltration and reverse osmosis technologies for the purpose of water purification.

Flow and Particle Transport in Micro/Nanonozzle

Nanotips with 80nm in aperture diameter, which are proposed to be the delivery component for gene delivery nanofluidic devices, have been fabricated for the purpose of gene delivery (Fei, 2007). To characterize the fluid flow and particle transport in such a tapering

205 geometry, micro-scale nozzles is fabricated and the transport of polystyrene beads in the

micronozzles has been investigated experimentally by Wang & Hu (2007). A theoretical

model is established in Chapter 4 to characterize the fluid flow and particle transport in a nozzle/diffuser. Based on the Debye-H¬uckel approximation and lubrication theory, the governing equations have been greatly simplified so that analytical solutions for potential and streamwise velocity are derived while the pressure equation is solved numerically. Re- sults have shown that a pressure gradient is induced by the electroosmotic flow and the magnitude of this induced pressure is O(), which is the ratio of the Debye length to the height of the nozzle/diffuser. The corresponding pressure-driven velocity (up) due to this induced pressure is thus small compared to the electroosmotic flow velocity (ueof ).

A non-linear calculation for the EOF in nozzles/diffusers is also discussed in Chapter

4 to give a more accurate predict to the potential and velocity ﬁeld when the zeta potential is high. Based on the lubrication theory, the non-linear Poisson-Boltzmann equation is simpliﬁed in a similar manner to the Debye-H¬uckel solutions. The only difference is that the equation for potential and the EOF streamwise velocity component are solved numerically using Finite Difference Methods (FDM). Comparison between the results calculated from the Debye-H¬uckel solution and the nonlinear numerical solution has shown that the

Debye-H¬uckel solutions over-predict the streamwise velocity and the induced pressure by

∼ 30% for a surface charge density −0.07C/m2. For the cases of thin EDLs ( 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-H¬uckel approximation by taking the limit → 0.

o =0 o = o The pressure outer solution (p ) indicates that the u ueof in the outer region and

206 knowing this can greatly simplify the calculations for EOF in any slowly varying channel

with constant wall potential/charge with 1. EOF in a wavy slowly varying channel is discussed as an example.

A reservoir-nozzle system (Figure 4.2 (a)) was established (Shin & Singer, 2009) to investigate fluid flow and ion transport at nano-scale using Molecular Dynamics (MD) simulations. In their simulations, the external applied electric field strength is very high

(∼ 108V/m). As discussed in Chapter 3, Boltzmann distribution cannot accurately describe the ionic distribution for such high ﬁeld strength and a full numerical solution based on the

Poisson-Nernst-Planck equations is necessary. The full numerical solutions are obtained using COMSOL and the compared with the results of the MD simulations. It is shown that the results for both the streamwise velocity and ionic concentration are qualitatively similar to each other, which can be used to validate the feasibility of continuum theories at such nano-scale.

Particle motion in a nozzle-reservoir system is investigated in Chapter 4 and it is found to be dependent on the wall charge, the particle charge and the parameter RCE. Here the particles are considered as point charges, and their general motion in the system is investigated. The analysis 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.

Two example of future work are discussed in Chapter 4 as an extension of the present work. To better resolve the particle motion in the nozzle-reservoir system, a two-phase flow model is set up. 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 inter-phase forces. Future work will

207 be focused on the numerical calculations for the ﬂuid ﬂow and particle motion based on the

two-phase model. On the other hand, 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 a dynamic assembly process, which is used to fabricate nanonozzles.

Predicted results from this model will help interpret the experimental data and obtain the

optimal combination of the controlling parameters for the experimental setup.

DNA Translocation

In Chapter 5, the forces that affects the DNA translocation are analyzed and it is shown

∗ that 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 (FU and FR), the

∗ viscous drag force acting on the linear DNA inside the nanopore (Fd ) and the viscous drag

∗ ∗ acting on the blob-like DNA outside the nanopore (Fblob1 and Fblob2). The magnitudes of these forces are calculated based on the parameters used in experiments. It is shown that the ∗ 100 ∗ electrical driving force (Fe ) is about pN, and the uncoiling and recoiling forces (FU and

∗ FR) 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 (Fblob1 and Fblob2) is about one or three orders of magnitude less. 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 (FU and FR) 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 (Fblob1) is considered as the force resisting DNA passage through the nanopore

208 by Storm et al. (2005b) and Fyta et al. (2006). Comparison between these forces and the electrical driving force has shown that they are not the key forces controlling DNA translocation.

The model derived in Chapter 4 based on the lubrication approximation is extended to calculate the ﬂow ﬁeld and the viscous drag force acting on a DNA immobilized in a

∗ nanopore (Fd ). Two kinds of numerical calculations are performed. One is based on the Poisson-Boltzmann equations and the lubrication approximation, using in-house written codes and the other is based on the Poisson-Nernst-Planck equations, using COMSOL. Re- sults 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. Therefore, the hydrodynamic interaction between the DNA and the nanopore

plays an important role in determining the DNA translocation velocity. The numerical re-

sults (both lubrication and COMSOL results) are in good agreement with the experimental

data given by Keyser et al. (2006) for the tethering force, validating the model. Note that in ∗ =0 Chapter 5, the DNA is assumed to be immobilized in the nanopore (uDNA ) and when the DNA is released, the viscous drag force acting on the DNA inside the nanopore will be even larger due to the movement of the DNA.

Based on the balance between the electrical driving force and the viscous drag force on the DNA inside the pore, the model derived in Chapter 5 is extended to account for the movement of DNA. The DNA velocity is calculated in an iterative manner to ensure that the force balance is satisfied, while the calculation of the flow field in the nanopore is similar to the model discussed in Chapter 5. Results have shown that the bulk flow moves in the opposite direction to the DNA translocation and the DNA motion against the bulk

ﬂow induces larger shear stress on the DNA surface as well as larger pressure gradient in

209 the nanopore. The resulting viscous drag force acting on the DNA surface is larger than that on a immobilized DNA as discussed in Chapter 5.

The DNA velocity is found to be dependent on the parameters that affect the ﬂow ﬁeld in the nanopore, for example the concentration of the electrolyte solution and the surface charge density of the nanopore. To validate the model, the DNA translocation velocity is compared to the experimental data and they are in good agreement. This has validated the assumption that the DNA velocity is determined by the force balance between the electrical driving force and the viscous drag force as well as the model used for the calculations.

In Chapter 6, modeling of a gene delivery device is proposed for future work. Differ- ent aspects that should be considered and integrated into the model have been discussed.

