Electrokinetic Flow in Micro- and Nano-Fluidic Components

ELECTROKINETIC FLOW IN MICRO- AND NANO-FLUIDIC COMPONENTS

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the Graduate

School of The Ohio State University

Zhi Zheng, M.S.

*****

The Ohio State University 2003

Dissertation Committee: Approved by Professor A. Terry Conlisk, Adviser

Professor Derek J. Hansford ______Adviser Professor L, James Lee Biomedical Engineering Center

ABSTRACT

In this work, electrokinetic flow through nano- and micro-channels is investigated. The governing equations for the flow in three dimensional channels are derived from Poisson-Nernst-Planck theory. The boundary conditions for the governing equations are obtained from the electrochemical equilibrium requirements. The results are compared with three sets of experimental data, provided by iMEDD inc., Oak Ridge

National Laboratory, and Georgia Institute of Technology. The results of comparison are extremely good. A finite difference method is used to solve the governing equations numerically. For asymptotic cases, the governing equations are solved analytically using singular perturbation method. The results show that multivalent counterions decrease the electroosmotic flow extraordinarily. Scaling laws for the governing equations are introduced which clarify that several dimensionless parameters determine the potential and velocity profiles and the distribution patterns of ion species. The relationship between the channel surface charge density and the ζ potential is explored. The diffusive fluxes, electric migrational fluxes and convective fluxes of different ion species in the electrokinetic flow are obtained from the governing equations. Results are also obtained for electroosmosis in 2D nanochannels. At the end of this work, the governing equations for unsteady flow are derived, the transport of macromolecules through micro- and nanochannels is discussed and the future work in these areas is suggested. ii Dedicated to my parents

iii ACKNOWLEDGMENTS

It is a big challenge for me to complete my PhD program at Ohio State. There are many people to whom I owe grateful thanks for helping me coming through the whole procedure.

I would like to express my gratitude to my advisor, Dr. A. Terry Conlisk, for the tremendous amount of support he has given me throughout this work, for his encouragement and enthusiasm which boosted my research work, and for his patience in correcting my scientific and linguistic errors. I am also grateful to the other members of my dissertation advisory committee: Dr. Derek Hansford and Dr. Jim Lee for their help with this work. I also want to thank all the other members of our research group, especially Kelly Evers, Jennifer McFerran, and Dr. Somnath Bhattacharyya.

This work is part of the project funded by DARPA. I am grateful to the contract monitors Dr. Anantha Krishnan (DARPA), Mr. Clare Thiem and Mr. Duane Gilmour

(IFTC) for their support. I also thank Dr. Mike Ramsey, Steve Jacobson, and J. P. Alarie of ORNL; Rob Walczak and Tony Boiarski of iMEDD; Dr. Minami Yoda and Reza Sadr of Georgia Tech; Dr. Sherwin Singer and Wei Zhu of OSU and Dr. Jim Coe of OSU for many discussions and for the use of their experimental results.

iv VITA

February 29, 1976……………………Born – Jinan, China

1998…………………………………B.S. Physics, Tsinghua University, Beijing.

June, 2003……………………………M.S. Biomedical Engineering, the Ohio State University.

1998 – present………………………Graduate Teaching and Research Associate, The Ohio State University

PUBLICATIONS

Research Publication

1. “Mass Transfer and Flow in Electrically Charged Micro- and Nanochannels”, A.T. Conlisk, Jennifer Mcferran, Zhi Zheng, Derek Hansford, Anal. Chem. 2002, 74, 2139-2150.

2. “Effects of Multivalent Ions on Electroosmotic Flow in Micro and Nanochannels”, Zhi Zheng, Derek J. Hansford, A. T. Conlisk, Electrophoresis, 2003, 24, 3006- 3017.

3. “Asymptotic Solutions for Electroosmotic Flow in Two-Dimensional Charged Micro- and Nano-channels”, S. Bhattacharyya, Zhi Zheng, A. T. Conlisk, 33rd AIAA Fluid Dynamics Conference, 2003.

4. “An Experimental and Modeling Study of Electroosmotic Bulk and Near-Wall Flows in Two-Dimensional Micro- and Nanochannels”, R. Sadr, Zhi Zheng, M. Yoda, A. T. Conlisk, IMECE, 2003-42917

FIELD OF STUDY

Major Field: Biomedical Engineering

v TABLE OF CONTENTS

Page

Dedication………………………………………………………………...…………iii

Acknowledgements………………………………………………………………….iv

Vita……………..…………………………………………………………………….v

LIST OF TABLES...... viii

LIST OF FIGURES ...... xi

CHAPTER 1 INTRODUCTION ...... 1 1.1 Background...... 1 1.2 The Basics of Electroosmosis ...... 5 1.3 Properties of Channel Surfaces...... 10 1.4 Previous Work on Modeling Electroosmotic Flow ...... 13 1.5 Outline of Dissertation...... 16

CHAPTER 2 GOVERNING EQUATIONS...... 19 2.1 Introduction...... 19 2.2 Classical Electric Double Layer Theories...... 20 2.3 The Three Dimensional Governing Equations ...... 33 2.4 The One Dimensional Governing Equations ...... 39 2.5 Singular Perturbation Equations ...... 41 2.6 Debye-Hückel Approximaton for Nanochannels where ε = Ο (1) ..... 47 2.7 Electrochemical Potential ...... 49 2.8 Electrochemical Consideration for a Channel-Reservoir System...... 51

CHAPTER 3 RESULTS FOR ONE DIMENSIONAL CHANNELS...... 58 3.1 Introduction...... 58 3.2 Finite Difference Method...... 59 3.3 Results for Monovalent Binary Electrolytes...... 64 3.4 Monovalent Binary Electrolytes In a Channel-Reservoir System ...... 73 3.5 Using Scaling Laws to Simplify Governing Equations ...... 81 3.6 Results for Multivalent Binary Electrolyte and Multi-component Electrolytes...... 90 vi CHAPTER 4 COMPARISON WITH EXPERIMENTS ...... 113 4.1 Introduction...... 113 4.2 Comparison with the iMEDD Experiments...... 114 4.3 Comparison with the ORNL Experiments...... 125 4.4 Comparison with the Georgia Tech Experiments...... 148 4.5 Issues of Surface Charge Density ...... 154

CHAPTER 5 ELECTROPHORESIS AND ION FLUXES...... 156 5.1 Introduction...... 156 5.2 Classical Theory of Capillary Electrophoresis ...... 157 5.3 Governing Equations ...... 159 5.4 Numerical Method ...... 164 5.5 Results...... 166

CHAPTER 6 EOF IN RECTANGULAR CHANNELS ...... 219 6.1 Introduction...... 219 6.2 Governing Equations ...... 220 6.3 Numerical Method ...... 222 6.4 Results For Symmetrical Cases ...... 223 6.5 Results for Asymmetrical cases...... 238

CHAPTER 7 UNSTEADY FLOW ...... 248

CHAPTER 8 SUMMARY AND FUTURE WORK ...... 252 8.1 Introduction...... 252 8.2 Transport of Macromolecules in Micro- and Nano- capillaries...... 253

APPENDICES…………………………………………………………………….257

BIBLIOGRAPHY...... 261

vii LIST OF TABLES

Page Table 2.1 Mean activity coefficients of aqueous salts at 25ºC. (http://www.psigate.ac.uk/newsite/reference/plambeck/chem2/p01193.htm, 07/18/03)...... 50 Table 2.2 Experimental activity coefficients of NaCl at 298K. [50]...... 50 Table 3.1 Molarity of ions in the reservoir, units=M; the ionic strength is 0.33288M. ... 93 Table 3.2 Average molarity of ions in the nanochannel. The channel height is 20nm..... 93 Table 3.3 Molarity of different ions at the walls of the channel. The channel height is 20nm. The thickness of the EDL is given in this table, and the volume flow rate is shown in the last column, for an imposed voltage of 0.05V over a channel length of 3.5 µm...... 94 Table 4.1 The electroosmotic volume flow rates measured by iMEDD, inc. The environmental temperature T=298K, the external voltage over the whole channel- reservoir system is 17V. (from Rob Walczak, iMEDD inc., 2001)...... 117 Table 4.2 The resistance of iMEDD membranes. The working buffer is PBS, ten times diluted PBS and a hundred times diluted PBS...... 121 Table 4.3 Voltage drops through the iMEDD nanochannel. The overall voltage drop between the two electrodes is 17 Volts. 0.1xPBS denotes 10 times diluted PBS, and 0.01xPBS denotes 100 times diluted PBS. Results are shown for channel heights of 4, 7, 13, 20 and 27 nm as in the experiments...... 122 Table 4.4 The volume flow rates for iMEDD configurations, calculated by the governing equations. The surface charge density σ= −0.2 C/m2, as suggested by Israelachvili [17]...... 122 Table 4.5 Geometry of the Oak Ridge National Laboratory (ORNL) channels...... 125 Table 4.6 The mobilities measured in ORNL experiments. The units of mobility is cm2/(V·s). There is a lack of experimental data for 0.02mM buffer at h=98, 290 and 1080nm. (Ramsey et. al. 2002)...... 126 Table 4.7 The standard deviations of the mobilities in Table 4.6. The units is cm2/(V·s). (Ramsey et. al. 2002)...... 127 Table 4.8 The field strength in ORNL experiments. For each channel height, there are three sets of data which are collected at different field strength. (Ramsey et. al. 2002) ...... 127 Table 4.9 The experimentally determined zeta potential for ORNL 83nm channel. (Ramsey et. al. 2002)...... 128 viii Table 4.10 The parameters of the governing equations for different buffer concentration in the reservoir for the ORNL 83nm channel. δ and γ are the scaling parameters 0 discussed in §3.5. ε, δ and γ are all dimensionless. X1 is the mole fraction of sodium 0 ions on the wall, and X2 is the mole fraction of tetraborate ions on the wall. β=c/I is defined along with equation (2.52), where c is the total concentration of the solvent and all ion species, and I is the ionic strength...... 130 Table 4.11 The parameters ε, δ and γ of the asymptotic equation for 20mM and 150mM buffer in the reservoir for the ORNL 83nm channel...... 134 Table 4.12 The equation parameters ε, δ and γ for h=83nm and h=290nm ORNL channels. The concentration of the buffer in the reservoir is 2mM...... 136 Table 4.13 Results for the average mobility and ζ potential for the indicated molarities in the ORNL 83nm×20.3µm channels. Both numerical and asymptotic values of the average mobility and ζ potential are shown. Yɍ=10. Superscripts e and a stand for experimental and asymptotic, respectively. The surface charge σ=−0.0154 C/m2 on the channel wall. For 0.02mM buffer, because the large ε, the asymptotic values are invalid...... 143 Table 4.14 The mobilities calculated from the model. For h=83nm, 98nm, 290nm and 300nm, the numeric model is used. For h=1080nm, the asymptotic model is used, and Yɍ=100 for 0.2mM solution, 30 for 2mM solution, 10 for 20mM and 150mM solutions. The surface charge density on the silicon channel wall is σ=−0.0154 C/m2. The units of mobility is cm2/(V·s).Not all cases are calculated, those have not been calculated are marked as ‘-‘, for which there is no experimental data...... 144 Table 4.15 The relative error on mobilities between the results of the model shown in Table 4.14 and the experimental results shown in Table 4.6. The entities marked ‘-‘ indicate that there is no experimental data available...... 144 Table 4.16 The mobilities calculated from the model. For h=83nm, 98nm, 290nm and 300nm, the numeric model is used. For h=1080nm, the asymptotic model is used, and Yɍ=100 for 0.2mM solution, 30 for 2mM solution, 10 for 20mM and 150mM solutions. The surface charge density on the silicon channel wall is σ=−0.007 C/m2 for the 0.02mM buffer. σ=−0.021 C/m2 for the 150mM buffer, and σ=−0.0154 C/m2 for other buffers. The units of mobility is cm2/(V·s)...... 145 Table 4.17 The relative error on mobilities between the results of the model shown in Table 4.16and the experimental results shown in Table 4.6. The entities marked ‘-‘ indicate that there is no experimental data available...... 145 µ e Table 4.18 Comparison of the average electrical mobility m , measured by the Georgia µ m ζ Tech experiments, and the mobility m , calculated by the model. potential, parameters ε, δ and γ calculated from the model are also shown. I is the ionic ix strength of the buffer in the reservoir, The surface charge densities being used in the model are: σ=−0.002C/m2 for 0.19mM buffer, σ=−0.0025C/m2 for 1.9mM buffer and 3.6mM buffer, σ=−0.0045C/m2 for 18.4mM buffer and 36mM buffer...... 149 Table 4.19 Comparison of the surface charge density calculated by using equation (4.5) from the ζ potential measured in ORNL experiments, with the surface charge density assumed in the numerical calculation...... 155 Table 5.1 The five channels being studied. Results for Long channels (L=3.6µm) are compared with shorter channels (L=36nm) for different external voltage drops (5V, 0.05V, and 0.0005V). The results for narrower channel (h=4nm) is compared with wider channels (h=20nm)...... 168 Table 5.2 Comparison of the mole fractions, potential, and velocity obtained from different mesh grids for channel I. The mesh point (2,2) near the corner of an 11×11 mesh corresponds to the mesh point (3,3) in a 21×21 mesh, (5,5) in a 41×41 mesh, and (9,9) in an 81×81 mesh. The mesh point (6,6) at the center of the 11×11 mesh corresponds to point (11,11) in a 21×21 mesh, (21,21) in a 41×41 mesh and (41,41) in an 81×81 mesh. The first and last row in these meshes represents the inlet and outlet, respectively...... 169 Table 6.1 The properties of the channels. The lengths of channels are 3.6µm, the wall concentration of cation is 0.154M, and the wall concentration of anion is 0.142M...... 224 µ µ a Table 6.2 The numerical mobility m ,and the analytical mobility m ...... 225 Table 6.3 Comparison of the numerical results for 41×41 mesh grids and the results for 81×81 meshes at three locations inside the channel. g=c1/I, and f=c2/I represents the concentration of cation species and anion species, respectively...... 226 Table 6.4 The boundary conditions for an asymmetric case. The four side walls are labeled with the corresponding y or z values there...... 239

x LIST OF FIGURES

Page Figure 1.1 NanoDrop® ND-1000 spectrophotometer. This new product of NanoDrop® technologies is a UV/VIS spectrophotometer for one microliter samples. The sample is directly pipetted onto the measurement surface and wiped out after the measurement, which eliminates the need of cuvettes or capillaries. (http://www.nanodrop.com/index.shtml, 06/24/2003)...... 2 Figure 1.2 Schematic of assay development microchips designed by Cohen et. al. [16]. Reservoirs are numbered and labeled with reagent solutions...... 3 Figure 1.3 Image from Gene-Chips (Microarray) The intensity and color of each spot encode information on a specific gene from the tested sample. (http://www.gene- chips.com/sample1.html, 06/24/2003, Authorized by Leming Shi.) ...... 4 Figure 1.4 A schematic figure showing the electroosmotic flow in a negatively charged channel with thin EDL. The real sizes of ions are magnified relative to the size of the channel. Instead of showing directly as molecules, the solvent are represent by the yellowish colored area...... 6 Figure 1.5 Counterions accumulate near the charged surface, while co-ions are depleted near the charged surface. In this plot, ρɍ is the electrolyte concentration in the bulk [17]...... 7 Figure 1.6 A schematic representation showing EDL and potential drop across the EDL. (http://www.geocities.com/CapeCanaveral/Hangar/5555/zeta.htm, 06/28/03)...... 8 Figure 1.7 Comparison of the pressure drop in pressure driving flow and the applied voltage in EOF. The flow rate is 1µl/min; the unit of channel height is meters...... 10 Figure 1.8 SEM pictures of a 20nm high nanochannel fabricated on a 3.5µm thick silicon membrane. The direction of flow is from the outside down to the inside of the page or vice versa. (Hansford, D.J., private communication 1999)...... 11 Figure 2.1 The electric potential in the EDL on a negatively charged surface...... 21 Figure 2.2 Geometry of one dimensional channel. W>>h, L>>h...... 23 Figure 2.3 The Debye-Hückel picture of the EDL for binary electrolyte solution where z1=−z2...... 27 Figure 2.4 Relative number densities of cation species (ρ1/ρ0) and anion species (ρ2/ρ0) as functions of ϖ, where -1<ϖ<1...... 29 Figure 2.5 Relative number densities of cation species (ρ1/ρ0) and anion species (ρ2/ρ0) as functions of ϖ. This is a blown up version of Figure 2.4 for -0.5<ϖ<0.5...... 29 xi ρ + ρ ρ − ρ Figure 2.6 The ratios 1 2 and 1 2 as a function of ϖ, for -1<ϖ<1...... 30 2ρ0 2ρ0 ρ + ρ ρ − ρ Figure 2.7 The ratios 1 2 and 1 2 as a function of ϖ. This is a blown up 2ρ0 2ρ0 version of Figure 2.6, for -0.5<ϖ<0.5...... 31 Figure 2.8 The Gouy-Chapman picture of the EDL for binary electrolyte solution where o z1=−z2=z . See also Figure 1.5...... 32 Figure 2.9 A schematic figure of the normal channel-reservoir system ...... 52 Figure 3.1 Function f=f(x,y,…) in a view of finite difference. ∆x<<1...... 60 Figure 3.2 The mole fraction distribution pattern across the channel for monovalent binary electrolyte. The wall mole fractions are constants, for sodium the wall mole fraction is 0.00276; for chloride, the wall mole fraction is 0.00254. Channel height h=20nm. ε=λ/h=0.040...... 66 Figure 3.3 The potential and velocity profile across the channel for monovalent binary electrolyte. The wall mole fractions are constants, for sodium the wall mole fraction is 0.00276; for chloride, the wall mole fraction is 0.00254. Channel height h=20nm. ε=λ/h=0.040...... 67 Figure 3.4 The shear stress profile across the channel for monovalent binary electrolyte. The wall mole fractions are constants, for sodium the wall mole fraction is 0.00276; for chloride, the wall mole fraction is 0.00254. Channel height h=20nm. ε=λ/h=0.040...... 67 Figure 3.5 The mole fraction distribution pattern across the channel for monovalent binary electrolyte. The wall mole fractions are known as constants, for sodium the wall mole fraction is 0.00276; for chloride, the wall mole fraction is 0.00254. Channel height h=4nm. ε=λ/h=0.200...... 68 Figure 3.6 The potential and velocity profile across the channel for monovalent binary electrolyte. The wall mole fractions are constants, for sodium the wall mole fraction is 0.00276; for chloride, the wall mole fraction is 0.00254. Channel height h=4nm. ε=λ/h=0.200...... 69 Figure 3.7 The shear stress profile across the channel for monovalent binary electrolyte. The wall mole fractions are constants, for sodium the wall mole fraction is 0.00276; for chloride, the wall mole fraction is 0.00254. Channel height h=4nm. ε=λ/h=0.200...... 70 Figure 3.8 The mole fraction distribution pattern near the wall for the asymptotic case for monovalent binary electrolyte. The wall mole fractions are constants, for sodium the wall mole fraction is 0.00276; for chloride, the wall mole fraction is 0.00254. Channel height h=20nm. ε=λ/h=0.040. Yɍ=6...... 70

xii Figure 3.9 The potential and velocity profile near the wall for the asymptotic case for monovalent binary electrolyte. The wall mole fractions are constants, for sodium the wall mole fraction is 0.00276; for chloride, the wall mole fraction is 0.00254. Channel height h=20nm. ε=λ/h=0.040. Yɍ=6...... 71 Figure 3.10 The shear stress near the wall for the asymptotic case for monovalent binary electrolyte. The wall mole fractions are constants, for sodium the wall mole fraction is 0.00276; for chloride, the wall mole fraction is 0.00254. Channel height h=20nm. ε=λ/h=0.040. Yɍ=6...... 71 Figure 3.11 Mole fractions across the channel for monovalent binary electrolytic solution. The channel height is 20nm, the width is 20µm. The surface charge density σ=−0.2C/m2, and the molarity of the solution is 0.1M in the reservoir. ε=λ/h= 0.0073...... 73 Figure 3.12 Potential and velocity profile across the channel for monovalent binary electrolytic solution. The channel height is 20nm, the width is 20µm. The surface charge density σ=−0.2C/m2, and the molarity of the solution is 0.1M in the reservoir. ε=λ/h= 0.0073...... 74 Figure 3.13 Shear stress across the channel for monovalent binary electrolytic solution. The channel height is 20nm, the width is 20µm. The surface charge density σ=−0.2C/m2, and the molarity of the solution is 0.1M in the reservoir. ε=λ/h= 0.0073...... 75 Figure 3.14 Dimensional plot of the potential across the channel for monovalent binary electrolytic solution. The channel height is 20nm, the width is 20µm. The surface charge density σ=−0.2C/m2, and the molarity of the solution is 0.1M in the reservoir. ε=λ/h= 0.0073...... 76 Figure 3.15 Dimensional plot of the mobility across the channel for monovalent binary electrolytic solution. The channel height is 20nm, the width is 20µm. σ=−0.2C/m2, and the molarity of the solution is 0.1M in the reservoir. ε=λ/h= 0.0073...... 77 Figure 3.16 Dimensional plot of the shear stress across the channel for monovalent binary electrolytic solution. The channel height is 20nm, the width is 20µm. σ=−0.2C/m2, and the molarity of the solution is 0.1M in the reservoir. ε=λ/h= 0.0073...... 77 Figure 3.17 Mole fractions across the channel for monovalent binary electrolytic solution. The channel height is 4nm, the width is 20µm. The surface charge density σ=−0.2C/m2, and the molarity of the solution is 0.1M in the reservoir. ε=λ/h=0.0331...... 78 Figure 3.18 Potential and velocity profile across the channel for monovalent binary electrolytic solution. The channel height is 4nm, the width is 20µm. The surface charge density σ=−0.2C/m2, and the molarity of the solution is 0.1M in the reservoir. ε=λ/h=0.0331. The straight line is the asymptotic result...... 78 xiii Figure 3.19 Shear stress across the channel for monovalent binary electrolytic solution. The channel height is 4nm, the width is 20µm. The surface charge density σ=−0.2C/m2, and the molarity of the solution is 0.1M in the reservoir. ε=λ/h=0.0331...... 79 Figure 3.20 Mole fractions near the wall for the asymptotic case for monovalent binary electrolyte. The channel heights h=25, 50 and 100 nm. The width of the channel is 100 µm, the surface charge density σ=−0.0154C/m2, and the concentration of electrolytes in the reservoir is 0.002M...... 79 Figure 3.21 Potential and velocity profiles near the wall for the asymptotic case for monovalent binary electrolyte. The channel heights h=25, 50 and 100 nm. The width of the channel is 100 µm, the surface charge density σ=−0.0154C/m2, and the concentration of electrolytes in the reservoir is 0.002M...... 80 Figure 3.22 Shear stress curves near the wall for the asymptotic case for monovalent binary electrolyte. The channel heights h=25, 50 and 100 nm. The width of the channel is 100 µm, the surface charge density σ=−0.0154C/m2, and the concentration of electrolytes in the reservoir is 0.002M...... 80 Figure 3.23 The mole fraction distribution pattern across the channel for monovalent 0 binary electrolyte. γ=0.92. The molarity of cation species c1 =0.154M at the wall. These cases agree with the Debye-Hückel theory of EDL...... 83 Figure 3.24 The potential and velocity profile across the channel for monovalent binary 0 electrolyte. γ=0.92. The molarity of cation species c1 =0.154M at the wall. The asymptotic outer solution φο=−(lnγ)/2 is plotted on the top as the horizontal line. These cases agree with the Debye-Hückel theory of EDL...... 84 Figure 3.25 The shear stress across the channel for monovalent binary electrolyte. 0 γ=0.92, The molarity of cation species c1 =0.154M at the wall. These cases agree with the Debye-Hückel theory of EDL...... 84 Figure 3.26 The mole fraction of ion species across the channel for monovalent binary 0 electrolyte. γ=0.1. The molarity of cation species c1 =0.154M at the wall. The Gouy-Chapman view of EDL is achieved...... 85 Figure 3.27 The potential and velocity profile across the channel for monovalent binary 0 electrolyte. γ=0.1. The molarity of cation species c1 =0.154M at the wall. The asymptotic outer solution φο=−(lnγ)/2 is plotted on the top as the horizontal line. The Gouy-Chapman view of EDL is achieved...... 86 Figure 3.28 The shear stress across the channel for monovalent binary electrolyte. γ=0.1, 0 The molarity of cation species c1 =0.154M at the wall. The Gouy-Chapman view of EDL is achieved...... 86 Figure 3.29 The mole fraction of ion species across the channel for monovalent binary 0 electrolyte. δ=0.1. The molarity of cation species c1 =0.154M at the wall...... 87

xiv Figure 3.30 The potential and velocity profile across the channel for monovalent binary 0 electrolyte. δ=0.1. The molarity of cation species c1 =0.154M at the wall. The asymptotic outer solution φο=−(lnγ)/2≈0.0417 for γ=0.92 is omitted in the plot...... 88 Figure 3.31 The shear stress across the channel for monovalent binary electrolyte. δ=0.1, 0 The molarity of cation species c1 =0.154M at the wall...... 88 Figure 3.32 The mole fraction of ion species across the channel for monovalent binary 0 electrolyte. δ=0.04. The molarity of cation species c1 =0.154M at the wall...... 89 Figure 3.33 The potential and velocity profile across the channel for monovalent binary 0 electrolyte. δ=0.04. The molarity of cation species c1 =0.154M at the wall. The asymptotic outer solution φο=−lnγ/2≈0.0417 for γ=0.92 is omitted in the plot...... 89 Figure 3.34 The shear stress across the channel for monovalent binary electrolyte. 0 δ=0.04, The molarity of cation species c1 =0.154M at the wall...... 90 Figure 3.35 The reservoir and the channels in iMEDDs’ experiments. The iMEDD membrane with nanochannels across it is located in the center of the reservoir. Through the nanochannels, the liquid in one side of the reservoir is driven to the other side by electroosmosis. The thickness of the membrane is 3.5µm. Parallel channels with widths of 44µm and heights in nanometer scale are etched on the membrane by microfabrication. The horizontal distance between two adjacent channels is 2µm...... 91 Figure 3.36 Dimensionless velocity and electric potential inside the channel, the external electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=20nm. ε=λ/h=0.0068 for solution 1, 2, and 3; ε=λ/h=0.0069 for solution 5 and PBS; ε=0.0317 for solution 4...... 96 Figure 3.37 Dimensionless velocity profiles and electric potential inside the channel for solution 1 through 5, the external electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=4nm. ε=λ/h=0.033 for solution 1, 2, 3, 5 and PBS; ε=λ/h=0.021 for solution 4...... 96 Figure 3.38 Mole fractions of Na+ and Cl- inside the channel for solution 1; the electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=20nm, ε=λ/h=0.0068...... 97 Figure 3.39 Mole fractions of Na+, Cl- and K+ inside the channel for solution 2; the electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=20nm, ε=λ/h=0.0068...... 99 + - - Figure 3.40 Mole fractions of Na , Cl and H2PO4 inside the channel for solution 3; the electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=20nm, ε=λ/h=0.0068...... 100 Figure 3.41 Mole fractions of sodium (monovalent cation) and tetraborate (monovalent anion) showing the Debye-Hückel picture of the EDL at small surface charge density. The channel height is 83nm, and its width is 20.3µm. The concentration of xv sodium tetraborate in the reservoir is assumed to be 2mM, and the surface charge density is assumed to be -0.001C/m2...... 100 Figure 3.42 Mole fractions of Na+, Cl- and Ca2+ inside the channel for solution 4; the electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=20nm, ε=λ/h=0.0317...... 101 + - - Figure 3.43 Mole fractions of Na , Cl and HPO4 inside the channel for solution 5; the electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=20nm, ε=λ/h=0.0069...... 101 + + - - 2- Figure 3.44 Mole fractions of Na , K , Cl , H2PO4 and HPO4 inside the channel for PBS. The electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=20nm and ε=λ/h=0.0069...... 102 Figure 3.45 A blown up version of the region near zero mole fraction in Figure 3.44. The electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=20nm and ε=λ/h=0.0069...... 102 Figure 3.46 Mole fractions of Na+ and Cl- inside the channel for solution 1; the electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=4nm, and ε=λ/h=0.0326...... 103 Figure 3.47 Mole fractions of Na+, Cl- and Ca2+ inside the channel for solution 4.The electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=4nm. ε=λ/h=0.021...... 103 Figure 3.48 The shear stress inside the channel for solution 1through 5. The electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=20nm. ε=λ/h=0.0068 for solutions 1, 2 and 3; and ε=λ/h=0.0069 for solution 5 and PBS; ε=λ/h=0.0317 for solution 4...... 104 Figure 3.49 The shear stress inside the channel for solution 1 through 5; the electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=4nm, ε=λ/h=0.033 for solutions 1, 2, 3, 5 and PBS, ε=λ/h=0.021 for solution 4...... 105 Figure 3.50 Dimensionless velocity profiles inside the channel for solution 4. The electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. ε=λ/h=0.021 for h=4nm, ε=0.0317 for h=20nm, ...... 106 Figure 3.51 The shear stress inside the channel for solution 4, with the electric field corresponding to 0.05 volts over a channel of length L=3.5 µm...... 107 Figure 3.52 The molarity of Na+, Ca2+ and Cl- at the channel wall for channels with different heights for solution 4. The external electric field corresponds to 0.05 volts over a channel of length L=3.5 µm...... 108 Figure 3.53 The electroosmotic volume flow rate as a function of channel height at a constant electric field, which corresponds to 0.05 volts over a channel of length

xvi L=3.5 µm. For all three solutions, their ionic strength in the reservoir is 0.33288M and the channel width is 44µ m...... 109 Figure 3.54 The effect of divalent counter-ions on the electroosmotic volume flow rate across the iMEDD membrane; the electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The molarity of charges of the ions in the reservoir is fixed at 0.31668M. The channel height is 20nm. The x-axis is the molarity of Ca2+ in the reservoir; the y-axis is the volume flow rate across the nanochannel-membrane with iMEDD geometry...... 110 Figure 3.55 Mole fractions of monovalent cations, divalent cations and monovalent anions near the wall. The channel heights h=25, 50 and 100 nm. The width of the channel is 100 µm, the surface charge density σ=−0.0154C/m2, and the concentration of electrolytes in the reservoir is 0.002M...... 111 Figure 3.56 Potential and velocity profiles of monovalent cations, divalent cations and monovalent anions near the wall...... 112 Figure 3.57 Shear stress of monovalent cations, divalent cations and monovalent anions near the wall...... 112 Figure 4.1 Nanopore membranes fabricated by iMEDD inc. Arrays of membranes are fabricated on a silicon wafer; on each membrane, there are vertical nanochannel arrays fabricated. The width of the channels is 44µm, and the distance between two arrays is 6µm, as shown in the picture at top right and in Figure 3.35. The bottom picture shows the cross section of the two nanochannels and their adjacent area. The material between the two channels is polysilicon. (http://www.imeddinc.com/technology%20platform.htm, Sept 30, 2003, from Mike Cohen et.al., iMEDD inc.)...... 115 Figure 4.2 The Ussing chamber used in iMEDD experiments...... 116 Figure 4.3 Resistance of the iMEDD 13nm membrane, measured by Jim Coe, et. al. [56] ...... 119 Figure 4.4 The simplified circuit for the iMEDD channel-reservoir system. Inside the membrane, the nanochannels are in parallel with the silicon wall; the membrane is in series with the reservoir. R1, R2 and R3 are the resistance of nanochannels, silicon wall, and all reservoirs, respectively. R12 is the resistance of the membrane. VT is the total voltage drop, in the iMEDD experiments, VT=17 Volts...... 120 Figure 4.5 Voltage drops through the iMEDD nanochannel. The overall voltage drop between the two electrodes is 17 Volts. 0.1xPBS denotes 10 times diluted PBS, and 0.01xPBS denotes 100 times diluted PBS. Results are shown for channel heights of 4, 7, 13, 20 and 27 nm as in the experiments...... 123 Figure 4.6 Comparison of our theoretical flow rates with the iMEDD experimental results. 1xPBS, 0.1xPBS and 0.01xPBS refer to original PBS, 10 folds diluted PBS and 100 folds diluted PBS, in respective...... 124

xvii Figure 4.7 Dimensionless potential and velocity profile across an ORNL 83nm channel. The buffer concentration in the reservoir varies from 0.2mM to 20mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0161 for 0.2mM buffer, ε=0.0158 for 2mM buffer, and ε=0.0133 for 20mM buffer...... 128 Figure 4.8 A Dimensionless plot of mole fractions across an ORNL 83nm channel. The buffer concentration in the reservoir varies from 2mM to 20mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0161 for 0.2mM buffer, ε=0.0158 for 2mM buffer, and ε=0.0133 for 20mM buffer...... 131 Figure 4.9 Dimensionless shear stress across an ORNL 83nm channel. The buffer concentration in the reservoir varies from 0.2mM to 20mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0161 for 0.2mM buffer, ε=0.0158 for 2mM buffer, and ε=0.0133 for 20mM buffer...... 131 Figure 4.10 Asymptotic results of the potential and velocity profile near the wall. The channel here is an ORNL 83nm-high channel. The buffer concentrations in the reservoir are 2mM and 20mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0147 for 2mM buffer, and ε=0.0126 for 20mM buffer. Yɍ=30...... 132 Figure 4.11 Asymptotic results of the mole fractions near the wall. The channel here is an ORNL 83nm-high channel. The buffer concentrations in the reservoir are 2mM and 20mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0147 for 2mM buffer, and ε=0.0126 for 20mM buffer. Yɍ=30...... 133 Figure 4.12 Asymptotic results of the shear stress near the wall. The channel here is an ORNL 83nm-high channel. The buffer concentrations in the reservoir are 2mM and 20mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0147 for 2mM buffer, and ε=0.0126 for 20mM buffer. Yɍ=30...... 133 Figure 4.13 Asymptotic results of the potential and velocity profile near the wall. The channel here is an ORNL 83nm-high channel. The buffer concentrations in the reservoir are 20mM and 150mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0125 for 20mM buffer, and ε=0.0072 for 150mM buffer. Yɍ=10...... 134 Figure 4.14 Asymptotic results of the mole fractions near the wall. The channel here is an ORNL 83nm-high channel. The buffer concentrations in the reservoir are 20mM and 150mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0125 for 20mM buffer, and ε=0.0072 for 150mM buffer. Yɍ=10...... 135 Figure 4.15 Asymptotic results of the shear stress near the wall. The channel here is an ORNL 83nm-high channel. The buffer concentrations in the reservoir are 20mM and