DNA dynamics simulations are necessary to simulate the electrophoretic transport of coiled

DNA in a nanonozzle. The results will help interpret experimental data and guide future experimental design.

In conclusion, the present work has investigated the electroosmotic ﬂow, ionic transport, particle transport and DNA translocation in micro/nano-scale channel/nozzles. A theoretical model is established for the electroosmotic ﬂow in a converging/diverging channel or pore based on lubrication theory. Numerical methods are used to solve the equations and the model is validated by the good agreement with the experimental data. Modeling results can help interpret experimental data and gain an insight into the nature of the problems, for example DNA translocation in a nanopore. The applications of the present work include transdermal drug delivery, gene delivery, biomolecular sensing and DNA sequencing along with some non-biomedical applications such as desalination. Future work include two-phase modeling of particle transport, modeling of the dynamic assembly fabrication process of the nanonozzle and gene delivery using these nanonozzles.

210 APPENDIX A

Discretising the Equations

A.1 Introduction

The numerical calculations for the electroosmotic ﬂow in a nanochannel with nonuniform surface potential (Chapter 3) are discussed in this appendix. As discussed in Chapter

2, 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 with each other and numerical calculations are necessary. The details of the numerical methods are discussed in this appendix.

A.2 Discritization

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 400 × 80. The nodes are located at the center of the cells and the subscript o denotes

the quantities at the nodes. Each node O corresponds to a coordinate (i, j) on the mesh.

The neighboring nodes of each node O is deﬁned as E(i +1,j), W (i − 1,j), N(i, j +1)

and S(i, j − 1). The faces of the cell are the boundaries of the node for example the east

boundary of node O is termed as east face and written in lowercase letter e. Usually the face

211 value of a quantity are represented by the values of that quantity at the center of the face ( + 1 ) ( − 1 ) ( + 1 ) ( − 1 ) which are located at n i, j 2 , s i, j 2 , e i 2 ,j , w i 2 ,j . For example for node

O, the x−axis velocity at O is denoted as uO and the value of u at its neighboring nodes are denoted as uN , uS, uE and uW as their direction to the node and the face velocities are un, us, ue and uw as shown in Figure 3.3. The length of the cell in the x−direction is deﬁned

as dx and in y-direction is deﬁned as dy.

The 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

∇•ρudV = ρu • ndS (A.1) V S

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 the control

volume surface is n. Using divergence theorem, the integral of the equation over the control

volume is related to the summation of some quantities at the surfaces.

Since the ﬁnite volume method is used in the calculation, the quantities at the faces of

the control volume are needed to approximate the difference equations. On a co-located

mesh, all the variables are stored at the cell centers, and we need to ﬁnd some way of

interpolating between the cell centers to obtain face value. Two ways of interpolating are

used in the calculation:

212 1. Take the value from the upwind cell center - upwind differencing;

2. Take an average of the two cell centers - central differencing.

The upwind differencing has been used to approximate all the inertial terms and central differencing has been used for other terms.

We consider a particular cell O as shown in Figure A.1 and its neighbor E. The east boundary of the cell is denoted as the east face e. Considering a quantity q, we do not know the value of the variables on the e face, just know the values stored at the cell centers E and O. We can interpolate between these values by take the value from the upstream cell.

If the ﬂow is O → E, the value of q at the e face is approximated by the upstream value qO

and if the ﬂow is in the opposite direction, qe = qE. Usually it is written in the way of

|ue| + ue |ue|−ue qe = qO − qE 2ue 2ue Alternatively we could average between the two cell center values :

q + q q = E O e 2 which is known as central differencing. Note that the central differencing is 2nd order accurate while the upwind differencing only obtain 1st order accuracy. However, upwind

finite-difference methods are more stable than centered techniques because they mimic the behavior of fluid flow by only using information taken from upstream in the fluid Gaskell

& C. (1988).

Inner Nodes

An inner node on the mesh is shown in Figure A.2. Considering the u velocity equation

(2.33) ﬁrst, here we rewrite the dimensionless velocity equation in the form of ∂v + ∂v + ∂v = −∂p + β 1 − Λ ∂φ + ∇•(∇ ) Re 1u v 2w 2 1 ziXi u (A.2) ∂x ∂y ∂z ∂x ∂x i 213 Figure A.1: The upwind difference method and the central difference method. Here qe is the velocity of east face using upwind difference method and qe is the face velocity approximated by the central difference.

214 Figure A.2: An inner node on the mesh.

Here u is the streamwise velocity and it is a scalar. This is a typical advection-diffusion equation. The term on the left hand side of the equation is the advection term and the last term on the right hand side is diffusion term. The first two terms on the right hand side are the source terms which represent the body forces acted on the fluid. The first source term is the source term due to pressure gradient. The second term on the right hand side is the source term due to the electrical field and here we have written it as SEx for simplicity. The vector u is the velocity vector: u = ui + vj. The left hand side of equation (A.2) can be

written as ∂v ∂v ∂v Re 1u + v + 2w = Re∇•(uu) ∂x ∂y ∂z

Integrate the equation over the control volume shows

∂p 2 Re ∇•(uu)dV = − + SEx + ∇ u dV (A.3) V V ∂x

215 Applying the divergence theorem for the advection term shows

Re ∇•(uu)dV = Re uu • ndS (A.4) V S

ue(uei + vej) •idy = ueuedy

Similarly the advection on the other faces can be obtained as

uw(uwi + vwj) • (−i)dy = −uwuwdy

un(uni + vnj) • j = unvndx

us(usi + vsj) • (−j)=−usvsdx and so the advection term becomes

Re uu • ndS =(ueue − uwuw)dy − (unvn − usvs)dx S

Similarly for the diffusion term we have

∇•(∇u)dV = ∇u • ndS V S and u u ∇u = A i + j x y

By substituting this term back we obtain the diffusion term

∂u ∂u ∂u ∂u ∇•(∇u)dV = |e − |w dy + |n − |s dx V ∂x ∂x ∂y ∂y

Here |e, |w, |n and |s denotes the quantity at the faces.