xviii 150mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0125 for 20mM buffer, and ε=0.0072 for 150mM buffer. Yɍ=10...... 135 Figure 4.16 The mole fractions of Na+ across the channel for the ORNL experiments. The buffer concentration in the reservoir is 2mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0158 for 83nm high channel, and ε=0.0050 for 290nm high channel...... 137 - Figure 4.17 The mole fractions of B(OH)4 across the channel for the ORNL experiments. The buffer concentration in the reservoir is 2mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0158 for 83nm high channel, and ε=0.0050 for 290nm high channel...... 138 Figure 4.18 The dimensionless potential and velocity profile across the channel for the ORNL experiments. The buffer concentration in the reservoir is 2mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0158 for 83nm high channel, and ε=0.0050 for 290nm high channel...... 138 Figure 4.19 The dimensionless shear stress across the channel for the ORNL experiments. The buffer concentration in the reservoir is 2mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0158 for 83nm high channel, and ε=0.0050 for 290nm high channel...... 139 Figure 4.20 Dimensional results of the electric potential across the channel for the ORNL experiments. The channel here is an ORNL 83nm-high channel. The buffer concentration in the reservoir is 2mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0158...... 139 Figure 4.21 Dimensional electric mobility of the fluid across the channel for the ORNL experiments. The channel here is an ORNL 83nm-high channel. The buffer concentration in the reservoir is 2mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0158...... 140 Figure 4.22 Dimensional results of the shear stress across the channel for the ORNL experiments. The channel here is an ORNL 83nm-high channel. The buffer concentration in the reservoir is 2mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0158...... 140 Figure 4.23 Dimensional results of the electric potential near the wall. The channel here is an ORNL 83nm-high channel. The buffer concentration in the reservoir is 20mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0125. Yɍ=10...... 141 Figure 4.24 Dimensional results of the mobility profile near the wall. The channel here is an ORNL 83nm-high channel. The buffer concentration in the reservoir is 20mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0125. Yɍ=10...... 141 xix Figure 4.25 Dimensional results of the shear stress near the wall. The channel here is an ORNL 83nm-high channel. The buffer concentration in the reservoir is 20mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0125. Yɍ=10...... 142 Figure 4.26 Comparison between the mobilities measured in ORNL experiments and the results from the model. The mobilities are drawn against the channel heights, which vary form 83nm to 1080nm, as shown in Table 4.5...... 146 Figure 4.27 Comparison between the mobilities measured in ORNL experiments and the results from the model. The molarity in the reservoir varies from 0.02mM to 150mM...... 147 Figure 4.28 The picture (left) and sketch (right) of the crossing microchannel chip used in the Georgia tech experiments, the region of interest (ROI) is indicated in the sketch plot; and the direction of flow is from top to bottom in the sketch plot, as shown by the arrow. (M. Yoda et. al., 2002) ...... 148 Figure 4.29 Comparison of the mobility measured in the Georgia Tech experiments and the average mobility calculated from the asymptotic model...... 150 Figure 4.30 The relations between factor γ and the ionicstrength of the solution in the µ 0 2 reservoir. The mobility coefficient m =0.00037cm /(V·s) can be calculated from the coefficient 3.7 in the power fit curve of ln(1/γ) vs. I...... 151 Figure 4.31 Simulated mole fractions of sodium ions and tetraborate ions for the Georgia Tech experiments. The channel height is 4.8µm, and the channel width is 17.9µm. The concentration of the sodium tetraborate solution in the reservoir is 18.4mM. The 2 surface charge density σ=−0.0045C/m . ε=0.000412, δ=0.000460, γ=0.243, Yɍ=10...... 152 Figure 4.32 Simulated potential and velocity profile for the Georgia Tech experiments. The channel height is 4.8µm, and the channel width is 17.9µm. The concentration of the sodium tetraborate solution in the reservoir is 18.4mM. The surface charge 2 density σ=−0.0045C/m . ε=0.000412, δ=0.000460, γ=0.243, Yɍ=10...... 153 Figure 4.33 Simulated shear stress for the Georgia Tech experiments. The channel height is 4.8µm, and the channel width is 17.9µm. The concentration of the sodium tetraborate solution in the reservoir is 18.4mM. The surface charge density 2 σ=−0.0045C/m . ε=0.000412, δ=0.000460, γ=0.243, Yɍ=10...... 153 Figure 5.1 A sketch figure of the electrophoresis in a channel. For convenience, the size of ions is exaggerated, and only one cation and one anion are shown in the figure. Cations are driven to the cathode by the electric field, and anions are driven to the anode. If the channel wall is negatively charged, the direction of electroosmotic flow is toward the cathode, as shown in the profile...... 157

xx Figure 5.2 The finite difference view of function f(x,y). For the electrophoresis problems discussed in this chapter, x direction is along the length of the channel, while y direction is along the height of the channel. The variations of f on z direction, which represents the width, is neglect, because h<

xxiii 0 0 c1 =0.154M, wall molarity of anion c2 =0.137M, molarity of cations or anions in the R R reservoir c1 =c2 =0.145M. ε=0.200, ξ=0.031, ε1=1/900...... 189 Figure 5.29 The mole fraction of anions across the channel. Channel length L=3.6µm, channel height h=4nm, voltage drop V=0.05Volts, wall molarity of cation 0 0 c1 =0.154M, wall molarity of anion c2 =0.137M, molarity of cations or anions in the R R reservoir c1 =c2 =0.145M. ε=0.200, ξ=0.031, ε1=1/900...... 190 Figure 5.30 Dimensionless electrical potential across the channel. Channel length L=3.6µm, channel height h=4nm, voltage drop V=0.05Volts, wall molarity of cation 0 0 c1 =0.154M, wall molarity of anion c2 =0.137M, molarity of cations or anions in the R R reservoir c1 =c2 =0.145M. ε=0.200, ξ=0.031, ε1=1/900...... 191 Figure 5.31 Dimensionless velocity profile in the channel. Channel length L=3.6µm, channel height h=4nm, voltage drop V=0.05Volts, wall molarity of cation 0 0 c1 =0.154M, wall molarity of anion c2 =0.137M, molarity of cations or anions in the R R reservoir c1 =c2 =0.145M. ε=0.200, ξ=0.031, ε1=1/900...... 192 Figure 5.32 Shear stress in the x direction. Channel length L=3.6µm, channel height 0 h=4nm, voltage drop V=0.05Volts, wall molarity of cation c1 =0.154M, wall 0 molarity of anion c2 =0.137M, molarity of cations or anions in the reservoir R R c1 =c2 =0.145M. ε=0.200, ξ=0.031, ε1=1/900...... 193 Figure 5.33 Diffusive fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y (bottom) directions. Channel length L=3.6µm, channel height h=4nm, voltage drop 0 V=0.05Volts, wall molarity of cation c1 =0.154M, wall molarity of anion 0 R R c2 =0.137M, molarity of cations or anions in the reservoir c1 =c2 =0.145M. ε=0.200, ξ=0.031, ε1=1/900. The spikes are caused by the discontinuity of mole fractions at the corners...... 194 Figure 5.34 Electric field strength in the x (left) and y (right) directions. L=3.6µm, 0 h=4nm, voltage drop V=0.05Volts, wall molarity of cations c1 =0.154M, wall 0 molarity of anions c2 =0.137M, molarity of cations or anions in the reservoir R R c1 =c2 =0.145M. ε=0.200, ξ=0.031, ε1=1/900...... 195 Figure 5.35 Electric migrational fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y (bottom) directions. L=3.6µm, h=4nm, voltage drop V=0.05Volts, wall 0 0 molarity of cations c1 =0.154M, wall molarity of anions c2 =0.137M, molarity of R R cations or anions in the reservoir c1 =c2 =0.145M. ε=0.200, ξ=0.031, ε1=1/900.. 196 Figure 5.36 Convective fluxes of Na+ (left side) and Cl- (right side) in the x direction. 0 L=3.6µm, h=4nm, voltage drop V=0.05Volts, wall molarity of cation c1 =0.154M, 0 wall molarity of anion c2 =0.137M, molarity of cations or anions in the reservoir R R c1 =c2 =0.145M. ε=0.200, ξ=0.031, ε1=1/900...... 197 Figure 5.37 Entire fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y (bottom) directions. L=3.6µm, h=4nm, voltage drop V=0.05Volts, wall molarity of 0 0 cation c1 =0.154M, wall molarity of anion c2 =0.137M, molarity of cations or anions xxiv R R in the reservoir c1 =c2 =0.145M. ε=0.200, ξ=0.031, ε1=1/900. The spikes are caused by the discontinuity of mole fractions and potential at the corners...... 198 Figure 5.38 Mole fraction of Na+ (upper left), Mole fraction of Cl- (upper right), potential (middle left), velocity (middle right), and shear stress (bottom left) in the channel. Channel length L=36nm, channel height h=20nm, voltage drop V=0.05Volts, wall 0 0 molarity of cation c1 =0.154M, wall molarity of anion c2 =0.137M, molarity of R R cations or anions in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/1.8. The color represents the magnitude of the function...... 200 Figure 5.39 Diffusive fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y (bottom) directions. Channel length L=36nm, channel height h=20nm, voltage drop 0 V=0.05Volts, wall molarity of cations c1 =0.154M, wall molarity of anions 0 R R c2 =0.137M, molarity of cations or anions in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/1.8...... 201 Figure 5.40 Electric field strength in the x (left) and y (right) directions. Channel length L=36nm, channel height h=20nm, voltage drop V=0.05Volts, wall molarity of 0 0 cations c1 =0.154M, wall molarity of anions c2 =0.137M, molarity of cations or R R anions in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/1.8...... 202 Figure 5.41 Electric migrational fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y (bottom) directions. L=36nm, h=20nm, voltage drop V=0.05Volts, wall 0 0 molarity of cations c1 =0.154M, wall molarity of anions c2 =0.137M, molarity of R R cations or anions in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/1.8... 203 Figure 5.42 Convective fluxes of Na+ (left side) and Cl- (right side) in the x direction. 0 L=36nm, h=20nm, voltage drop V=0.05Volts, wall molarity of cations c1 =0.154M, 0 wall molarity of anions c2 =0.137M, molarity of cations or anions in the reservoir R R c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/1.8...... 204 Figure 5.43 Entire fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y (bottom) directions. Channel length L=36nm, channel height h=20nm, voltage drop 0 V=0.05Volts, wall molarity of cations c1 =0.154M, wall molarity of anions 0 R R c2 =0.137M, molarity of cations or anions in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/1.8...... 205 Figure 5.44 Mole fraction of Na+ (upper left), Mole fraction of Cl- (upper right), potential (middle left), velocity (middle right), and shear stress (bottom left) in the channel. Channel length L=36nm, channel height h=20nm, voltage drop V=0.0005Volts, wall 0 0 molarity of cations c1 =0.154M, wall molarity of anions c2 =0.137M, molarity of R R cations or anions in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=3.1, ε1=1/1.8...... 206 Figure 5.45 Diffusive fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y (bottom) directions. L=36nm, h=20nm, voltage drop V=0.0005Volts, wall molarity 0 0 of cations c1 =0.154M, wall molarity of anions c2 =0.137M, molarity of cations or R R anions in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=3.1, ε1=1/1.8...... 207

xxv Figure 5.46 Electric field strength in the x (left) and y (right) directions. L=36nm, 0 h=20nm, voltage drop V=0.0005Volts, wall molarity of cations c1 =0.154M, wall 0 molarity of anions c2 =0.137M, molarity of cations or anions in the reservoir R R c1 =c2 =0.145M. ε=0.040, ξ=3.1, ε1=1/1.8...... 208 Figure 5.47 Electric migrational fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y (bottom) directions. L=36nm, h=20nm, voltage drop V=0.0005Volts, wall 0 0 molarity of cations c1 =0.154M, wall molarity of anions c2 =0.137M, molarity of R R cations or anions in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=3.1, ε1=1/1.8...... 209 Figure 5.48 Convective fluxes of Na+ (left side) and Cl- (right side) in the x direction. L=36nm, h=20nm, voltage drop V=0.0005Volts, wall molarity of cations 0 0 c1 =0.154M, wall molarity of anions c2 =0.137M, molarity of cations or anions in R R the reservoir c1 =c2 =0.145M. ε=0.040, ξ=3.1, ε1=1/1.8...... 210 Figure 5.49 Entire fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y (bottom) directions. Channel length L=36nm, channel height h=20nm, voltage drop 0 V=0.0005Volts, wall molarity of cations c1 =0.154M, wall molarity of anions 0 R R c2 =0.137M, molarity of cations or anions in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=3.1, ε1=1/1.8...... 211 Figure 5.50 Mole fraction of Na+ (upper left), Mole fraction of Cl- (upper right), potential (middle left), velocity (middle right), and shear stress (bottom left) in the channel. Channel length L=3.6µm, channel height h=20nm, voltage drop V=5Volts, wall 0 0 molarity of cations c1 =0.154M, wall molarity of anions c2 =0.137M, molarity of R R cations or anions in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.00031, ε1=1/180...... 213 Figure 5.51 Diffusive fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y (bottom) directions. L=3.6µm, h=20nm, voltage drop V=5Volts, wall molarity of 0 0 cations c1 =0.154M, wall molarity of anions c2 =0.137M, molarity of cations or R R anions in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.00031, ε1=1/180...... 214 Figure 5.52 Electric field strength in the x (left) and y (right) directions. Channel length L=3.6µm, channel height h=20nm, voltage drop V=5Volts, wall molarity of cations 0 0 c1 =0.154M, wall molarity of anions c2 =0.137M, molarity of cations or anions in R R the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.00031, ε1=1/180...... 215 Figure 5.53 Electric migrational fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y (bottom) directions. Channel length L=3.6µm, channel height h=20nm, 0 voltage drop V=5Volts, wall molarity of cations c1 =0.154M, wall molarity of 0 R R anions c2 =0.137M, molarity of cations or anions in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.00031, ε1=1/180...... 216 Figure 5.54 Convective fluxes of Na+ (left side) and Cl- (right side) in the x direction. Channel length L=3.6µm, channel height h=20nm, voltage drop V=5Volts, wall 0 0 molarity of cations c1 =0.154M, wall molarity of anions c2 =0.137M, molarity of

xxvi R R cations or anions in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.00031, ε1=1/180...... 217 Figure 5.55 Entire fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y (bottom) directions. Channel length L=3.6µm, channel height h=20nm, voltage drop 0 V=5Volts, wall molarity of cations c1 =0.154M, wall molarity of anions 0 R R c2 =0.137M, molarity of cations or anions in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.00031, ε1=1/180...... 218 Figure 6.1 Rectangular channel where h/W=O(1), and h<

xxvii 0 anion species: c2 =0.142M. The walls are at zero potential, and no-slip condition is applied. ε=0.200, ε2=1...... 234 Figure 6.10 Mole fractions of ions for monovalent binary electrolyte solution. W=20nm, 0 h=4nm, wall molarity of cation species: c1 =0.154M, wall molarity of anion species: 0 c2 =0.142M. The walls are at zero potential, and no-slip condition is applied. ε=0.200, ε2=0.2...... 235 Figure 6.11 Dimensionless electrical potential and velocity profile for monovalent binary 0 electrolyte solution. W=20nm, h=4nm, wall molarity of cation species: c1 =0.154M, 0 wall molarity of anion species: c2 =0.142M. The walls are at zero potential, and no- slip condition is applied. ε=0.200, ε2=0.2...... 236 Figure 6.12 Dimensionless shear stress τyx for monovalent binary electrolyte solution. 0 W=20nm, h=4nm, wall molarity of cation species: c1 =0.154M, wall molarity of 0 anion species: c2 =0.142M. The walls are at zero potential, and no-slip condition is applied. ε=0.200, ε2=0.2...... 237 Figure 6.13 Dimensionless shear stress τzx for monovalent binary electrolyte solution. 0 W=20nm, h=4nm, wall molarity of cation species: c1 =0.154M, wall molarity of 0 anion species: c2 =0.142M. The walls are at zero potential, and no-slip condition is applied. ε=0.200, ε2=0.2...... 238 Figure 6.14 Mole fractions of ions for monovalent binary electrolyte solution. W=20nm, 1 2 3 h=20nm, wall molarity of cation species: c1 =0.154M, c1 =0.160M, c1 =0.166M, 4 1 2 3 c1 =0.143M; wall molarity of anion species: c2 =0.142M, c2 =0.137M, c2 =0.131M, 4 1 2 3 c2 =0.153M. The potentials on the side walls are φ =0, φ =0.001V, φ =0.002V, 4 φ =−0.002V, and no-slip condition is applied. ε=0.200, ε2=1...... 241 Figure 6.15 The net charge (X1-X2) inside the channel for monovalent binary electrolyte 1 solution. W=20nm, h=20nm, wall molarity of cation species: c1 =0.154M, 2 3 4 1 c1 =0.160M, c1 =0.166M, c1 =0.143M; wall molarity of anion species: c2 =0.142M, 2 3 4 1 c2 =0.137M, c2 =0.131M, c2 =0.153M. The potentials on the side walls are φ =0, 2 3 4 φ =0.001V, φ =0.002V, φ =−0.002V, and no-slip condition is applied. ε=0.200, ε2=1...... 242 Figure 6.16 Dimensionless potential for monovalent binary electrolyte solution. 1 2 W=20nm, h=20nm, wall molarity of cation species: c1 =0.154M, c1 =0.160M, 3 4 1 2 c1 =0.166M, c1 =0.143M; wall molarity of anion species: c2 =0.142M, c2 =0.137M, 3 4 1 2 c2 =0.131M, c2 =0.153M. The potentials on the side walls are φ =0, φ =0.001V, 3 4 φ =0.002V, φ =−0.002V, and no-slip condition is applied. ε=0.200, ε2=1...... 243 Figure 6.17 A dimensionless velocity profile for monovalent binary electrolyte solution. 1 2 W=20nm, h=20nm, wall molarity of cation species: c1 =0.154M, c1 =0.160M, 3 4 1 2 c1 =0.166M, c1 =0.143M; wall molarity of anion species: c2 =0.142M, c2 =0.137M, 3 4 1 2 c2 =0.131M, c2 =0.153M. The potentials on the side walls are φ =0, φ =0.001V, 3 4 φ =0.002V, φ =−0.002V, and no-slip condition is applied. ε=0.200, ε2=1...... 244 xxviii Figure 6.18 Dimensionless shear stress τyx for monovalent binary electrolyte solution. 1 2 W=20nm, h=20nm, wall molarity of cation species: c1 =0.154M, c1 =0.160M, 3 4 1 2 c1 =0.166M, c1 =0.143M; wall molarity of anion species: c2 =0.142M, c2 =0.137M, 3 4 1 2 c2 =0.131M, c2 =0.153M. The potentials on the side walls are φ =0, φ =0.001V, 3 4 φ =0.002V, φ =−0.002V, and no-slip condition is applied. ε=0.200, ε2=1...... 245 Figure 6.19 Dimensionless shear stress τzx for monovalent binary electrolyte solution. 1 2 W=20nm, h=20nm, wall molarity of cation species: c1 =0.154M, c1 =0.160M, 3 4 1 2 c1 =0.166M, c1 =0.143M; wall molarity of anion species: c2 =0.142M, c2 =0.137M, 3 4 1 2 c2 =0.131M, c2 =0.153M. The potentials on the side walls are φ =0, φ =0.001V, 3 4 φ =0.002V, φ =−0.002V, and no-slip condition is applied. ε=0.200, ε2=1...... 246 Figure 8.1 A spacefill model of the human serum albumin, from the protein data bank (database code: 1AO6). oxygen, nitrogen, carbon and sulfur atoms are shown as red, blue, charcoal and yellow balls, hydrogen atoms are not shown...... 254 Figure 8.2 A Wireframe model of a duplex DNA segment CGCAAATTGGCG, from the protein data bank, (database code:111D). Carbon, oxygen, nitrogen and phosphorus atoms are represented by charcoal, red, blue and yellow wires...... 255

xxix CHAPTER 1

INTRODUCTION

1.1 Background

Miniaturization has been one of the swiftest revolutions in the scientific and industrial world during last century. The term “microfluidics” was invented about 30 years ago when microfabricated fluid systems were developed at Stanford (gas chromatography) and at IBM (ink jet printer nozzles) [1]. In recent years, the advances in microelectromechanical systems, aka MEMS, have led to the development of even smaller devices, nanoelectromechanical systems, or NEMS. An example of such a device is depicted in Figure 1.1.

The name MEMS, was officially adopted at a meeting in Salt Lake City in

1989.[2] Originally, the term MEMS referred to very small systems which have moving parts and built-in electronics. In Recent years, the concept of MEMS has been expanded to cover many other types of small things, including thermal, magnetic, fluidic and optical devices and systems.[3] Furthermore, the perhaps most important field of MEMS,

BioMEMS, has attracted much attention in the beginning of the new millennium.

Microfluidics, the study of fluid flow in micrometer sized devices, is one of the disciplines on which the operation of MEMS depends. In the last decade, many 1 microfluidics devices have been fabricated, including micropumps [4] [5], microvalves

[5], micro flow sensors [2] [5] - [7], microactuators [2] [8]. These components have been integrated into microfluidics systems [9] - [14], for instance, micro total analysis systems

(µ−TAS) [15].

Figure 1.1 NanoDrop® ND-1000 spectrophotometer. This new product of NanoDrop® technologies is a UV/VIS spectrophotometer for one microliter samples. The sample is directly pipetted onto the measurement surface and wiped out after the measurement, which eliminates the need of cuvettes or capillaries. (http://www.nanodrop.com/index.shtml, 06/24/2003)

One example of the µ-TAS, enzyme assay development microchip, is provided by

Cohen, et. al. [16] A scheme on the left side of Figure 1.2 illustrates a simple microfluidic system for enzyme assays. The width of the separation channel and all other channels is 70µm, and the depth is 15µm. Electroosmotic flow is generated from reservoir 2 to reservoir 8 in the first step, then from reservoir 1 to reservoir 7 in the second step. The detection is performed at the bottom of the separation channel. Another

2 design, which is shown on the right side of Figure 1.2, is capable of carrying on-chip dilution of the reagent (substrate) in reservoir 1 by varying the electric current passing through reservoir 1 and 8. An incubation channel is used between reservoir 3 and reservoirs 1, 2, 7 and 8.

Figure 1.2 Schematic of assay development microchips designed by Cohen et. al. [16]. Reservoirs are numbered and labeled with reagent solutions.

The analytical MEMS devices as shown in Figure 1.2 have been referred to as a

“Lab on a chip”. About ten years ago, “Lab on a Chip” was still a concept, but today there exist many products performing a wide array of tasks, such as DNA sequencing, protein analysis, drug delivery, etc. These devices are used in hospitals, research institutes as well as in industry.

Figure 1.3 Image from Gene-Chips (Microarray) The intensity and color of each spot encode information on a specific gene from the tested sample. (http://www.gene- chips.com/sample1.html, 06/24/2003, Authorized by Leming Shi.)

Figure 1.3 shows an image produced by a Gene-Chip, the color of each dot contains information about the corresponding gene tested. “Lab on a Chip” is small, easy to carry; it uses a small amount of sample; it is cheaper and can be largely produced; the operation is easier, which also minimizes the possibility of operational mistakes.

In this work, we study the transport of biofluids through micro and nanochannels, analyze aqueous solutions with a number of electrolytes, and calculate the electric field, flow field and ion distribution. We will also compare the results with experimental data

4 from several institutes. The other topics this work covers are: electrophoresis, transport of biomolecules, unsteady flow in nanochannels, and the applications of this research in

MEMS.

1.2 The Basics of Electroosmosis

Operation of a “Lab on a chip” requires fluid transport through micro or nanochannels. As we will see, transporting fluids by the imposition of an electric field is an effective method of accomplishing this task.

Electroosmosis is the movement of solvent together with solute under an applied electric field. The direction and flow rate of electroosmotic flow is determined by many factors, such as the electric field strength, the concentration of electrolytes, the surface charge density on the inner surface of the channel or capillary, temperature, pressure, viscosity etc.

A typical electroosmotic flow system is shown in Figure 1.4. The inner surface of the channel wall is assumed negatively charged. The surface charge is positive or neutral for some materials and the polarity and intensity of the surface charge depends on the material and the environment such as pH and temperature. This will be discussed in §1.3.

In the case shown in Figure 1.4, counter-ions (ions have opposite charges to the surface) are cations, and co-ions (ions have same charges as the surface) are anions. Anions are expelled away from the surface by interaction with the negative surface charges, while cations are attracted to the surface by interaction with the surface charges. Therefore, cations are adsorbed and accumulate on the inner surface of the channel, while anions are driven out of this region. Usually, the strongly adsorbed cations cannot fully offset the

5 surface charges, so there are some loosely adsorbed cations exist outside them, and the total surface charge is fully balanced by the cations at certain distance.

Channel Wall

Solvent + Anode Cathode

Cation Anion Surface charges

Figure 1.4 A schematic figure showing the electroosmotic flow in a negatively charged channel with thin EDL. The real sizes of ions are magnified relative to the size of the channel. Instead of showing directly as molecules, the solvent are represent by the yellowish colored area.

As the result, two layers of cations are formed upon the layer of negative surface charges. This structure of two layers containing more counter ions than co-ions is called the electrical double layer, or EDL. Because the electrical double layer model was first introduced by Stern (1924) by combining the Helmholtz and the Gouy-Chapman models,

6 the EDL is also called Stern layer. For channels having negatively charges on surfaces, the solution inside the channel flows toward the cathode, which is known as electroosmosis. The velocity profile of a fully developed electroosmotic flow (EOF) in channels with thin EDL is shown in Figure 1.4.

Figure 1.5 Counterions accumulate near the charged surface, while co-ions are depleted near the charged surface. In this plot, ρ∞ is the electrolyte concentration in the bulk [17].

Figure 1.5 also shows the concentration difference between counterions and co- ions near the interface. This concentration difference results in the building up of a new 7 electric field. Figure 1.6 shows the electrical double layer and the potential drop in this field.

Figure 1.6 A schematic representation showing EDL and potential drop across the EDL. (http://www.geocities.com/CapeCanaveral/Hangar/5555/zeta.htm, 06/28/03)

The electric potential in the solution drops exponentially with the distance from the interface. Zeta potential (ζ) is the electrical potential drop between the interface and the bulk solution outside the electric double layer. Zeta potential is mainly determined by the surface charge density and the dielectric constant of the solution. Because zeta

8 potential can be measured experimentally, it is widely used to represent the electrical factors of the double layer.

For micro- or nanotechnologies, such as capillary electrophoresis, electroosmotic effects can be significant. The flow rate of electroosmotic flow is

Qe ~U 0 hW (1.1) and the flow rate of pressure driven flow is

Wh 3 Q ~∆P (1.2) p µL where L, W and h are the length, width and height of the channel, respectively; and h<

From equations (1.1) and (1.2), the electroosmotic flow rate is proportional to the channel height, while the pressure driven flow rate is proportional to the cube of channel height. Therefore, as channel height decreases, the electroosmotic flow rate drops much slower than the pressure driven flow rate. In another words, to maintain the same flow rate at very small channel height relative to large channel height, electroosmotic flow requires a much less increase on inputting energy than pressure driven flow does.

Figure 1.7 shows the dependence of pressure drop and applied voltage on the channel height. The flow rate remains constant at 1µl/min. As the channel height decreases from 70nm to 10nm, the pressure drop increases tremendously, however the applied voltage in EOF does not increase much, for 10nm high channel, the applied voltage is still less than 0.25Volts.

In microfluidics and many applications, µl/min is a typical unit of the flow rate.

9 4 Pressure Drop(Atm) Applied Potential(Volts) 3.5

2.5

1.5

0.5 Pressure Drop(Atm) and Applied Potential(Volts) Applied and Drop(Atm) Pressure

0 1 2 3 4 5 6 7 8 Channel Height -8 x 10

Figure 1.7 Comparison of the pressure drop in pressure driving flow and the applied voltage in EOF. The flow rate is 1µl/min; the unit of channel height is meters.

1.3 Properties of Channel Surfaces

From the discussion in last section, we know that surface charges on the channel are required for the formation of the EDL. The density of surface charges, the distribution pattern of surface charges, the surface roughness, and chemical composition of the surfaces, all account for the influences that the channel wall exerts on the EDL. In this work, all the channels being discussed are silicon channels because most nanochannels being used today are silicon channels; this fact is due to several reasons. First of all, silicon and its compound such as silicon dioxide have excellent mechanical and electric properties. Secondly, much information of silicon has been accumulated during the development of semiconductor industry. Thirdly, silicon material is cheap [2]. For silicon 10 channels, the surface smoothness and charge distribution depend largely on the protocol and the environment of the fabrication. In this work, we assume that the channel surfaces are ideally smooth, and the surface charges are distributed equally on every surface area.

Statistically, this assumption is valid for real channels, unless the channel height is very small. From Figure 1.8, we can see the roughness on the nanochannel surfaces.

Nevertheless, the surfaces are still “smooth”, relative to the size of the channel. Finally, the only surface property we must specify is the surface charge density.

Figure 1.8 SEM pictures of a 20nm high nanochannel fabricated on a 3.5µm thick silicon membrane. The direction of flow is from the outside down to the inside of the page or vice versa. (Hansford, D.J., private communication 1999).

11 The surface of silicon channel consists of silanol groups (SiOH). The silicon atom in SiOH is covalently bonded with the oxygen atom and other silicon atoms either on the surface or under the surface; on the other hand, the hydroxide group (OH) sticks out of the surface. If the silicon surfaces are put into water or an aqueous solution, silanol

+ groups will react with water (H2O), and the products will be deprotonated (removing H )

- + silanol groups (SiO ) and hydronium (H3O ).

+ ↔ − + + SiOH H 2O SiO H 3O (1.3)

The isoelectric point of silanol group in aqueous solution is pI = 2.4, which means at pH2.4, the reaction shown in equation (1.3) is in equilibrium. If the pH of the solution is higher than 2.4, the process of deprotonating SiOH will overwhelm the protonation of SiO-. If the pH of solution is higher than 4, almost all SiOH are deprotonated, thus the silicon surface is highly negatively charged, and the charge density, which is the density of deprotonated silanol groups, approaches its maximum limit, and can be treated as a constant.

The pH of pure water is 7, the pH of most physiological fluids, such as blood serum and saline, is 7.4. Under these conditions, the channel surfaces are all negatively charged, and the surface charge density is a constant.

Experimentally, the surface charge density σ depends on the deposition and cleaning condition during microfabrication. From Israelachvili [17], for pH higher than 4, the suggested value of σ for silicon surfaces is 0.2 Coulombs per meter square for aqueous solution.

12 1.4 Previous Work on Modeling Electroosmotic Flow

Essentially electroosmotic flow is mass transfer due to an external electrical potential coupled with classical Fickian diffusion. The system under consideration is an aqueous solution of strong electrolytes, in which all the solutes are entirely dissociated into cations and anions. If the mixture does not dissociate entirely, the two most common mass transfer forms are Fickian diffusion and convection [18]. Undissociated high concentration salt solutions have been studied by Conlisk et. al. [19] - [22]. For the case of dissociated ions, if the walls are charged, a relatively low imposed electric field can generate significant volume flow rates at very low voltages; this phenomenon is known as electroosmosis.

The earliest models of the EDL were given by Helmholtz [23], then by Debye and

Hückel [24]. They all assumed that the EDL’s influences to cation and anion species are opposite and equal in strength. The Gouy-Chapman model came out later and allowed more counterions to bind to the surface charges, which explained the accumulation of counterions near the interface [25]. However, both models consider only a single surface and assume there is no flow. Verwey and Overbeek discussed Debye-Hückel and Gouy-

Chapman theories in their book [26].

The first work on electroosmotic flow problem appears to have been done by

Burgeen and Nakache [27]. They considered channel heights in the order of the double layer thickness, by making an assumption that the number density of ions in the solution follows the Boltzmann distribution. Then Rice and Whitehead [28] gave an analysis of electroosmotic flow in round capillaries. Levine et. al. advanced theories for both types of electroosmotic flow: in cylindrical capillaries [29] and in narrow parallel-plate 13 channels [29]. Additional work has been done by other researchers [30] - [32]. Now there are several textbooks in which the classical theory of electroosmotic flow field is discussed [30] - [33].

Barcilon et. al. [34] employed singular perturbation and numerical simulation methods to solve the Poisson-Nernst-Planck equation system for the ion-channel problem. They compared between the results of simulation and singular perturbation

(singular perturbation is a method to solve differential equations if there is a small parameter appears at the highest derivative [35]), and found that they were in closed form. They also found that the quotient of the Debye length and the characteristic length scales, served as the singular perturbation parameter.

Qu and Li [36] consider channels with overlapped double layers. They established a model for overlapped EDL fields between two infinitely large flat plates. Both the electrical potential and ionic concentration distribution were included in their model.

They claimed that the classical theory may lead to an inaccurate description of EOF due to the misuse of Boltzmann equation. Then they set up new governing equations for the overlapped EDL field. In comparison of their model with the results of the classical theory, they found differences especially at small channel height.

Conlisk et. al. [37] consider the case of ion species Na+ and Cl- for channel heights varying from micrometer scale to nanometer scale. They calculate results for velocity, potential and solute mole fractions. At lower electrolyte concentrations, the

Debye-Hückel picture of the electric double layer was recovered, while the Gouy-

Chapman picture of EDL arose naturally as the concentration increases. Both symmetric and asymmetric geometries were investigated, and they found that the velocity field and

14 potential field were similar in the symmetric case. Excellent agreement between the numerical results and the analytical singular perturbation results is observed and compares well with experimental data. Conlisk et. al. [37] also consider the case where a potential difference across the channel is imposed corresponding to, in one case, oppositely charged walls.

The case of multivalent ions has been investigated by Friedl et. al. [38] who develop an empirical model to calculate the effective mobilities of multivalent ions in capillary electrophoresis. They perform a number of experiments and from the data they develop an empirical formula of mobility for a number of different acidic mixtures.

Gavryushov et. al. [39] compute the potential due to a pair of 1:1 and 1:2 ions around a cylindrical polyion. They solve a nonlinear Boltzmann-like equation that includes an excluded volume effect. They are able to validate their model against data for the case of a 2:2 electrolyte solution. They find that there are slight differences in the results for the

1:2 electrolytic system depending on whether the Poisson-Boltzmann equation is used for the ion distribution. The differences are especially large at high molarity as expected.

Vlachy [40] finds that the co-ion concentration in the capillary may become lower than its concentration in the reservoir for the 3:3 electrolytes, if the surface charge density on the capillary wall is high enough and the electrolyte concentration in the reservoir is low enough. This situation is unusual because in the standard case co-ions, in the case of a negatively charged wall, the cations would be drawn in great numbers to the surface of the capillary.

It also has been noted in the literature that the Poisson-Boltzmann description of the electric double layer may not be appropriate for solutions containing multivalent ionic

15 species. Adamczyk et. al. [41] [42] discuss this possibility and note that corrections to the

Poisson-Boltzmann description are most important at high electrical field strength (~106

V/cm) and high ionic strength (~ 1M). Cuvillier and Rondelez [43] measure the EDL profile near a positively charged Langmuir monolayer. They find that the trivalent anions are drawn to the surface in far greater quantities, which is necessary to preserve electroneutrality, and thus the picture of the EDL is much different from that of Poisson-

Boltzmann description. This result is similar to the effect of the divalent ions that we see in this work.

Beskok et. al. [44] consider the slip-flow in microgeometries and present a numerical algorithm appropriate for simulations of heat and momentum transfer in time- dependent slip-flows in microgeometries. They validate their method by comparing the numerical simulation results in channel slip-flows with analytical results. They also consider model inlet flows past a microcylinder.

1.5 Outline of Dissertation

In the beginning of Chapter 2, previous theories of the EDL, the Debye-Hückel model and Gouy-Chapman model, the Poisson-Boltzmann equations are discussed. Next, the three dimensional governing equations for mole fractions, electrical potential and velocity are derived, and then the 3D governing equations are simplified for one dimensional case. Next, singular perturbation solution of the governing equations is obtained for asymptotic cases. At the end of this chapter, the boundary conditions of the governing equations are obtained from electrochemical consideration by using an iterative method.

16 In Chapter 3, the basics of finite difference method are introduced, and the governing equations are discretized with finite difference method. Then the results are shown and discussed. The results for monovalent binary electrolytes are discussed first, then the multivalent ions and multi-component electrolytes, such as PBS. The analytical results of singular perturbation for asymptotic cases are also discussed in this chapter.

Finally, a scaling law for the EOF in nano- and microchannels is established, which attributed the variation of ion distribution and potential field to a series of parameter numbers.

Comparison of the model with experimental results from three different sources is discussed in Chapter 4. The experimental results from iMEDD inc. fit the model very well; and the comparison of the model with the results from Oak Ridge National

Laboratory and the results from Georgia Tech is excellent.

In Chapter 5, electrophoresis in nanochannels is investigated. Electrophoresis is the movement of charged particles in fluid where an external electric field is applied.

Electrophoresis and electroosmosis are both electrokinetic phenomena, and they occur simultaneously. Many studies of electroosmosis are for electrophoresis applications. In this chapter, the governing equations of electrophoresis are derived, and then the numerical method is discussed. At the end of this chapter, the electrophoretic fluxes of different ion species in nanochannels are computed numerically, and the results are shown as figures.

In Chapter 6, the electroosmotic flow in rectangular nanochannels is studied. The governing equations are derived in a similar way as in Chapter 5. Then the mole fraction

17 distribution, potential field, velocity profile and shear stress in rectangular channels are shown as the results.

In Chapter 7, the governing equations for unsteady flow are discussed. In Chapter

8, the future work is discussed; this includes transport of biomolecules, such as protein and DNA, effects of channel properties and fluid properties such as surface roughness, channel geometry and fluid viscosity.

18 CHAPTER 2

GOVERNING EQUATIONS

2.1 Introduction

Water is an efficient solvent for most polar molecules and electrolytes, although it does not dissolve many organic substances. Many experiments discussed in this work are done using aqueous solution; however, the model is also effective for other solvents. All the solutes we discuss in this chapter are strong electrolytes, which are fully ionized in the solvent.

In this chapter, the general knowledge for modeling electroosmotic flow is discussed. The model is a continuum, no-slip model. In other words, we ignore the molecular effects of solvent and assume there is no slip at the interfaces between the fluid and channel wall. The excellent comparison with the experiments in Chapter 5 strongly supports that these assumptions are effective even at nanometer scale.

It is also assumed that the solution in nanochannels is incompressible, which is generally accepted because of the properties of liquids. Furthermore, the electroosmotic flow in nanochannels is a laminar (non-turbulent) flow, because the Reynolds number Re in this case will be very small. The definition of Reynolds number is

Re=ρU0d/µ (2.1) 19 where ρ is the density of fluid, U0 is the characteristic velocity of flow, d is the characteristic length of the channel and µ is the dynamic fluid viscosity. It is well known that turbulent flow only exists for Re higher or around 2000. For example, if the channel height is 10 nm, the characteristic flow speed is 1mm/s, the solution is a dilute aqueous solution which has similar density and viscosity as water, then the Reynolds number is

Re =ρU0h/µ

=(1.0×103kg/m3)(10-3m/s)(10-8m)/(10-3kg/m/s)

=0.00001

Therefore this electroosmotic flow in a 10nm high channel is a laminar flow.