Integrating the pressure source term shows

∂p − dxdy =(p − p )dy ∂x w e

216 Organizing all the terms together shows

Re(ueue − uwuw + unvn − usvs)=(pw − pe)dy + SEx|Odxdy ∂u ∂u ∂u ∂u + | − | dy + | − | dx (A.5) ∂x e ∂x w ∂y n ∂y s

Now we can rewrite the equation (A.5) in terms of ﬂux at the faces

− + − = x Je Jw Jn Js SO

where the ﬂux are deﬁned as ∂u J = u u − | dy e e e ∂x e ∂u J = u u − | dy w w w ∂x w ∂u J = u v − | dx n n n ∂y n ∂u J = u v − | dx s s s ∂y s and

x = +( | − | ) SO Sxdxdy p w p e dy

The inertial terms on the faces are approximated using the ﬁrst-order upwind difference scheme as following + + = uO uE = uO uW ue 2 ,uw 2 + + = vO vN = vO vS vn 2 ,vs 2 and | | + | |− = ue ue − ue ue ueue 2 uO 2 uE | | + | |− = uw uw − uw uw uwuw 2 uW 2 uO

217 | | + | |− = vn vn − vn vn unvn 2 uO 2 uN | | + | |− = vs vs − vs vs unvs 2 uS 2 uO

The diffusion terms are approximated by

− ∂u| = uE uO ∂x e dx − ∂u| = uO uW ∂x w dx − ∂u| = uN uO ∂y n dy − ∂u| = uO uS ∂y s dx and the pressure from central difference

+ | = pO pW p w 2 + | = pO pE p e 2

Substituting all of these into equation (A.5) and write the ﬁnal equation in a ﬁve-band fashion

∗ + ∗ + ∗ + ∗ + ∗ = x AN uN AS uS AE uE AW uW AO uO SO (A.6) where AN , AS, AE,AW and AO are the link coefﬁcients of the equation and are given by

|u |−u dy A = −Re e e dy − E 2 dx |u | + u dy A = −Re w w dy − W 2 dx |v |−v dx A = −Re n n dx − N 2 dy |v | + v dx A = −Re s s dx − S 2 dy |u | + u |u |−u |v | + v |v |−v dy dx A = Re e e dy + w w dy + n n dx + s s dx +2 +2 O 2 2 2 2 dx dy

218 The source term is given by

− x = PW PE + ∗ SO 2 dy SEx dxdy

β 1 − Λ ∂φ where SEx is the value of 2 1 ∂x i ziXi at the cell center. Writing Sx in the form of the cell center values shows − = β 1 − Λ φE φW | SEx 2 1 2 ziXi O dx i

Now we have discretized velocity u equation and prepared all the coefﬁcients for solving for an inner node. The link coefﬁcients of the v velocity equation is the same as u equation except for the source terms

− y = PN PS + ∗ SO 2 dx SEy dxdy where the electrical source term is

Λ − = −β φN φS | SEy 2 2 ziXi O dy i

Considering the equation of species i, we write the dimensionless mole fraction equation in a vector form:

∇•(−∇Xi + ziXiE + PeXAu)=0 (A.7) where E is the electrical ﬁeld 1 ∂φ E = − A x Λ ∂x ∂φ E = − y ∂y

Integrating over the control volume and applying the divergence theorem shows

(−∇Xi + ziXiE + PeXAu) • ndS =0 (A.8) S 219 Written in the ﬂux form it is

Je − Jw + Jn − Js =0 (A.9) where ∂X J = − i | + z X | E | + PeX | u| e ∂x e i i e x e i e e ∂X J = − i | + z X | E | + PeX | u| w ∂x w i i w x w i w w ∂X J = − i | + z X | E | + PeX| v| n ∂y n i i n y n i n n ∂X J = − i | + z X | E | + PeX| v| n ∂y s i i s y s i s s Again we use the upwind difference scheme to approximate the inertial term

For the ﬁrst term which is the ﬂux due to diffusion, the derivative of mole fractions are

approximated by | − | ∂Xi | = Xi E Xi O ∂x e dx | − | ∂Xi | = Xi O Xi W ∂x w dx | − | ∂Xi | = Xi N Xi O ∂y n dy | − | ∂Xi | = Xi O Xi S ∂y s dy And for the second term which is the ﬂux due to the electric ﬁeld, we have 1 ∂φ X | + X | 1 φ − φ X | E | = X | − A | = i E i O − A E O i e x e i e Λ ∂x e 2 Λ dx

220 1 ∂φ X | + X | 1 φ − φ X | E |w = X | − A | = i W i O − A O W i w x i w Λ ∂x w 2 Λ dx ∂φ X | + X | φ − φ X | E | = −X | | = − i N i O N O i n y n i n ∂y n 2 dy ∂φ X | + X | φ − φ X | E | = −X | | = − i S i O O S i s y s i s ∂y s 2 dy

Substituting all of these terms into equation (A.9) and organizing it into the ﬁve band fashion shows

where the link coefﬁcients are given by dy z dy 1 φ − φ |u |−u A = − + i − A E O + Pe e e dy E dx 2 Λ dx 2 dy z dy 1 φ − φ |u | + u A = − − i − A O W + Pe w w dy W dx 2 Λ dx 2 dx z dx φ − φ |v |−v A = − − i N O + Pe n n dx N dy 2 dx 2 dx z dx φ − φ |v | + v A = − + i O S + Pe s s dx S dy 2 dx 2 dy dx dy φ − 2φ + φ dx φ − 2φ + φ A =2 + − z A E O W − N O S O dx dy i dx 2 dy 2 | | + | | + | |− | | + − ue ue + uw uw + vn vn + vs vs Pe 2 dy 2 dy 2 dx 2 dx

and the source term is SO =0.

Substituting these terms back and organizing them shows

AN ∗ φN + AS ∗ φS + AE ∗ φE + AW ∗ φW + AO ∗ φO = SO

where the link coefﬁcients are given by

dy A = −A E dx dy A = −A W dx dx A = − N dy dx A = − S dy dy dx A =2A +2 O dx dy and the source term is = − β | SO 2 ziXi Odxdy i

Pressure correction

The pressure correction equation is derived based on the continuity equation. The purpose is to ﬁnd an expression of the continuity equation in terms of the pressure correction

222 of cell O and its neighbor cells. Supposing we have guessed a velocity ﬁeld uˆ, vˆ and the actual velocity ﬁeld is uˆ, vˆ which satisfy the continuity equation:

ˆ ˆ ˆ ˆ (uˆe − uˆw)dy +(vˆn − vˆ)dx =0

If we deﬁne u = uˆ − uˆ, the continuity equation can be written as

(ˆ − ˆ ) +(ˆ − ˆ ) = − [( − ) +( − ) ] ue uw dy vn vs dx ue uw dy vn vs dx and we define the left hand side of the equation as the imbalance of the cell m˙ . The imbalance m˙ shows the mass balance of the cell and it should goes to 0 if the continuity is satisfied. But for a guessed velocity field, it is usually not equal to zero and so we define a quantity m˙ to show how far it is from current velocity field to the actual velocity field. The mass imbalance can calculated based on the guessed velocity field

m˙ =(ˆuE − uˆw)dy +(ˆvn − vˆs)dx and the continuity equation becomes

− ˙ =( − ) +( − ) m ue uw dy vn vs dx (A.11)