2.2 Classical Electric Double Layer Theories

At electrochemical equilibrium, the density of electrolyte i follows a Boltzmann distribution:

− φ ∗ ()∗ zi e r ρ ()∗ = ρ 0 kT i r i e (2.2)

∗ where r*(x*,y*,z*) is the vector of location in Cartesian coordinates, φ is the electrical potential in Volts, zi is the valence of ion species i, e is the elementary charge, i.e. the charges one electron carries, k is the Boltzmann constant, and T is the absolute

ρ 0 temperature in Kelvins, i is the number density (i.e. numbers per cube meter) of ion

∗ species i at the location where φ =0. In this section, for channels or capillaries with

∗ ∗ ∗0 dimensions much larger than the Debye length, we define φ =0 in the core and φ =φ on the wall, as shown in Figure 2.1. For the case where the wall is negatively charged,

φ∗0≤φ∗≤0, φ∗ ρ 0 = and increases as y increases. zi i 0 , because the solution is i 20 electrically neutral in the core. For binary electrolyte solution where z1=-z2,

ρ 0 = ρ 0 = ρ 0 ρ ρ = ()ρ 0 2 1 2 , and we have 1 2 .

φ∗

y 0

φ∗0

Figure 2.1 The electric potential in the EDL on a negatively charged surface.

According to Poisson’s equation,

∗ ∗ 1 ∗ 1 ∗ ∇ 2φ ()r = − ρ ()r = − e z ρ ()r (2.3) ε ε e ε ε ∑ i i r 0 r 0 i where εr is the dielectric constant of the substance, and ε0 is the permittivity of free space.

Combining the two equations above, we have

φ ∗ ()∗ − zie r ∗ ∗ 1 ∇ 2φ ()r = − e z ρ 0e kT (2.4) ε ε ∑ i i r 0 i which is the well-known Poisson-Boltzmann equation.[17] [45]

21 z eφ ∗0 For cases where the Debye-Hückel limit i << 1 is satisfied, we have kT

z eφ ∗ i << φ∗0≤φ∗≤0. ρ 0 = 1, since Using Taylor series, noticing that ∑ zi ι 0 , and kT i equation (2.4) can be simplified to [24]

∗ ∗ ∗ ∗ 1  z eφ ()r  ∇ 2φ ()r = − e z ρ 0 1− i  ε ε ∑ i i  kT  r 0 i   (2.5) z 2e 2 ρ 0φ ∗ ()r ∗ = i i ∑ ε ε i r 0 kT

∗ ∗ z oeφ ()r ∗ and from << 1, we have φ << 0.026 Volts kT

kT RT Now we define the potential scale φ = = , and nondimensionalizing 0 e F

φ ∗ ()r * x * y * z * equation (2.5) with φ()r(x, y,z) = , x = , y = , z = , we get φ 0 L h W

∂ 2φ ∂ 2φ ∂ 2φ 2 2 2 2 ε1 + + ε 2 = h κ φ (2.6) ∂x 2 ∂y 2 ∂z 2

2 ρ 0 2 ∑ zi i e i where κ = , ε1=h/L, ε =h/W. ε ε 2 r 0 kT

The reciprocal of κ has dimension of length. Thus it is a characteristic thickness of the electric double layer. From κ we defined EDL thickness, or Debye length, λ, as

1 ε ε kT ε ε RT 1 ε ε RT λ = = r 0 = r 0 = r 0 (2.7) κ 2 2 ρ 0 2 2 e ∑ zi i F ∑ zi ci F I i i

22 where R is the universal ideal gas constant, T is the temperature, F is the Faraday’s constant, NA is Avogadro number, I is the ionic strength of the solution at the location

φ∗ = 2 where =0, I zi ci , and ci is the concentration of ion species i in moles per cube

∗ meter at the location where φ =0.

The ratio λ/h is a quantity which represents the relative thickness of EDL to the height of channel, and we define ε=λ/h. Thus we have hκ = h/λ=1/ε.

L v y x u W w z

h (nm)

Figure 2.2 Geometry of one dimensional channel. W>>h, L>>h.

For a one dimensional channel as shown in Figure 2.2, ε1<<1, ε2<<1, equation

(2.6) becomes

2 2 d φ ε = φ (2.8) dy 2 and the boundary conditions are

23 0 0∗ * φ(y=0)=φ =φ (y =0)/φ0 (2.9)

φ(y=∞)=0 (2.10) as shown in Figure 2.1. The solution of equation (2.8) is

φ = φ 0 exp()− y / ε (2.11)

∗ and the dimensional potential φ is

φ * = φ ∗0 exp(− κy* ) (2.12)

∗ this is the Debye-Hückel solutions of φ in the EDL.

From equation (2.2), using Taylor series, we have

− φ ∗ ()∗ zie r ρ ()∗ = ρ 0 kT i r i e m ∞ 1  − z eφ ∗  = ρ 0  i  i ∑   (2.13) m=0 m! kT  2 3  z eφ ∗ 1  − z eφ ∗  1  − z eφ ∗   = ρ 0 1− i +  i  +  i  +  i  kT 2  kT  6  kT  L      

For binary electrolytes and z1=−z2=zº >0. Since the solution is electrically neutral

ρ 0 = ρ 0 = ρ 0 in the core, we have 1 2 , and from equation (2.13), we have

2   − z oeφ ∗   ρ + ρ = ρ 0 2 +   +  (2.14) 1 2   kT  L     and

3  − o φ ∗ − o φ ∗  0  2z e 1 z e   ρ1 − ρ 2 = ρ +   + (2.15)  kT 3  kT  L    

For cases where the Debye-Hückel limit is satisfied, equations (2.14) and (2.15) become

ρ + ρ 1 2 ≈ 1 (2.16) 2ρ 0 24 ρ − ρ − z oeφ ∗ − z oeφ ∗0 1 2 ≈ = exp()− κy* (2.17) 2ρ 0 kT kT

At the wall where y*=0, equation (2.17) becomes

ρ ()0 − ρ ()0 − z oeφ ∗0 1 2 ≈ << 1 (2.18) 2ρ 0 kT because the Debye-Hückel limit is satisfied.

ρ ()0 − ρ ()0 On the other hand, suppose 1 2 << 1 is valid; then 2ρ 0

ρ (0) − ρ (0) 1 >> 1 2 2ρ 0

∗ 2n+1 ∞ 1  − z oeφ 0  =   ∑ ()+   n=0 2n 1 ! kT  (2.19) 2n z oeφ ∗0 ∞ 1  − z o eφ ∗0  = ⋅ 1+   ∑ ()+   kT n=1 2n 1 ! kT 

o φ ∗0 > z e kT

z oeφ ∗0 so the Debye-Hückel limit << 1 is satisfied. So we have kT

z oeφ ∗0 ρ ()0 − ρ ()0 << 1⇔ 1 2 << 1 (2.20) kT 2ρ 0

ρ ()0 − ρ ()0 in other words, for 1 2 << 1, the Debye-Hückel approximation is valid for the 2ρ 0

EDL and vice versa.

The charge density difference ρ1−ρ2 is affected by the surface charge density on the wall. If there are no surface charges, the solution is electrically neutral, and

25 ρ1−ρ2=0 . As the surface charge density increases, more counter-ions are adsorbed to the wall, while co-ions are driven out of the EDL, thus ρ1−ρ2 increases.

The number density of an ion species is directly proportional to the molar concentration of that ion species.

ρ = ⋅ i N A ci (2.21) where NA is the Avogadro number, and ci is the molar concentration of ion species i in moles per cube meter.

0 2 ρ1 ()0 − ρ 2 ()0 Since ρ ρ 2 = ()ρ , from equation (2.21), << 1 becomes 1 2ρ 0

N c 0 − N c 0 N cX 0 − N cX 0 X 0 − X 0 A 1 A 2 = A 1 A 2 = 1 2 << 1 (2.22) 0 ⋅ 0 0 ⋅ 0 0 0 2 N Ac1 N Ac2 2 N AcX 1 N AcX 2 2 X 1 X 2

= 0 + where c is the total molar concentration of all ions and solvent c ∑ci csolvent at the i wall, which is a constant for the system in equilibrium. Xi is the mole fraction of ion

ci 0 0 species i X = , and X1 and X2 are the mole fractions of cation and anion species at i c the wall, respectively. If equation (2.22) is valid, the Debye-Hückel limit will be satisfied, and the Debye-Hückel view of the EDL will be achieved, as shown in Figure 2.3.

On the other hand, for cases where the Debye-Hückel limit may not be satisfied, the Gouy-Chapman theory emerges [25]. For binary electrolytes, if z1 = -z2 = zº, we will

ρ 0 = ρ 0 = ρ 0 have 1 2 , therefore equation (2.4) becomes

∗ 0 o o ∗ 2eρ z  z eφ ()r *  ∇2φ ()r* = sinh  (2.23) ε ε   r 0  kT 

ρ1 ρ0 ρ2

Figure 2.3 The Debye-Hückel picture of the EDL for binary electrolyte solution where z1=−z2.

For a one dimensional channel as shown in Figure 2.2, equation (2.23) becomes

∂ 2φ * 2eρ 0 z o  z o eφ ∗ ()r *  = sinh  (2.24) ∂ *2 ε ε   y r 0  kT  and the boundary conditions (2.9) and (2.10) are still valid. The equation (2.24) can be solved [46] to give

 o φ ∗   o φ ∗0  z e = z e ()− κ * tanh  tanh exp y (2.25)  4kT   4kT  from which we have

 −κ *  2kT 1+ ωe y  φ * = ln (2.26) o   z e  −κy*  1− ωe  27 o φ ∗0 oφ 0 ω =  z e  =  z  φ∗=0 where tanh  tanh  . Because in the bulk, for negatively charged  4kT   4 

∗0 ∗0 channel surfaces, φ <0, thus -1<ω<0; for positively charged surfaces, φ >0 and 0<ω<1.

The nondimensional electric potential φ is

2 1+ ωe− y /ε  φ = ln  (2.27)  − y /ε  z o 1− ωe 

and the cation charge density ρ1 becomes

−2 − φ ∗ ()∗  −κ *  z1e r 1+ ωe y  ρ = ρ 0 kT = ρ 0 1 1 e   (2.28)  −κy*  1− ωe  and for the anion species, we have

2  + ω −κy*  ρ = ρ 0 1 e  2   (2.29)  −κy*  1− ωe 

ρ 0 = ρ 0 = ρ 0 κ>0, 0≤ *≤ 0< -κy*≤1. where 1 2 , and y h, thus e For negative surface charges,

κ κ −1<ωe- y*<0; for positive surface charges, 0<ωe- y*<1. Next we define a new variable

κ ϖ=ωe- y*. Τhe relative number densities are plotted in Figure 2.4 as the ratio between

ρ1 and ρ0, and the ratio between ρ2 and ρ0. Figure 2.4 shows that for channels with negatively charged walls, the number density of cations is much higher than the number density of anions, except at ϖ≈0. Figure 2.5 is a blown up plot of Figure 2.4 for the region -0.5<ϖ<0.5, which shows that ρ1 and ρ2 equal to ρ0 for ϖ=0, and as ϖ decreases,

ρ1 increases and ρ2 decreases.

28 1600 ρ /ρ 1 0 1400 ρ /ρ 2 0

1200

1000

800 Ratios

600

400

200

0 -1 -0.5 0 0.5 1 ϖ

Figure 2.4 Relative number densities of cation species (ρ1/ρ0) and anion species (ρ2/ρ0) as functions of ϖ, where -1<ϖ<1.

9 ρ /ρ 1 0 8 ρ /ρ 2 0

Ratios 4

0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 ϖ

Figure 2.5 Relative number densities of cation species (ρ1/ρ0) and anion species (ρ2/ρ0) as functions of ϖ. This is a blown up version of Figure 2.4 for -0.5<ϖ<0.5.

29 From equations (2.28) and (2.29), we have

−2 2 1+ϖ  1+ϖ  ρ0   + ρ   ρ + ρ 1−ϖ 0 1−ϖ 1+ 6ϖ 2 +ϖ 4 1 2 =     = (2.30) ρ ρ 2 2 2 0 2 0 ()1−ϖ and

−2 2 1+ϖ  1+ϖ  ρ0   − ρ   ρ − ρ 1−ϖ 0 1−ϖ − 4ϖ ()1+ϖ 2 1 2 =     = (2.31) ρ ρ 2 2 2 0 2 0 ()1−ϖ

ρ + ρ ρ − ρ as shown in Figure 2.6 and Figure 2.7. Figure 2.6 shows that 1 2 ≈ 1 2 for ρ ρ 2 0 2 0

ρ1 + ρ 2 ρ1 − ρ 2 ϖ<−0.5, and ≈ − for ϖ>0.5, because ρ1>>ρ2 for ϖ<−0.5, and ρ ρ 2 0 2 0

ρ1<<ρ2 for ϖ>0.5, which are shown in Figure 2.4.

800 (ρ +ρ )/(2ρ ) 1 2 0 (ρ -ρ )/(2ρ ) 600 1 2 0

400

200

0 Ratios

-200

-400

-600

-800 -1 -0.5 0 0.5 1 ϖ

ρ + ρ ρ − ρ Figure 2.6 The ratios 1 2 and 1 2 as a function of ϖ, for -1<ϖ<1. 2ρ0 2ρ0 30 5 (ρ +ρ )/(2ρ ) 1 2 0 4 (ρ -ρ )/(2ρ ) 1 2 0 3

0 Ratios -1

-2

-3

-4

-5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 ϖ

ρ + ρ ρ − ρ Figure 2.7 The ratios 1 2 and 1 2 as a function of ϖ. This is a blown up 2ρ0 2ρ0 version of Figure 2.6, for -0.5<ϖ<0.5.

ρ + ρ ρ − ρ For ϖ≈0, Figure 2.7 shows that 1 2 ≈ 1 and 1 2 << 1. In fact, for y*=0, ρ ρ 2 0 2 0

ϖ=ωe0=ω, and from equation (2.31), we have

ρ ()0 − ρ ()0 − 4ω()1+ ω 2 1 2 = (2.32) ρ 2 2 2 0 ()1− ω which has the same format as equation (2.31), if we substitute ρ1(0), ρ2(0) and ω with

ρ1, ρ2 and ϖ, respectively. Therefore, according to equation (2.20), (2.32) and Figure 2.7,

ρ ()0 − ρ ()0 for ω<0.1, 1 2 << 1 is valid, and the Debye-Hückel theory is valid. ρ 2 0

31 ∗ z oeφ 0 On the other hand, for cases where the Debye-Hückel limit << 1 is kT

∗ ∗ ∗ ∗ o φ 0 o φ 0 o φ o φ ω =  z e  ≈ z e <<  z e  ≈ z e satisfied, we have tanh  1 and tanh  . Thus  4kT  4kT  4kT  4kT equation (2.25) becomes

o ∗0 4kT z eφ ∗ φ * = ⋅ exp()− κy* = φ 0 exp ()− κy* (2.33) z oe 4kT which is the potential equation (2.12) in the Debye-Hückel theory. So for binary electrolyte solution and z1=−z2, the Debye-Hückel theory is a special case of the Gouy-

∗ z oeφ 0 Chapman theory in which << 1 is valid. A general Gouy-Chapman view of the kT

EDL is shown in Figure 2.8.

ρ1 ρ0 ρ2

Figure 2.8 The Gouy-Chapman picture of the EDL for binary electrolyte solution where o z1=−z2=z . See also Figure 1.5. 32 2.3 The Three Dimensional Governing Equations

The governing equations for electroosmotic flow are based on three general equations, the Poisson equation for the electric field, the Navier-Stokes equation for the flow field, and the mass transfer equation.

The Poisson equation (2.3) is also one of the four Maxwell equations. We transform equation (2.3) with Faraday’s constant and mole fractions of each ion species as below

∗ ∗ 1 ∗ ∇ 2φ ()r = − e z ρ ()r ε ε ∑ i i r 0 i

1 ∗ = − eN z c ()r (2.34) ε ε A ∑ i i r 0 i

1 ∗ = − Fc z X ()r ε ε ∑ i i r 0 i

= + where c is the total molar concentration of all ions and solvent c ∑ci csolvent , and Xi is i

c the mole fraction of ion species i X = i , and the total mole fractions of all ions and i c solvent is 1. In our model, c is assumed constant for each system. This assumption is

4 3 valid for dilute solutions. For example, in an aqueous solution, csolvent=5.56×10 moles/m

3 is constant at 25ºC, for cases where each ci is less or around 1000moles/m , the variation of ci does not change c much.

As we do in §2.2, we nondimensionalize equation (2.34) with

φ ∗ ()r * x * y * z * φ()r(x, y,z) = , x = , y = , z = , and we get φ 0 L h W

Fch 2 ∇ 2φ = − z X (2.35) ε φ ∑ i i e 0 i 33 kT RT where ε1=h/L, ε2=h/W, εe=εrε0, and φ = = . Later in this section, we will use 0 e F

φ RT E = 0 = as the scale of electric field strength. For the dimensionless equations in 0 h Fh

∂ ∂ ∂ ∂ 2 ∂ 2 ∂ 2 this work, ∇ = ε xˆ + yˆ + ε zˆ and ∇ 2 = ε 2 + + ε 2 . Equation (2.35) 1 ∂x ∂y 2 ∂z 1 ∂x 2 ∂y 2 2 ∂z 2 is the Poisson equation for electroosmotic flow in a channel. Additional assumption for equation (2.35) is the volume of ions can be neglect, i. e. the point charge assumption, so

that the charge densities can be calculated as Fc zi X i .

kT RT Applying φ = = and equation (2.7) into Equation (2.35), equation (2.35) 0 e F becomes

ε 2 ∇ 2φ = −β ∑ zi X i (2.36) i where ε=λ / h and β = c / I, λ is the Debye length, c is the total concentration and I is the ionic strength. The boundary conditions are

0 φ=φ y=0 (2.37) and

1 φ=φ y=1 (2.38)

Where φ0 is the dimensionless potential at y=0, and φ1 is the dimensionless potential at y=1. Similar boundary conditions are hold at z=0 and z=1. The boundary conditions at x=0 and x=1 will be discussed later.

Next let us look at the equation of mass transfer. In this work, we focus on electroosmotic flow, so for convenience, we consider only the case where an externally

34 imposed pressure gradient does not exist. At this case, pressure gradient still exists, which is built up by the electroosmotic flow; however, for small channels connecting large upstream and downstream reservoirs, this pressure gradient is negligible because the relative volume changes in the reservoirs are negligible. Furthermore, for nanochannels, even there is external pressure, as long as the pressure is not extremely large, the pressure diffusion is still a minor factor comparing with the electroosmosis, see the discussion in §1.2. Thus the driving forces for mass transfer of fluid are ordinary Fick diffusion, convection, and migration due to the presence of an electric field. In dimensional form, the molar flux of ion species i for a dilute mixture is a vector given by

[30]

r = − ∇ + v ∗ + ∗ ni Di ci ziui Fci E ciu (2.39) where Di, ci, zi and ui are the diffusion coefficient, molarity, valence and mobility

∗ = ∗ + ∗ + ∗ coefficient of ion species i, respectively. E Ex xˆ E y yˆ Ez zˆ is the electric field strength, and uv ∗ = u ∗ xˆ + v∗ yˆ + w∗ zˆ is the mass average velocity of the fluid. Here we assume that the diffusion coefficient Di is independent of the concentration of ion species i and other ion species, which is the normal case. The effects of mass fraction on diffusion coefficients are discussed by Conlisk[21]. The diffusion coefficient and mobility coefficient are related by the Nernst-Einstein equation

Di=RTui (2.40)

Further discussion of the equation (2.40) can be found in “Electrochemical Systems” by J.

S. Newman [30]. Therefore, equation (2.39) becomes

D v ∗ ∗ nr = −cD ∇X + c i z FX E + cX uv (2.41) i i i RT i i i 35 ci where Xi is the mole fraction of ion species i, which is obtained by . For static fluidic c system, equation (2.41) reduces to the Boltzmann distribution. See Appendix A for the discussion.

At steady state, the mass transport equation is

∇ ⋅ v = ni 0 (2.42)

Substituting equations (2.41) into equation (2.42), and nondimensionalizing

* * * * * * * equation (2.42) with x=x /L, y=y /h, z=z /W, u=u /U0, v=v /U0, w=w /U0, Ex=Ex /E0,

* * Ey=Ey /E0, and Ez=Ez /E0,where U0 and E0 are the velocity and field strength scales, r respectively. Next E = −∇φ is applied for steady states, and equation (2.42) becomes

∇ 2 + ∇ • ()⋅∇φ − ⋅ ⋅∇ • ()= X i zi X i Re Sc X i u 0 i=1,2,…M (2.43)

ρU h µ where Re = 0 is the Reynolds number, and Sc = is the Schmidt number. The µ ρ Di velocity scale U0 will be discussed later.

Equation (2.43) is the dimensionless mass transfer equation for electroosmotic flow. For the electroosmotic flow in micro- and nanochannels, the Reynolds number Re is very small, see §2.1; so equation (2.43) becomes

∇ • ()∇ + ∇φ = X i zi X i 0 (2.44)

The boundary conditions are

0 Xi=Xi y=0 (2.45) and

1 Xi=Xi y=1 (2.46)

36 0 1 Where Xi is the mole fraction of ion species i at y=0, and Xi is the mole fraction of i at y=1. Similar boundary conditions are hold at z=0 and z=1. The boundary conditions at x=0 and x=1 will be discussed later. For static EDL, equations (2.36) and (2.44) lead to the Poisson-Boltzmann equation (See Appendix A).

Now let us look at the Navier-Stokes equation. The three dimensional momentum equation for an incompressible, steady flow is

Fcφ h z X Re⋅ vr • ∇vr = −∇p − 0 i i ⋅ ∇φ ⋅ +∇ 2 vr (2.47) µ U 0

* where Re=ρU0h/µ, is the Reynolds number and p=p /(µU0/h) is the dimensionless pressure. Equation (2.47) is the Navier-Stokes equations for constant density and viscosity. An assumption being made here is that the flow is continuum. For non- continuum flow, the Navier-Stokes equations are not valid. Chung’s work [46] shows that the continuum theories of EOF, including Poisson-Boltzmann theories as we discussed in previous section, and the Poisson-Nernst-Planck theories such as this model, are no longer valid for nanopores less than 2nm. [46]

The boundary conditions (2.37), (2.38), (2.45), and (2.46) all hold for equation

(2.47). For no-slip flow, the boundary conditions for u is u = 0 at the wall. Equation

(2.47) are highly nonlinear and its three components on x y and z directions are coupled with each other, so approximations need to be made to solve them. Because the electroosmotic flow is a laminar flow and x is the main direction of flow, we assume v<

Navier-Stokes equation in the x direction

37  ∂u ∂u ∂u  ∂p Fcφ h ∂φ Reε u + v + ε w  = −ε − ε 0 z X + ∇2u (2.48)  1 ∂ ∂ 2 ∂  1 ∂ 1 µ ∂ ∑ i i  x y z  x U 0 x i is the major equation being used in this work.

For steady flow, we assume the external electric field is in x direction and is a constant

∂φ ∗ ∂φ φ ∂φ ε v * * * 0 1 E = E xˆ . Then there is E = = ⋅ = ⋅φ ⋅ , and we set the velocity scale x x ∂x ∗ ∂x L ∂x 0 h as

ε E∗φ U = e x 0 (2.49) 0 µ

Then equation (2.48) becomes

 ∂u ∂u ∂u  ∂p β Reε u + v + ε w  = −ε + z X + ∇ 2u (2.50)  1 ∂ ∂ 2 ∂  1 ∂ ε 2 ∑ i i  x y z  x i

For the electroosmotic flow in micro- and nanochannels, the Reynolds number Re is very small, see §2.1; the pressure driving flow can be neglected, see §1.2; so equation

(2.50) can be simplified:

ε 2 ∇ 2 = −β u ∑ zi X i (2.51) i

Comparing equations (2.51) with equation (2.36), we find that the dimensionless potential and dimensionless velocity are governed by same differential equation.

However, the boundary conditions of these two equations are not always the same.

38 2.4 The One Dimensional Governing Equations

For a long and wide channel as shown in Figure 2.2, the classical governing equations

(2.36), (2.44) and (2.51) can be simplified. Because h<

∂ 2φ ε 2 = −β z X (2.52) 2 ∑ i i ∂y i

Equation (2.44) becomes

∂  ∂X ∂φ  i + ⋅ =  zi X i  0 i=1,2,…M (2.53) ∂y  ∂y ∂y 

Equation (2.51) becomes

∂ 2u ε 2 = −β z X (2.54) 2 ∑ i i ∂y i

The shear stress in the direction of flow, which is the x direction, is calculated by

∂u τ = (2.55) ∂y

∗ Once the governing equations are solved, the dimensional electric potential φ can be calculated as

∗ RTφ φ = φ ⋅φ0 = (2.56) F

The dimensional velocity in the x direction becomes

uε E ∗φ uε RTE ∗ u * = u ⋅U = e 0 0 = e 0 (2.57) 0 µ µF

39 * * where u is proportional to the electric field intensity E0 . Thus the electric mobility of the fluid becomes

u * uε RT µ = = e (2.58) m * µ E0 F and the dimensional shear stress in x direction is

∂u * ∂()u ⋅U ∂u U τε RTE * τ * = = 0 = ⋅ 0 = e 0 (2.59) ∂y* ∂()y ⋅ h ∂y h µFh

In summary, the boundary conditions for equations (2.52), (2.53) and (2.54) are

0 φ=φ y=0 (2.60)

1 φ=φ y=1 (2.61)

0 Xi=Xi y=0 i=1,2,…M (2.62)

1 Xi=Xi y=1 i=1,2,…M (2.63)

u = 0 y=0 and y=1 (2.64)

Equation (2.64) originates from the non-slip conditions at the channel wall. For a real microchannel or nanochannel, if the inner surfaces of the channel are fabricated at the same environment and at the same time, the electrochemical properties of difference surfaces will be the same. Thus the channel is symmetric. For symmetrical channels, we

0 1 0 1 take φ =φ =0, and we also have Xi =Xi . Thus equations (2.60)-(2.63) become

φ=0 y=0 and y=1 (2.65)

0 Xi=Xi y=0 and y=1 i=1,2,…M (2.66)

0 In §2.8, we will show how we find Xi based on considerations of electrochemical potential.

40 2.5 Singular Perturbation Equations

The solution of equations (2.52) and (2.54) is a function of ε and β. In the case of

2 2 2 ∂ φ 2 ∂ u microchannels where h>>λ, and ε=λ/h<<1; ε ≈ 0 and ε ≈ 0 are valid in the ∂y 2 ∂y 2 region far outside the EDL. Thus, the governing equations (2.52), (2.53) and (2.54) in the region far outside the EDL become

∂  ∂X ∂φ  io + ⋅ o =  zi X io  0 i=1,2,…M (2.67) ∂y  ∂y ∂y 

= ∑ zi X io 0 (2.68) i

= φ uo o (2.69) where the subscript o indicates that Xio, φο, and uo are the outer solutions of the governing equations. Equation (2.68) shows that the fluid is electrically neutral in the core region of microchannels.

y For the region near y=0, for ε<<1, we define a new variable Y = , and if ε

ε→0, there will be Y→∞. In this case, equation (2.52), (2.53) and (2.54) become

∂ ∂X ∂φ  i + ⋅  =  zi X i  0 (2.70) ∂Y  ∂Y ∂Y 

∂ 2φ = −β z X (2.71) 2 ∑ i i ∂Y i

∂ 2u = −β z X (2.72) 2 ∑ i i ∂Y i

41 These singular perturbation equations are the governing equations for fluids near the wall, and the solutions of equations (2.70), (2.71), and (2.72) are the inner solutions of the governing equation, corresponding to the outer solutions mentioned previously.

The boundary conditions for equations (2.70), (2.71), and (2.72) are

0 Xi=Xi Y=0 (2.73)

Xi=Xio Y=Y∞→∞ (2.74)

φ=0 Y=0 (2.75)

φ=φo Y= Y∞→∞ (2.76)

u=0 Y=0 (2.77)

u=uo Y= Y∞→∞ (2.78) where Xio , φo and uo are the outer solutions, which are unknowns. Y∞→∞ represents the thickness of EDL, and the inner solutions equal the outer solutions at Y=Y∞, because this is the interface between the EDL and the core, for which the inner equations and the outer equations apply, respectively. In this section, we focus on solving Xi and φ, because equations (2.71), (2.72), and (2.75)-(2.78) imply that the solution for velocity can be obtained by substituting all φο in the solution of φ with uo.

The requirement ε=λ/h<<1, shows this is the case for microchannels. For small nanochannels where ε=Ο(1), the core region is not electrically neutral, and the whole problem must be solved in the entire domain 0≤y≤1. In the rest of this section ε=λ/h<<1 is assumed for all discussion.

First of all, let us look at the outer solution. In the core region, if we add equation

(2.67) for i from 1 to N, and substitute equation (2.68) into it, equation (2.67) becomes

42 ∂ 2 X io = 0 (2.79) ∑ 2 i ∂y

Multiply both sides of equation (2.67) by zi and add them for i from 1 to N, then apply equation (2.68) into it, equation (2.67) becomes

∂  ∂φ   o ⋅ z 2 X  = 0 (2.80) ∂  ∂ ∑ i io  y  y i 

Integrating equation (2.67), we have

∂X ∂φ io + z ⋅ X o = C (2.81) ∂y i io ∂y 1 where C1 is a constant. Whenever there is no mass flux, we have C1=0, thus equation

(2.81) can be solved easily, and the solutions are

− φ = C zi o X io X i e (2.82)

C where Xi is a constant depends on i. Equation (2.82) shows that the mole fractions of ion species in the outer solution follow a Boltzmann distribution. Substituting equation

(2.82) into equation (2.79), equation (2.79) becomes

2 ∂ 2φ  ∂φ  o z X −  o  z 2 X = 0 (2.83) ∂ 2 ∑∑i io  ∂  i io y ii y 

= According to equation (2.68), ∑ zi X io 0 . Thus equation (2.83) turns out to be i

2  ∂φ   o  z 2 X = 0 . X is always a positive number, because it represents the mole  ∂  ∑ i io i  y  i

∂φ fraction of ion species i. Therefore equation (2.83) becomes o = 0 . So the outer ∂y solution φo is a constant, and equation (2.82) becomes

43 − φ = C zi o X io X i e (2.84) which shows that the outer solution Xio is also a constant. From equation (2.69), uo=φo.

Next, in order to solve the outer solutions φο, uo and Xio, let us look at the inner solutions of the governing equations. For the cases where there is no mass flux, integrating equation (2.70), and we have

∂X ∂φ i + z ⋅ X = 0 (2.85) ∂Y i i ∂Y from which we solve

− φ = 0 zi X i X i e (2.86)

0 where Xi is the mole fraction of ion species i at the wall where φ=0, see equations (2.73) and (2.75). Thus at Y=Y∞, from equation (2.74), and (2.76)

− φ = 0 zi ο X io X i e (2.87) which shows that the mole fractions of all ion species also agree with the Boltzmann distribution. Generally, the outer solutions cannot be solved analytically, however there are several special cases in which the analytical solution of the outer solution exists. For instance, if there are only two ion species in the electrolytic solution, combining equations (2.68) and (2.87), we have

− z φ − z φ 0 1 o + 0 2 o = z1 X 1 e z2 X 2 e 0 (2.88)

Solve equation (2.88), we have

1  − z X 0  φ = ln 1 1  (2.89) o −  0  z1 z2  z2 X 2 

From equations (2.84) and (2.89), the outer solution X1o and X2o are

44 z1 0 −  − z X  z1 z2 X = X 0  2 2  (2.90) 1o 1  0   z1 X 1 

z2 0 −  − z X  z1 z2 X = X 0  2 2  (2.91) 2o 2  0   z1 X 1 

For a special case, z1=-z2=1, equations (2.89), (2.90) and (2.91) become

1  X 0  φ = ln 1  (2.92) o  0  2  X 2 

= = = 0 0 X 1o X 2o X o X 1 X 2 (2.93) where Xo is the outer solution of the cation or the anion species, which are the same.

If there are three ion species, combining equations (2.68) and (2.84), we have

(z − z )⋅ φ (z − z )⋅ φ 0 + 0 1 2 o + 0 1 3 o = z1 X 1 z2 X 2 e z3 X 3 e 0 (2.94)

There are no analytical solutions for equation (2.94) generally. However, for several cases, equation (2.94) can be solved analytically. First of all, if z3=z1, we have

1  z ()X 0 + X 0  φ = ln 1 1 3  (2.95) o −  − 0  z1 z2  z2 X 2 

For z3=z2 or z2=z1, equation (2.94) can be solved similarly. Secondly, if z1+z2=2z3, the answer is

 − 0 ± ()0 2 − 0 0  1  z3 X 3 z3 X 3 4z1 z2 X 1 X 2  φ = ln (2.96) 0  0  z − z  2z X  1 3  2 2  similarly, cases such as z1+z3=2z2 or z2+z3=2z1 can also be solved. Thirdly, if

2z1+z3=3z2, we have

45 1  t 2z X 0  φ = ln − 2 2  (2.97) 0 −  0  z1 z2  6z3 X 3 t 

1 3  0 3 0 2 0    4()z X + 27 ()z X z X 2  where t = −108z X 0 +12 3 2 2 1 1 3 3  ⋅ ()z X 0 .  1 1 0 3 3   z X    3 3  

After discussing the outer solutions, let us go back to the inner solutions. The governing equation (2.70) has been converted to Boltzmann equation (2.86), substituting equation (2.86) into governing equation (2.71), we have

2 ∂ φ − φ = −β z X 0e zi (2.98) 2 ∑ i i ∂Y i

φ ∗ − φ * zi e o zi e o If the Debye-Hückel limit << 1 is valid, − z φ ≤ − z φο = << 1. kT i i kT

− φ zi ≈ − φ Therefore, e 1 zi , and equation (2.98) becomes

∂ 2φ = −β z X 0 ()1− z φ = φ − β z X 0 (2.99) 2 ∑ i i i ∑ i i ∂Y i ι

2 0 zi ci β 2 0 = c 2 0 = i = I = because of ∑ zi X i ∑ zi X i 1. The solution of equation (2.99) is i I i I I

φ = −Y + Y + β 0 Ae Be ∑ zi X i (2.100) i where A and B are constants. From (2.100) and boundary conditions (2.75), (2.76), we

= −β 0 φ = β 0 φ solve A ∑ zi X i , B=0, and o zi X i . So the solution of the governing i i equations is

φ = β 0 ( − −Y ) ∑ zi X i 1 e (2.101) i

46 which is valid for arbitrary number of ion species with arbitrary number of valences as long as the Debye-Hückel limit is satisfied. From equations (2.86) and (2.101), the inner solutions of mole fractions are

−β 2 0 ()− −Y ∑ zi X i 1 e −()− −Y = 0 i = 0 1 e X i X i e X i e (2.102) and it is easier to see that the inner solution of fluid velocity u is same as the electric potential φ

= φ = β 0 ( − −Y ) u zi X i 1 e (2.103) i

z X 0 c i i where β z X 0 = z X 0 = i . For instance, for monovalent binary ∑ i i ∑ i i 2 0 i I i ∑ zi X i i

X 0 − X 0 electrolyte solution where z =-z =1, β z X 0 = 1 2 . 1 2 ∑ i i 0 + 0 i X 1 X 2

z eφ ∗ For cases where the Debye-Hückel limit i o << 1 may not be valid, equation kT

(2.98) can be solved numerically.

2.6 Debye-Hückel Approximation for Nanochannels where ε = Ο (1)

For small nanochannels where ε=λ/h=O(1), assuming there is no mass flux in the channel, equation (2.53) become

∂X ∂φ i + z ⋅ X = 0 (2.104) ∂y i i ∂y and the boundary conditions for symmetrical channel are equations (2.66) and (2.65), from which we solve the mole fractions Xi 47 − φ = 0 zi X i X i e (2.105)

Thus equation (2.52) becomes

2 ∂ φ − φ ε 2 = −β z X 0e zi (2.106) 2 ∑ i i ∂y i

z eφ for cases where the Debye-Hückel limit i << 1 is valid, equation (2.106) becomes kT

∂ 2φ ε 2 = −β z X 0 ()1− z φ (2.107) 2 ∑ i i i ∂y i and the solution of equation (2.107) is given by

 y   y  φ = Αcosh + B sinh + β z X 0 (2.108)  ε   ε  ∑ i i     i where A and B are constants. The boundary condition (2.65) φ(0)=0 leads to

0 ∑ zi X i A = −β z X 0 = − i ∑ i i 2 0 i ∑ zi X i i

∂φ and in the symmetry case, we apply the condition = 0 at y=1/2 and obtain ∂y

 1   1  B = −A ⋅ tanh = tanh β z X 0  ε   ε  ∑ i i  2   2  i so the solution is

 y   y   1  φ = −β z X 0 cosh + sinh tanh β z X 0 + β z X 0 ∑ i i  ε   ε   ε  ∑ i i ∑ i i i      2  i i   y 1    cosh −   (2.109) ε 2ε = β z X 0 1−    ∑ i i   1   i  cosh     2ε  

48 and the mole fractions Xi can be calculated from equation (2.105). [48]

2.7 Electrochemical Potential

For the boundary conditions of the governing equations, mole fractions at the wall are unknown and we need to develop a technique to calculate these values. §2.7 and §2.8 show how we apply electrochemical equilibrium conditions into the channel-reservoir system to calculate the wall mole fractions based on the concentrations in the upstream reservoir.

th µ~ The electrochemical potential of the i electrolyte species i is defined by:

µ~ = µ o + + φ * i i RT ln ai zi F , (2.110)

µ o th where i is the standard chemical potential of the i species. Defined as the chemical potential of the ith species for the case where the activity of this species is 1. R is the

th universal gas constant, T is the temperature, ai is the activity of the i species; zi is the

∗ valence of the ith species, F is Faraday’s constant, and φ is the electric potential in Volts.

The SI unit of the electrochemical potential is J/mol [49].

th The activity of the i species, ai, is related to molality mi by

= γ ai imi (2.111) where γi is a function of concentration as well as temperature and pressure, which is known as the activity coefficient. As shown in Table 2.1, the activity coefficient of the solute approaches 1 as the concentration, which is in the form of molality in Table 2.1, approaches zero [49].

γ = solute 1 (2.112) lim→ xsolute 0

49 Therefore, mole fractions and other concentration formats, such as molarity and molality, can be used in place of activity for dilute solutions. The activity coefficient of

NaCl is shown in Table 2.2.

Concentration Coefficient Coefficient Coefficient Coefficient (mol/kg H2O) KCl (1:1) H2SO4 (2:1) CaCl2 (1:2) ZnSO4 (2:2)

0.0001 0.989 -- 0.962 -- 0.0005 0.975 0.885 0.918 0.780 0.001 0.965 0.830 0.887 0.700 0.005 0.928 0.639 0.783 0.477 0.01 0.901 0.544 0.724 0.387 0.05 0.816 0.340 0.574 0.202 0.10 0.769 0.265 0.518 0.150 0.50 0.649 0.154 0.448 0.063 1.00 0.603 0.130 0.500 0.044 5.00 0.590 0.212 5.89 (0.07) 10.00 -- 0.553 43.0 --

Table 2.1 Mean activity coefficients of aqueous salts at 25ºC. (http://www.psigate.ac.uk/newsite/reference/plambeck/chem2/p01193.htm, 07/18/03)

I(M) γ 0.000997 0.98462 0.001994 0.978827 0.004985 0.967829 0.009969 0.95638 0.01993 0.945634 0.049891 0.917686 0.09953 0.898346 0.1987 0.877393 0.4940 0.85274 0.9788 0.846115 1.921 0.864763 3.696 0.95342 5.305 1.082096

Table 2.2 Experimental activity coefficients of NaCl at 298K. [50] 50 Debye and Hückel showed that in dilute solution the activity coefficient of species i with a charge number of zi is given by

γ = − 2 1/ 2 log i Azi I (2.113) where I is the ionic strength and

1/ 2 3 / 2 1  2πN m   e 2  A =  A solv    (2.114)  πε κ  2.303  V   4 0 kT  where msolv is the mass of solvent in volume V. [49]

2.8 Electrochemical Consideration for a Channel-Reservoir System

The classical Poisson-Boltzmann EDL theories define the boundary conditions as

φ=0, Xi=Xo at y>>ε

λ for ε = << 1. Qu and Li [36] work on small channel height which may cause h overlapped EDL. They obtain new boundary conditions at the middle of the channel by combining the mass conservation of all ion species in the solution inside the channel and a converging condition

h/2 h 1 φh/2=0, Xi =Xo, for κ ⋅ = → ∞ . 2 2ε

However, for a channel-reservoir system, because of the ion fluxes between the channel and the reservoir, the amount of the ith ion inside the channel is not a constant.

Additionally, considering the external voltage drop in the x direction, φ=0 for y=h/2 is not valid for all x positions. Therefore, another way is required for obtaining the boundary conditions of the governing equations. In this section, we will show how the boundary conditions are determined by considerations on electrochemical equilibrium. 51 Figure 2.9 shows a typical single-channel and reservoir system. For microchannels and nanochannels, the volume of the upstream reservoirs and downstream reservoirs are both much larger than the volume of the channel. We assume that in a finite time period, the concentrations of all ion species in the upstream reservoir are same as their original concentrations before the external voltage is applied.

Direction of Flow

Upstream Downstream Reservoir Channel Reservoir

Figure 2.9 A schematic figure of the normal channel-reservoir system

Since the molarity in the upstream reservoir is assumed constant and uniform, and since the flow is assumed fully developed, it is reasonable to assume that the electrochemical potential of any ion species in the channels is same as the electrochemical potential of the same ion species in the reservoir. Note that in fully developed flow the electrochemical potential does not change with distance down the channel. Thus the only place where the electrochemical potential can change is in the entrance region. Since the flow rates under consideration are so small it is unlikely that the electrochemical potential will deviate significantly from equilibrium conditions. This 52 assumption is validated later in this chapter by the excellent comparison with experimental data and has also been used in the work of Nonner [7].

µ~ According to this assumption, for ion species i in equation (2.110), i in the

µ~ µ o µ o channel is same as i in the reservoirs. The definition of i tells us that i is a constant.

Therefore, we have

R + φ * = C + φ * … RT ln ai zi F R RT ln ai zi F C i = 1, 2, N (2.115) which leads to

RT X R γ R RT X R RT c R ∆φ * = φ * −φ * = ln i i ≈ ln i = ln i (2.116) C R C γ C C C zi F X i i zi F X i zi F ci

C R where ∆φ is the Nernst potential, zi is the charge number of ion specie i, Xi and Xi are the mole fractions of ion species i in the fully developed region of the channel and in the

γ C γ R C reservoir, respectively; i and i are the corresponding activity coefficients; ci and

R ci are the corresponding concentrations, usually in moles per liter. From equation

γ R γ C (2.112), i and i are about 1 for dilute solutions. There are also arguments that in the micro and nano regime, concentrations rather than activities should be used, which are supported by the work of Kohonen et. al. [51].

We assume a negatively charged wall as is customary for a silicon channel in which the negative charge is due to deprotonated silanol groups (SiOH) on the surface at pH > 4. The surface charge density σ depends on the deposition and cleaning condition during microfabrication, and in the computational process must be assumed an iterated until a calculated value is equal to the assumed value. For the PBS solution results it turns 53 out that the surface charge density is about 0.2 C/m2, which is about the value suggested for silicon surfaces by Israelachvili [17].

To describe the procedure for calculating the wall mole fractions, electroneutrality in the channel requires

+ C = … z f c f ∑ zi ci 0 (Channels) i=1, 2, N (2.117) i where zf and cf are the charge number and concentration of the charged groups on the wall. Electroneutrality in the reservoir requires

R = … ∑ zi ci 0 (Reservoir), i=1, 2, N (2.118) i and it is assumed that the concentrations in the reservoir are known, so that equation

(2.118) is identically satisfied. In equation (2.117), the concentration of the fixed charge groups, cf is converted from surface charge density σ by assuming that the charges are distributed evenly over the whole channel. Therefore we obtain

Aσ c = (2.119) f 1000VF where A is the surface area, V is the volume of the channel and F is Faraday’s constant.

3 3 The conversion factor 1000liter/m is used to convert the units of cf from moles/m to moles/liter. For rectangular channels, A=2·L·(h+W), V=L·W·h.

R As mentioned above, zi, and ci are known for each species in experiments. Then if the surface charge density is known, equations (2.116) and (2.117) are N+1 equations

C with N+1 unknowns in Nernst potential ∆Ψ and the average molarity ci , i =1,2, … N.

Equating Nernst Potentials for each species in equation (2.116) lead to

54 zi C  c  z1 cC =  1  c R i = 2, 3, … N (2.120) i  R  i  c1 

Substituting equation (2.120) into (2.117), we obtain

N R zi C zi ci C z c + z c + ()c z1 = 0 (2.121) f f 1 1 ∑ z 1 = i i 2 ()R z c1 1

C This is a single equation for c1 , the average concentration of ion species 1 in the channel.

The ion species 1 is arbitrarily chosen and in this paper species 1 corresponds to the sodium ions. It is important to note that there may be multiple solutions for equation

(2.121); however, in all the cases considered here, only one solution is physically

C C reasonable. Next, ci (i = 2, 3, … N) are calculated based on equation (2.120). ci can be

N C = C + converted to average mole fractions Xi by dividing by total molarity c ∑ci csolvent . i=1 where csolvent is the molarity of the solvent, commonly, water. Simultaneously, the average concentration of ion species i is also defined in dimensional form as

1 h cC = cydy()* * (2.122) ii∫ h 0

* * where molarity ci(y ) is a function of y , and h is the channel height. In dimensionless form, we have for the mole fraction

1 X C = X ()y dy (2.123) i ∫ i 0

C * where Xi is the average mole fraction, and mole fraction Xi(y) is a function of y=y /h.

Therefore, if the wall mole fractions of all ion species are known, and the governing equations (2.53) and (2.52) are solved, the channel average concentration can be

55 calculated based on equations (2.122) or (2.123), which should equal the average concentration obtained from equations (2.120) and (2.121). Thus an iteration procedure is

C followed to obtain the wall mole fractions from ci obtained from equations (2.120) and

(2.121)

At the first iteration, the wall mole fractions are assumed to be the average mole fractions as calculated from the concentrations in equations (2.120) and (2.121). After convergence of the governing equations, the average mole fractions are calculated with equation (2.123). Clearly the only way that these two average mole fractions can be equal is if the mole fractions across the channel are constant. This is not the case and so the governing equations are solved again with the new wall mole fractions defined by

X C m+1 = m i X i (0) X i (0) m (2.124) X i

m where m denotes the iteration number, Xi (0) is the wall mole fraction at y=0 of species i

th m for the m iteration, and X i is the average mole fraction for specie i obtained from

m equation (2.123) after solving the governing equations with Xi (0) as the wall mole fractions. The form of this equation is motivated by the fact that a higher value of the wall mole fraction will lead to a higher average mole fraction, and vice versa. The iteration procedure is said to converge if successive iterations of the wall mole fractions differ by less than 10-4.

As is mentioned above, the surface charge density, σ, is assumed. To check whether the iterations are consistent with the assumed surface charge density, we recalculate the concentration of charged groups on the wall from the formula

56 1 h c =− zcdy* (2.125) fii∑ ∫ i zhf 0

If the recalculated cf is equal to the cf calculated from equation (2.119) with the assumed surface charge density, the iteration is consistent with the assumed surface charge density and the solution process is complete. Otherwise, the solution is recalculated with a different charge density. For more discussions of the EOF in channel-reservoir system, see Zheng et. al. [52].

57 CHAPTER 3

RESULTS FOR ONE DIMENSIONAL CHANNELS

3.1 Introduction

Binary electrolytes refer to an electrolyte which consists of one cation species and one anion species after ionization. Many common electrolytes such as NaCl, CaCl2, and

K2SO4 are all binary electrolytes. There are other electrolytes that consist of three or more ion species after ionization, these electrolytes are named accordingly, for example, ternary electrolytes. Naturally, our efforts of building a model of electroosmosis starts with the aqueous solution of binary electrolytes, since they are easier to model than other electrolytes mixture.

The governing equations and boundary conditions of the model have been discussed in last chapter. In this chapter, the numerical tools for this model, finite difference method, is introduced; and the continuous space governing equations and boundary conditions are converted into difference equations by using finite difference method.

The equations are solved numerically using MATLAB programs. The results for monovalent binary electrolytes are also discussed in this chapter.

58 3.2 Finite Difference Method

Finite difference method is used in the numerical calculation of this work.

Consider a function f=f(x,y,…), in which variable x varies from xmin to xmax, if we divide this region by a large number N, we have

x1=xmin

xj+1=xj+∆x j=1,2,…,N

− xmax xmin where ∆x = << 1. If other variables y, z,… are maintained as constants, for xj N

(j=1,2,…N), the corresponding values of f are

f1=f(x1,y,…)

…

fj=f(xj,y,…)

…

fN+1=f(xN+1,y,…)

Figure 3.1 shows a function f=f(x,y,…) in the region between xi-2 and xi+1, where

∆x<<1. Using Taylor series, the central difference approximation of derivatives of f are:

−  ∂f  f j+1 f j−1 = (3.1)  ∂  ∆  x  j 2 x

2 − +  ∂ f  f + 2 f f −   = j 1 j j 1 (3.2)  ∂ 2  ()∆ 2  x  j x

59 f

fi+1

fi-1

fi-2

xi-2 xi-1 xi x xi+2 ∆ ∆ ∆ i+1 ∆ x x x x x

Figure 3.1 Function f=f(x,y,…) in a view of finite difference. ∆x<<1.

Substitute equations (3.1) and (3.2) into equations (2.52), (2.53) and (2.54), for j=2,3,…N, we have

()− ()+ () X i j+1 2 X i j X i j−1 ()∆y 2 ()− () X + X − φ + −φ − +z ⋅ i j 1 i j 1 j 1 j 1 (3.3) i 2∆y 2∆y

φ + − 2φ + φ − +z ()X j 1 j j 1 = 0 i i j ()∆y 2

()∆y 2 φ − 2φ + φ = −β z ()X (3.4) j−1 j j+1 2 ∑ i i j ε i

()∆y 2 u − 2u + u = −β z ()X (3.5) j−1 j j+1 2 ∑ i i j ε i

60 where Xi is the mole fraction of ion species i. Both Xi and φ are functions of y, for example, (Xi)j is the mole fraction of ion species i at y=yj.

Substituting equation (3.4) into equation (3.3), after simplification, equation (3.3) becomes

 z  1− i ()φ −φ ()X  j+1 j−1  i j−1  4  β +− − 2 ()()()∆ 2  2 zi X i y X i  ε 2 j  j (3.6)  z  + + i ()φ −φ ()X 1 j+1 j−1  i j+1  4  β ()∆y 2 = z z ()()X X 2 ∑ i k i j k j ε k≠i

Equations(2.64),(2.66) and (2.65), the boundary conditions for equations (2.52),

(2.53) and (2.54), become

u1=0, uN+1=0 (3.7)

φ1=0, φΝ+1=0 (3.8)

0 0 (Xi)1=Xi , (Xi)N+1=Xi (3.9)

Equation (3.4), (3.5) and (3.6) are all in form bj fj-1+aj fj+cj fj+1=dj, which can be solved by Thomas algorithm, a procedure based on Gaussian elimination.

In order to explain the Thomas algorithm, let us consider a system of equations

+ + = b2 f1 a2 f 2 c2 f3 d 2 (3.10)

+ + = b3 f 2 a3 f3 c3 f 4 d3 (3.11)

…

+ + = b j−1 f j−2 a j−1 f j−1 c j−1 f j d j−1 (3.12)

61 + + = b j f j−1 a j f j c j f j+1 d j (3.13)

…

+ + = bN f N −1 aN f N cN f N +1 d N (3.14) where aj, bj, cj and dj are coefficients, fj=f(xj) (j=2,3,…N) are the unknown values of function f in the internal mesh points, and f1, fN+1 are known as the boundary values of function f. For unknown fj, this system of equations is known as a tridiagonal system of equations. From equation (3.10) , we have

γ c f f = 2 − 2 3 (3.15) 2 α α 2 2

α = γ = − where 2 a2 , 2 d 2 b2 f1 . Substitute equation (3.15) into equation (3.11), we have

α + = γ 3 f3 c3 f 4 3 (3.16)

b c b γ where α = a − 3 2 , γ = d − 3 2 . From equation (3.16), we have 3 3 α 3 3 α 2 2

γ c f = 3 − 3 4 f3 (3.17) α 3 α 3

Analogically, we have

α + = γ j f j c j f j+1 j j= 3,4, … N (3.18)

b c − b γ − where α = a − j j 1 and γ = d − j j 1 . For j=N, from equation (3.18), we have j j α j j α j−1 j−1

γ c f f = N − N N +1 (3.19) N α α N N where fN+1 is known as the boundary values. Substitute equation (3.19) into equation

(3.18) for j=N-1, fN-1 is solved. Analogically, we have

62 γ c f + f = j − j j 1 j=N, N-1,…2,1 (3.20) j α α j j

For this work, from equations (3.4)-(3.6), it can be seen that the discretized equations for u, φ, Xi are all tridiagonal system of equations. Equation (3.7)-(3.9) are the corresponding boundary equations which provide the value of f1 and fN+1. However, some coefficients in equations (3.4)-(3.6) contain variables such as φj+1 and (Xi)j. In order to solve this problem, iteration is a necessary. First of all, let

0 0 (Xi)j=(Xi)j , φj=φj j=2,3,…N (3.21)

0 0 where (Xi)j and φj are the values guessed based on the boundary conditions. The superscript 0 shows that it takes zero iteration to calculate these (Xi)j and φj. Then (Xi)j and φj for j=1,2,…N+1 are used to calculated the coefficients in equations (3.4)-(3.6), and then Thomas algorithm can be applied to solve new (Xi)j and φj

1 1 (Xi)j=(Xi)j , φj=φj j=2,3,…N (3.22)

1 1 where the superscript 1 means that (Xi)j and φj are the results after 1 iteration. Then the new (Xi)j and φj values are used calculate the coefficients in equations (3.4)-(3.6) again, and the Thomas algorithm can be applied again to solve new (Xi)j and φj

2 2 (Xi)j=(Xi)j , φj=φj j=2,3,…N (3.23) and so on.

The iteration procedure is said to converge if after k iterations, we have

− ()X k − ()X k 1 i j i j < δ i=1,2,…M, j=2,3,…N (3.24) ()k X i j and

63 φ k −φ k −1 j j < δ j=2,3,…N (3.25) φ k j where δ is a small number, and δ=10-4 is used in this work. In some programs of this work, a less rigid criterion

− ()X k − ()X k 1 1 j 1 j < δ j=2,3,…N (3.26) ()k X i j is used, where X1 is the mole fraction of ion specie 1. The ion specie 1 is arbitrarily chosen, however, for convenience, the most popular cation is the default ion specie 1 in this work.

If the iteration converges, the results

k k (Xi)j=(Xi)j , φj=φj j=2,3,…N (3.27) is the solution of equation (3.4) and (3.6) with boundary conditions (3.8) and (3.9). Then equation (3.27) is substituted into equation (3.5), and uj for j=2,3,…N is solved with the

Thomas algorithm. (Xi)j, φj and uj for j=1,2,…N+1 are the numerical solutions of equations (2.52)-(2.54). For more information about finite difference methods, please refer to G. D. Smith. [53]

3.3 Results for Monovalent Binary Electrolytes

In nanochannels shown in Figure 2.2, if the electrolyte consists of monovalent cation and monovalent anion, such as sodium chloride, the governing equations (2.52)-

(2.54) become

∂ ∂ ∂φ  X + +  =  X +  0 (3.28) ∂y  ∂y ∂y 

64 ∂ ∂ ∂φ  X − −  =  X −  0 (3.29) ∂y  ∂y ∂y 

∂ 2φ 2 ε = −β ()X + − X − (3.30) ∂y 2

∂ 2 2 u ε = −β ()X + − X − (3.31) ∂y 2 where X+ is the mole fraction of cation, and X- is the mole fraction of anion. The boundary conditions, (2.65) and (2.66) become

u = 0 y=0 and y=1 (3.32)

φ=0 y=0 and y=1 (3.33)

0 X+=X+ y=0 and y=1 (3.34)

0 X-=X- y=0 and y=1 (3.35)

0 0 where X+ is the mole fraction of the cation at the channel wall, and X- is the wall mole fraction of the anion. Governing equations (3.28)-(3.31) are solved numerically, and the results will be discussed in this section.

If the wall mole fractions of both cation and anion are known as constants, and

ε<<1, the plots in Figure 3.2-Figure 3.4 show the mole fraction distribution, potential and

0 0 velocity profile, and shear stress, respectively, for the case X+ /X- =O(1),. Debye-Hückel picture of EDL is achieved for this case.

65 -3 x 10 2.8 cations anions

2.75

2.7

2.65 Mole Fractions 2.6

2.55

2.5 0 0.2 0.4 0.6 0.8 1 y

Figure 3.2 The mole fraction distribution pattern across the channel for monovalent binary electrolyte. The wall mole fractions are constants, for sodium the wall mole fraction is 0.00276; for chloride, the wall mole fraction is 0.00254. Channel height h=20nm. ε=λ/h=0.040.

Figure 3.2 shows the mole fraction distribution of cations and anions across the channel. Both mole fractions and y are dimensionless. In the bulk of the channel, the mole fractions of cations and anions are the same, which implies that the electrolytic solution is neutral in the bulk. In the EDL, the Debye-Hückel picture of EDL is achieved: the mole fraction of cations increases as the distance to the wall decreases, while the mole fraction of anions decreases as the distance to the wall decreases, the increment of the cation concentration equals the decrement of the anion concentration. The concentration difference between cation and anion species reaches its maximum at the wall.

66 0.045

0.04

0.035

0.03

0.025

0.02

0.015 Potential and Velocity

0.01

0.005 potential and velocity Outer solution

0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.3 The potential and velocity profile across the channel for monovalent binary electrolyte. The wall mole fractions are constants, for sodium the wall mole fraction is 0.00276; for chloride, the wall mole fraction is 0.00254. Channel height h=20nm. ε=λ/h=0.040.

0.8

0.6

0.4

0.2

shear stress -0.2

-0.4

-0.6

-0.8

-1 0 0.2 0.4 0.6 0.8 1 y

Figure 3.4 The shear stress profile across the channel for monovalent binary electrolyte. The wall mole fractions are constants, for sodium the wall mole fraction is 0.00276; for chloride, the wall mole fraction is 0.00254. Channel height h=20nm. ε=λ/h=0.040. 67 Figure 3.3 shows the potential and velocity profile across the channel. The same format of governing equations (3.30) and (3.31) and boundary conditions (3.32) and

(3.33) imply that the potential curve of φ and velocity profile of u are the same, which is shown as the blue curve in Figure 3.3. The circles in Figure 3.3 shows the outer solution solved from the singular perturbation equation (2.92). In the bulk region, the potential and velocity curve are superposed to the outer solution. The potential and velocity decrease as approaching the wall. Figure 3.4 shows the shear stress across the channel.

The shear stress is null in the core, which corresponds to the flat portion of the velocity profile. At the wall, the shear stress reaches maximum.

-3 x 10 2.8 cations anions

2.75

2.7

2.65 Mole Fractions 2.6

2.55

2.5 0 0.2 0.4 0.6 0.8 1 y

Figure 3.5 The mole fraction distribution pattern across the channel for monovalent binary electrolyte. The wall mole fractions are known as constants, for sodium the wall mole fraction is 0.00276; for chloride, the wall mole fraction is 0.00254. Channel height h=4nm. ε=λ/h=0.200.

68 For the case where ε=O(1), the plots in Figure 3.5-Figure 3.7 show the mole fraction distribution, potential and velocity profile, and shear stress, respectively. Figure

3.5 shows the mole fractions of cations and anions across the channel at ε=O(1). In the bulk, the mole fractions of cations is higher than the mole fraction of anions, which is the same as at the wall. Therefore, across the channel, the solution is not electrically neutral.

Figure 3.6 shows the potential and velocity profile for ε=O(1). The profile is a parabolic curve which has a maximum in the middle of the channel. The singular perturbation solution solved from equation (2.92) is far off from the curve, which shows that the singular perturbation solution is not valid for the case ε=O(1). Figure 3.7 shows the shear stress for ε=O(1). The shear stress in the bulk changes distinctly, and there is no zero shear stress region in the bulk.

0.045

0.04 potential and velocity Outer solution 0.035

0.03

0.025

0.02

0.015 Potential and Velocity and Potential

0.01

0.005

0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.6 The potential and velocity profile across the channel for monovalent binary electrolyte. The wall mole fractions are constants, for sodium the wall mole fraction is 0.00276; for chloride, the wall mole fraction is 0.00254. Channel height h=4nm. ε=λ/h=0.200. 69 0.25

0.2

0.15

0.1

0.05

shear stress -0.05

-0.1

-0.15

-0.2

-0.25 0 0.2 0.4 0.6 0.8 1 y

Figure 3.7 The shear stress profile across the channel for monovalent binary electrolyte. The wall mole fractions are constants, for sodium the wall mole fraction is 0.00276; for chloride, the wall mole fraction is 0.00254. Channel height h=4nm. ε=λ/h=0.200.

-3 x 10 2.8 Anion Numerical Anion Analytical Cation Numerical 2.75 Cation Analytical

2.7

2.65 Mole Fractions Mole 2.6

2.55

2.5 0 1 2 3 4 5 6 Y

Figure 3.8 The mole fraction distribution pattern near the wall for the asymptotic case for monovalent binary electrolyte. The wall mole fractions are constants, for sodium the wall mole fraction is 0.00276; for chloride, the wall mole fraction is 0.00254. Channel height h=20nm. ε=λ/h=0.040. Y∞=6. 70 0.045

0.04 Potential Velocity 0.035

0.03

0.025

0.02

0.015 Potential and Velocity

0.01

0.005

0 0 1 2 3 4 5 6 y

Figure 3.9 The potential and velocity profile near the wall for the asymptotic case for monovalent binary electrolyte. The wall mole fractions are constants, for sodium the wall mole fraction is 0.00276; for chloride, the wall mole fraction is 0.00254. Channel height h=20nm. ε=λ/h=0.040. Y∞=6.

0.045

0.04

0.035

0.03

0.025

0.02 Shear Stress 0.015

0.01

0.005

0 0 1 2 3 4 5 6 y

Figure 3.10 The shear stress near the wall for the asymptotic case for monovalent binary electrolyte. The wall mole fractions are constants, for sodium the wall mole fraction is 0.00276; for chloride, the wall mole fraction is 0.00254. Channel height h=20nm. ε=λ/h=0.040. Y∞=6. 71 Figure 3.8-Figure 3.10 show the mole fraction, potential and velocity, shear stress near the wall for ε<<1. The mole fraction of cations is 0.154M at the wall, and the mole fraction of anions is 0.142 at the wall. Here we solve only for the region inside EDL having outer solutions solved from equations (2.92) and (2.93).

Figure 3.8 shows the mole fractions near the wall for the asymptotic case. The analytical results are calculated according to equations (2.86), (2.92) and (2.93). For both cation species and anion species, the analytical results are same as the numerical results.

y y ∗ Y= = is the horizontal axis, where y* is the dimensional location on the direction ε λ of the channel width. In order to show a complete picture of the EDL, results must be plotted from Y=0 to Y=Y∞, where Y∞ is a number large enough, which represents the ratio of the thickness of the EDL to the Debye-length. Here we use Y∞=6, which is shown

* at the bottom right corner of Figure 3.8. At Y=Y∞=6, i.e. y =6λ, the concentrations of cations and anions both equal the corresponding outer solution, and the solution becomes electrically neutral as predicted by equation (2.93). In other words, the EDL effects vanish in the region which is away from the wall by more than 6 times Debye-length.

Figure 3.9 shows the potential and velocity profile near the wall. As Y increases, the slope of the curve decreases. At Y=6, the slope becomes null, and the potential and velocity reach maximum. Since the slope of velocity profile is the shear stress, this plot also shows the trend of shear stress in the region near the wall, as shown in Figure 3.10.

72 3.4 Monovalent Binary Electrolytes In a Channel-Reservoir System

For a channel-reservoir system as shown in Figure 2.2, the mole fractions of ionic species at the wall are unknown. The known is the concentration of ion species in the reservoir and the surface charge density on the wall. For this case, the electrochemical considerations we discussed previously can be applied. Figure 3.11-Figure 3.13 are the results, which show the mole fractions, potential and velocity, and shear stress across the channel. Here the electrolyte is still monovalent binary, and the concentration of both cations and anions are 0.1M in the reservoir. The channel height is 20nm, width is 20µm, and the surface charge density is -0.2C/m2.

0.14 Cation Anion 0.12

0.1

0.08

0.06 Mole Fraction

0.04

0.02

0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.11 Mole fractions across the channel for monovalent binary electrolytic solution. The channel height is 20nm, the width is 20µm. The surface charge density σ=−0.2C/m2, and the molarity of the solution is 0.1M in the reservoir. ε=λ/h= 0.0073.

73 Figure 3.11 shows the mole fractions of cations and anions across the channel. In the bulk, the concentration of cation species is same as the concentration of anion species, which implies the electrical neutrality there. At the wall, the mole fraction of cation species is as high as 0.13, however, the mole fraction of anion species is close to zero.

Inside the EDL, the mole fractions of cations and anions represent a Gouy-Chapman picture of EDL. This plot clearly shows that for negative surface charges, the cations are attracted to the wall, while the anions are repelled away from the wall, which causes the dominance of cations in the EDL.

2 Potential and Velociy and Potential

0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.12 Potential and velocity profile across the channel for monovalent binary electrolytic solution. The channel height is 20nm, the width is 20µm. The surface charge density σ=−0.2C/m2, and the molarity of the solution is 0.1M in the reservoir. ε=λ/h= 0.0073.

74 Figure 3.12 shows the potential and velocity profile across the channel. The curve is flat in the bulk, and the maximum of potential and velocity is hundreds times stronger than the maximum shown in Figure 3.3. This is caused by the much larger

[cation]/[anion] ratio at the wall for the real channel-reservoir system. Figure 3.13 shows the shear stress across the channel. For this case, the shear stress decreases very fast away from the wall, and in most part of the channel, the shear stress is zero.

200

150

100

Shear Stress -50

-100

-150

-200 0 0.2 0.4 0.6 0.8 1 y

Figure 3.13 Shear stress across the channel for monovalent binary electrolytic solution. The channel height is 20nm, the width is 20µm. The surface charge density σ=−0.2C/m2, and the molarity of the solution is 0.1M in the reservoir. ε=λ/h= 0.0073.

Figure 3.14-Figure 3.16 are the dimensional plots of the electric potential, electric mobility and shear stress across the channel, which can be calculated from equations

(2.56), (2.58) and (2.59). Figure 3.17-Figure 3.19 show the mole fraction, potential and

75 velocity, and shear stress for the case ε=O(1). The channel height is 4nm, other parameters are same as parameters in Figure 3.11-Figure 3.13. The solution is not electrically neutral in the bulk, and the potential and velocity profile is a parabolic curve.

Figure 3.20-Figure 3.22 show the mole fractions, potential and velocity, and shear stress near the wall for asymptotic cases. The surface charge density is assumed to be

σ=−0.0154 C/m2, which is consistent with the ORNL experimental data, as will be shown in Chapter 4. As the channel height decreases, the Debye length increases, and the volume of bulk portion where solution is electrically neutral decreases.

0.14

0.12

0.1

0.08

0.06 Potential (V) Potential

0.04

0.02

0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.14 Dimensional plot of the potential across the channel for monovalent binary electrolytic solution. The channel height is 20nm, the width is 20µm. The surface charge density σ=−0.2C/m2, and the molarity of the solution is 0.1M in the reservoir. ε=λ/h= 0.0073.

76 -3 x 10 1

0.8

0.6

0.4 Mobility (cm*cm/V/sec)

0.2

0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.15 Dimensional plot of the mobility across the channel for monovalent binary electrolytic solution. The channel height is 20nm, the width is 20µm. σ=−0.2C/m2, and the molarity of the solution is 0.1M in the reservoir. ε=λ/h= 0.0073.

2500

2000

1500

1000

500

-500

Shear Stress(N/m/m) -1000

-1500

-2000

-2500 0 0.2 0.4 0.6 0.8 1 y

Figure 3.16 Dimensional plot of the shear stress across the channel for monovalent binary electrolytic solution. The channel height is 20nm, the width is 20µm. σ=−0.2C/m2, and the molarity of the solution is 0.1M in the reservoir. ε=λ/h= 0.0073.

77 0.18 Cation Anion 0.16

0.14

0.12

0.1

0.08 Mole Fraction Mole 0.06

0.04

0.02

0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.17 Mole fractions across the channel for monovalent binary electrolytic solution. The channel height is 4nm, the width is 20µm. The surface charge density σ=−0.2C/m2, and the molarity of the solution is 0.1M in the reservoir. ε=λ/h=0.0331.

2 Potential and Velociy and Potential

0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.18 Potential and velocity profile across the channel for monovalent binary electrolytic solution. The channel height is 4nm, the width is 20µm. The surface charge density σ=−0.2C/m2, and the molarity of the solution is 0.1M in the reservoir. ε=λ/h=0.0331. The straight line is the asymptotic result. 78 50

-10 Shear Stress

-20

-30

-40

-50 0 0.2 0.4 0.6 0.8 1 y

Figure 3.19 Shear stress across the channel for monovalent binary electrolytic solution. The channel height is 4nm, the width is 20µm. The surface charge density σ=−0.2C/m2, and the molarity of the solution is 0.1M in the reservoir. ε=λ/h=0.0331.

Figure 3.20 Mole fractions near the wall for the asymptotic case for monovalent binary electrolyte. The channel heights h=25, 50 and 100 nm. The width of the channel is 100 µm, the surface charge density σ=−0.0154C/m2, and the concentration of electrolytes in the reservoir is 0.002M. 79

Figure 3.21 Potential and velocity profiles near the wall for the asymptotic case for monovalent binary electrolyte. The channel heights h=25, 50 and 100 nm. The width of the channel is 100 µm, the surface charge density σ=−0.0154C/m2, and the concentration of electrolytes in the reservoir is 0.002M.

Figure 3.22 Shear stress curves near the wall for the asymptotic case for monovalent binary electrolyte. The channel heights h=25, 50 and 100 nm. The width of the channel is 100 µm, the surface charge density σ=−0.0154C/m2, and the concentration of electrolytes in the reservoir is 0.002M. 80 3.5 Using Scaling Laws to Simplify Governing Equations

Equations (2.52)-(2.54) can be simplified further by using scaling laws below

X ∗ = i X i 0 (3.36) X 1

ε 2 δ 2 = (3.37) β 0 X 1

∗ ∂  ∂X ∗ ∂φ   i +  =  zi X i  0 (3.38) ∂y  ∂y ∂y 

∂ 2φ N δ 2 = − z X ∗ (3.39) 2 ∑ i i ∂y i=1

2 N ∂ u ∗ δ 2 = − z X (3.40) 2 ∑ i i ∂y i=1

0 where X1 is the wall mole fraction of ion species 1. For convenience, we usually choose the most populous ions species as ion species 1, for example, the sodium ions in Na2SO4.

The boundary conditions (2.64)-(2.66) become

u=0 y=0 and y=1 (3.41)

φ=0 y=0 and y=1 (3.42)

∗ = γ X i i y=0 and y=1 (3.43)

X 0 γ = i where i 0 is the ratio of wall mole fraction of ion species i to the wall mole fraction X 1 of ion species 1.

After applying the scaling laws to governing equations, we find that the solution of governing equations (3.38)-(3.40) only depends on factors zi, γi and δ. For monovalent

81 binary electrolytes such as sodium chloride, there are z1=1, γ1=1 for sodium, z2=-1, γ2=γ for chloride. Thus only two parameters δ and γ are required for the governing equations.

On the other hand, this means that the mole fraction distribution, potential and velocity profile etc. only depends on δ and γ .

From equation (3.37), we have

ε 2 λ2 δ 2 = = (3.44) β 0 β 0 2 X 1 X 1 h

0 2 and so δ is inversely proportional to X1 and h . Thus, the solution of governing equations

(3.38)-(3.40) for small channels can be mimicked in large channels at much lower concentrations, which is much easier to achieve for current experimental techniques.

X 0 c 0 γ = i β = c 0 = i Substituting i 0 , and X i into equation (3.37), X 1 I c

ε 2 ε 2 ε 2 δ 2 = = = = ε 2 z 2γ (3.45) 0 0 ∑ i i βX c 0 X 1 ⋅ X 1 i I 1 2 0 ∑ zi ci c i

For monovalent binary electrolytes, equation (3.45) becomes

δ 2 = ε 2 ()1+ γ (3.46) which implies that for γ<<1, δ ≈ ε. The singular perturbation solution, equations (2.92) and (2.93) become

1  X 0  1 φ = ln 1  = − lnγ (3.47) o  0  2  X 2  2 and

82 X 0 X = X 0 X 0 = X 0 γ = 2 (3.48) o 1 2 1 γ

For the case where γ=0.92, Figure 3.23-Figure 3.25 show the mole fractions, potential and velocity, and shear stress across the channel for monovalent binary electrolytic solution by using the scaling laws.