First we consider the left hand side of equation (A.11), the imbalance of the cell. The pressure weighted interpolation method (PWIM) (Rhie & Chow, 1983) is used to compute the face velocities. Ordinarily a simple linear interpolation for face velocities would be considered as shown before. However, the calculation shows that the diagonal dominance of the pressure correction equation is lost if the central difference is used and the solution can not be obtained (Tannehill et al., 1997). Therefore some other method need to be found to approximate the face velocities and inspection of the momentum equations reveals that the cell-face velocity also depends on the pressure gradient terms at the cell center and its

223 neighboring nodes. From the equation (A.6) we have

∂pk AO|euê = − Am|euˆm|e − |edxdy + SEx|edxdy (A.14) m ∂x By linear interpolation, we define 1 1 1 1 = + (A.15) AO|e 2 AO|O AO|E and A | uˆ | 1 A | uˆ | A | uˆ | m m e m e = m m O m O + m m E m E (A.16) AO|e 2 AO|O AO|E From the equation (A.12) and (A.13) we obtain | ˆ | 1 k m Am Oum O ∂p = − uO + |O + Sx|O dxdy AO|O AO|O ∂x and | ˆ | 1 k m Am Eum E ∂p = − uE + |E + Sx|O dxdy AO|E AO|E ∂x We can substitute them into equation (A.16) | ˆ | 1 1 k 1 k m Am eum e dxdy ∂p ∂p = (ûO +ûE)+ |O + |E AO|e 2 2 AO|O ∂x AO|E ∂x Here we have neglected the electrical source terms. Substituting back into equation (A.14)

along with the definition given in equation (A.15) shows 1 dxdy 1 ∂pk 1 ∂pk 1 1 ∂pk uê = (ûO +ûE) |O + |E − + |e 2 2 AO|O ∂x AO|E ∂x AO|O AO|E ∂x (A.17)

224 In the same way uˆw, vˆn and vˆs can be obtained.

Now we need to deal with the right hand side of the equation (A.11). Note that the velocity ﬁeld uˆ and vˆ is unknown so that the face velocities can not be obtained in the

same way as we derive the face velocities for the guessed velocity ﬁeld. But using the

continuity equation we can estimate how much the guess is off and it can be corrected

using a correcting pressure ﬁeld p. The pressure correction is deﬁned as p = pk+1 − pk

and so the purpose is to ﬁnd ue, uw, vn and vs in terms of the pressure correction p at the

cells. Again taking ue as an example:

= ˆ − ˆ ue ue ue

we have the formula for uê as shown in the equation (A.17) and similarly we have +1 +1 ˆ 1 ˆ ˆ dxdy 1 ∂pk 1 ∂pk uê = uÔ + uÊ + |O + |E 2 2 AO|O ∂x AO|E ∂x dxdy 1 1 ∂pk+1 − + |e 2 AO|O AO|E ∂x and so 1 1 1 1 1 = ( + )+dxdy ∂p | + ∂p | − + ∂p | ue uO uE O E e 2 2 AO|O ∂x AO|E ∂x AO|O AO|E ∂x (A.18)

1 dxdy ∂p u = − A | u | − | O | m O m O | O AO O m AO O ∂x 225 When the results are converged, u → 0 and so we can neglect the term of

1 − A | u | | m O m O AO O m

In the same way the other three face velocities can be obtained 1 1 1 = + ( − ) uw pW pO dy 2 AO|O AO|W 1 1 1 = + ( − ) vn pO pN dx 2 AO|O AO|N 1 1 1 = + ( − ) vs pS pO dx 2 AO|O AO|S and substituting them back into equation (A.11) ﬁnally gives the pressure correction equation, again in ﬁve band fashion:

∗ + ∗ + ∗ + ∗ + ∗ = AN pN AS pS AE pE AW pW AO pO SO

where dy2 1 1 AE = − + 2 AO|O AO|E dy2 1 1 AW = − + 2 AO|O AO|W dx2 1 1 AN = − + 2 AO|O AO|N 226 dx2 1 1 AS = − + 2 AO|O AO|S

AO = − (AE + AW + AN + AS) and

SO =0

Boundary Nodes

All the equations have been discretized for an inner node as shown before. The link coefﬁcients of the boundary nodes will be different from the inner node depending on the boundary conditions. Here the treatment of the boundary nodes will be discussed. Note that all the boundary values are always located at the faces of its nearby nodes. For simplicity, we have written all the equations in the form of

Je − Jw + Jn − JS = SO (A.19) and we will discuss how to treat the flux at the boundary faces in the following. After we obtain the flux at the boundary faces, we just substitute them back into the equation (A.19) and organize it again in a five band fashion to obtain the link coefficients.

First we consider the nodes in the ﬁrst column in the grids as shown in Figure A.3.

Since we have assumed that the flow is fully developed at the inlet, all the inlet value of u, v, Xi and φ at the inlet are fixed to the value denoted as in. For the nodes in the first column, all of the quantities at the west face are known.

For the velocity equations, the ﬂux at the west faces of the nodes in the ﬁrst column is

∂u J =(Reu u − | )dy w w w ∂x w

Here uw is just the inlet velocity and so

uwuw = uinuin

227 Figure A.3: A boundary node at the inlet.

∂u| The second term ∂x w can be obtained by the Taylor’s series. We expand uO and uE at the inlet: dx ∂u dx2 ∂2u u = u − | + | + O(dx3) (A.20) O in 2 ∂x w 8 ∂x2 w

3dx ∂u 9dx2 ∂2u u = u − | + | + O(dx3) (A.21) E in 2 ∂x w 8 ∂x2 w

If we (A.20) × 9 − (A.21) we can get

9 − − 8 ∂u| = uO uE uin ∂x w 3dx

The ﬂux at the inlet will be

9u − u − 8u J =(Reu u − O E in )dy w in in 3dx

The ﬂux at the other three faces remain the same as the inner nodes. Substituting Jw back will gives the link coefﬁcients of velocity equations at the inlet. For the pressure source

Sp = pe − pw, the inlet pressure pw can be approximated by the cell center pressure pO and

= pE−pO so the source term becomes Sp 2 .

Similarly, for the species equation, the ﬂux at the inlet

∂X J = − i | + z X | E | + PeX | u| w ∂x w i i w x w i w w

228 Figure A.4: A boundary node at the outlet.

And for the potential equation, the ﬂux at the inlet is

∂φ 9φ − φ − 8φ J = − | = O E in w ∂x w 3dx

The last column of nodes are shown in Figure A.4. The boundary condition at the outlet is ∂φ ∂u =0, =0; ∂x ∂x ∂p ∂X =0, i =0; ∂x ∂x and thus we can approximate the face values at e to the cell center values of o which gives

Je = uOuO

for velocity equation and 1 = | + | Je ΛziXi O PeXi OuO

for species and

Je =0 for potential.

229 Figure A.5: A boundary node at the lower wall.

Near the walls we have deﬁne the boundary condition for velocity as u =0,v =0and

for the velocity equation 9u − u J =0− O N s 3dy

The pressure source term Sp = pn − ps and the pressure at the s face can be approximated

as following. From Taylor’s series we have

dx ∂p dy2 ∂2p p = p − | + | + O(dy3) O s 2 ∂y s 8 ∂y2 s

3dx ∂p 9dy2 ∂2p p = p − | + | + O(dy3) N s 2 ∂y s 8 ∂y2 s which give us 3 − = pO pN + ( 2) ps 2 O dy

However, it is found that the numerical calculation is not accurate using the approximation due to the large gradient of the electrical source term near the patch; thus a higher order

230 approximation is used. We add another node NN(i, j +2)and expand the pressure at NN

node 5dx ∂p 25dy2 ∂2p p = p − | + | + O(dy3) NN s 2 ∂y s 8 ∂y2 s

The face value of ps can be obtained by the interpolation of the three nodes

15 − 10 +3 = pO pN pNN + ( 3) ps 8 O dy

For mass transport equations, we have assume the net ﬂux into the walls is zero as the boundary condition which is

Js =0

For the potential equation the ﬂux at the face of the wall is

9φ − φ − 8φ J = O N wall s 3dy

For the upper wall of the channel, we can use the same method to obtain the ﬂux at the n face.

A.3 The SIMPLE Algorithm

The calculation precedure has beed listed in the flowchart shown in Figure 3.4. First we read in the fully developed profiles of φ, u and Xi and define the initial guess as:

= k = k = |k φ φin,u uin,Xi Xi in

here k is the number of the outer iteration which is the iteration to calculate all of the variables and k =1initially. In the second step, the face velocities are calculated by linear interpolation: + + = uO uE = uO uW ; ue 2 ,uw 2

231 + + = vO vN = vO vS vn 2 ,vs 2 .

Then we compute the link coefﬁcients of the momentum equations using the guessed value

k k k of p , φ and Xi . In the fourth step, the momentum equation is solved by x − + + + SO AEuE AW uW AN uN ASuS uO = AO y − + + + SO AEvE AW vW AN vN ASvS vO = AO Here the alternating direction implicit (ADI) scheme is used to compute the solution o the

equations iteratively. The iterations which only calculate one variable are called the inner

iteration and usually for velocities the number of inner iterations is very low for example 2 in our calculation.

In step 7, the pressure correction equation is solved by − + + + = pO AEpE AW pW AN pN ASpS pO AO

and here the inner iteration has been performed for 150 to obtain a better predict of velocity

ﬁeld at each outer iteration in our calculation. Otherwise the calculation of the scalars will

blow up due to the imbalance of mass in the domain.

In step 12, the residuals are calculated to check for convergence (discussed in the next

k+1 k+1 k+1 k+1 k+1 section in detail). If it is not converged, u , v , p , φ and Xi are used as the guess value for the next step k +1. If it is converged, output the results.

The numerical parameters such as ωu, ωv and ωp are the relaxation factors and are chosen to make the calculation more stable and efﬁcient.

A.4 Residuals

The residual is deﬁned as the number that results when the difference equations, written in a form giving zero on the right-hand side, is evaluated for an intermediate of provisional

232 solution Tannehill et al. (1997). For example for streamwise velocity equation, we have

∗ + ∗ + ∗ + ∗ + ∗ − x =0 AO uO AE uE AW uW AN uN AS uS SO and at k step the residual of node O, the residual will be

k = ∗ k + ∗ k + ∗ k + ∗ k + ∗ k − ( x )k res AO uO AE uE AW uW AN uN AS uS SO

and the least of the residuals of all the nodes Rk

⎛ ⎞ 1 2 k = ⎝ ( k )2⎠ R resi,j i,j

gives the residual of the equation at step k.

Figure A.6: The residuals of the numerical calculation.

If the velocity ﬁeld satisfy the Navier-Stokes equation, the residual Rk will vanish.

Usually the residual will approach 0 if the calculation is converging and we use a small value as the criteria of convergence. If the residual is larger than , the calculation is not converged and the results are used to be the guess for the next iteration. If the residual is smaller than , the calculation is converged and the results are a good approximation to the

233 actually values. In our calculation the convergence criteria is =10−6 and the residual of all the equations are calculated and compared to the criteria, which ensure that all of the equations has been satisﬁed by the results. The calculation is converging slowing due to the large source terms in the momentum equations. The residuals are shown in Figure A.6 as an example. It is shown that the residuals are oscillating and approaching to 0 slowly.

234 BIBLIOGRAPHY

AJDARI, A. 1995 Electro-osmosis on inhomogeneously charged surfaces. Physics Review

Letters 75, 755Ð758.

AJDARI, A. 1996 Generation of transverse ﬂuid currents and forces by an electric ﬁeld:

Electro-osmosis on charge-modulated and undulat surface. Physics Review E. 53, 4996Ð

5005.

ALBERTS,B.&RAFF, M. 2003 An introducton to the Molecular Biology of the Cell, 2nd

edn. Garland Science.

ALPER, J. 1999 Biotechnology industry organization meeting:from the bioweapons

trenches, new tools for battling microbes. Science 284, 1754.

ANDERSON, D. J. 1995 Computational Fluid Dynamics: The Basics with Applications,

3rd edn. McGraw-Hill.

ANDERSON,J.L.&IDOL, W. K. 1985 Anderson, j. l. and idol, w. k. (1985) “electroos-

mosis through pores with nonuniformly charged walls. Chemical Engineering Commu-

nication 38, 93Ð106.

ASHOKE, K., GUPTA,S.&PAPADOPOULOS, K. D. 1997 Electrokinetic transport of a

slender particle through a pore. J. Chem. Soc., Faraday Trans 93 (20), 3695Ð3701.

235 BAYLEY, H. 2006 Sequencing single molecules of dna. Current Opinion in Chemical Bi-

ology 10, 628Ð637.

BAYLEY,H.&CREMER, P. S. 2001 Stochastic sensors inspired by biology. Nature 413,

226Ð230.

BAYLEY,H.&MARTIN, C. R. 2000 Resistive-pulse sensings from microbes to molecules.