The similarity between these plots and Figure 3.2-Figure 3.4 verify the effort of using scaling laws to simplify the governing equations. Figure 3.23 Figure 3.25 represent a Debye-Hückel view of EDL.

-3 x 10 5.3 cations 5.25 anions

5.2 δ=0.5

5.15

δ=0.25 5.1 δ=0.0075 δ=0.075 δ =0.0075 δ 5.05 δ=0.1 δ=0.075 δ=0.25 Mole fraction 5

δ=0.5 4.95

4.9

4.85 0 0.2 0.4 0.6 0.8 1 y

Figure 3.23 The mole fraction distribution pattern across the channel for monovalent 0 binary electrolyte. γ=0.92. The molarity of cation species c1 =0.154M at the wall. These cases agree with the Debye-Hückel theory of EDL.

83 0.045 φ=-lnγ/2 δ 0.04 =0.0075 δ=0.1 δ=0.075

0.035 δ=0.25

0.03

0.025

0.02 δ=0.5

0.015 Potential and velocity

0.01

0.005

0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.24 The potential and velocity profile across the channel for monovalent binary 0 electrolyte. γ=0.92. The molarity of cation species c1 =0.154M at the wall. The asymptotic outer solution φο=−(lnγ)/2 is plotted on the top as the horizontal line. These cases agree with the Debye-Hückel theory of EDL.

0.8

δ=0.075 0.6

δ=0.1 0.4 δ=0.25

0.2 δ=0.5

Shear Stress -0.2

-0.4

-0.6 γ=0.92

-0.8 0 0.2 0.4 0.6 0.8 1 y

Figure 3.25 The shear stress across the channel for monovalent binary electrolyte. 0 γ=0.92, The molarity of cation species c1 =0.154M at the wall. These cases agree with the Debye-Hückel theory of EDL. 84 From equation (2.22), we have

X 0 − X 0 X 0 ()1− γ 1− γ 1 2 = 1 = ≈ 0.042 << 1 0 0 X 0 γ γ 2 X 1 X 2 2 1 2

which indicates that the Debye-Hückel theory is valid for γ=0.92, as discussed in

§2.2.For cases where γ=0.1, Figure 3.26-Figure 3.28 show the mole fractions, potential and velocity, and shear stress across the channel for monovalent binary electrolytic solution, respectively. Figure 3.26-Figure 3.28 represents a Gouy-Chapman picture of the

X 0 − X 0 X 0 ()1− γ 1− γ EDL, because 1 2 = 1 = ≈ 1.42 . 0 0 X 0 γ γ 2 X 1 X 2 2 1 2

-3 x 10 3 cations anions γ=0.1 2.5

1.5 δ=0.25

Mole Fraction δ=0.1 1 δ=0.05

δ=0.1 0.5 δ=0.25

0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.26 The mole fraction of ion species across the channel for monovalent binary 0 electrolyte. γ=0.1. The molarity of cation species c1 =0.154M at the wall. The Gouy- Chapman view of EDL is achieved.

85 1.4

φ=-lnγ/2~1.15 1.2

δ=0.05 1 δ=0.1 0.8

δ=0.25 0.6

Potential and Velocity 0.4

0.2 γ=0.1

0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.27 The potential and velocity profile across the channel for monovalent binary 0 electrolyte. γ=0.1. The molarity of cation species c1 =0.154M at the wall. The asymptotic outer solution φο=−(lnγ)/2 is plotted on the top as the horizontal line. The Gouy-Chapman view of EDL is achieved.

15 δ=0.05

5 δ=0.1

δ=0.25 0

Shear Stress -5

-10

-15 γ=0.1

-20 0 0.2 0.4 0.6 0.8 1 y

Figure 3.28 The shear stress across the channel for monovalent binary electrolyte. γ=0.1, 0 The molarity of cation species c1 =0.154M at the wall. The Gouy-Chapman view of EDL is achieved.

86 For cases where δ=0.1, Figure 3.29-Figure 3.31 show the mole fractions, potential and velocity, and shear stress across the channel for monovalent binary electrolytic solution, respectively. As γ decreases from 0.92 to 0.02, the Debye-Hückel limit is violated, and the Gouy-Chapman theory emerges. For cases where δ=0.04, similar changes happen for γ decreases from 0.92 to 0.02, as shown in Figure 3.32 - Figure 3.34.

However, the EDL shrinks for δ decreases from 0.1 to 0.04, because the Debye length

0 1/2 λ=εh=δh(βX1 ) become less.

-3 x 10 3

γ=0.92 2.5

γ=0.5 2

cations 1.5 δ =0.1 anions

Mole Fraction Mole γ 1 =0.1

0.5 γ=0.02

0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.29 The mole fraction of ion species across the channel for monovalent binary 0 electrolyte. δ=0.1. The molarity of cation species c1 =0.154M at the wall.

87 2 φ=-lnγ/2~1.96 1.8 δ=0.1 1.6 γ=0.02

1.4

φ=-lnγ/2~1.15 1.2

γ=0.1 1

0.8

Potential and Velocity 0.6

φ=-lnγ/2~0.35 γ=0.5 0.4

0.2 γ=0.92 0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.30 The potential and velocity profile across the channel for monovalent binary 0 electrolyte. δ=0.1. The molarity of cation species c1 =0.154M at the wall. The asymptotic outer solution φο=−(lnγ)/2≈0.0417 for γ=0.92 is omitted in the plot.

γ=0.02 10

γ=0.1

5 γ=0.5

0 γ=0.92 Shear Stress -5

-10

-15 0 0.2 0.4 0.6 0.8 1 y

Figure 3.31 The shear stress across the channel for monovalent binary electrolyte. δ=0.1, 0 The molarity of cation species c1 =0.154M at the wall.

88 -3 x 10 3

γ=0.92 2.5

2 cations δ δ=0.04 anions 1.5 Mole Fraction Mole 1 γ=0.1

0.5 γ=0.02

0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.32 The mole fraction of ion species across the channel for monovalent binary 0 electrolyte. δ=0.04. The molarity of cation species c1 =0.154M at the wall.

2 φ=-lnγ/2~1.96 γ=0.02 1.8

1.6

1.4

φ=-lnγ/2~1.15 1.2

γ=0.1 1

0.8

Potential and Velocity and Potential 0.6 δ=0.04

0.4

0.2 γ=0.92 0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.33 The potential and velocity profile across the channel for monovalent binary 0 electrolyte. δ=0.04. The molarity of cation species c1 =0.154M at the wall. The asymptotic outer solution φο=−lnγ/2≈0.0417 for γ=0.92 is omitted in the plot.

89 40

20 γ=0.02

10 γ=0.1

0 γ=0.92

Shear Stress -10

-20

-30

-40 0 0.2 0.4 0.6 0.8 1 y

Figure 3.34 The shear stress across the channel for monovalent binary electrolyte. 0 δ=0.04, The molarity of cation species c1 =0.154M at the wall.

3.6 Results for Multivalent Binary Electrolyte and Multi-component

Electrolytes.

The characteristics of electroosmosis in a channel for monovalent binary electrolytes are discussed in the previous section. One example of such solutions is phosphate buffered saline (PBS). If multivalent ion species exist in the solution, or if the number of ion species is higher than 2, the general governing equations (2.36), (2.44) and (2.51), the governing equations (2.52)-(2.54) for a one-dimensional channel, the singular perturbation equations (2.70)-(2.72) for ε<<1, and the “after scaling laws” equations (3.38)-(3.40) are still valid. The governing equations are solved using the

Thomas Algorithm same as before.

Reservoir C hannels

(a)

L=3.5µm

6µm

µ W=44 m

2µm (b)

Figure 3.35 The reservoir and the channels in iMEDDs’ experiments. The iMEDD membrane with nanochannels across it is located in the center of the reservoir. Through the nanochannels, the liquid in one side of the reservoir is driven to the other side by electroosmosis. The thickness of the membrane is 3.5µm. Parallel channels with widths of 44µm and heights in nanometer scale are etched on the membrane by microfabrication. The horizontal distance between two adjacent channels is 2µm.

91 In general, four-digit accuracy is achieved in solutions for 161 points across the channel in all of the variables for all of the runs made. In all cases, a unique, physically reasonable solution for the wall mole fractions was found. We investigate specifically

+ - + - 2+ 2- ions such as Na , Cl , K , H2PO4 , Ca and HPO4 , however the numerical results apply equally well to any ionic species of the same valence. Note that each of the ions being considered here are small inorganic ions, which have ionic radii on the order of 3Å.

Figure 3.35(a) shows the geometry of the channel-reservoir system in iMEDD’s’ experiments. A microfabricated silicon membrane is located between the upstream and downstream reservoirs. Across the membrane, there are vertical nanochannels connecting the upstream reservoir to the downstream reservoir. Figure 3.35(b) shows the geometry of an iMEDD membrane. More details of the iMEDD membrane and iMEDD’s experiments will be discussed in next chapter. Here in this section, the geometry of this channel- reservoir system is adopted as the default. The surface charge density on the iMEDD membrane is −0.2C/m2, the minus sign indicates that the surface charges are negative.

Table 3.1 lists the molarity of different ionic species in the reservoirs for the five mixtures of interest. The ionic strength of all these mixtures is a constant, which is

0.333M. Mixture 1 is NaCl aqueous solution and other ionic species are added to produce

+ - 2+ 2- mixtures 2, 3, 4 and 5, which contain K , H2PO4 , Ca and HPO4 , respectively. For any of these four solutions, the molarity of the third ionic species in the reservoir is 0.008M,

2- which is the molarity of HPO4 in PBS.

92 + - + - 2+ 2- Na Cl K H2PO4 Ca HPO4 Solution 1 0.16644 0.16644 - - - - Solution 2 0.15834 0.16644 0.00810 - - - Solution 3 0.16644 0.15834 - 0.00810 - - Solution 4 0.14214 0.15834 - - 0.00810 - Solution 5 0.15834 0.14214 - - - 0.00810 PBS 0.15420 0.14067 0.00414 0.00147 - 0.00810

Table 3.1 Molarity of ions in the reservoir, units=M; the ionic strength is 0.33288M.

Table 3.2 shows the channel average molarity of all ionic species in the nanochannels, obtained from the electrochemical equilibrium conditions for h=20nm.

The table shows that the concentration of each cation species in the nanochannel increases substantially, while the concentration of each anion decreases due to the negatively charged channel walls.

+ - + - 2+ 2- Na Cl K H2PO4 Ca HPO4 Solution 1 0.2997 0.0924 - - - - Solution 2 0.2851 0.0924 0.0146 - - - Solution 3 0.2997 0.0879 - 0.0045 - - Solution 4 0.2484 0.0906 - - 0.0247 - Solution 5 0.2897 0.0777 - - - 0.0024 PBS 0.2822 0.0769 0.0076 0.0008 - 0.0024

Table 3.2 Average molarity of ions in the nanochannel. The channel height is 20nm.

93 Flow rate + - + - 2+ 2- Na Cl K H2PO4 Ca HPO4 λ (nm) ε (µl/min) Solution 10.1235 0.0011 - - - - 0.136 0.0068 2.98 1 Solution 9.6308 0.0011 0.3961 - - - 0.136 0.0068 2.99 2 Solution 10.1222 0.0011 - 0.0001 - - 0.136 0.0068 2.98 3 Solution 1.4258 0.0109 - - 0.4358 - 0.634 0.0317 1.41 4 Solution 9.6962 0.0009 - - - 0 0.138 0.0069 2.98 5

PBS 9.6871 0.0009 0.2095 0 - 0.0001 0.138 0.0069 3.02

Table 3.3 Molarity of different ions at the walls of the channel. The channel height is 20nm. The thickness of the EDL is given in this table, and the volume flow rate is shown in the last column, for an imposed voltage of 0.05V over a channel length of 3.5 µm.

Table 3.3 shows the molarity of all of the ionic species at the channel walls for h=20nm. From this table, we note that there are major differences between the results for solution 4 and the other solutions. The concentration of Na+ at the wall in solution 4 is much less than in the other solutions, while the Cl- concentration is much higher in solution 4 than in the other solutions. The thickness of the EDL in solution 4 is 0.634nm, which is more than 4 times larger than the EDL thickness in other solutions.

Simultaneously, the flow rate for solution 4 is less than a half of the flow rate for the other solutions. The results for 4nm high channel are shown in Appendix B.

These differences can be explained by the existence of Ca2+ in solution 4. Because a Ca2+ ion has two unit charges, and the Na+ has one, the electrical attracting forces

94 exerted by the negatively charged channel wall is two fold stronger for a Ca2+ than a Na+ at the same distance. Therefore Ca2+ ions are accumulated in the EDL and crowd out the

Na+ ions. On the other hand, at the same distance, the electrical repelling force between the Ca2+ and Na+ ions is two-fold stronger than the repelling force between two Na+ ions, and the repelling force between two Ca+ ions is four-fold stronger than that between two

Na+ ions. As the result, the charge density inside the EDL is much lower for solution 4 than for other solutions, and the EDL thickness in solution 4 is much larger than others.

On the other hand, if the surface charge is positive, a corresponding calculation also indicates that multivalent anions will be the dominating ion species. Therefore, in the general case multivalent counter-ions dominate the formation of the EDL, which leads to a decrease of charge density in EDL. On the other hand, multivalent co-ions have much less effect, because most co-ions are repelled from the channel wall, and therefore far away from the EDL.

Figure 3.36 and Figure 3.37 show the electric potential and velocity profile in

20nm and 4nm- high channels for the mixtures 1 through 5 and for PBS. In these dimensionless plots, the potential curve coincides with the velocity profile as the governing equations predict. In both figures, the existence of divalent Ca2+ ions in solution 4 reduces the potential and velocity of the flow by more than 50 percent. The horizontal line above the curves is the asymptotic value of the dimensionless potential φo in the outer solution, which is valid for ε<<1 for a two component solution.

95 5 1/2ln(g0/f0) Solution 1,2,3,5 and PBS 4

Solution 4 2

Potential and Velociy 1

0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.36 Dimensionless velocity and electric potential inside the channel, the external electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=20nm. ε=λ/h=0.0068 for solution 1, 2, and 3; ε=λ/h=0.0069 for solution 5 and PBS; ε=0.0317 for solution 4.

5 1/2ln(g0/f0) 4

Solution 1,2,3,5 and PBS 3

Solution 4 Potential and Velociy 1

0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.37 Dimensionless velocity profiles and electric potential inside the channel for solution 1 through 5, the external electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=4nm. ε=λ/h=0.033 for solution 1, 2, 3, 5 and PBS; ε=λ/h=0.021 for solution 4.

96 0.14 Na Cl 0.12

0.1

0.08

0.06 Mole Fraction

0.04

0.02

0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.38 Mole fractions of Na+ and Cl- inside the channel for solution 1; the electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=20nm, ε=λ/h=0.0068.

The mole fractions of Na+ and Cl- ions in the channel for mixture 1 are shown in

Figure 3.38 as a function of y. The mole fraction of Na+ is higher than that of Cl-, therefore at a 20nm-high channel, the system is not electrical neutral except in the region near the center of the channel. At the channel wall, the mole fraction of Na+ is significantly higher than Cl-, and the picture of ion concentration at the wall represents a

Gouy-Chapman [43] description of the EDL.

Figure 3.39 and Figure 3.40 show that mole fraction plots obtained for mixtures 2 and 3 are similar to mixture 1, which implies that monovalent ions and multivalent co- ions do not affect the ionic distribution much. In order to check whether electrical neutrality is satisfied in the channel, we calculate the net charge density of the solution,

- and compare it with cf, which is the volume charge density of SiO calculated from

97 equation (15). The net charge density we get from integration is 0.2071M, and the net charge is positive. The assumed volume charge density cf for the negative surface charges is 0.2073M. Therefore, the result is consistent with the assumption of electrical neutrality in the channel system.

Figure 3.38 represents the Gouy-Chapman picture of the EDL. According to equation (2.22), for cases where the surface charge density is low, for instance

σ=−0.001C/m2, the Debye-Hückel limit is obtained, and the mole fraction distribution calculated from this theory turns out to be a Debye-Hückel view of EDL, as shown in

Figure 3.41.

For solution 4, where Ca2+ is added, the mole fractions of the ions are shown in

Figure 3.42. The mole fraction of Na+ at the wall is much lower than that in Figure 3.38, which is consistent with the explanation given above. Simultaneously, in the region near the wall, the slope of the cation mole fraction curve in Figure 3.42 is less than the slopes in Figure 3.38.

+ - - Figure 3.43 shows the mole fractions of Na , Cl and HPO4 inside the channel for solution 5. Compared Figure 3.43 with Figure 3.38 and Figure 3.42, it is clearly seen that

2- multivalent co-ions (HPO4 ) do not affect the EDL thickness and ionic distribution as much as multivalent counter-ions do.

For PBS, the mole fractions of different ion species in the 20nm channel are shown in Figure 3.44 and Figure 3.45. Both cations, Na+ and K+ are more concentrated in the EDL than in the bulk, and all anions are less concentrated in the EDL than in the bulk.

Figure 3.45 also shows that even the dilute cation species, K+, is much more concentrated in the EDL than Cl-, the major anion species.

98 Figure 3.46 shows the mole fractions of Na+ and Cl- across the channel. The channel height is 4nm and ε=λ/h=0.0326. Comparing Figure 3.46 with Figure 3.38, it can be seen that as the channel height decreases, counter-ions in the channel become more concentrated, especially at the wall; simultaneously, co-ions become less concentrated. the wall mole fraction of cations higher in the smaller channel, and the concentration of anions in the bulk is less for a smaller channel. In Figure 3.46, the mole fraction of sodium ions in the bulk is higher than the mole fractions of chloride ions, which implies that there are positive net charges for solution in the core.

0.14 Na K 0.12 Cl

0.1

0.08

0.06 Mole Fraction

0.04

0.02

0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.39 Mole fractions of Na+, Cl- and K+ inside the channel for solution 2; the electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=20nm, ε=λ/h=0.0068.

99 0.14 Na Cl 0.12 H2PO4

0.1

0.08

0.06 Mole Fraction Mole

0.04

0.02

0 0 0.2 0.4 0.6 0.8 1 y

+ - - Figure 3.40 Mole fractions of Na , Cl and H2PO4 inside the channel for solution 3; the electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=20nm, ε=λ/h=0.0068.

-4 x 10 4.2 Sodium Tetraborate 4

3.8

3.6

Mole Fraction Mole 3.4

3.2

3 0 0.2 0.4 0.6 0.8 1 y

Figure 3.41 Mole fractions of sodium (monovalent cation) and tetraborate (monovalent anion) showing the Debye-Hückel picture of the EDL at small surface charge density. The channel height is 83nm, and its width is 20.3µm. The concentration of sodium tetraborate in the reservoir is assumed to be 2mM, and the surface charge density is assumed to be -0.001C/m2. 100 0.025 Na Cl Ca 0.02

0.015

0.01 Mole Fraction Mole

0.005

0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.42 Mole fractions of Na+, Cl- and Ca2+ inside the channel for solution 4; the electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=20nm, ε=λ/h=0.0317.

0.16 Na Cl 0.14 HPO4

0.12

0.1

0.08

Mole Fraction 0.06

0.04

0.02

0 0 0.2 0.4 0.6 0.8 1 y

+ - - Figure 3.43 Mole fractions of Na , Cl and HPO4 inside the channel for solution 5; the electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=20nm, ε=λ/h=0.0069.

101 0.16 Na K 0.14 Cl H2PO4 HPO4 0.12

0.1

0.08

Mole Fraction 0.06

0.04

0.02

0 0 0.2 0.4 0.6 0.8 1 y

+ + - - 2- Figure 3.44 Mole fractions of Na , K , Cl , H2PO4 and HPO4 inside the channel for PBS. The electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=20nm and ε=λ/h=0.0069.

-3 x 10 3.5 K Cl 3 H2PO4 HPO4

2.5

1.5 Mole Fraction

0.5

0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.45 A blown up version of the region near zero mole fraction in Figure 3.44. The electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=20nm and ε=λ/h=0.0069.

102 0.18 Na Cl 0.16

0.14

0.12

0.1

0.08 Mole Fraction 0.06

0.04

0.02

0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.46 Mole fractions of Na+ and Cl- inside the channel for solution 1; the electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=4nm, and ε=λ/h=0.0326.

0.035 Na Cl 0.03 Ca

0.025

0.02

0.015 Mole Fraction

0.01

0.005

0 0 0.2 0.4 0.6 0.8 1 y

Figure 3.47 Mole fractions of Na+, Cl- and Ca2+ inside the channel for solution 4.The electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=4nm. ε=λ/h=0.021.

103 Figure 3.47 shows the mole fractions of Na+, Cl- and Ca2+ in the channel for solution 4 for a channel height h=4 nm. Comparing Figure 3.47 with Figure 3.43, it can be seen that as the channel height decreases, counter-ions in the channel become more concentrated, especially at the wall; simultaneously, co-ions become less concentrated.

Comparing Figure 3.47 with Figure 3.46, it can be seen that the existence of divalent cation Ca2+ inside the channel distinctly reduces the concentration of monovalent sodium ions in the channel, especially near the wall.

3000

2000

1000 Solution 1, 2, 3, 5 and PBS Solution 4 (Na+, Cl-, Ca2+) 0

-1000 Sheer Stress(N/m/m) Sheer

-2000

-3000 0 0.2 0.4 0.6 0.8 1 y

Figure 3.48 The shear stress inside the channel for solution 1through 5. The electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=20nm. ε=λ/h=0.0068 for solutions 1, 2 and 3; and ε=λ/h=0.0069 for solution 5 and PBS; ε=λ/h=0.0317 for solution 4.

104 Figure 3.48 shows the shear stress inside a 20nm-high channel. In the center of the channel, there is a flat portion on the shear stress curve, where the velocity is constant. This non-shear region is good for transport of particles such as globular protein.

For solution 1, 2, 3, 5 and PBS, there are large shear stress in the region near the wall, and this region is good analysis of fluidic properties. For solution 4 which contains multivalent counter-ions, the shear stress near the channel wall is much less than the shear stress in other solutions, which implies that electrolyte solution containing multivalent counter-ions is a better medium for transporting particles.

3000

2000

Solution 1, 2, 3, 5 and PBS 1000

+ - 2+ 0 Solution 4 (Na , Cl , Ca )

-1000 Sheer Stress(N/m/m)

-2000

-3000 0 0.2 0.4 0.6 0.8 1 y

Figure 3.49 The shear stress inside the channel for solution 1 through 5; the electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The channel height h=4nm, ε=λ/h=0.033 for solutions 1, 2, 3, 5 and PBS, ε=λ/h=0.021 for solution 4.

105 The shear stress in the 4 nm channel is shown in Figure 3.49 and it is notable that the shear stress in the bulk portion of the channel is not zero, which makes this channel not suitable for transporting macromolecules. For small channels as shown in this case, more concentrated working fluid is required to maintain a non-shear region at the core, which is implied in equation (3.44). Figure 3.50 shows the effect of channel height on the potential and velocity profile for solution 4. As seen in the figures, the velocity in the bulk portion increases as the channel height increases. However, the increase becomes less and less as the channel height increases.

2.5 h=20nm h=13nm 2

1.5 h=4nm

Potential and Velociy and Potential 0.5

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y

Figure 3.50 Dimensionless velocity profiles inside the channel for solution 4. The electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. ε=λ/h=0.021 for h=4nm, ε=0.0317 for h=20nm,

106 800

600

400 h=4nm

200 h=13nm 0 h=20nm

-200

Sheer Stress(N/m/m) -400

-600

-800 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y

Figure 3.51 The shear stress inside the channel for solution 4, with the electric field corresponding to 0.05 volts over a channel of length L=3.5 µm.

Figure 3.51 shows that the shear stress in a smaller channel is higher than the shear stress in a large channel. For same fluids, larger channel is better for transport of macromolecules than smaller channel is.

The ionic concentrations at the wall are shown in Figure 3.52(a) and Figure

3.52(b). The wall molarity of co-ions such as Cl- in this case tends to decrease as the channel height decreases, while the wall molarity of counter-ions such as Na+ and Ca2+ here increases as the channel height decreases. Furthermore, the molarity of divalent counter-ions increases faster than monovalent counter-ions which is the reason for the cross-over on Figure 3.52(a). Note that the concentrations on this figure reach an asymptotic limit for channel heights greater than 25 nm.

107 4 Na+ 3.5 Ca++ 3

2.5

1.5 Wall Molarity (M) Molarity Wall 1

0.5

0 0 5 10 15 20 25 30 Channel height (nm)

(a)

0.014

Cl- 0.012

0.01

0.008

0.006

Wall Molarity (M) Molarity Wall 0.004

0.002

0 0 5 10 15 20 25 30 Channel height (nm)

(b)

Figure 3.52 The molarity of Na+, Ca2+ and Cl- at the channel wall for channels with different heights for solution 4. The external electric field corresponds to 0.05 volts over a channel of length L=3.5 µm.

108 4.5 Na+, Cl- and Ca++ 4 NaCl 3.5 PBS

2.5

1.5

Flow rate (microliter/min) 1

0.5

0 0 5 10 15 20 25 30 Channel height (nm)

Figure 3.53 The electroosmotic volume flow rate as a function of channel height at a constant electric field, which corresponds to 0.05 volts over a channel of length L=3.5 µm. For all three solutions, their ionic strength in the reservoir is 0.33288M and the channel width is 44µ m.

The electroosmotic flow rates across membranes of differing heights are calculated for three different mixtures: solution 1, solution 4 and PBS. The results shown here are based on a channel width of 44 µm as in the iMEDD configuration. From Table

2, the ionic strength of the solution in the reservoir is 0.333M for all three cases, and the voltage drop across the membrane is assumed to be a constant 0.05V. The results are shown in Figure 3.53. The magnitude of the volume flow rates is of the order of 1µl/min.

As the channel height increases, the flow rate increases linearly for all three mixtures as expected. Note that for all channel heights, about 50 percent less flow is always produced

109 with solution 4 as the working fluid, compared with the results of using the other mixtures.

The volume flow rates for mixtures containing divalent counter-ions and monovalent counter-ions at different molarities are also calculated, and the results are shown in Figure 3.54. Here again the wall is negatively charged, Na+ is chosen to represent monovalent counter-ions, and Ca2+ is chosen to represent divalent counter-ions.

The volume charge density in the reservoir is kept as same as that in solution 4. From this figure, we can see that the volume flow rate drops significantly at very low divalent counter-ions concentration.

3.5

2.5

1.5

1 Solution 4 Flow rate (microliter/min) rate Flow 0.5

0 0 0.010.020.030.040.050.06 [Ca++] (M)

Figure 3.54 The effect of divalent counter-ions on the electroosmotic volume flow rate across the iMEDD membrane; the electric field corresponds to 0.05 volts over a channel of length L=3.5 µm. The molarity of charges of the ions in the reservoir is fixed at 0.31668M. The channel height is 20nm. The x-axis is the molarity of Ca2+ in the reservoir; the y-axis is the volume flow rate across the nanochannel-membrane with iMEDD geometry.

110 For multi-component or multivalent electrolytic solution, if ε<<1, the singular perturbation method works as well as for the monovalent binary system. For instance, for a tertiary system, the outer solution φ0 can be solved according to equations (2.94)-(2.97), and then the outer solutions Xi0 can be solved from equation (2.84). Then the inner solutions can be solved asymptotically.

Figure 3.55-Figure 3.57 show the asymptotic solutions for a +1:+2:−1 electrolytic system in a single nanochannel. The surface charge density σ=−0.0154C/m2 on the channel wall.

-3 x 10 7

100nm 6 50nm 25nm

Monovalent Cation 3 Mole Fraction

1 Monovalent Anion

Divalent Cation 0 0 0.05 0.1 0.15 0.2 0.25 y

Figure 3.55 Mole fractions of monovalent cations, divalent cations and monovalent anions near the wall. The channel heights h=25, 50 and 100 nm. The width of the channel is 100 µm, the surface charge density σ=−0.0154C/m2, and the concentration of electrolytes in the reservoir is 0.002M.

111

Figure 3.56 Potential and velocity profiles of monovalent cations, divalent cations and monovalent anions near the wall.

Figure 3.57 Shear stress of monovalent cations, divalent cations and monovalent anions near the wall.

112 CHAPTER 4

COMPARISON WITH EXPERIMENTS

4.1 Introduction

The results of the EOF model are discussed in the previous chapter for both monovalent binary electrolytes and multivalent multi-component electrolytes. In this chapter, the parameters in the model are specified according to the systems in the experiments of iMEDD inc., Oak Ridge National Laboratory, and Georgia Institute of

Technology. Then the results of the model are compared with the experimental results from iMEDD, ORNL and Georgia Tech. The working fluid in iMEDD experiments is

+ + - - phosphate buffered saline, which contains five ion species: Na , K , Cl , H2PO4 and

2- HPO4 . We find good agreement between the results of the model and the experiments, which indicates that this model represents well the EOF of multivalent multi-component electrolytes. The working fluid in ORNL experiments is sodium tetraborate aqueous-

2- methanol solution. At normal concentration, the tetraborate ions exist as B4O5(OH)4 in

- the solution. However, for dilute solutions, the tetraborate ions become B(OH)4 after hydration. The dissociation equation of sodium tetraborate in water is:

+ ↔ + + ()+ ()− Na2 B4O7 7H 2O 2Na 2B OH 3 2B OH 4

113 In the ORNL experiments, the molarity of sodium tetraborate varies from 150mM to

0.02mM, which can be treated as dilute solutions. Therefore, the ion species in the ORNL

+ - working fluid are Na and B(OH)4 . The excellent results of the comparison between the results of the model and the ORNL experimental data indicate the accuracy of the model for monovalent binary electrolytes solutions. The working fluid in Georgia Tech experiments is a sodium tetraborate aqueous solution, and the ionicstrength of the sodium tetraborate solution in the reservoirs varies from 0.2mM to 36mM. The model and the experimental data agree very well.

4.2 Comparison with the iMEDD Experiments

The first set of data that will be compared with the model is originally taken by iMEDD, Inc. of Columbus, Ohio. Using techniques developed by Hansford. et. al. [54], iMEDD fabricate a variety of silicon membranes consisting of a series of nanochannels, as show in Figure 4.1, and they measure electroosmotic flow rate across those membranes. The geometry of the iMEDD nanochannel membrane and the testing apparatus are shown in Figure 3.35.The membranes are 3.5mm long, 1.5mm wide, and

3.5µm thick. In iMEDD’s configuration, there are 47,500 nanochannels on each of these membranes. The nanochannels themselves are 3.5 µm in length and 44 µm in width. iMEDD inc. fabricate and made tests to channels with different heights: 4nm, 7nm,

13nm, 20nm, 27nm and 49nm in particular.

The surfaces of the channel wall in a silicon membrane are negatively charged

[17], which causes the accumulation of cations near the channel wall to form the electrical double layer. The aqueous solutions being considered in this work are dilute, so

114 the mole fraction of any ionic species is much less than the mole fraction of the water.

For large channel height, electro-neutrality in the core of the channel is preserved as shown in Figure 3.11. The symmetrical case in which the two walls have the same surface charge density is considered at constant temperature and atmospheric pressure. In the comparison with experimental data, we also assume that the reservoir is electrically neutral as is common practice in experiments.

Nanopore Wafer Channel Array

Channel Cross Section

Figure 4.1 Nanopore membranes fabricated by iMEDD inc. Arrays of membranes are fabricated on a silicon wafer; on each membrane, there are vertical nanochannel arrays fabricated. The width of the channels is 44µm, and the distance between two arrays is 6µm, as shown in the picture at top right and in Figure 3.35. The bottom picture shows the cross section of the two nanochannels and their adjacent area. The material between the two channels is polysilicon. (http://www.imeddinc.com/technology%20platform.htm, Sept 30, 2003, from Mike Cohen et.al., iMEDD inc.) 115

Figure 4.2 The Ussing chamber used in iMEDD experiments.

iMEDD has performed several experiments to test the efficiency of using nanopore membranes to transport fluid containing multivalent ionic species. The major instrument in iMEDD’s experiment is a snap Ussing chamber produced by WPI, as shown in Figure 4.2. In their typical experimental apparatus, a nanopore membrane is placed in the center of an Ussing chamber. The two reservoirs are filled with buffer solution. The buffer used in iMEDD’s experiments is PBS, ten times diluted PBS, and

100 times diluted PBS. 17 Volts DC is applied on the two electrodes at the far ends of the two reservoirs. It should be noted that the voltage drop across the membrane is much less than 17 Volts, because most of the 17 Volts is dissipated outside the nanopore membrane, due to the resistance of the buffer in the reservoir. The experimental parameters include channel dimensions, the environmental temperature, and the mixture properties: density,

116 viscosity and dielectric constant and the molarity in the upstream reservoir. The voltage drop across the membrane is also required and this value is calculated from a resistance model for each channel height. The electroosmotic flow rates are measured in iMEDD’s experiments to the accuracy of 0.1µl/min [55].

Table 4.1 shows the electroosmotic volume flow rates measured in the iMEDD experiments. The buffer being used inside the reservoirs are PBS (column two and column three), ten times diluted PBS (column four), and 100 times diluted PBS (column five). The concentration of different ion species in the PBS is shown in Table 3.1. In the ten times diluted PBS, the concentration of all ion species are 10 times less than their concentration in the PBS. In the one hundred times diluted PBS solution, the concentration of all ions species are 100 times less.

Channel PBS×1, Flow PBS×1, Flow PBS×0.1, Flow PBS×0.01,Flow Height (nm) Rate, (µl/min) Rate, (µl/min) Rate, (µl/min) Rate, (µl/min)

4 N/A 0.37 0.56 N/A 7 1.05 0.98 0.48 0.01 13 1.4 1.33 0.81 0.176 20 1.23 1.15 0.77 0.101 27 0.98 N/A N/A N/A 49 0.52 0.455 0.98 2.28

Table 4.1 The electroosmotic volume flow rates measured by iMEDD, inc. The environmental temperature T=298K, the external voltage over the whole channel- reservoir system is 17V. (from Rob Walczak, iMEDD inc., 2001) 117 Because PBS and diluted PBS solution are all at low molarity, the densities, viscosities and dielectric constants of these solutions are quite same as those properties of water at 298K, which are 1.0×103kg/m3, 0.001kg/(m·s), and 78.54, respectively. The channel heights are listed in Table 4.1, and the channel width and length are shown in

Figure 3.35(b). Thus the only parameter which remains unknown is the voltage drop across the iMEDD membrane.

In the iMEDD experiments, the total voltage drop across the channel-reservoir system is 17 Volts. Because the upstream reservoir, nanopore membrane and the downstream reservoir are in series, if the resistance of the membrane and the reservoirs are known, then the voltage drop across the membrane can be calculated according to

Ohm’s law.

U I = (4.1) R

Figure 4.2 shows that the upstream and downstream reservoirs can be view as two cylinders, in which the electrolyte solution flows along the axis. From physical chemistry

[49], at constant temperature, the resistance of an electrolyte solution in a container can be expressed as

L L R = = (4.2) Aκ AFcu z c ∑ i i i where L and A are the length and cross area of the container, κc is the conductivity of the

solution, F is Faraday’s constant, zi, ci , and ui are the valence, average molarity and average velocity of the ion species i in the channel. Note here we assume the direction of the ions’ movement is along the length of the container. Equation (4.2) shows that the

118 resistance of the solution is directly proportional to the length of container, and inversely proportional to the cross area of the container and the charge density in the solution. The manual of the Ussing chamber for WPI indicates that the resistance of reservoir is

5kΩ for saline. Thus for ten times diluted PBS, and one hundred times diluted PBS, the resistance of the reservoir are 50kΩ and 500kΩ, respectively.

The resistance data of iMEDD membrane having 13nm high channels in it has been determined by Jim Coe et. al. in the Ohio State University, as shown in Figure 4.3.

The results show that the resistance of iMEDD 13nm membrane is

0.032V/0.0035A=9.1Ω. The working buffer in their experiment is PBS. However the resistance of other iMEDD membranes, and the resistance of iMEDD 13nm membrane for diluted PBS solutions remain unknown, and have to be calculated.

Coe Group 7/12/01

iMEDD 13 nm iMEDD 13 nm 0.07

0.004 0.05

0.03 0.003 0.01 V (V) -0.01 ∆ 0.002

-0.03

0.001 -0.05 Cell Current Cell Current (A)

Membrane Membrane -0.07 0.000 -0.09

-0.11 -0.001

-0.0001 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 -0.0001 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 time (s) time (s) 32 mV Steady State across Membrane 3.5 mA Steady State Current

Figure 4.3 Resistance of the iMEDD 13nm membrane, measured by Jim Coe, et. al. [56]

119 From Figure 3.35(a), the membrane and reservoirs are in series; from Figure

3.35(b), the iMEDD membrane is a parallel system of 47500 nanochannels and the silicon between them. Therefore, the whole system can be simplified as shown in Figure

4.4. The resistance of silicon wall R2 is a constant, and this constant is assumed to be

100Ω. R1, the resistance of channels can be calculated by applying equation (4.2).

R1=Rchannels

R3=Rreservoir R R R = 1 2 12 + R1 R2

R2=Rsilicon

V T

Figure 4.4 The simplified circuit for the iMEDD channel-reservoir system. Inside the membrane, the nanochannels are in parallel with the silicon wall; the membrane is in series with the reservoir. R1, R2 and R3 are the resistance of nanochannels, silicon wall, and all reservoirs, respectively. R12 is the resistance of the membrane. VT is the total voltage drop, in the iMEDD experiments, VT=17 Volts.

From Coe’s results, For iMEDD 13nm membrane, R12=9.1Ω . Therefore R1=10Ω can be calculated from R2 and R12 for 13nm membrane. From the electrochemical equilibrium considerations, we calculate the average concentrations of ion species inside the channel, from which the charge density can be calculated. Assuming the distribution patterns of charges inside the channel are the same for different membranes at different

120 buffer concentration, because R1=10Ω for 13nm membrane in PBS, using equation (4.2)

R R and R = 1 2 , we calculate R for all circumstance in the experiment, which are 12 + 12 R1 R2 shown in Table 4.2

Channel Height PBS, R12 0.1×PBS, R12 0.01×PBS, R12

(nm) (Ω) (Ω) (Ω) 4 24 76 97 7 16 65 95 13 9.1 50 91 20 6.1 39 87 27 4.6 32 83 49 2.6 21 73

Table 4.2 The resistance of iMEDD membranes. The working buffer is PBS, ten times diluted PBS and a hundred times diluted PBS.

The voltage drop across the iMEDD membrane V12 is calculated from

V R V = T 12 (4.3) 12 + R12 R3 and the results are shown in Table 4.3 and Figure 4.5. Based on the calculated voltage drops, the volume flow rates in iMEDD experiment are calculated by using governing equations as shown in last chapter. The results are shown in Table 4.4.

121 Channel Height PBS, V12 0.1×PBS, V12 0.01×PBS, V12

(nm) (Ω) (Ω) (Ω) 4 0.083 0.026 0.0033 7 0.053 0.022 0.0032 13 0.031 0.017 0.0031 20 0.021 0.013 0.0029 27 0.016 0.011 0.0028 49 0.0088 0.0071 0.0025

Table 4.3 Voltage drops through the iMEDD nanochannel. The overall voltage drop between the two electrodes is 17 Volts. 0.1xPBS denotes 10 times diluted PBS, and 0.01xPBS denotes 100 times diluted PBS. Results are shown for channel heights of 4, 7, 13, 20 and 27 nm as in the experiments.

Channel Height PBS, Volume Flow 0.1× PBS, Volume 0.01× PBS, Volume

(nm) Rate (µl/min) Flow Rate (µl/min) Flow Rate (µl/min) 4 0.73 0.23 0.029 7 1.0 0.44 0.065 13 1.2 0.79 0.14 20 1.2 1.1 0.24 27 1.2 0.25 N/A 49 1.2 N/A N/A

Table 4.4 The volume flow rates for iMEDD configurations, calculated by the governing equations. The surface charge density σ= −0.2 C/m2, as suggested by Israelachvili [17].

122 0.09 PBS 0.08 0.1xPBS 0.07 0.01xPBS 0.06 0.05 0.04 0.03 0.02 Voltage Drop (Volts) 0.01 0 0 5 10 15 20 25 30 Channel Height (nm)

Figure 4.5 Voltage drops through the iMEDD nanochannel. The overall voltage drop between the two electrodes is 17 Volts. 0.1xPBS denotes 10 times diluted PBS, and 0.01xPBS denotes 100 times diluted PBS. Results are shown for channel heights of 4, 7, 13, 20 and 27 nm as in the experiments.

According to Table 4.1 and Table 4.4, the comparison of the calculated flow rates with the iMEDD experimental results can be made. The results are shown in Figure 4.6.

The data of iMEDD experiments agree very well with the results of the model, except at

4nm. It is possible that at this channel height, some non-continuum effects are present which cannot be captured by the purely continuum theory described here. It is possible that the finite-size of the ions must be included in the calculation (ion-ion interactions), as well as an explicit description of the interaction of the ions with the walls. This

123 possibility is currently under consideration. Also there is some disagreement at larger heights; we believe these discrepancies are due to the estimate of the voltage drop.

For constant voltage drop across the membrane, the volume flow rates vary near linearly as shown in Figure 3.53. For constant voltage drops across the whole channel- reservoir system as the iMEDD experiments, the rapid drop in the cross membrane voltage leads to a maximum in the volume flow rate curve, as shown in Figure 4.6. This phenomenon is unusual and does not occur in pressure-driven flow.

1xPBS Model 0.1x PBS Model 0.01x PBS Model 1x PBS Experiment 0.1x PBS Experiment 0.01x PBS Experiment 1.6

1.4

l/min) 1.2 µ µ µ µ 1

0.8

0.6

0.4

Volume Flow Rate ( Flow Volume 0.2

0 0 102030405060 Channel Height (nm)

Figure 4.6 Comparison of our theoretical flow rates with the iMEDD experimental results. 1xPBS, 0.1xPBS and 0.01xPBS refer to original PBS, 10 folds diluted PBS and 100 folds diluted PBS, in respective.

124 4.3 Comparison with the ORNL Experiments

The second set of data for comparison is the experimental results obtained by

Ramsey et. al. [57] at Oak Ridge National Laboratory. Ramsey et. al. measure the electroosmotic mobility of sodium tetraborate aqueous/methanol solution flowing through different single rectangular silicon nanochannels. The geometry of the channels in their experiment is shown in Table 4.5. For the 10.4µm high channel, the height:width ratio is about 1:4, which represents a 2D rectangular channel rather than a 1D channel, the 1D model cannot by applied. For all other cases, the mobilities Ramsey et.al obtained from their experiment agreed very well with the model.

Channel height (nm) Channel width (µm) 83 20.3 98 18.4 290 18.3 300 20.2 1080 20.1 10.4µm 41.8

Table 4.5 Geometry of the Oak Ridge National Laboratory (ORNL) channels.

Ramsey et. al. has taken average mobility data and ζ-potential on the electroosmotic flow of a mixture of 50% sodium tetraborate aqueous solution and 50% 125 methanol. Comparison with the results of that work is a good test of the model, in particular, of the procedure to calculate the wall mole fractions. At lower concentrations,

- tetraborate anions exist in the form of B(OH)4 . The average mobility defined as the average velocity divided by the electric field has been measured.

u * uε RT µ = = e (4.4) m * µ E0 F

µ * where m , u and u are the average mobility, average dimensional velocity and average dimensionless velocity, respectively. This definition originates from the definition of electric mobility, as shown in equation (2.58) in chapter 2. Information required for the comparison with the ORNL data are the channel dimensions, the environmental temperature, the mixture properties: density, viscosity and dielectric constant and the molarity in the upstream reservoir. The voltage drops across channels are not necessary, since the average mobility is independent of the electric field.

Buffer Molarity 83 nm 98 nm 290 nm 300 nm 1080 nm 10.4 µm (mM) 0.02 2.57E-04 - - 3.80E-04 - 5.51E-04 0.2 3.23E-04 3.08E-04 4.28E-04 4.36E-04 4.42E-04 4.60E-04 2 2.85E-04 3.25E-04 3.16E-04 3.15E-04 3.14E-04 3.32E-04 20 1.89E-04 1.87E-04 1.68E-04 1.73E-04 1.58E-04 1.71E-04 150 7.99E-05 7.57E-05 4.32E-05 4.98E-05 3.81E-05 4.40E-05

Table 4.6 The mobilities measured in ORNL experiments. The units of mobility is cm2/(V·s). There is a lack of experimental data for 0.02mM buffer at h=98, 290 and 1080nm. (Ramsey et. al. 2002) 126 The dielectric constant of the ORNL working fluid is 59.24, the viscosity is

0.00168kg/(m·s) [57], the mobilities measured are shown in Table 4.6. The standard deviations are shown in Table 4.7. The electric field strength in the ORNL experiments is shown in Table 4.8. The zeta potential measured in ORNL experiments are shown in

Table 4.9.

Buffer Molarity 83 nm 98 nm 290 nm 300 nm 1080 nm 10.4 µm (mM) 0.02 5.00E-06 - - 4.19E-06 - 9.67E-06 0.2 1.23E-05 1.27E-05 1.27E-05 2.70E-05 7.31E-06 8.71E-06 2 9.85E-06 8.02E-06 8.02E-06 1.84E-05 4.87E-06 4.64E-05 20 1.06E-05 3.96E-06 3.96E-06 1.54E-05 1.99E-06 1.36E-05 150 1.12E-05 7.23E-07 7.23E-07 1.05E-05 7.77E-07 5.56E-06

Table 4.7 The standard deviations of the mobilities in Table 4.6. The units is cm2/(V·s). (Ramsey et. al. 2002)

Channel Height 1st set (kV/cm) 2nd set (kV/cm) 3rd set (kV/cm) 83 nm 0.227 0.116 0.058 98 nm 0.235 0.119 0.060 290 nm 0.237 0.119 0.060 300 nm 0.205 0.103 0.051 1080 nm 0.192 0.096 0.048 10.4 µm 0.381 0.191 0.095

Table 4.8 The field strength in ORNL experiments. For each channel height, there are three sets of data which are collected at different field strength. (Ramsey et. al. 2002) 127 Concentration ζe (V) (mM) 0.02 -0.173 0.2 -0.138 2 -0.098 20 -0.054 150 -0.017

Table 4.9 The experimentally determined zeta potential for ORNL 83nm channel. (Ramsey et. al. 2002)

0.2mM 5

4 2mM

Potential andVelociy 2 20mM

0 0 0.2 0.4 0.6 0.8 1 y

Figure 4.7 Dimensionless potential and velocity profile across an ORNL 83nm channel. The buffer concentration in the reservoir varies from 0.2mM to 20mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0161 for 0.2mM buffer, ε=0.0158 for 2mM buffer, and ε=0.0133 for 20mM buffer.

128 The monovalent binary model is applied to the ORNL configuration, and the results are shown in Figure 4.7 through Figure 4.25. The discussions in §3.3 and §3.4 are valid for any specific monovalent binary electrolyte solution, including sodium tetraborate solution at low concentration. Thus, similar analysis can be made to the

ORNL experiments.

Figure 4.7 ― Figure 4.9 illustrates the potential and velocity profile, mole fractions, and shear stress across the ORNL 83nm channel for different buffer concentrations. The surface charge density is assumed to be -0.0154C/m2. The good agreement between the model and the experimental results discussed below supports this assumption, and agreement between the iterated and the assumed surface charge density verifies the validity of this value. In Figure 4.7, for 0.2mM buffer, the parabolic velocity profile implies that the asymptotic solution is not valid. For 2mM buffer, the velocity profile at the core only has a short flat portion between y=0.4 and y=0.6, which implies that asymptotic solution can be achieved. From §2.5, the dimension scale Y=y/ε for the asymptotic solution. The values of ε are shown in Table 4.10, thus for 2mM buffer, the asymptotic solution is good only for Y∞>(0.4/0.0158)≈25.3. For 20mM buffer, the long and flat velocity profile at the core shows that the asymptotic solution will be good at this case. Figure 4.8 and Figure 4.9 show the mole fractions and shear stress in the ORNL

83nm channel. The effects of buffer concentration on the mole fractions of ion species in the channel and the effects on shear stress has been discussed in §3.3 and §3.4. Using the scaling laws defined in §3.5, we calculate the two parameters δ and γ which determine the governing equations. The parameters ε, δ and γ are shown in Table 4.10. For

129 γ<<1, from equation (3.46), δ = ε 1+ γ ≅ ε which is shown in Table 4.10. Thus for concentration varies from 0.2mM to 20mM, δ and ε does not change much. On the other hand, parameter γ varies up to 105 as the buffer concentration varies from 0.2mM to

20mM. As the discussion in §3.5 shows, γ≈1 represents a Debye-Hückel picture of EDL,

γ<<1 represents a Gouy-Chapman picture of EDL, and the dimensionless double layer thickness Y∞·ε increases as δ increases or γ decreases. For the cases shown in Figure

4.7―Figure 4.9, the large difference in γ is the major cause of the large differences in the plots.

0 1/2 0 0 Conc. (mM) ε = λ/h δ = ε/(β·X1 ) γ = X2 /X1 0.2 0.0161 0.0161 6.32×10-7 2 0.0158 0.0158 1.38×10-4 20 0.0133 0.0134 0.0209

Table 4.10 The parameters of the governing equations for different buffer concentration in the reservoir for the ORNL 83nm channel. δ and γ are the scaling parameters discussed 0 in §3.5. ε, δ and γ are all dimensionless. X1 is the mole fraction of sodium ions on the 0 wall, and X2 is the mole fraction of tetraborate ions on the wall. β=c/I is defined along with equation (2.52), where c is the total concentration of the solvent and all ion species, and I is the ionic strength.

130 -3 x 10 2.5 Sodium Tetraborate

1.5

1 Mole Fraction

0.5 20mM

2mM 0 0 0.2 0.4 0.6 0.8 1 y

Figure 4.8 A Dimensionless plot of mole fractions across an ORNL 83nm channel. The buffer concentration in the reservoir varies from 2mM to 20mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0161 for 0.2mM buffer, ε=0.0158 for 2mM buffer, and ε=0.0133 for 20mM buffer.

100

20 0.2mM 2mM 0 20mM -20 Shear Stress

-40

-60

-80

-100 0 0.2 0.4 0.6 0.8 1 y

Figure 4.9 Dimensionless shear stress across an ORNL 83nm channel. The buffer concentration in the reservoir varies from 0.2mM to 20mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0161 for 0.2mM buffer, ε=0.0158 for 2mM buffer, and ε=0.0133 for 20mM buffer.

131 The results of the asymptotic model for the 2mM and 20mM buffer are shown in

Figure 4.10 ― Figure 4.12. As expected in the previous discussion, Y∞ larger than 25 is required for the asymptotic solution for 2mM buffer in the reservoir. Here we use Y∞=30.

The channel height in these plots is still 83nm. For same concentrations, the value ε=λ/h in these plots are less than the corresponding values in the symmetric plots Figure 4.7,

Figure 4.8 and Figure 4.9, which can be explained by the less λ in the asymptotic solution caused by a higher wall mole fraction of Na+, the co-ions. Nonetheless, the asymptotic figures also demonstrate the effects of buffer concentration, as the symmetry figures

Figure 4.7, Figure 4.8 and Figure 4.9 do.

4.5

4 2mM 3.5

2.5

2 20mM 1.5 Potential and Velociy and Potential

0.5

0 0 5 10 15 20 25 30 Y

Figure 4.10 Asymptotic results of the potential and velocity profile near the wall. The channel here is an ORNL 83nm-high channel. The buffer concentrations in the reservoir are 2mM and 20mM. The surface charge density on the silicon channel wall 2 σ=−0.0154C/m . ε=λ/h=0.0147 for 2mM buffer, and ε=0.0126 for 20mM buffer. Y∞=30.

132 -3 x 10 2.5 Sodium Tetraborate

1.5

1 Mole Fraction

0.5 20mM

2mM 0 0 5 10 15 20 25 30 Y

Figure 4.11 Asymptotic results of the mole fractions near the wall. The channel here is an ORNL 83nm-high channel. The buffer concentrations in the reservoir are 2mM and 20mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0147 for 2mM buffer, and ε=0.0126 for 20mM buffer. Y∞=30.

100

40 Shear Stress

20 2mM

10 20mM 0 0 5 10 15 20 25 30 Y

Figure 4.12 Asymptotic results of the shear stress near the wall. The channel here is an ORNL 83nm-high channel. The buffer concentrations in the reservoir are 2mM and 20mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0147 for 2mM buffer, and ε=0.0126 for 20mM buffer. Y∞=30. 133

0 1/2 0 0 Conc. (mM) ε = λ/h δ = ε/(β·X1 ) γ = X2 /X1 20 0.0125 0.0126 0.0226 150 0.0072 0.0080 0.2215

Table 4.11 The parameters ε, δ and γ of the asymptotic equation for 20mM and 150mM buffer in the reservoir for the ORNL 83nm channel.

1.8

1.6 20mM 1.4

1.2

150mM 0.8

Potential and Velociy and Potential 0.6

0.4

0.2

0 0 2 4 6 8 10 y

Figure 4.13 Asymptotic results of the potential and velocity profile near the wall. The channel here is an ORNL 83nm-high channel. The buffer concentrations in the reservoir are 20mM and 150mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0125 for 20mM buffer, and ε=0.0072 for 150mM buffer. Y∞=10.

134 -3 x 10 6 Sodium Tetraborate

3 150mM Mole Fraction Mole 2

1 20mM

0 0 2 4 6 8 10 y

Figure 4.14 Asymptotic results of the mole fractions near the wall. The channel here is an ORNL 83nm-high channel. The buffer concentrations in the reservoir are 20mM and 150mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0125 for 20mM buffer, and ε=0.0072 for 150mM buffer. Y∞=10.

1.4

1.2

0.8

0.6

Shear Stress Shear 20mM

0.4

0.2 150mM

0 0 2 4 6 8 10 Y

Figure 4.15 Asymptotic results of the shear stress near the wall. The channel here is an ORNL 83nm-high channel. The buffer concentrations in the reservoir are 20mM and 150mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0125 for 20mM buffer, and ε=0.0072 for 150mM buffer. Y∞=10. 135 Table 4.11 shows the equation parameters for the asymptotic equation for 20mM buffer and 150mM buffer. The channel heights are both 83nm. Y∞=10 is good enough for these concentrations. From equation (2.22), we have

X 0 − X 0 X 0 ()1− γ 1− γ 1 2 = 1 = (4.5) 0 0 X 0 γ γ 2 X 1 X 2 2 1 2

Figure 4.13 ― Figure 4.15 show the asymptotic results for 20mM buffer and 150mM 1− γ buffer. h=83nm. Y∞=10. For 20mM buffer, = 3.25 , and for 150mM buffer, 2 γ 1− γ = 0.827 . Thus, as the buffer concentration increases, the ratio decreases. For 2 γ X 0 − X 0 150mM buffer, the ratio 1 2 <1, and thus the results are close to the Debye-Hückel 0 0 2 X1 X 2 view of EDL, as shown in Figure 4.14.

Channel 0 1/2 0 0 ε = λ/h δ = ε/(β·X1 ) γ = X2 /X1 height (nm) 83 0.0158 0.0158 1.38×10-4 290 0.0050 0.0050 5.24×10-5

Table 4.12 The equation parameters ε, δ and γ for h=83nm and h=290nm ORNL channels. The concentration of the buffer in the reservoir is 2mM.

The channel heights also have an effect on the EOF. For instance, the mole fractions of sodium ions and tetraborate ions for h=83nm and h=290nm are shown in

Figure 4.16 and Figure 4.17, respectively. The buffer concentration in the reservoir is

2mM for all cases. For both cations and anions, their concentrations at the core are higher for a larger channel (290nm) than for a smaller channel (83nm). The asymptotic solution 136 works well for the 290nm channel, while it works for 83nm channel only for large Y∞.

Figure 4.18 and Figure 4.19 show the potential and velocity profile and shear stress at different channel heights. For transporting macromolecules, the 290nm channel has these advantages over the smaller channel (83nm). First of all, the velocity profile for 290nm channel represents a plug flow, while the plug flow in the 83nm channel is not significant. Secondly, the non-shear region in the 290nm channel is also wider than the

83nm channel. The equation parameters are shown in Table 4.12. γ for 83nm channel is higher than γ for 290nm channel, so the asymptotic potential for the outer solution

1 φο = − lnγ is higher in the 83nm channel than in the 290nm channel, which explains 2 the crossover of the two potential curves in Figure 4.18.

-3 x 10 2.5

h=290nm h=83nm 2

1.5

1 Mole Fraction

0.5 h=290nm

h=83nm

0 0 0.2 0.4 0.6 0.8 1 y

Figure 4.16 The mole fractions of Na+ across the channel for the ORNL experiments. The buffer concentration in the reservoir is 2mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0158 for 83nm high channel, and ε=0.0050 for 290nm high channel. 137 -5 x 10 4.5

4 h=290nm

3.5

3 h=83nm 2.5

2 Mole Fraction 1.5

h=290nm 0.5 h=83nm

0 0 0.2 0.4 0.6 0.8 1 y

- Figure 4.17 The mole fractions of B(OH)4 across the channel for the ORNL experiments. The buffer concentration in the reservoir is 2mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0158 for 83nm high channel, and ε=0.0050 for 290nm high channel.

4.5 h=83nm 4 h=290nm 3.5

2.5

1.5 Potential and Velociy and Potential

0.5

0 0 0.2 0.4 0.6 0.8 1 y

Figure 4.18 The dimensionless potential and velocity profile across the channel for the ORNL experiments. The buffer concentration in the reservoir is 2mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0158 for 83nm high channel, and ε=0.0050 for 290nm high channel. 138 250

200

150

100

50 h=83nm 0 h=290nm -50 Shear Stress

-100

-150

-200

-250 0 0.2 0.4 0.6 0.8 1 y

Figure 4.19 The dimensionless shear stress across the channel for the ORNL experiments. The buffer concentration in the reservoir is 2mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0158 for 83nm high channel, and ε=0.0050 for 290nm high channel.

0.12

0.1

0.08

0.06 Potential (V) Potential 0.04

0.02

0 0 0.2 0.4 0.6 0.8 1 y

Figure 4.20 Dimensional results of the electric potential across the channel for the ORNL experiments. The channel here is an ORNL 83nm-high channel. The buffer concentration in the reservoir is 2mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0158. 139 -4 x 10 3.5

2.5

1.5

Mobility (cm*cm/V/sec) 1

0.5

0 0 0.2 0.4 0.6 0.8 1 y

Figure 4.21 Dimensional electric mobility of the fluid across the channel for the ORNL experiments. The channel here is an ORNL 83nm-high channel. The buffer concentration in the reservoir is 2mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0158.

250

200

150

100

-50

Shear Stress(N/m/m) -100

-150

-200

-250 0 0.2 0.4 0.6 0.8 1 y

Figure 4.22 Dimensional results of the shear stress across the channel for the ORNL experiments. The channel here is an ORNL 83nm-high channel. The buffer concentration in the reservoir is 2mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0158. 140 0.05

0.045

0.04

0.035

0.03

0.025

Potential (V) Potential 0.02

0.015

0.01

0.005

0 0 2 4 6 8 10 Y

Figure 4.23 Dimensional results of the electric potential near the wall. The channel here is an ORNL 83nm-high channel. The buffer concentration in the reservoir is 20mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0125. Y∞=10.

-4 x 10 1.6

1.4

1.2

0.8

0.6 Mobility (cm*cm/V/sec) 0.4

0.2

0 0 2 4 6 8 10 Y

Figure 4.24 Dimensional results of the mobility profile near the wall. The channel here is an ORNL 83nm-high channel. The buffer concentration in the reservoir is 20mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0125. Y∞=10. 141 3

2.5

1.5

1 Shear Stress(N/m/m)

0.5

0 0 2 4 6 8 10 Y

Figure 4.25 Dimensional results of the shear stress near the wall. The channel here is an ORNL 83nm-high channel. The buffer concentration in the reservoir is 20mM. The surface charge density on the silicon channel wall σ=−0.0154C/m2. ε=λ/h=0.0125. Y∞=10.

The dimensional potential, mobility and shear stress are shown in Figure 4.20,

Figure 4.21, Figure 4.22, Figure 4.23, Figure 4.24 and Figure 4.25. The first three figures are results of the symmetric model, and the next three figures are results of the asymptotic model.

For the ORNL 83nm channel, the average mobility and ζ potential measured in

ORNL experiments are compared with the results of the symmetric numerical model and the asymptotic model, and the results of the comparison are shown in Table 4.13. For

2mM or higher concentrations, both the numerical model and the asymptotic model agree well with the experiments; for 0.02mM and 0.2mM buffer, the numerical model still works well, while the asymptotic model produces large error. This is because for 83nm

142 channel, if the concentration is lower than 0.2mM, the potential profile is parabolic and the core is no longer electrically neutral, which are shown in Figure 4.7.

e -4 -4 a -4 Conc. µ µ µ ζe (V) ζ (V) ζa (V) m (×10 m (×10 m (×10 (mM) cm2/(V·s)) cm2/(V·s)) cm2/(V·s)) 0.02 2.57 3.35 invalid -0.173 -0.142 invalid 0.2 3.23 3.40 7.53 -0.138 -0.141 -0.264 2 2.85 2.94 3.28 -0.098 -0.111 -0.125 20 1.89 1.56 1.21 -0.054 -0.0464 -0.0490 150 0.800 0.580 0.530 -0.017 -0.0237 -0.0195

Table 4.13 Results for the average mobility and ζ potential for the indicated molarities in the ORNL 83nm×20.3µm channels. Both numerical and asymptotic values of the average mobility and ζ potential are shown. Y∞=10. Superscripts e and a stand for experimental and asymptotic, respectively. The surface charge σ=−0.0154 C/m2 on the channel wall. For 0.02mM buffer, because the large ε, the asymptotic values are invalid.

The calculated average mobility from the model is listed in Table 4.14. For 83 nm, 98 nm, 290 nm and 300 nm channels, the results are calculated from the numerical model. For 1080 nm channel, ε=λ/h is too small for the numerical model to handle; therefore the asymptotic model is utilized for these two channel heights. Comparing

Table 4.14 with Table 4.6, the results are fairly good, the percentage relative error

µ − µ e δ = m m ×100% are shown in Table 4.15. The relatively large error for 0.02mM and µ e m

150mM buffer results from the assumed value of surface charge density if the molarity of

143 the working fluid is too low or too high. Therefore, we adjust the surface charge density being assumed and recalculate the mobilities. The results are shown in Table 4.16. The relative errors are shown in Table 4.17, and we note the results of comparison are generally better.

Buffer Molarity 83 nm 98 nm 290 nm 300 nm 1080 nm (mM) 0.02 3.35E-04 3.77E-04 - 5.26E-04 - 0.2 3.40E-04 3.72E-04 4.56E-04 4.57E-04 5.30E-04 2 2.94E-04 3.00E-04 3.05E-04 3.03E-04 3.05E-04 20 1.56E-04 1.68E-04 1.54E-04 1.52E-04 1.52E-04 150 5.80E-05 5.80E-05 5.10E-05 5.10E-05 5.40E-05

Table 4.14 The mobilities calculated from the model. For h=83nm, 98nm, 290nm and 300nm, the numeric model is used. For h=1080nm, the asymptotic model is used, and Y∞=100 for 0.2mM solution, 30 for 2mM solution, 10 for 20mM and 150mM solutions. The surface charge density on the silicon channel wall is σ=−0.0154 C/m2. The units of mobility is cm2/(V·s).Not all cases are calculated, those have not been calculated are marked as ‘-‘, for which there is no experimental data.

Buffer Molarity 83 nm 98 nm 290 nm 300 nm 1080 nm (mM) (%) (%) (%) (%) (%) 0.02 30.4 - - 38.4 - 0.2 5.3 20.8 6.5 4.8 20.0 2 3.2 -7.7 -3.5 -3.8 -2.9 20 -17.5 -10.2 -8.3 -12.1 -3.8 150 -27.4 -23.4 18.1 2.4 41.7

Table 4.15 The relative error on mobilities between the results of the model shown in Table 4.14 and the experimental results shown in Table 4.6. The entities marked ‘-‘ indicate that there is no experimental data available.

144 Buffer Molarity 83 nm 98 nm 290 nm 300 nm 1080 nm (mM) 0.02 2.49E-04 2.70E-04 4.20E-04 4.24E-04 8.08E-04 0.2 3.40E-04 3.72E-04 4.56E-04 4.57E-04 5.30E-04 2 2.94E-04 3.00E-04 3.05E-04 3.03E-04 3.05E-04 20 1.56E-04 1.68E-04 1.54E-04 1.52E-04 1.52E-04 150 7.40E-05 7.54E-05 5.08E-05 4.96E-05 6.43E-05

Table 4.16 The mobilities calculated from the model. For h=83nm, 98nm, 290nm and 300nm, the numeric model is used. For h=1080nm, the asymptotic model is used, and

Y∞=100 for 0.2mM solution, 30 for 2mM solution, 10 for 20mM and 150mM solutions. The surface charge density on the silicon channel wall is σ=−0.007 C/m2 for the 0.02mM buffer. σ=−0.021 C/m2 for the 150mM buffer, and σ=−0.0154 C/m2 for other buffers. The units of mobility is cm2/(V·s).

Buffer Molarity 83 nm 98 nm 290 nm 300 nm 1080 nm (mM) (%) (%) (%) (%) (%) 0.02 -3.1 - - 11.6 - 0.2 5.3 20.8 6.5 4.8 15.2 2 3.2 -7.7 -3.5 -3.8 -2.9 20 -17.5 -10.2 -8.3 -12.1 -3.8 150 -7.4 -0.4 17.6 -0.4 68.8

Table 4.17 The relative error on mobilities between the results of the model shown in Table 4.16and the experimental results shown in Table 4.6. The entities marked ‘-‘ indicate that there is no experimental data available.

145 The results in Table 4.6 and Table 4.14 are plotted in Figure 4.26, which shows the comparison between the results of the model and the experiment more directly. For

83nm and 300nm channels, the average mobilities from the model and the experiment are both plotted against the concentration of the buffer, and the plot is shown in Figure 4.27.

For these two channel height, large errors only exist at 0.02mM.

6.00E-04 150mM Exp 150mM Num 0.02mM 5.00E-04 0.2mM 20mM Exp 20mM Num 4.00E-04 2mM Exp 2mM Num 2mM 3.00E-04 0.2mM Exp 0.2mM Num 0.02mM Exp

Mobility (cm*cm/V/sec) 2.00E-04 0.02mM Num 20mM

1.00E-04

150mM

0.00E+00 1 10 100 1000 10000 Channel Height (nm)

Figure 4.26 Comparison between the mobilities measured in ORNL experiments and the results from the model. The mobilities are drawn against the channel heights, which vary form 83nm to 1080nm, as shown in Table 4.5.

146 6.00E-04 83nm (Exp) 83nm (Model)

5.00E-04 300nm(Exp) 300nm (Model)

4.00E-04

3.00E-04

2.00E-04 Mobility (cm*cm/V/s)

1.00E-04

0.00E+00 0.01 0.1 1 10 100 1000 Molarity (mM)

Figure 4.27 Comparison between the mobilities measured in ORNL experiments and the results from the model. The molarity in the reservoir varies from 0.02mM to 150mM.

147 4.4 Comparison with the Georgia Tech Experiments

Figure 4.28 The picture (left) and sketch (right) of the crossing microchannel chip used in the Georgia tech experiments, the region of interest (ROI) is indicated in the sketch plot; and the direction of flow is from top to bottom in the sketch plot, as shown by the arrow. (M. Yoda et. al., 2002)

In Georgia Institute of Technology, Dr. M. Yoda et. al. uses Nano-PIV technique to study the flows at the nano- to micrometer scale experimentally. The channels they test are shown in Figure 4.28. The velocity profile and mobility of fluids are measured in their experiments. The spatial resolution of their experiment is 100nm. According to the scaling laws discussed in §3.5, channels less than 100nm can be mimicked by applying much less concentrated solutions in microchannels [58] [59].

148 I (mM) 0.19 1.9 3.6 18.4 36 h (µm) 4.9 10.2 24.7 4.8 24.7 W (µm) 17.3 26.1 51.2 17.9 51.2 µ e 2 -1 -1 m (cm ·V ·s ) 4.82E-4 2.37E-4 2.04E-4 1.24E-4 1.16E-4 µ m 2 -1 -1 m (cm ·V ·s ) 4.92E-4 2.52E-4 2.00E-4 1.39E-4 1.15E-4 ζ (V) -0.0636 -0.0324 -0.0256 -0.0178 -0.0147 λ (nm) 8.82 4.97 4.02 1.98 1.46 ε 1.80E-3 4.87E-4 1.63E-4 4.12E-4 5.93E-5 δ 1.81E-3 5.05E-4 1.73E-4 4.60E-4 6.8E-5 γ 0.00647 0.0770 0.131 0.243 0.311

µ e Table 4.18 Comparison of the average electrical mobility m , measured by the Georgia µ m ζ Tech experiments, and the mobility m , calculated by the model. potential, parameters ε, δ and γ calculated from the model are also shown. I is the ionic strength of the buffer in the reservoir, The surface charge densities being used in the model are: σ=−0.002C/m2 for 0.19mM buffer, σ=−0.0025C/m2 for 1.9mM buffer and 3.6mM buffer, σ=−0.0045C/m2 for 18.4mM buffer and 36mM buffer.

Table 4.18 and Figure 4.29 show the average mobility measured in the Georgia

Tech experiments, and the average mobility calculated from the model. The asymptotic model is used because the channel height is in micrometer scale, which is much large than the Debye length, as shown in Table 4.18. The experimental and model values for mobility agree within 10% over a 200 fold change in ionic strength, as shown in Figure

4.29. Figure 4.29 also implies that the average mobility

149 µ = µ 0 ⋅ ()−η m m I (4.5) where I, in millimolar, is the ionicstrength of the electrolyte solution in the reservoir. A

µ 0 2 η curve fit of the experimental data points gives m =0.00036cm /(V·s), and =0.28. A

µ 0 2 η curve fit of the numerical data points gives m =0.00037cm /(V·s), and =0.28, which is in excellent agreement with the experimental results.

1.00E-03 Experiment Model Model Experiment

µ m -0.28 m = 0.00037I R2 = 0.9957 Mobility (cm*cm/V/sec) Mobility

µ e -0.28 m = 0.00036I R2 = 0.9926

1.00E-04 0.1 1 10 100 I [mM]

Figure 4.29 Comparison of the mobility measured in the Georgia Tech experiments and the average mobility calculated from the asymptotic model.

150 10 ) γ ln(1/γ) = 3.7I -0.28 ln(1/ R2 = 0.9954

µ 0= -4 m 3.7×10 1 0.1 1 10 100 Ionicstrength (mM)

Figure 4.30 The relations between factor γ and the ionicstrength of the solution in the µ 0 2 reservoir. The mobility coefficient m =0.00037cm /(V·s) can be calculated from the coefficient 3.7 in the power fit curve of ln(1/γ) vs. I.

For asymptotic cases, the average mobility can be calculated analytically as

u * U  1  ε φ  1  µ = = 0 ln  = e 0 ln  (4.6) m * *  γ  µ  γ  E0 2E0   2  

RT where µ is the viscosity of the fluid, φ0 = is the potential scale, ε = ε ⋅ε 0 is the F e r

0 γ = X − γ permittivity of the fluid, and nondimensional factor 0 . Figure 4.30 shows ln(1/ ) X + vs. I and the corresponding power curve fit which supports Equation (4.6) and leads to

151 ε φ µ 0 = e 0 ⋅ 3.7 = 3.7 ×10−4 cm 2 /()V ⋅ s , which agrees with the experimental and m 2µ numerical mobility coefficients obtained in Figure 4.29. The mole fractions, potential and velocity, and shear stress for the 18.4mM buffer are shown in Figure 4.31 - Figure 4.33.

-4 x 10 8 Sodium Tetraborate 7

4 Mole Fraction Mole

1 0 2 4 6 8 10 Y

Figure 4.31 Simulated mole fractions of sodium ions and tetraborate ions for the Georgia Tech experiments. The channel height is 4.8µm, and the channel width is 17.9µm. The concentration of the sodium tetraborate solution in the reservoir is 18.4mM. The surface 2 charge density σ=−0.0045C/m . ε=0.000412, δ=0.000460, γ=0.243, Y∞=10.

152 0.8 Potential φ =(-1/2)lnγ=0.707 a 0.7 Velocity

0.6

0.5

0.4

0.3 Potential and Velociy and Potential 0.2

0.1

0 0 2 4 6 8 10 Y

Figure 4.32 Simulated potential and velocity profile for the Georgia Tech experiments. The channel height is 4.8µm, and the channel width is 17.9µm. The concentration of the sodium tetraborate solution in the reservoir is 18.4mM. The surface charge density 2 σ=−0.0045C/m . ε=0.000412, δ=0.000460, γ=0.243, Y∞=10.

1600

1400

1200

1000

800

Shear Stress 600

400

200

0 0 2 4 6 8 10 Y

Figure 4.33 Simulated shear stress for the Georgia Tech experiments. The channel height is 4.8µm, and the channel width is 17.9µm. The concentration of the sodium tetraborate solution in the reservoir is 18.4mM. The surface charge density σ=−0.0045C/m2. ε=0.000412, δ=0.000460, γ=0.243, Y∞=10.