Chem. Rev. 100, 2575Ð2594.

BINNING,P.&CELIA, M. A. 1999 Practical implementation of the fractional ﬂow ap-

proach to multi-phase ﬂow simulation. Advances in Water Resources 22 (5), 461Ð478.

BIRD,R.B.,STEWART,W.E.&LIGHTFOOT, E. N. 2001 Transport phenomena, 2nd

edn. New York: Wiley.

BOAL, D. 2002 Mechanics of the Cell. Cambridge.

BONTHUIS,D.J.,MEYER,C.,STEIN,D.&DEKKER, C. 2009 Conformation and dy-

namics of dna conﬁned in slitlike nanoﬂuidic channels. PHYSICAL REVIEW LETTERS

101, 108303.

BURGEEN,D.&NAKACHE, F. 1964 Electrokinetic ﬂow in ultraﬁne capillary slits. The

Journal of Physical Chemistry 68, 1084Ð1091.

BUSTAMANTE,C.,BRYANT,Z.&SMITH, S. B. 2003 Ten years of tension:single-

molecule dna mechanics. Nature 421, 423Ð427.

BUTLER,T.Z.,GUNDLACH,J.H.&TROLLY, M. 2007 Ionic current blockades from dna

and rna molecules in the a-hemolysin nanopore. Biophysical Journal 93, 3229Ð3240.

CALLADINE,C.&DREW, H. 1992 in Understanding DNA. Academic Press.

236 CALLADINE,C.R.,DREW,H.R.,LUISI,B.F.&TRAVERS, A. A. 2004 Understanding

DNA: the molecule and how it works, 3rd edn. Academic Press.

CERVERA,J.,SCHIEDT,B.&RAMIREZ, P. 2005 A poisson/nernst-planck model for ionic

transport through synthetic conical nanopores. Europhysics Letters 71 (1), 35Ð41.

CHAN, E. 2005 Advances in sequencing technology. Mutation Research 573, 13Ð40.

CHANG,C.C.&YANG, R. J. 2004 Computational analysis of electrokinetically driven

ﬂow mixing in microchannels with patterned blocks. Journal of Micromechanics and

Microengineering 14, 550Ð558.

CHANG, H. 2005 Characterization of dna translocation through a silicon based nanopore.

PhD thesis, PURDUE UNIVERSITY, US.

CHANG,H.,KOSARI,F.,ANDREADAKIS,G.,ALAM,M.A.,VASMATZIS,G.&

BASHIR, R. 2004 Dna-mediated ﬂuctuations in ionic current through silicon oxide

nanopore channels. Nanoletters 4 (8), 1551Ð1556.

CHAPMAN, D. L. 1913 A contribution to the theory of electrocapillarity. Philos. Mag. 25,

475Ð481.

CHEN,P.,GU,J.,BRANDIN, E., KIM, Y.-R., WANG,Q.&BRANTON, D. 2004 Probing

single dna molecule transport using fabricated nanopores. Nano Letters 4 (11), 2293Ð

2298.

CHEN, Y.-C., WANGA,C.&LUO, M.-B. 2007 Simulation study on the translocation of

polymer chains through nanopores. Journal Of Chemical Physics 127, 044904.

237 CONLISK, A. T. 2005 The debye-huckel approximation: Its use in describing electroos-

motic ﬂow in micro- and nanochannels. Electrophoresis 26, 1896Ð1912.

CONLISK,A.T.,MCFERRAN,J.,ZHENG,Z.&HANSFORD, D. 2002 Mass transfer and

ﬂow in electrically charged micro- and nano-channels. Analytical Chemistry 74, 2139Ð

2150.

COULTER, W. 1953 Means for counting particles suspended in a ﬂuid. US Patent 2656508.

DATTA,S.,CONLISK, A., LIB,H.&YODA, M. 2009 Effect of divalent ions on elec-

troosmotic ﬂow in microchannels. Mechanics Research Communications 36 (1), 65Ð74.

DEBLOIS,R.&BEAN, C. 1970 Counting and sizing of submicron particles by resistive

pulse technique. Review of Scientiﬁc Instruments 41, 909.

DEBLOIS, R., BEAN,C.&WESLEY, R. 1977 Electrokinetic measurements with submi-

cron particles and pores by resistive pulse technique. Journal of Colloid and Interface

Science 61, 323.

DUTCHER,J.&MARANGONI, A. G. 2004 Soft Materials: Structural and Dynamics.

CRC Press.

EHRFELD,W.&SCHMIDT, A. 1999 Recent developments in deep x-ray lithography. Jour-

nal of Vacuum Science and Technology B16, 3526.

ERICKSON &LI, D. 2002a Inﬂuence of surface heterogeneity on electrokinetically driven

microﬂuidic mixing. Langmuir 18, 1883Ð1892.

ERICKSON,D.&LI, D. 2002b Inﬂuence of surface heterogeneity on electrokinetically

driven microﬂuidic mixing. Langmuir 18, 1883Ð1892.

238 FAN,R.,KARNIK,R.,YUE,M.,LI,D.,MAJUMDAR,A.&YANG, P. 2005 Dna translo-

cation in inorganic nanotubes. Nanoletters 5 (9), 1633Ð1637.

FEI, Z. 2007 Private communication.

FORREY,C.&MUTHUKUMARA, M. 2007 Langevin dynamics simulations of ds-dna

translocation through synthetic nanopores. Journal Of Chemical Physics 127, 015102.

FREDLAKE,C.,HERT,D.,MARDIS,E.&BARRON, A. 2006 What is the future of

electrophoresis in large-scale genomic sequencing? Electrophoresis 27 (19), 3689Ð3702.

FRIEDL,W.,REIJENGA,J.C.&KENNDLER, E. 1995 Ionic strength and charge num-

ber correction for mobilities of multivalent organic anions in capillary electrophoresis.

Journal of Chromatography A 709, 163Ð170.

FYTA,M.G.,MELCHIONNA, S., KAXIRAS,E.&SUCCI, S. 2006 Multiscale coupling

of molecular dynamics and hydrodynamics: Application to dna translocation through a

nanopore. Multiscale Modeling and Simulation 5 (4), 1156Ð1173.

GASKELL, P. H. & C., L. A. K. 1988 Curvature-compensated convective transport:

Smart, a new boundedness-preserving transport algorithm. International Journal for Nu-

merical Methods in Fluids 8, 617Ð641.

GHOSAL, S. 2002 Lubrication theory for electroosmotic ﬂow in a microﬂuidic channel of

slowly varing cross-section and wall charge. Journal of Fluid Mechanics 459, 103Ð128.