153 4.5 Issues of Surface Charge Density

Other models of the EOF are based on the ζ potential, because the ζ potential can be measured experimentally. The surface charge density is derived from the ζ potential then. For instance, Healy and White [60] have developed an equation to calculate the surface charge density σ from the ζ potential. The equation is adapted by Qu and Li [36] as

δ φ + ζ eN s sinh( Ν 0 ) σ 0 = (4.5) 1+ δ cosh(φΝ + ζ 0 )

18 -2 where Ns is the site density on the surface. For silicon dioxide surface, Ns=5×10 m

[60]. ζ0 is the nondimensional ζ potential

ζ ζF ζ = = (4.6) 0 φ 0 RT

φN is the nondimensional Nernst potential which is given by

φN=ln(10)×(pHz-pH) (4.7) where pHz is the pH value at which the surface reaches the point of zero charge. For silica surfaces, we use the isoelectric point pI as pHz, so pHz=pI=2.4, see §1.3. The pH of

PBS solution is 7.4, the pH of sodium tetraborate, or sodium chloride solutions are both

7. Here we use the δ in equation (4.5) is defined as

−∆pK δ = 2×10 2 (4.8) where ∆pK=pK−-pK+ is the dissociation constant difference. For SiO2/H2O interface,

∆pK is no less than 6. Here we use a value of 10 for ∆pK, as indicated by Healy and

White [60]. 154 For the ORNL experiments, we apply the above equations to calculate the surface charge density from the ζe, which is the zeta potential measured in the ORNL experiments.

Concentration σ (C/m2) Assumed ζe (V) (mM) σ (C/m2) 0.02 -0.173 -0.0004 -0.0154 0.2 -0.138 -0.0015 -0.0154 2 -0.098 -0.0071 -0.0154 20 -0.054 -0.038 -0.0154 150 -0.017 -0.14 -0.0154

Table 4.19 Comparison of the surface charge density calculated by using equation (4.5) from the ζ potential measured in ORNL experiments, with the surface charge density assumed in the numerical calculation.

There are errors between the assumed surface charge density and σ calculated from equation (4.5). The cause of the error is not clear. Notice that the calculated surface charge density varies with the change of concentration. One possible source of error is the value of ∆pK, which is assumed to be 10 in previous calculation.

155 CHAPTER 5

ELECTROPHORESIS AND ION FLUXES

5.1 Introduction

Electrophoresis is the movement of charged particles in fluids where electric field exists. In an electrokinetic process, electrophoresis and electroosmosis usually occur simultaneously. The electroosmotic characters of the flow have been considered in the last three chapters; in this chapter, we focus on the electrophoretic properties such as mass fluxes of ions and the intensity of electric current.

Capillary electrophoresis is the electrophoresis in nano- or micro- capillaries. As a successful experimental technique for chromatography, capillary electrophoresis has been widely studied in the past twenty years, and there are books [61] [62] and papers [63]

[64] [65] [66] [67]published on this subject. Ion fluxes in micro- and nanochannels have also been studied mathematically [68] [69] [67] [70].

The transport of ions through biological ion channels is another application area for the models of nanopores. The work has been done by many researchers [71] [72] [73]

[74]-[80], [81], [82].

In this chapter, the governing equations of electrophoresis in micro- or nanochannels where h<

5.2 Classical Theory of Capillary Electrophoresis

E + − Cation Anion

Figure 5.1 A sketch figure of the electrophoresis in a channel. For convenience, the size of ions is exaggerated, and only one cation and one anion are shown in the figure. Cations are driven to the cathode by the electric field, and anions are driven to the anode. If the channel wall is negatively charged, the direction of electroosmotic flow is toward the cathode, as shown in the profile.

For an electrolyte solution in the presence of an electric field, electric forces acting on the ions drive the ions in opposite directions. The cations are driven down the field strength to lower potential, i. e. toward the cathode, and the anions are driven to higher potential, i. e. toward the anode, as shown in Figure 5.1. The migration of ions in a fluid caused by external electric field is called electrophoresis. For cases where the

157 channel walls are charged, electroosmosis also exists, as discussed in §1.2.

Electrophoresis and electroosmosis are two aspects of the electrokinetic phenomenon.

The differences are: 1. If the channel wall is neutral, there is no electroosmosis, while the electrophoresis still exists. The wall concentrations of the ion species in this case equal the reservoir concentrations. 2. Electroosmosis is defined by the motion of the entire solution, i.e. both the solute and the solvent are involved in the electroosmotic flow; however, electrophoresis is defined by the motion of the ions only, the solvent and other neutral solutes are not affected by the electrophoresis.

The electrophoresis considered in this work is capillary electrophoresis, in which the diameter of the capillary or the height of the channel is in nano- or micrometers. The surfaces of the channel are assumed to be negatively charged, so that the previous discussion on electroosmosis can be applied here. This assumption is supported by the fact that the glass capillaries and silicon channels being used in the experiments on capillary electrophoresis have negatively charged surfaces. The term solution being used here is a general definition of solution, which includes colloids; and the ions include charged macromolecules, such as protein and DNA.

In capillary electrophoresis, the electrokinetic velocity vEK of an ion species is

([61] [62])

= + vEK vEOF vEP (5.1) where vEOF is the velocity of the electroosmotic flow, and vEP is the electrophoretic migration velocity of this ion species. The electrokinetic mobility of this ion species is

µ = µ + µ EK EOF EP (5.2)

158 v v where µ = EK is the electrokinetic mobility of the ion species, µ = EOF is the EK E EOF E

v electroosmotic mobility of the solution, and µ = EP is the electrophoretic mobility of EP E the ion species. E is the field strength.

For different ion species, the electrophoretic mobilities are different. According to

vEK equation (5.2) and µ = , the electrokinetic velocities vEK for different ion species EK E are different. If solution is added at the inlet, different ion species will be separated by the electrokinetic flow in the channel. For charged macromolecules such as different proteins, electrophoresis is an effective method for separation.

5.3 Governing Equations

Equation (2.36) is the nondimensional Poisson equation for a 3D channel.

ε 2 ∇ 2φ = −β ∑ zi X i (2.36) i

For the electrophoresis in one dimensional channel as shown in Figure 2.2,

h ∂ 2φ ε 2 = << 1, thus the ε 2 part in equation (2.36) can be neglect, and equation (2.36) W ∂z 2 becomes

 ∂ 2φ ∂ 2φ  ε 2 ε 2 +  = −β z X (5.3)  1 ∂ 2 ∂ 2  ∑ i i  x y  i

159 λ ∂2φ h c 2 where ε = , ε1 = , and β = . Although ε1<<1, ε cannot be neglected in the h L I 1 ∂x2

∂φ discussion of electrophoresis, because E = − is required for solving the x ∂x electrophoretic velocity and ion flux of any ion species. The boundary conditions of electrical potential φ on the inlet and outlet are

φ = E x=0 (inlet) (5.4)

φ = 0 x=1 (outlet) (5.5)

Equations (5.5) shows that the electric potential at the outlet of the channel are chosen as the ground, equation (5.4) shows that the potential drop across the channel is E.

For the mass transport in electrokinetic flow, the flux equation of ion species i is shown in equation (2.41) as

D v ∗ ∗ nr = −cD ∇X + c i z FX E + cX uv (2.41) i i i RT i i i

v ∗ where cX iu is the convective flux, and u represents the electroosmotic velocity.

D v ∗ D v ∗ c i z FX E is the electric migration flux, and i z FE represents the electrophoretic RT i i RT i

− ∇ velocity. cDi X i is the diffusive flux caused by the concentration gradient. The flux caused by pressure gradient are assumed to be null, which is supported by the fact that the pressure driven flow is negligible as comparing with the electrokinetic flow in capillaries (§1.2). The x components of Equation (2.41) reduces to equation (5.1) for cases where the concentration gradient is negligible.

At steady state, the mass transport equation is

∇ ⋅ v = ni 0 (2.42) 160 which can be expanded to nondimensional equation (2.43) as shown in §2.3

∇ 2 + ∇ • ()⋅ ∇φ − ⋅ ⋅ ∇ • ()= X i zi X i Re Sc X i u 0 i=1,2,…M (2.43)

ρU h µ where Re = 0 is the Reynolds number, and Sc = is the Schmidt number. In this µ ρ Di

RT work, we choose E = as the scale of the nondimensional electric field strength, and 0 hF equation (2.43) becomes

∂ ∂X ∂φ ε 2  i +  1  zi X i  ∂x  ∂x ∂x  ∂  ∂X ∂φ  + i +  zi X i  ∂y  ∂y ∂y  (5.6) ∂ ∂X ∂φ +ε 2  i +  2  zi X i  ∂z  ∂z ∂z   ∂X ∂X ∂X  = ⋅ ⋅ ε i + i + ε i Re Sc  1u v 2 w   ∂x ∂y ∂z 

⋅ ⋅ ∇ • ()r ≈ For the steady flow in micro- or nanochannels, Re Sc X i u 0 because the

Reynolds number is small, see §2.1. Therefore equation (5.6) can be simplified to

∂  ∂X ∂φ  2 ∂ ∂X ∂φ i + + ε  i +  =  zi X i  1  zi X i  0 (5.7) ∂y  ∂y ∂y  ∂x  ∂x ∂x  and the boundary conditions for the mole fraction of ion species i is

R Xi=Xi x=0 (5.8)

R Xi=Xi x=1 (5.9)

0 Xi=Xi y=0 (5.10)

0 Xi=Xi y=1 (5.11)

0 R Where Xi is the concentration of ion species i on the wall, and Xi is the concentration of ion species i in the reservoir. 161 The Navier-Stokes equation (2.51) becomes

2 2  2 ∂ u ∂ u  ε 2 ε +  = −β z X (5.12)  1 ∂ 2 ∂ 2  ∑ i i  x y  i and the boundary conditions for u are assumed to be [83]

u=0 y=0 (5.13)

u=0 y=1 (5.14)

∂u = 0 x=0 (5.15) ∂x

∂u = 0 x=1 (5.16) ∂x

The boundary conditions of φ on the channel wall are derived by the following method. At the wall, the flux of cations on the y direction is zero, and at steady state, v, the y-component of the convection velocity, is zero too. Thus equation (2.41) becomes

∂ X D v * − cD i + c i z FX E = 0 y=0,1 (5.17) i ∂y RT i i y

∂φ v * Noticing that E = − , and coefficients c≠0, and Di≠0, thus after y ∂y nondimensionalization, equation (5.17) is reduced to

∂φ −1 ∂X = ⋅ i y=0,1 (5.18) ∂ ∂ y zi X i y

This is the boundary conditions of φ at the channel walls.

For the solved governing equations (5.3), (5.7) and (5.12), the following parameters can be derived from φ, u, and Xi.

Electrical field strength in the x direction

162 ∂φ E = − (5.19) x ∂x

Electrical field strength in the y direction

∂φ E = − (5.20) y ∂y

Shear stress in the x direction

∂u τ = (5.21) x ∂y

Convective flux of ion species i in the x direction

c ()= ⋅ ⋅ ⋅ nx i Re Sc X i u (5.22)

Diffusive flux of ion species i in the x direction

∂ d X i n ()i = −ε1 (5.23) x ∂x where ε1=h/L is the aspect ratio of the channel height and length.

Diffusive flux of ion species i in the y direction

∂X n d ()i = − i (5.24) y ∂y

Electric migration flux of ion species i in the x direction

∂φ n m ()i = −ε z X (5.25) x 1 i i ∂x

Electric migration flux of ion species i in the y direction

∂φ n m ()i = −z X (5.26) y i i ∂y

Total molar flux of ion species i in the x direction

163 ∂ ∂φ X i n ()i = −ε1 − ε z X + Re⋅ Sc ⋅ X ⋅u (5.27) x ∂x 1 i i ∂x i

Total molar flux of ion species i in the y direction

∂X ∂φ n ()i = − i − z X (5.28) y ∂y i i ∂y

Equations (5.19)-(5.28) are nondimensional equations. The corresponding dimensional fluxes can be obtained by multiplied with cDi/h.

5.4 Numerical Method

f(k,j+1)

f(k-1,j) f(k,j) f(k+1,j)

∆y ∆x f(k,j-1)

Figure 5.2 The finite difference view of function f(x,y). For the electrophoresis problems discussed in this chapter, x direction is along the length of the channel, while y direction is along the height of the channel. The variations of f on z direction, which represents the width, is neglect, because h<

164 Equations (5.3), (5.7) and (5.12) are elliptic equations. Successive over-relaxation method (SOR) is used to solve these equations numerically. SOR method, as an evolution of Gauss-Seidel method, is one of the systematic iterative methods for solving large linear systems of algebraic equations. For more details about SOR, please see the discussion by G.D. Smith [52]

As shown in Figure 5.2, a function of x and y can be discretized as values on every 2D mesh points by the finite difference method. Similar as equations (3.1) and

(3.2) for a function f=f(x); for a function f=f(x,y), the first and secondary derivatives at mesh point (i,j) are

−  ∂f  f k +1, j f k −1, j = (5.29)  ∂  ∆  x  k, j 2 x

−  ∂f  f + f −   = k , j 1 k , j 1 (5.30)  ∂  ∆  y k, j 2 y

2 − +  ∂ f  f + 2 f f −   = k 1, j k, j k 1, j (5.31)  ∂ 2  ()∆ 2  x  k, j x

2 − +  ∂ f  f + 2 f f −   = k , j 1 k, j k, j 1 (5.32)  ∂ 2  ()∆ 2  y  k, j y

For equations (5.3), (5.7) and (5.12), variables x,y and functions Xi, u and φ are all dimensionless, and x, y both varies from 0 to 1. M+1 grids on x axis and N+1 grids on y axis are chosen for discretizing equations (5.3), (5.7) and (5.12). For convenience, we

1 1 choose M=N so that ∆x = = = ∆y . Therefore, equation (5.3) becomes M N

2 1  2 ()∆y  φ = φ + φ + ε 2φ + ε φ + β ()z ()X  (5.33) k, j  k, j−1 k, j+1 1 k −1, j 1 k+1, j ε 2 ∑ i i k, j  4  i  165 equation (5.12) becomes

2 1  2 ()∆y  u = u + u + ε 2u + ε u + β ()z ()X  (5.34) k, j  k, j−1 k, j+1 1 k−1, j 1 k +1, j ε 2 ∑ i i k, j  4  i  and equation (5.7) becomes

 2 β 2  2 + 2ε + z 2 ()()()X ∆y X  1 2 i i k , j  i k, j  ε  = ()+ ()+ ε 2 ()+ ε 2 () X i k, j−1 X i k, j+1 1 X i k −1, j 1 X i k+1, j

zi + []()X − ()X ⋅ ()φ + −φ − (5.35) 4 i k , j+1 i k, j−1 k , j 1 k, j 1

zi 2 + ε1 []()X − ()X ⋅ ()φ + − φ − 4 i k +1, j i k −1, j k 1, j k 1, j β ()∆y 2 − z z ()()X X 2 ∑ i l i k, j l k, j ε l≠i

Equations (5.33), (5.34) and (5.35) are correlated, and can be solved numerically by using

SOR method.

M=N=40 or 80 are used in this work. The iteration procedure is said to converge if after m iterations, we have

φ m −φ m−1 k , j k , j < δ k=2,3,…M, j=2,3,…N (5.36) φ m k, j where δ is a small number, and δ=10-4 is the default number used in this work.

5.5 Results

In this work, electrophoresis of monovalent binary electrolyte solution in nanochannels is numerically solved. Nonetheless, the governing equations and numerical method discussed in last two sections are valid for multi-component electrolyte solution with multivalent ions in it, and this work will be done in the future. NaCl is chosen as a

166 representative of monovalent binary electrolytes. At infinite dilution in water at 25ºC, the diffusion coefficient of Na+ is 1.334×10-5cm2/sec, and the diffusion coefficient of Cl- is

2.032×10-5cm2/sec. [30] The concentration of Na+ and Cl- in the reservoir are assumed to be 0.145M, and the concentration of Na+ at the channel wall is assumed as 0.154M, and the concentration of Cl- at the wall is arbitrarily assumed to be 0.137M. the Debye-length is calculated by

78.54 ⋅ ⋅ 9 8.315 300 1 ε RT 1 π ⋅ − λ = e = 36 10 = 8.0×10 10 ()m = 0.80 (nm ) F I 96500 ()0.154 + 0.137 ⋅1000

Because the electric potential at the outlet is chosen as the ground, the external electric potential drop across the channel is represented by the electric potential at the inlet, which is E. Three different voltage drops: 5V, 0.05V and 0.0005V are used to test the model. The intensity of the internal electrical field produced by the channel surface charges are represented by the asymptotic potential of the outer solution (See §2.5 ), obtained as

φ X 0 RT c 0 8.315× 300 0.154 φ = 0 ln 1 = ln 1 = ln = 0.00156()V o 0 0 × 2 X 2 2F c2 2 96500 0.137

The ratio ξ=φο/E is a dimensionless parameter which shows the relative impaction of the external electric field to the channel. The five channel-voltage combinations being studied are shown in Table 5.1. Three dimensionless parameters: ε1, ε, ξ determines the solution of the governing equations. The Reynolds number is calculated as shown in §2.1

ρhε ERT Re = ρhU / µ = ρh()ε ERT / µFL / µ = e 0 e µ 2 FL

167 where ρ and µ are the density and viscosity of the fluid, h and L are the channel height and length, E is the external voltage drop, R is the universal gas constant, T is temperature, F is Faraday’s constant and εe is the permittivity of the fluid. As shown in

Table 5.1, the Reynolds number for all cases are less than 1, so flow in all these cases are laminar flow.

Channel Channel Voltage length height ε1=h/L ε=λ/h ξ=φo/E Re drop (E) (µm) (nm) Channel I 3.6 20 0.05 1/180 0.040 0.031 0.0050 Channel II 3.6 4 0.05 1/900 0.200 0.031 0.0010 Channel III 0.036 20 0.05 1/1.8 0.040 0.031 0.50 Channel IV 0.036 20 0.0005 1/1.8 0.040 3.1 0.0050 Channel V 3.6 20 5 1/180 0.040 0.00031 0.50

Table 5.1 The five channels being studied. Results for Long channels (L=3.6µm) are compared with shorter channels (L=36nm) for different external voltage drops (5V, 0.05V, and 0.0005V). The results for narrower channel (h=4nm) is compared with wider channels (h=20nm).

Numerical accuracy of the results is checked by comparing the results for channel

I at different meshes: 11×11, 21×21, 41×41 and 81×81. The results of comparison are shown in Table 5.2. Excellent agreement is achieved for the mole fractions. Some errors exist for the results of potential and velocity due to the boundary conditions at the inlet

168 and outlet. However, the numerical accuracy for large meshes is still verified. For example, for the velocity at the corner, we have (0.05278-0.05452):(0.05452-

0.05506):(0.05506-0.05521)~16:4:1, which shows that the numerical error decreases with a power of two as the mesh points increases.

11×11 21×21 41×41 81×81 X1=2.615E-3 X1=2.610E-3 X1=2.608E-3 X1=2.607E-3 X =2.577E-3 X =2.580E-3 X =2.581E-3 X =2.582E-3 2,2 2 3,3 2 5,5 2 9,9 2 φ=4.550 φ=2.600 φ=1.915 φ=1.789 u=5.278E-2 u=5.452E-2 u=5.506E-2 u=5.521E-2 X1=2.596.E-3 X1=2.595E-3 X1=2.595E-3 X1=2.594E-3 X =2.596E-3 X =2.595E-3 X =2.595E-3 X =2.594E-3 6,6 2 11,11 2 21,21 2 41,41 2 φ=4.086 φ=1.835 φ=1.144 φ=1.018 u=6.020E-2 u=6.021E-2 u=6.022E-2 u=6.022E-2

Table 5.2 Comparison of the mole fractions, potential, and velocity obtained from different mesh grids for channel I. The mesh point (2,2) near the corner of an 11×11 mesh corresponds to the mesh point (3,3) in a 21×21 mesh, (5,5) in a 41×41 mesh, and (9,9) in an 81×81 mesh. The mesh point (6,6) at the center of the 11×11 mesh corresponds to point (11,11) in a 21×21 mesh, (21,21) in a 41×41 mesh and (41,41) in an 81×81 mesh. The first and last row in these meshes represents the inlet and outlet, respectively.

The results for Channel I are shown in Figure 5.3-Figure 5.27. Figure 5.3 and

Figure 5.4 shows the mole fractions of Na+ and Cl- across the channel, respectively. The color represents the magnitude of the functions. The larger the wavelength of the color, the higher the value of the function. First of all, both plots indicate that the entry length is small, which corresponds to ε1=1/180<<1. Secondly, the mole fractions do not vary in the x direction beyond the entry length, which shows that the ions are distributed equally in the direction of flow. Thirdly, the cross section to the x axis represents the mole fraction

169 distribution pattern of ions in 1D electroosmosis, as shown in Figure 3.2. Fourthly, the mole fractions of Na+ and Cl- in the core are the same, as implied by

ε=0.040<<1. Therefore the electrolyte solution is neutral in the core. Fifthly, the mole fractions at the inlet, outlet and the wall are constants, due to the boundary conditions we used. Thus large concentration gradients are obtained at the four corners. Sixthly, in the boundary conditions we used, the mole fractions at the inlet and outlet is same as the asymptotic concentration in the channel, which is the square root of the product of the wall mole fractions of cation and anions. So the mole fractions in the core do not change at the inlet and the outlet, and the entrance length is not shown in the mole fraction plot.

Figure 5.5 shows the electric potential in the channel. The potential decreases linearly in the x direction. The potential does not varies in the y direction in the core, as implied by ε=0.040<<1. It can be seen that the potential at wall is lower than the potential in the core at the same x position, due to the ζ potential caused by the negative surface charges. The potential difference is much less than the external voltage drop, which can be explained by ξ=0.031<<1. Nonetheless, it is found that the cross section of the potential profile to the x axis represents the potential profile of electroosmosis in 1D channel, as shown in Figure 3.3.

Figure 5.6 shows the velocity profile in the channel. Due to the boundary conditions at the inlet (equation (5.15)) and outlet (equation (5.16)) that we used, there is no entry length in this plot. Instead, the velocity becomes a constant from x=0 to x=1 for any y position. The cross section is the velocity profile we obtained for 1D electroosmosis, as shown in Figure 3.3. In the core of the channel, the velocity becomes a constant, which is the property of channels where ε<<1. 170 The shear stress in the direction of flow is shown in Figure 5.7. The larges shear stress is at the wall, and the shear stress becomes less and less as getting away from the wall. In the core region, the shear stress is zero. The cross section of the contour is the shear stress plot for 1D electroosmosis, like Figure 3.4.

Figure 5.3 Mole fraction of cations across the channel. Channel length L=3.6µm, channel 0 height h=20nm, voltage drop V=0.05Volts, wall molarity of cation c1 =0.154M, wall 0 molarity of anion c2 =0.137M, molarity of cations or anions in the reservoir R R c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180.

171

Figure 5.4 Mole fraction of anions across the channel. Channel length L=3.6µm, channel 0 height h=20nm, voltage drop V=0.05Volts, wall molarity of cation c1 =0.154M, wall 0 molarity of anion c2 =0.137M, molarity of cations or anions in the reservoir R R c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180.

Figure 5.5 Electric potential across the channel as a function of x and y. Channel length L=3.6µm, channel height h=20nm, voltage drop V=0.05Volts, wall molarity of cation 0 0 c1 =0.154M, wall molarity of anion c2 =0.137M, molarity of cations or anions in the R R reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180. 172

Figure 5.6 Velocity of the electroosmotic flow across the channel. Channel length L=3.6µm, channel height h=20nm, voltage drop V=0.05Volts, wall molarity of cation 0 0 c1 =0.154M, wall molarity of anion c2 =0.137M, molarity of cations or anions in the R R reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180.

Figure 5.7 Shear stress as a function of x and y. Channel length L=3.6µm, channel height 0 h=20nm, voltage drop V=0.05Volts, wall molarity of cation c1 =0.154M, wall molarity of 0 R R anion c2 =0.137M, molarity of cations or anions in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180. 173

Figure 5.8 The x component of the flux of cations due to diffusion. Channel length L=3.6µm, channel height h=20nm, voltage drop V=0.05Volts, wall molarity of cation 0 0 c1 =0.154M, wall molarity of anion c2 =0.137M, molarity of cations or anions in the R R reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180. The large spikes at the corners are caused by the discontinuity of mole fractions in the corners.

Figure 5.8-Figure 5.11 show the diffusive fluxes of Na+ and Cl- in the x and y directions, respectively. The diffusive fluxes are calculated according to equations (5.23) and (5.24). From these plots, first of all, the diffusive flux of both ion species in the x direction is zero across the channel, except at the four corners; this is because the mole

174 fractions in the upstream and downstream reservoirs are assumed to be the same, which leads to the constant mole fractions in the x direction of the channel, as shown in Figure

5.3 and Figure 5.4. Secondly, the diffusive flux of both ion species in the y direction is zero in the core, and increase as getting closer to the wall, which corresponds to the constant mole fractions of both ion species in the core, as shown in Figure 5.3. The large spikes of diffusive fluxes at the four corners are caused by the discontinuities of mole fractions at the four corners, as shown in Figure 5.3. In summary, for negatively charged walls, the cations diffuse away from the wall, while the anions diffuse towards the wall.

Figure 5.9 The x component of the flux of anions due to diffusion. Channel length L=3.6µm, channel height h=20nm, voltage drop V=0.05Volts, wall molarity of cation 0 0 c1 =0.154M, wall molarity of anion c2 =0.137M, molarity of cations or anions in the R R reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180. 175

Figure 5.10 The y component of the flux of cations due to diffusion. Channel length L=3.6µm, channel height h=20nm, voltage drop V=0.05Volts, wall molarity of cation 0 0 c1 =0.154M, wall molarity of anion c2 =0.137M, molarity of cations or anions in the R R reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180.

Figure 5.11 The y component of the flux of anions due to diffusion. Channel length L=3.6µm, channel height h=20nm, voltage drop V=0.05Volts, wall molarity of cation 0 0 c1 =0.154M, wall molarity of anion c2 =0.137M, molarity of cations or anions in the R R reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180. 176

Figure 5.12 The x component of the electrical field strength. Channel length L=3.6µm, 0 channel height h=20nm, voltage drop V=0.05Volts, wall molarity of cation c1 =0.154M, 0 wall molarity of anion c2 =0.137M, molarity of cations or anions in the reservoir R R c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180.

Figure 5.12 and Figure 5.13 show the x and y components of the electric field strength vector inside the channel. In most part of the channel, the field strength is a constant in x direction, and zero in the y direction. In the region near the wall, the y component of field strength is nonzero, and reaches its maximum at the wall. At the outlet and inlet, the profile of the x component of field strength is very steep. 177

Figure 5.13 The y component of the electrical field strength. Channel length L=3.6µm, 0 channel height h=20nm, voltage drop V=0.05Volts, wall molarity of cation c1 =0.154M, 0 wall molarity of anion c2 =0.137M, molarity of cations or anions in the reservoir R R c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180. The spikes in the corners are caused by the discontinuity of potential in the corners.

178

Figure 5.14 The x component of the flux of cations due to electrical migration. Channel length L=3.6µm, channel height h=20nm, voltage drop V=0.05Volts, wall molarity of 0 0 cation c1 =0.154M, wall molarity of anion c2 =0.137M, molarity of cations or anions in R R the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180.

The electric migrational fluxes are calculated with equations (5.25) and (5.26).

The results are shown in Figure 5.14-Figure 5.17. The fluxes are proportional to the corresponding electric field strength and mole fractions. Except at the inlet and outlet, the migrational flux of Na+ in the x direction is a constant, and the direction of the flux is toward the cathode; on the other hand, the migrational flux of Cl- is a constant but toward 179 the anode. For 41×41 mesh grids, the entrance length is (4/41)×L=0.35µm, calculated by using the 99% of the constant flux in the channel, The migrational flux of Na+ in the y direction is zero in the core, so does the migrational flux of Cl- in the y direction. In the region near the wall, the Na+ is driven toward the wall by the migrational forces, while the Cl- is driven away from the wall. So the y component of the migrational flux for an ion species is in the opposite direction from the diffusion flux of that ion species in the y direction, and these two types of fluxes canceled with each other.

In the core region, the intensity of the flux of Cl- in the x direction is same as the intensity of Na+ flux in the x direction. Same things happen in the y direction too. This is caused by the same mole fraction of Na+ and Cl- in the core. However, for dimensional plots, these fluxes are no longer same, because Na+ and Cl- have different diffusion coefficients.

The convective fluxes of Na+ and Cl- in the x direction are calculated from equation (5.22), and the results are shown in Figure 5.18 and Figure 5.19, respectively.

The convective flux in the core region is constant, because the mole fractions and electroosmotic velocity are all constant in the core. The fluxes for Na+ and Cl- are different because the Schmitt number is different for different ions.

Figure 5.20-Figure 5.23 show the entire fluxes in the x and y directions for different ion species. The entire flux is the sum of the diffusive flux, migrational flux and convective flux, which shows the real movement of ion species in the channel. For Na+, the flux in the x direction inside the channel is constant and its direction is forward to the cathode. For Cl-, the flux in the x direction inside the channel is a constant but backwards to the anode, because the migrational flux, which is backwards, is larger than the

180 convective flux, which is forward. The fluxes of both ion species in the y direction are nonzero in the channel. The reason is not clear so far. The dimensional fluxes are shown in Figure 5.24-Figure 5.27. They are obtained by multiply the dimensionless fluxes with

2 cDi/h, and the units are moles/(m ·s).

Figure 5.15 The x component of the flux of anions due to electrical migration. Channel length L=3.6µm, channel height h=20nm, voltage drop V=0.05Volts, wall molarity of 0 0 cation c1 =0.154M, wall molarity of anion c2 =0.137M, molarity of cations or anions in R R the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180. 181

Figure 5.16 The y component of the flux of cations due to electrical migration. Channel length L=3.6µm, channel height h=20nm, voltage drop V=0.05Volts, wall molarity of 0 0 cation c1 =0.154M, wall molarity of anion c2 =0.137M, molarity of cations or anions in R R the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180. The spikes in the corners are caused by the discontinuity of potential in the corners.

Figure 5.17 The y component of the flux of anions due to electrical migration. Channel length L=3.6µm, channel height h=20nm, voltage drop V=0.05Volts, wall molarity of 0 0 cation c1 =0.154M, wall molarity of anion c2 =0.137M, molarity of cations or anions in R R the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180. 182

Figure 5.18 The flux of cations in the x direction due to convection. Channel length L=3.6µm, channel height h=20nm, voltage drop V=0.05Volts, wall molarity of cation 0 0 c1 =0.154M, wall molarity of anion c2 =0.137M, molarity of cations or anions in the R R reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180.

Figure 5.19 The flux of anions in the x direction due to convection. Channel length L=3.6µm, channel height h=20nm, voltage drop V=0.05Volts, wall molarity of cation 0 0 c1 =0.154M, wall molarity of anion c2 =0.137M, molarity of cations or anions in the R R reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180. 183

Figure 5.20 The entire flux of cations in the x direction. Channel length L=3.6µm, 0 channel height h=20nm, voltage drop V=0.05Volts, wall molarity of cation c1 =0.154M, 0 wall molarity of anion c2 =0.137M, molarity of cations or anions in the reservoir R R c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180. The spikes in the diffusive flux are not shown in this plot because the electrical migrational flux is dominant.

Figure 5.21 The entire flux of anions in the x direction. Channel length L=3.6µm, 0 channel height h=20nm, voltage drop V=0.05Volts, wall molarity of cation c1 =0.154M, 0 wall molarity of anion c2 =0.137M, molarity of cations or anions in the reservoir R R c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180. 184 -5 x 10 6

-2 1 -4 0.8 -6 0.6

Flux of cationsin the y direction 1 0.8 0.4 0.6 0.4 0.2 0.2 0 0 x (length) y (height)

Figure 5.22 The entire flux of cations in the y direction. Channel length L=3.6µm, 0 channel height h=20nm, voltage drop V=0.05Volts, wall molarity of cation c1 =0.154M, 0 wall molarity of anion c2 =0.137M, molarity of cations or anions in the reservoir R R c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180. The spikes at the four corners are removed by assuming the function values to be zero at the corners.

Figure 5.23 The entire flux of anions in the y direction. Channel length L=3.6µm, 0 channel height h=20nm, voltage drop V=0.05Volts, wall molarity of cation c1 =0.154M, 0 wall molarity of anion c2 =0.137M, molarity of cations or anions in the reservoir R R c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180. 185

Figure 5.24 The dimensional flux of cations in the x direction. Channel length L=3.6µm, 0 channel height h=20nm, voltage drop V=0.05Volts, wall molarity of cation c1 =0.154M, 0 wall molarity of anion c2 =0.137M, molarity of cations or anions in the reservoir R R 2 c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180. The units of flux is [moles/(m ·s)].

Figure 5.25 The dimensional flux of anions in the x direction. Channel length L=3.6µm, 0 channel height h=20nm, voltage drop V=0.05Volts, wall molarity of cation c1 =0.154M, 0 wall molarity of anion c2 =0.137M, molarity of cations or anions in the reservoir R R 2 c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180.The units of flux is [moles/(m ·s)]. 186

Figure 5.26 The dimensional flux of cations in the y direction. Channel length L=3.6µm, 0 channel height h=20nm, voltage drop V=0.05Volts, wall molarity of cation c1 =0.154M, 0 wall molarity of anion c2 =0.137M, molarity of cations or anions in the reservoir R R 2 c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180. The units of flux is [moles/(m ·s)].

Figure 5.27 The dimensional flux of anions in the y direction. Channel length L=3.6µm, 0 channel height h=20nm, voltage drop V=0.05Volts, wall molarity of cation c1 =0.154M, 0 wall molarity of anion c2 =0.137M, molarity of cations or anions in the reservoir R R 2 c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/180. The units of flux is [moles/(m ·s)]. 187 The results for channel II are shown in Figure 5.28-Figure 5.37. For the small change height h=4nm, the parameter ε=0.200 is not much less than 1, thus according to the discussions in previous chapters, the mole fractions, potential and velocity profiles for a given x should be in parabolic shape. The results in Figure 5.28-Figure 5.31 confirm this reasoning. Because ε1=1/900 is 5 times less than the ε1 for channel I, the entry length for channel II is also less than the entry length for channel I. However, because the grids we used here is 41×41, the change of entry length is not shown clearly in these plots. The other parameter ξ=0.031 is the same as for channel I. Thus the external potential in the x direction is still dominating. As the results, the potential is still decreasing linearly in the x direction, and the potential change in the y direction is much less than the external voltage drop across the channel. The plot of shear stress, Figure 5.32, indicates that the shear stress is no longer zero in the core of the channel.

The diffusive fluxes, field strength, migrational fluxes, convective fluxes and entire fluxes are shown in Figure 5.33-Figure 5.37, respectively. The x components of these vectors are similar to previous case for channel I, while things are different for the y components of these vectors. Because of the parabolic shape of mole fractions, potential and velocity profiles at given x position in the channel, the y components of mole fraction gradients, potential gradients and velocity gradients are nonzero in the core of the channel. Thus the diffusive fluxes, migrational fluxes are no longer zero in the core either.

188

Figure 5.28 The mole fraction of cations across the channel. Channel length L=3.6µm, 0 channel height h=4nm, voltage drop V=0.05Volts, wall molarity of cation c1 =0.154M, 0 wall molarity of anion c2 =0.137M, molarity of cations or anions in the reservoir R R c1 =c2 =0.145M. ε=0.200, ξ=0.031, ε1=1/900.

189

Figure 5.29 The mole fraction of anions across the channel. Channel length L=3.6µm, 0 channel height h=4nm, voltage drop V=0.05Volts, wall molarity of cation c1 =0.154M, 0 wall molarity of anion c2 =0.137M, molarity of cations or anions in the reservoir R R c1 =c2 =0.145M. ε=0.200, ξ=0.031, ε1=1/900.

190

Figure 5.30 Dimensionless electrical potential across the channel. Channel length L=3.6µm, channel height h=4nm, voltage drop V=0.05Volts, wall molarity of cation 0 0 c1 =0.154M, wall molarity of anion c2 =0.137M, molarity of cations or anions in the R R reservoir c1 =c2 =0.145M. ε=0.200, ξ=0.031, ε1=1/900.

191

Figure 5.31 Dimensionless velocity profile in the channel. Channel length L=3.6µm, 0 channel height h=4nm, voltage drop V=0.05Volts, wall molarity of cation c1 =0.154M, 0 wall molarity of anion c2 =0.137M, molarity of cations or anions in the reservoir R R c1 =c2 =0.145M. ε=0.200, ξ=0.031, ε1=1/900.

192

Figure 5.32 Shear stress in the x direction. Channel length L=3.6µm, channel height 0 h=4nm, voltage drop V=0.05Volts, wall molarity of cation c1 =0.154M, wall molarity of 0 R R anion c2 =0.137M, molarity of cations or anions in the reservoir c1 =c2 =0.145M.