GHOSAL, S. 2006 Electrophoresis of a polyelectrolyte through a nanopore. Physical Re-

view E 74, 041901.

239 GHOSAL, S. 2007a Effect of salt concentration on the electrophoretic speed of a polyelec-

trolyte through a nanopore. Physical Review Letters 98, 238104.

GHOSAL, S. 2007b Electrokinetic-ﬂow-induced viscous drag on a tethered dna inside a

nanopore. Physical Review E 76, 061916.

GRAHAME, D. C. 1953 Diffuse double layer theory for electrolytes of unsymmetrical

valence types. Journal of Chemical Physics 21, 1054.

GUOY, G. 1910 About the electric charge on the surface of an electrolyte. Journal of

Physics A 9, 457Ð468.

GAD-EL HAK, M. 1999 The ﬂuid mechanics of microdevices. Journal of Fluids Engineer-

ing 121, 5Ð33.

HASSANALI,A.&SINGER, S. 2006 Private communication.

HELMHOLTZ, H. L. F. 1879 Uber den einﬂu der elektrischen grenzschichten bei galvanis-

cher spannung und der durch wasserstromung erzeugten potentialdiffernz. Ann. Physik.

3 (7), 337Ð387.

HENRY,S.,MCALLISTER, D., ALLEN,M.&PRAUSNITZ, M. 1998 Microfabricated

microneedles: a novel method to increase transdermal drug delivery. J. Pharm. Sci. 87,

922Ð925.

VAN DER HEYDEN,F.H.J.,STEIN,D.&DEKKER, C. 2005 Streaming currents in a

single nanoﬂuidic channel. Physical Review Letters 95, 116104.

240 HIBIKI,T.&ISHII, M. 2003 One-dimensional drift-ﬂux model and constitutive equations

for relative motion between phases in various two-phase ﬂow regimes. International

Journal of Heat and Mass Transfer 46, 49354948.

HUBBARD, A. T. 2002 Encyclopedia of surface and colloid science. New York: Marcel

Dekker.

JOHNSON,T.J.,ROSS,D.,GAITAN,M.&LOCASCIO, L. E. 2001 Laser modiﬁcation of

preformed polymer microchannels: Application to reduce band broadening around turns

subject to electrokinetic ﬂow. Analytical Chemistry 73, 3656Ð3661.

KANTOR,Y.&KARDAR, M. 2004 Anomalous dynamics of forced translocation. Physical

Review E 69, 021806.

KASIANOWICZ, J. J., BRANDIN,E.&DEAMER, D. B. D. W. 1996 Characterization of

individual polynucleotide molecules using a membrane channel. The Proceedings of the

National Academy of Sciences USA 93, 13770Ð13773.

KEYSER,U.F.,KOELEMAN, B. N., DORP,S.V.,KRAPF, D., SMEETS,R.M.M.,

LEMAY,S.G.,DEKKER,N.H.&DEKKER, C. 2006 Direct forcemeasurements on dna

in a solid-state nanopore. Nature Physics 2, 473Ð477.

KIM, S. H., PANWAR, A. S., KUMAR, S., AHN,K.H.&LEE, S. J. 2004 Electrophore-

sis of a bead-rod chain through a narrow slit: A brownian dynamics study. Journal of

Chemical Physics 121 (18), 9116Ð9122.

KIRBY,B.J.&HASSELBRINK, E. F. 2004 Zeta potential of microﬂuidic substrates: 2.

data for polymers. Electrophoresis 25, 203Ð213.

241 KLUSHIN,L.I.,SKVORTSOV,A.M.,HSU, H.-P. & BINDER, K. 2008 Dragging a poly-

mer chain into a nanotube and subsequent release. Macromolecules 18 (15), 5890Ð5898.

KUMAR,A.&MAZUMDER, S. 2007 Assessment of the dilute approximation for the

prediction of combined heat and mass transfer rates in multi-component systems. Heat

and Mass Transfer 43 (12), 1329Ð1337.

LEVINE, M., R., J. & KENNETH, R. 1975 Theory of electrokinetic ﬂow in a narrow

parallel-plate channel. Faraday Transactions II 71, 1Ð11.

LI,J.,GERSHOW,M.,STEIN2, D., BRANDIN,E.&GOLOVCHENKO, J. A. 2003 Dna

molecules and conﬁgurations in a solidstate nanopore microscope. Nature Materials 2,

611Ð615.

LI, J., STEIN,D.,MCMULLAN,C.,BRANTON, D., AZIZ,M.J.&GOLOVCHENKO,

J. A. 2001 Ion-beam sculpting at nanometre length scales. Nature 412, 166Ð169.

LIN,W.,CORMIER,M.,SAMIEE, A., GRIFFIN,A.,JOHNSON,B.,TENG,C.,HARDEE,

G. & DADDONA, P. 2001 Transdermal delivery of antisense oligonucleotides with mi-

croprojection patch (macroﬂux) technology. Pharmaceutical Research 18 (12), 1789Ð

1793.

M, M., J, E., D, E. & D, C. 1981 The dimensions of dna in solution. Journal of Molecular

Biology 152, 153Ð161.

MADOU, M. 2002 Fundamentals of Microfabrication: The Science of Miniaturization, 2nd

edn. New York: CRC Press.

MARBLE, F. E. 1970 Dynamics of dusty gases. Annual Review of Fluid Mechanics 2,

397Ð446.

242 MELLER 2001 Voltage-driven dna translocations through a nanopore. Physical Review Let-

ters 86 (15), 3435Ð3438.

MURPHY,R.J.&MUTHUKUMARA, M. 2007 Threading synthetic polyelectrolytes

through protein pores. Journal Of Chemical Physics 126, 051101.

MUTHUKUMAR, M. 1999 Polymer translocation through a hole. Journal Of Chemical

Physics 111 (12), 10371Ð10374.

MUTHUKUMAR, M. 2007 Mechanism of dna transport through pores. The Annual Review

of Biophysics and Biomolecular Structure 36, 435Ð450.

ODIJK, T. 1983 Ten years of tension:single-molecule dna mechanics. Macromolecules

16 (8), 13401344.

OHSAWA,K.,MURATA,M.&OHSHIMA), H. 1986 Zeta potential and surface charge

density of polystyrene-latex; comparison with synaptic vesicle and brush border mem-

brane vesicle. Colloid and Polymer Science 264, 1005Ð1009.

O’KEEFFE,J.,COZMUTA,I.&STOLC, V. 2003 Polymer translocation through a

nanopore: A geometry dependence study. In Third IEEE Conference on Nanotechnol-

ogy. IEEE.

OLESIK, S. 2006 Private communication.

PARK,S.Y.,RUSSO,C.J.,BRANTON,D.&STONE, H. A. 2006 Eddies in a bottle

neck: An arbitrary debye length theory for capillary electroosmosis. Journal of Colloid

and Interface Science 197, 832Ð839.

243 PATANKAR,S.V.&SPALDING, D. 1972 A calculation procedure for heat, mass and

momentum transfer in three-dimensional parabolic ﬂows. International Journal of Heat

Mass Transfer 15, 1787Ð1806.

PRINSEN,P.,FANG,L.T.,YOFFE, A. M., KNOBLER,C.M.,&GELBART,W.M.

2009 The force acting on a polymer partially conﬁned in a tube. the Journal of Physical

Chemistry B 113 (12), 3873Ð3879.

PROBSTEIN, R. F. 2003 Physicochemical Hydrodynamics: An Introduction, 2nd edn.

Wiley-Interscience.

RAMIREZ,J.C.&CONLISK, A. T. 2006 Formation of vortices near abrupt nano-channel

height changes in electro-osmotic ﬂow of aqueous solutions. Biomed Microdevices 8,

325Ð330.

RAPAPORT, D. C. 1995 The Art of Molecular Dynamics Simulation. Cambridge: Univer-

sity Press.

RHIE,C.M.&CHOW, W. L. 1983 Numerical study of the turbulent ﬂow past a airfoil

with trailing edge seperation. AIAA Journal 21, 1525Ð1532.

RICE,C.L.&WHITEHEAD, R. 1965 Electrokinetic ﬂow in a narrowcapillary. Journal of

Physical Chemistry 69 (11), 4017Ð4024.

SALAMONE, J. C. 1996 Polymeric Materials Encyclopedia. CRC Press.

SANGER,F.,NICKLEN,S.&COULSON, A. R. 1977 Dna sequencing with chain-

terminating inhibitors. The Proceedings of the National Academy of Sciences U. S. A.

74 (12), 5463Ð5467.

244 SCHIEDT, B. 2007 Characterization and application of ion track-etched nanopores. Mas-

ter’s thesis, The Ruperto-Carola University of Heidelberg, Germany.

SEXTON,L.T.,HORNE,L.P.&MARTIN, C. R. 2007 Developing synthetic conical

nanopores for biosensing applications. Molecular BioSystems 3 (10), 667Ð685.

SHIN,Y.K.&SINGER, S. J. 2009 Private communication.

SILVERSTEIN,A.&NUNN, L. S. 1002 DNA. Twenty-First Century Books.

SMEETS,R.M.M.,KEYSER,U.F.,KRAPF, D., WU, M.-Y., DEKKER,N.H.&

DEKKER, C. 2006 Salt dependence of ion transport and dna translocation through solid-

state nanopores. Nano Letters 6 (1), 89Ð95.

SMITH, S. H. 1987 Unsteady stokes’ ﬂow in two dimensions. Journal of Engineering

Mathematics 21, 271Ð285.

SMOLUCHOWSKI, M. V. 1903 Contributiona la theorie de l’endosmoseelectrique et de

quelques phenomenes correlatifs. Bull. Intl Acad. Sci. Cracovie p. 184.

SMOLUCHOWSKI, M. V. 1912 Experimentall nachweisbare, der ublichen thermodynamic

widersprechende molekular phanomene. Physikalische Zeitschrift 13, 1069Ð1080.

STERN, O. Z. 1924 The theory of the electrolytic double layer. Elektrochem 30, 508Ð516.

STORM,A.J.,CHEN, J. H., ZANDBERGEN,H.W.&DEKKER, C. 2003 Fabrication of

solid-state nanopores with single-nanometre precision. Nature Materials 2, 537Ð540.

STORM,A.J.,CHEN,J.H.,ZANDBERGEN,H.W.&DEKKER, C. 2005a Translocation

of double-strand dna through a silicon oxide nanopore. Physical Review E 71, 051903.

245 STORM, A. J., STORM,C.,CHEN,J.,ZANDBERGEN,H.,JOANNY, J.-F. & DEKKER,C.

2005b Fast dna translocation through a solid-state nanopore. Nano Letters 5 (7), 1193Ð

1197.

STROOCK, A. D., WECK, M., CHIU,D.T.,HUCK,W.T.S.,KENIS,P.J.A.,ISMAG-

ILOV,R.F.&WHITESIDES, G. M. 2000 Patterning electro-osmotic ﬂow with patterned

surface charge. Physics Review Letters 84, 3314Ð3317.

TANNEHILL, J. C., ANDERSON,D.A.&PLETCHER, R. H. 1997 Computational Fluid

Mechanics and Heat Transfer, 3rd edn. Taylor and Francis.

WANG,S.&HU, X. 2007 Private communication.

WANG, S., ZENG,C.,LAI, S., JUANG, Y.-J., YANG,Y.,&LEE, L. 2005 Polymer

nanonozzle array fabricated by sacriﬁcial template imprinting. Advanced Materials 17,

1182Ð1186.

WILKE,N.,MULCAHY, A., YE, S.-R. & MORRISSEY, A. 2005 Process optimization

and characterization of silicon microneedles fabricated by wet etch technology. Micro-

electronics Journal 36, 650Ð656.

WILLIAMS,R.A.,ELIMELECH,M.,GREGORY,J.&JIA, X. 1995 Particle Deposition

and Aggregation: Measurement, Modelling and Simulation, 1st edn. Elsevier.

YARIV, E. 2004 Electroosmotic ﬂwo near a surface charge discontinuity. Journal of Fluid

Mechanics 521, 181Ð189.

ZENG, C., WANG,S.&LEE, L. J. 2005 Dynamic silica assembly for fabrication of

nanoscale polymer channels. Materials Letters 59, 3095Ð3098.

246 ZHENG,Z.,HANSFORD,D.J.&CONLISK, A. T. 2003 Effect of multivalent ions on

electroosmotic ﬂow in micro- and nanochannels. Electrophoresis 24, 30063017.

ZHU,W.,SINGER, S. J., ZHENG,Z.&CONLISK, A. T. 2005 Electro-osmotic ﬂow of a

model electrolyte. Physics Review E. 71, 041501Ð1 to 041501Ð12.

ZWOLAK, M. 1809 Sur un nouvel effet de l’ctricit galvanique. Memoires de la Societ.

Imperiale de Naturalistes de Moscou 2, 327Ð337.

ZWOLAK, M. 2008 Colloquium: Physical approaches to dna sequencing and detection.

Reviews of Modern Physics 80, 141Ð165.

247