ε=0.200, ξ=0.031, ε1=1/900.

193

Figure 5.33 Diffusive fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y (bottom) directions. Channel length L=3.6µm, channel height h=4nm, voltage drop 0 0 V=0.05Volts, wall molarity of cation c1 =0.154M, wall molarity of anion c2 =0.137M, R R molarity of cations or anions in the reservoir c1 =c2 =0.145M.

ε=0.200, ξ=0.031, ε1=1/900. The spikes are caused by the discontinuity of mole fractions at the corners.

194

Figure 5.34 Electric field strength in the x (left) and y (right) directions. L=3.6µm,

0 h=4nm, voltage drop V=0.05Volts, wall molarity of cations c1 =0.154M, wall molarity of

0 R R anions c2 =0.137M, molarity of cations or anions in the reservoir c1 =c2 =0.145M.

ε=0.200, ξ=0.031, ε1=1/900.

195

Figure 5.35 Electric migrational fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y (bottom) directions. L=3.6µm, h=4nm, voltage drop V=0.05Volts, wall molarity of

0 0 cations c1 =0.154M, wall molarity of anions c2 =0.137M, molarity of cations or anions in

R R the reservoir c1 =c2 =0.145M. ε=0.200, ξ=0.031, ε1=1/900.

196

Figure 5.36 Convective fluxes of Na+ (left side) and Cl- (right side) in the x direction.

0 L=3.6µm, h=4nm, voltage drop V=0.05Volts, wall molarity of cation c1 =0.154M, wall

0 molarity of anion c2 =0.137M, molarity of cations or anions in the reservoir

R R c1 =c2 =0.145M. ε=0.200, ξ=0.031, ε1=1/900.

197

Figure 5.37 Entire fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y

(bottom) directions. L=3.6µm, h=4nm, voltage drop V=0.05Volts, wall molarity of cation

0 0 c1 =0.154M, wall molarity of anion c2 =0.137M, molarity of cations or anions in the

R R reservoir c1 =c2 =0.145M. ε=0.200, ξ=0.031, ε1=1/900. The spikes are caused by the discontinuity of mole fractions and potential at the corners.

198 The results for channel III are shown in Figure 5.38-Figure 5.43. The judgment criteria for the convergence is δ=10-5. Parameter ε=0.040, are the same as the ε for channel I, so the mole fraction contours are similar as Figure 5.3-Figure 5.4. The third parameter ε1=1/1.8, so the entry length is comparable with the length of the channel. As the result, the velocity contour is not flat in the regions near the inlet or the outlet, as are the convective fluxes, as shown in Figure 5.38 and Figure 5.42. The potential inside the channel is also affected by the large ε1. Instead of decreasing linearly in the x direction as in channel I, the potential in channel III decreases along a curve in the x direction.

Because ξ=0.031<<1, the curvature is very little, as shown in Figure 5.38.

However, from Figure 5.40, it can be seen that the field strength in the x direction,

∂φ which equals − , increases from 1.9 to 2 as x varies from 0 to 1. As the result, the ∂x migrational flux of Na+ in the x direction increases as x increases, and the migrational flux of Cl- in the x direction, which is negative, decreases as x increases, as shown in

Figure 5.41. The diffusive fluxes in the x direction are still zero in the core, as shown in

Figure 5.39. The y components of different kinds of fluxes are similar to those in channel

I, since the y component is mainly determined by the channel height and zeta potential, which are the same for channel I and III. Finally, the entire fluxes in the x direction, as the sum of diffusive, migrational and convective fluxes, are shown in Figure 5.43, from which we know that the fluxes in the x direction varies with x position, for channels where h/L=O(1).

199

Figure 5.38 Mole fraction of Na+ (upper left), Mole fraction of Cl- (upper right), potential (middle left), velocity (middle right), and shear stress (bottom left) in the channel. Channel length L=36nm, channel height h=20nm, voltage drop V=0.05Volts, wall 0 0 molarity of cation c1 =0.154M, wall molarity of anion c2 =0.137M, molarity of cations or R R anions in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/1.8. The color represents the magnitude of the function. 200

Figure 5.39 Diffusive fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y (bottom) directions. Channel length L=36nm, channel height h=20nm, voltage drop 0 0 V=0.05Volts, wall molarity of cations c1 =0.154M, wall molarity of anions c2 =0.137M, R R molarity of cations or anions in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031,

ε1=1/1.8.

201

Figure 5.40 Electric field strength in the x (left) and y (right) directions. Channel length L=36nm, channel height h=20nm, voltage drop V=0.05Volts, wall molarity of cations 0 0 c1 =0.154M, wall molarity of anions c2 =0.137M, molarity of cations or anions in the R R reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/1.8.

202

Figure 5.41 Electric migrational fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y (bottom) directions. L=36nm, h=20nm, voltage drop V=0.05Volts, wall molarity of

0 0 cations c1 =0.154M, wall molarity of anions c2 =0.137M, molarity of cations or anions in

R R the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/1.8.

203

Figure 5.42 Convective fluxes of Na+ (left side) and Cl- (right side) in the x direction.

0 L=36nm, h=20nm, voltage drop V=0.05Volts, wall molarity of cations c1 =0.154M, wall

0 molarity of anions c2 =0.137M, molarity of cations or anions in the reservoir

R R c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/1.8.

204

Figure 5.43 Entire fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y (bottom) directions. Channel length L=36nm, channel height h=20nm, voltage drop 0 0 V=0.05Volts, wall molarity of cations c1 =0.154M, wall molarity of anions c2 =0.137M, R R molarity of cations or anions in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.031, ε1=1/1.8.

The results for channel IV are shown in Figure 5.44-Figure 5.49, which are very similar to the results for channel III. The major difference is ξ=3.1 in this case, which is

100 times higher than the ξ for channel III. So the curvature of the potential contour is distinct, as shown in Figure 5.44.

205

Figure 5.44 Mole fraction of Na+ (upper left), Mole fraction of Cl- (upper right), potential (middle left), velocity (middle right), and shear stress (bottom left) in the channel. Channel length L=36nm, channel height h=20nm, voltage drop V=0.0005Volts, wall 0 0 molarity of cations c1 =0.154M, wall molarity of anions c2 =0.137M, molarity of cations R R or anions in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=3.1, ε1=1/1.8.

206

Figure 5.45 Diffusive fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y

(bottom) directions. L=36nm, h=20nm, voltage drop V=0.0005Volts, wall molarity of

0 0 cations c1 =0.154M, wall molarity of anions c2 =0.137M, molarity of cations or anions in

R R the reservoir c1 =c2 =0.145M. ε=0.040, ξ=3.1, ε1=1/1.8.

207

Figure 5.46 Electric field strength in the x (left) and y (right) directions. L=36nm,

0 h=20nm, voltage drop V=0.0005Volts, wall molarity of cations c1 =0.154M, wall

0 molarity of anions c2 =0.137M, molarity of cations or anions in the reservoir

R R c1 =c2 =0.145M. ε=0.040, ξ=3.1, ε1=1/1.8.

208

Figure 5.47 Electric migrational fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y (bottom) directions. L=36nm, h=20nm, voltage drop V=0.0005Volts, wall molarity

0 0 of cations c1 =0.154M, wall molarity of anions c2 =0.137M, molarity of cations or anions

R R in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=3.1, ε1=1/1.8.

209

Figure 5.48 Convective fluxes of Na+ (left side) and Cl- (right side) in the x direction.

0 L=36nm, h=20nm, voltage drop V=0.0005Volts, wall molarity of cations c1 =0.154M,

0 wall molarity of anions c2 =0.137M, molarity of cations or anions in the reservoir

R R c1 =c2 =0.145M. ε=0.040, ξ=3.1, ε1=1/1.8.

210

Figure 5.49 Entire fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y

(bottom) directions. Channel length L=36nm, channel height h=20nm, voltage drop

0 V=0.0005Volts, wall molarity of cations c1 =0.154M, wall molarity of anions

0 R R c2 =0.137M, molarity of cations or anions in the reservoir c1 =c2 =0.145M.

ε=0.040, ξ=3.1, ε1=1/1.8.

211 The results for channel V are shown in Figure 5.50-Figure 5.55. Channel V is different from Channel I on the external voltage drop, which is 5V for channel V and

0.05V for Channel I. As the result, the parameter ξ=0.00031 for Channel V, which is 100 times less than the ξ for Channel I. To cope with such a small ξ, a 100 times less

−6 judgment criteria for convergence δ=10 is used for the numerical calculations on

Channel V. Figure 5.50-Figure 5.52 show that the mole fractions, potential, velocity, shear stress, diffusive fluxes and field strength are quite similar to the results for Channel

The electric migrational fluxes in the x direction, as shown in Figure 5.53, is quite different from the corresponding picture for Channel I, on the distinct surface fluxes near the wall. The migrational flux of an ion species is proportional to the field strength and the mole fraction of the ion species. Since the x component of the electric field is constant in the y direction, and the mole fraction is higher near the wall, it is naturally that there should be higher migrational fluxes near the wall. The reason it is not shown in previous cases is because the parameter ξ is not small enough, so that the large variances of fluxes at the inlet and the outlet shadow the relative smaller variances of fluxes near the wall. For channel V, ξ=0.00031 is100 times less, and correspondingly, the field strength in the x direction is 100 times stronger, so that the migrational fluxes inside the channel is 100 times higher, which made the variance of fluxes near the wall comparable with the variances at the inlet and outlet. As the results, the wall flux is shown in Figure

5.53. From Figure 5.55, we know that the entire flux in the x direction is also higher in the region near the wall, which agrees with the traditional capillary electrophoresis theory which states that the total fluxes of ions consists of volume flux and surface flux. 212

Figure 5.50 Mole fraction of Na+ (upper left), Mole fraction of Cl- (upper right), potential (middle left), velocity (middle right), and shear stress (bottom left) in the channel. Channel length L=3.6µm, channel height h=20nm, voltage drop V=5Volts, wall molarity 0 0 of cations c1 =0.154M, wall molarity of anions c2 =0.137M, molarity of cations or anions R R in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.00031, ε1=1/180. 213

Figure 5.51 Diffusive fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y

(bottom) directions. L=3.6µm, h=20nm, voltage drop V=5Volts, wall molarity of cations

0 0 c1 =0.154M, wall molarity of anions c2 =0.137M, molarity of cations or anions in the

R R reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.00031, ε1=1/180.

214

Figure 5.52 Electric field strength in the x (left) and y (right) directions. Channel length

L=3.6µm, channel height h=20nm, voltage drop V=5Volts, wall molarity of cations

0 0 c1 =0.154M, wall molarity of anions c2 =0.137M, molarity of cations or anions in the

R R reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.00031, ε1=1/180.

215

Figure 5.53 Electric migrational fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y (bottom) directions. Channel length L=3.6µm, channel height h=20nm, voltage

0 drop V=5Volts, wall molarity of cations c1 =0.154M, wall molarity of anions

0 R R c2 =0.137M, molarity of cations or anions in the reservoir c1 =c2 =0.145M. ε=0.040,

ξ=0.00031, ε1=1/180.

216

Figure 5.54 Convective fluxes of Na+ (left side) and Cl- (right side) in the x direction.

Channel length L=3.6µm, channel height h=20nm, voltage drop V=5Volts, wall molarity

0 0 of cations c1 =0.154M, wall molarity of anions c2 =0.137M, molarity of cations or anions

R R in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.00031, ε1=1/180.

217

Figure 5.55 Entire fluxes of Na+ (left side) and Cl- (right side) in the x (top) and y

(bottom) directions. Channel length L=3.6µm, channel height h=20nm, voltage drop

0 0 V=5Volts, wall molarity of cations c1 =0.154M, wall molarity of anions c2 =0.137M,

R R molarity of cations or anions in the reservoir c1 =c2 =0.145M. ε=0.040, ξ=0.00031,

ε1=1/180.

218 CHAPTER 6

EOF IN RECTANGULAR CHANNELS

6.1 Introduction

h h For rectangular micro- and nanochannels where ε = << 0 and ε = = O()1 , 1 L 2 W as shown in Figure 6.1, the electroosmotic flow cannot be reduced to the one-dimensional model discussed in Chapter 3 and 4. The 2D model of electrokinetic flow has been investigated by Li [83] and Bhattacharyya et. al.[85]. In this chapter, the electroosmotic flow in rectangular channels is obtained with a 2D model.

The governing equations for 2D model are derived from the general 3D governing equations obtained in §2.3. The SOR method is used to solve the governing equations numerically, and the procedure is similar to the procedure of solving governing equations of fluxes in Chapter 5.

The model is valid for both symmetric and asymmetric cases. In the governing equations for asymmetric cases, we add parameters for the slip conditions which will

219 change the velocity of fluid at the wall. Future work on slip conditions will be discussed in the last chapter.

Direction of Flow y,v h

x,u W z,w L

Figure 6.1 Rectangular channel where h/W=O(1), and h<

6.2 Governing Equations

Equation (2.36) is the nondimensional Poisson equation for a 3D channel.

ε 2 ∇ 2φ = −β ∑ zi X i (2.36) i

h If ε = << 1, equation (2.36) becomes 1 L

 ∂ 2φ ∂ 2φ  ε 2  + ε 2  = −β z X (6.1)  ∂ 2 2 ∂ 2  ∑ i i  y z  i

λ h c where ε = , ε = = O()1 , and β = . The boundary conditions are h 2 W I

1 φ = φ y=0 (6.2)

220 2 φ = φ z=0 (6.3)

3 φ = φ y=1 (6.4)

4 φ = φ z=1 (6.5)

1 2 3 4 where φ ,φ ,φ , and φ are the potential on the four side walls. For symmetrical cases, the potentials on different side walls are the same, and it is convenient to set the side walls to be at zero potential, thus the boundary conditions of φ can be simplified:

0 φ = φ =0 y=0, z=0, y=1 and z=1 (6.6)

0 where constant φ =0 is the electrical potential on all side walls.

For the mass transport in electrokinetic flow, the 3D nondimensional equation is equation (2.43), as shown below:

∇ 2 + ∇ • ()⋅ ∇φ − ⋅ ⋅ ∇ • ()= X i zi X i Re Sc X i u 0 i=1,2,…M (2.43)

ρU h µ where Re = 0 is the Reynolds number, and Sc = is the Schmidt number. In this µ ρ Di

RT work, we choose E = as the scale of the nondimensional electric field strength. For 0 hF

⋅ ⋅ ∇ • ()≈ the steady flow in micro- or nanochannels, Re Sc X i u 0 , because the Reynolds

h number is small, see §2.1. Since ε = << 1, equation (2.43) can be simplified to 1 L

∂  ∂X ∂φ  2 ∂ ∂X ∂φ i + + ε  i +  =  zi X i  2  zi X i  0 (6.7) ∂y  ∂y ∂y  ∂z  ∂z ∂z  and the boundary conditions for the mole fractions of ion species i are

1 Xi=Xi y=0 (6.8)

2 Xi=Xi z=0 (6.9)

221 3 Xi=Xi y=1 (6.10)

4 Xi=Xi z=1 (6.11)

1 2 3 4 Where Xi , Xi , Xi and Xi are the concentration of ion species i on the four side walls.

For symmetrical cases, the boundary conditions become

0 Xi=Xi y=0, z=0, y=1 and z=1 (6.12)

Similarly, the Navier-Stokes equation (2.51) becomes

2 2  ∂ u 2 ∂ u  ε 2  + ε  = −β z X (6.13)  ∂ 2 2 ∂ 2  ∑ i i  y z  i and the boundary conditions for u are

u=u1 y=0 (6.14)

u=u2 z=0 (6.15)

u=u3 y=1 (6.16)

u=u4 z=1 (6.17) where u1, u2, u3 and u4 are the velocities of fluid on the four side walls. For cases where no-slip condition applies, the boundary conditions for u become

u=0 y=0, z=0, y=1 and z=1 (6.18)

6.3 Numerical Method

The successive over-relaxation method (SOR) is used to solve the governing equations numerically. The whole procedure is similar to the way we solve the governing equations for fluxes in Chapter 5, because these two sets of governing equations have same form. The only thing that need to be changed is to use z instead of x, and ε2 instead of ε1. 41×41 mesh grids are used, and ∆z=∆y. The discretized governing equations are

222 2 1  2 ()∆y  φ = φ + φ + ε 2φ + ε φ + β ()z ()X  (6.19) k, j  k, j−1 k, j+1 2 k −1, j 2 k+1, j ε 2 ∑ i i k, j  4  i 

2 1  2 ()∆y  u = u + u + ε 2u + ε u + β ()z ()X  (6.20) k, j  k, j−1 k, j+1 2 k−1, j 2 k +1, j ε 2 ∑ i i k, j  4  i 

 2 β 2  2 + 2ε + z 2 ()()()X ∆y X  2 2 i i k , j  i k, j  ε  = ()+ ()+ ε 2 ()+ ε 2 () X i k, j−1 X i k, j+1 2 X i k −1, j 2 X i k+1, j

zi + []()X − ()X ⋅ ()φ + −φ − (6.21) 4 i k , j+1 i k, j−1 k , j 1 k, j 1

zi 2 + ε []()X − ()X ⋅ ()φ + − φ − 4 2 i k +1, j i k −1, j k 1, j k 1, j β ()∆y 2 − z z ()()X X 2 ∑ i l i k, j l k, j ε l≠i where subscript j and k denote the grid on y and z axis, respectively. The iteration procedure is said to converge if after m iterations, we have

φ m −φ m−1 k , j k , j < δ k=2,3,…M, j=2,3,…N (6.22) φ m k, j where M=N=40, δ is a small number, and δ=10-4 is the default criteria used in this work.

6.4 Results For Symmetrical Cases

The model assumes that h/W=O(1) and h/L<<1. In the numerical calculations, we use L=3.6µm, and h, W in nanometers, as shown in Table 6.1, thus h/W=O(1) and h/L<<1 are satisfied. Three channels with different geometry are studied. Channel 1 is a

20nm×20nm square channel, channel 2 is a 4nm×4nm square channel, and channel 3 is a

4nm×20nm rectangular channel. The electrolytes being studied here are monovalent binary electrolytes, such as NaCl. For all these cases, the concentration of cations on the

223 side walls is 0.154M, and the concentration of anions on the walls is 0.142M. Then the

Debye length is the same for all cases, which is 0.79nm. The Reynolds number is in the order of 10-6, as shown in Table 6.1, which supports the assumption

 ∂X ∂X ∂X  ⋅ ⋅ ε i + i + ε i ≈ Re Sc  1u v 2 w  0 made in §6.2.  ∂x ∂y ∂z 

h W (nm) λ (nm) Re ε=λ/h (nm) Channel 1 20 20 0.79 5.0×10-6 0.040 Channel 2 4 4 0.79 1.0×10-6 0.200 Channel 3 4 20 0.79 1.0×10-6 0.200

Table 6.1 The properties of the channels. The lengths of channels are 3.6µm, the wall concentration of cation is 0.154M, and the wall concentration of anion is 0.142M.

µ In Table 6.2, m is the average electric mobility of the solution calculated

µ a numerically, and m is the analytical mobility calculated by using the asymptotic solution of the governing equations:

u ⋅ L 1  X 0  µ a = 0 ⋅ ln 1  (6.23) m φ ∗  0  2  X 2 

The analytical and numerical velocity can be calculated by multiplying the corresponding mobility with the electric field in the x direction. For instance, if 5 Volts is applied on a

3.6µm long channel, the electric field is 1.4×106 (V/m), and the average numerical velocity for channel 1 is 0.85mm/s.

224

µ µ a m m (cm2·V-1·s-1) (cm2·V-1·s-1) Channel 1 6.1×10-6 7.3×10-6 Channel 2 3.0×10-6 7.3×10-6 Channel 3 4.1×10-6 7.3×10-6

µ µ a Table 6.2 The numerical mobility m ,and the analytical mobility m .

The asymptotic solution is valid for large channels only. The less the channel height and width, the more error there is between the asymptotic solution and the true values, which can be represented by the numerical solution at present accuracy. For

µ a µ channel 1, m / m ~1.2, the 20% error is caused by the higher estimation of the

µ a µ asymptotic solution for velocity in the EDL. For channel 2, m / m ~2.4, the much larger

µ a µ error was due to the EDL overlapping. For channel 3, m / m ~1.8, and the relative error is less than the error for channel 2 but larger than the error for channel 1.

The numerical accuracy is checked by comparing the results with the results obtained if 81×81 meshes are used. If the numerical results for 41×41 meshes are absolutely accurate, the results at grid point (m,n) should equals the results at grid point

(2m-1,2n-1) for 81×81 meshes. The comparison being made for channel 1 is shown in

Table 6.3. The errors between two meshes do not exceed 0.0002 for all variables, so the results for 41×41 meshes are accurate enough.

225 41×41 81×81

X1=0.5089 X1=0.5088 X =0.4904 X =0.4906 (3,3) 2 (5,5) 2 φ=0.0220 φ=0.0224 u=0.0220 u=0.0224

X1=0.5026 X1=0.5025 X =0.4966 X =0.4967 (5,5) 2 (9,9) 2 φ=0.0345 φ=0.0347 u=0.0345 u=0.0347

X1=0.4996 X1=0.4996 X =0.4996 X =0.4996 (21,21) 2 (41,41) 2 φ=0.0406 φ=0.0408 u=0.0406 u=0.0408

Table 6.3 Comparison of the numerical results for 41×41 mesh grids and the results for 81×81 meshes at three locations inside the channel. g=c1/I, and f=c2/I represents the concentration of cation species and anion species, respectively.

The mole fractions of both ion species in channel 1 are shown in Figure 6.2. In the core region, the mole fractions of cation species and anion species are same, so the solution is electrically neutral in the core. In the regions near the wall, the EDL exists, where the cations become more concentrated than the anions.

The dimensionless potential and velocity contours inside the channel are shown in

Figure 6.3. These two contours overlap other, because the governing equations and boundary conditions for φ and u are the same. In the core region, the potential and velocity are constant, which can be estimated by the asymptotic solution: 226 1  X 0  1  c 0  1  0.154  φ = u = ln 1  = ln 1  = ln = 0.0406 .  0   0    2  X 2  2  c2  2  0.142  which is very close to the numerical value shown in Figure 6.3. Because channel 1 is a square channel, the potential and velocity changes in the z direction are the same as their changes in the y direction, as shown in Figure 6.3. Similar relationships also exist in

Figure 6.2 for the mole fractions of both ion species.

Figure 6.2 Mole fractions of ions for monovalent binary electrolyte solution. W=20nm, 0 h=20nm, wall molarity of cation species: c1 =0.154M, wall molarity of anion species: 0 c2 =0.142M. The walls are at zero potential, and no-slip condition is applied.

ε=0.040, ε2=1.

227

Figure 6.3 Dimensionless electrical potential and velocity profile for monovalent binary 0 electrolyte solution. W=20nm, h=20nm, wall molarity of cation species: c1 =0.154M, 0 wall molarity of anion species: c2 =0.142M. The walls are at zero potential, and no-slip condition is applied. ε=0.040, ε2=1.

In the 2D model, the shear stress in the x direction, which is the direction of flow, has two components. τyx and τzx. τyx is produced by the gradient of u in the y direction:

∂u ∂u τ = , and τzx is caused by the gradient of u in the z direction: τ = . τyx is shown yx ∂y zx ∂z

in Figure 6.4, and τzx is shown in Figure 6.5. In the core of the channel, both τyx and τzx 228 are zero. The magnitude of τyx is the largest on walls y=0 and y=1, except at the four corners, where τyx~0. The magnitude of τzx is the largest on walls z=0 and z=1, except at the corners, where τzx~0.

Figure 6.4 Dimensionless shear stress τyx for monovalent binary electrolyte solution. 0 W=20nm, h=20nm, wall molarity of cation species: c1 =0.154M, wall molarity of anion 0 species: c2 =0.142M. The walls are at zero potential, and no-slip condition is applied. ε=0.040, ε2=1.

229

Figure 6.5 Dimensionless shear stress τzx for monovalent binary electrolyte solution. 0 W=20nm, h=20nm, wall molarity of cation species: c1 =0.154M, wall molarity of anion 0 species: c2 =0.142M. The walls are at zero potential, and no-slip condition is applied.

ε=0.040, ε2=1.

The mole fractions of ion species for channel 2 are shown in Figure 6.6. ε=0.200 implies that the EDL on one side wall is overlapped with the EDL on the opposite wall.

As the results, the concentration of cation species is always higher than the concentration of anion species, and the core is not electrically neutral. The shape of the contours of mole fractions is close to paraboloid.

230 The dimensionless potential and velocity contours are shown in Figure 6.7. The paraboloidal contours have their maximum in the center of the channel, and the contours are not flat in the core. Nonetheless, the dimensionless potential and velocity are still the same, which is determined by their governing equations and boundary conditions.

Shear stresses τyx and τzx are shown in Figure 6.8 and Figure 6.9, respectively.

Both τyx and τzx are nonzero in the core.

Figure 6.6 Mole fractions of ions for monovalent binary electrolyte solution. W=4nm, 0 h=4nm, wall molarity of cation species: c1 =0.154M, wall molarity of anion species: 0 c2 =0.142M. The walls are at zero potential, and no-slip condition is applied. ε=0.200, ε2=1.

231

Figure 6.7 Dimensionless electric potential and velocity profile for monovalent binary 0 electrolyte solution. W=4nm, h=4nm, wall molarity of cation species: c1 =0.154M, wall 0 molarity of anion species: c2 =0.142M. The walls are at zero potential, and no-slip condition is applied. ε=0.200, ε2=1.

232

Figure 6.8 Dimensionless shear stress τyx for monovalent binary electrolyte solution. 0 W=4nm, h=4nm, wall molarity of cation species: c1 =0.154M, wall molarity of anion 0 species: c2 =0.142M. The walls are at zero potential, and no-slip condition is applied.

ε=0.200, ε2=1.

233

Figure 6.9 Dimensionless shear stress τzx for monovalent binary electrolyte solution. 0 W=4nm, h=4nm, wall molarity of cation species: c1 =0.154M, wall molarity of anion 0 species: c2 =0.142M. The walls are at zero potential, and no-slip condition is applied.

ε=0.200, ε2=1.

234 The mole fractions, potential and velocity, and shear stresses for rectangular channel 3 are shown in Figure 6.10-Figure 6.13. The width of channel 3 is 20nm, same as the width of channel 1; the height of channel 3 is 4nm, same as the height of channel 2.

As the result, Specifically, in the y direction, Figure 6.10-Figure 6.13 are similar to

Figure 6.6-Figure 6.9; and in the z direction, Figure 6.10-Figure 6.13 are similar to Figure

6.2-Figure 6.5.

Figure 6.10 Mole fractions of ions for monovalent binary electrolyte solution. W=20nm, 0 h=4nm, wall molarity of cation species: c1 =0.154M, wall molarity of anion species: 0 c2 =0.142M. The walls are at zero potential, and no-slip condition is applied. ε=0.200, ε2=0.2. 235

Figure 6.11 Dimensionless electrical potential and velocity profile for monovalent binary 0 electrolyte solution. W=20nm, h=4nm, wall molarity of cation species: c1 =0.154M, wall 0 molarity of anion species: c2 =0.142M. The walls are at zero potential, and no-slip condition is applied. ε=0.200, ε2=0.2.

236

Figure 6.12 Dimensionless shear stress τyx for monovalent binary electrolyte solution. 0 W=20nm, h=4nm, wall molarity of cation species: c1 =0.154M, wall molarity of anion 0 species: c2 =0.142M. The walls are at zero potential, and no-slip condition is applied.

ε=0.200, ε2=0.2.

237

Figure 6.13 Dimensionless shear stress τzx for monovalent binary electrolyte solution. 0 W=20nm, h=4nm, wall molarity of cation species: c1 =0.154M, wall molarity of anion 0 species: c2 =0.142M. The walls are at zero potential, and no-slip condition is applied.

ε=0.200, ε2=0.2.

6.5 Results for Asymmetrical cases

The symmetrical cases are the specially cases for the model. More generally, the model is also valid for asymmetrical cases. In this section, the asymmetric boundary conditions are considered for monovalent binary electrolytes, such as NaCl. The example

238 being considered here is for a square channel where width and height are both 20nm. The length of the channel being used in numerical calculation remains at 3.6µm. The molarities and the potential on each side wall are related by setting the electrochemical potential on each wall to be equal. Thus according to equation (2.110), we have

1 ∗1 2 ∗2 3 ∗3 4 ∗4 RTln(ai )+ziFφ =RTln(ai )+ziFφ =RTln(ai )+ziFφ =RTln(ai )+ziFφ (6.24)

Where i=1 for cation species and i=2 for anion species. According to equations (2.111) and (2.112), the activity of an ion species equals its molarity at low concentration. Thus equation (6.24) becomes

1 ∗1 2 ∗2 3 ∗3 4 ∗4 RTln(ci )+ziFφ =RTln(ci )+ziFφ =RTln(ci )+ziFφ =RTln(ci )+ziFφ (6.25)

1 2 3 4 In the example shown below, the potential φ , φ , φ , φ and molarities at y=0 are arbitrarily chosen, then the molarities on the other side walls are calculated according to equation (6.25). The no-slip condition is assumed valid, so the velocities remain 0 on the walls. The boundary conditions on the four side walls are shown in Table 6.4

y=0 z=0 y=1 z=1 Conc. Of Cation (M) 0.154 0.160 0.166 0.143 Conc. Of Anion (M) 0.142 0.137 0.131 0.153 Potential (V) 0 0.001 0.002 -0.002 Velocity (m/s) 0 0 0 0

Table 6.4 The boundary conditions for an asymmetric case. The four side walls are labeled with the corresponding y or z values there.

239 The Debye length, Reynolds number and ε are same as the values for the channel

µ = × −6 2 −1 −1 1 of the symmetrical cases. m 7.6 10 cm V s . The asymptotic solutions solved based on the wall mole fractions on any wall are invalid for the whole channel, because the wall mole fractions are different from wall to wall.

The mole fractions of cation and anion species are shown in Figure 6.14.

Although the shape of the contours is much more complex than the symmetrical cases, the system is still neutral in most part of the core, as shown in Figure 6.15. This implies that for channels in which ε is small enough, the core region, which is outside the EDL, is always neutral no matter what boundary conditions are. The cation contour is above the anion contour at most grids, except those near z=1, where the wall concentration of anions is higher. The electric potential and velocity contour are shown in Figure 6.16 and

Figure 6.17 separately. These two contours do not overlap with each other, because the boundary conditions of φ and u are different in this case. However, the shape of these two contours is basically the same, which is determined by the similarity between the governing equations of φ and u. The shear stresses τyx and τzx, which are derived from the velocity gradients, also vary significantly differently in the regions near the different walls, as shown in Figure 6.18 and Figure 6.19.

240

Figure 6.14 Mole fractions of ions for monovalent binary electrolyte solution. W=20nm, 1 2 3 h=20nm, wall molarity of cation species: c1 =0.154M, c1 =0.160M, c1 =0.166M, 4 1 2 3 c1 =0.143M; wall molarity of anion species: c2 =0.142M, c2 =0.137M, c2 =0.131M, 4 1 2 3 c2 =0.153M. The potentials on the side walls are φ =0, φ =0.001V, φ =0.002V, 4 φ =−0.002V, and no-slip condition is applied. ε=0.200, ε2=1.

241

Figure 6.15 The net charge (X1-X2) inside the channel for monovalent binary electrolyte 1 2 solution. W=20nm, h=20nm, wall molarity of cation species: c1 =0.154M, c1 =0.160M, 3 4 1 2 c1 =0.166M, c1 =0.143M; wall molarity of anion species: c2 =0.142M, c2 =0.137M, 3 4 1 2 c2 =0.131M, c2 =0.153M. The potentials on the side walls are φ =0, φ =0.001V, 3 4 φ =0.002V, φ =−0.002V, and no-slip condition is applied. ε=0.200, ε2=1.

242

Figure 6.16 Dimensionless potential for monovalent binary electrolyte solution. 1 2 W=20nm, h=20nm, wall molarity of cation species: c1 =0.154M, c1 =0.160M, 3 4 1 2 c1 =0.166M, c1 =0.143M; wall molarity of anion species: c2 =0.142M, c2 =0.137M, 3 4 1 2 c2 =0.131M, c2 =0.153M. The potentials on the side walls are φ =0, φ =0.001V, 3 4 φ =0.002V, φ =−0.002V, and no-slip condition is applied. ε=0.200, ε2=1.

243

Figure 6.17 A dimensionless velocity profile for monovalent binary electrolyte solution. 1 2 W=20nm, h=20nm, wall molarity of cation species: c1 =0.154M, c1 =0.160M, 3 4 1 2 c1 =0.166M, c1 =0.143M; wall molarity of anion species: c2 =0.142M, c2 =0.137M, 3 4 1 2 c2 =0.131M, c2 =0.153M. The potentials on the side walls are φ =0, φ =0.001V, 3 4 φ =0.002V, φ =−0.002V, and no-slip condition is applied. ε=0.200, ε2=1.

244

Figure 6.18 Dimensionless shear stress τyx for monovalent binary electrolyte solution. 1 2 W=20nm, h=20nm, wall molarity of cation species: c1 =0.154M, c1 =0.160M, 3 4 1 2 c1 =0.166M, c1 =0.143M; wall molarity of anion species: c2 =0.142M, c2 =0.137M, 3 4 1 2 c2 =0.131M, c2 =0.153M. The potentials on the side walls are φ =0, φ =0.001V, 3 4 φ =0.002V, φ =−0.002V, and no-slip condition is applied. ε=0.200, ε2=1.

245

Figure 6.19 Dimensionless shear stress τzx for monovalent binary electrolyte solution. 1 2 W=20nm, h=20nm, wall molarity of cation species: c1 =0.154M, c1 =0.160M, 3 4 1 2 c1 =0.166M, c1 =0.143M; wall molarity of anion species: c2 =0.142M, c2 =0.137M, 3 4 1 2 c2 =0.131M, c2 =0.153M. The potentials on the side walls are φ =0, φ =0.001V, 3 4 φ =0.002V, φ =−0.002V, and no-slip condition is applied. ε=0.200, ε2=1.

246 This example shows how the 2D model works for asymmetrical cases. The properties inside the channel are mainly affected by the closest wall, although other walls also have effects. For instance, the concentration of cation species is less than the concentration of anions at the wall z=1, so the net charge in the EDL near this wall is negative, which implies that the surface charges on this wall is positive. As the result, the velocity in this region is less than zero, which means there is a reverse flow there, as shown in Figure 6.17.

A comparison has been made between the mole fractions and potential computed with this model with the results from Bhattacharyya et. al., who use Thomas algorithm and 41×41 meshes [85] to solve the EOF in 5nm×5nm square channels. The results show excellent agreement. The relative error between the two models is of no more than 0.5% on all mesh grid points.

247 CHAPTER 7

UNSTEADY FLOW

The work we have done in previous chapters assumes that the flow in micro- and nanochannels are in steady state, so the potential field, velocity field and ion distribution does not change with time. For unsteady flow, such as the transient flow before the system reaches steady state, time should be incorporated into the model [86] [87] [88].

For unsteady flow, the mole fraction of ion species, electric field strength and velocity of flow are keeping changing, until the steady state is reached. Therefore, the governing equations of unsteady flow in three dimensional nano- and microchannels are different from the governing equations for steady flow, which are discussed in §2.3.

Because the electric field is not static, the Maxwell’s equations should be used to describe the electromagnetic field in the channel.

r F zi ci ∇ • E * = ∑ (7.1) ε 0

r r ∂B * ∇× E * = − (7.2) ∂t *

r ∇ • B * = 0 (7.3)

248 r ∂E* ∇ × B* = µ F z nr + ε µ (7.4) 0 ∑ i i 0 0 ∂t* r where E* and B* are the electric field in Volt per meter and magnetic field in Tesla,

* respectively. t is time in seconds; ε0 is permittivity of free space; and µ0 is the

3 r permeability of free space. F∑ zici is the charge density in C/m , and F∑ zini is the vector current density in A/m2.

∂B * For varying electromagnetic field, ≠ 0 , thus we cannot define a scalar ∂t * potential parameter φ which satisfies E * = −∇φ [89]. Therefore equation (7.1) cannot be converted to equation (2.36). Instead, after nondimensionalization, equation (7.1) becomes

ε 2∇ • r = β E ∑ zi X i (7.5) i

RT c λ where E = is the scale of electric field, and β = , and ε = . For one 0 Fh I h dimensional channel as shown in Figure 2.2, equation (7.5) becomes

∂E ε 2 y = β ∑ zi X i (7.6) ∂y i

The flux definition equation (2.41) is valid for both the steady state and unsteady state.

D v ∗ ∗ nr = −cD ∇X + c i z FX E + cX uv (2.41) i i i RT i i i

For the unsteady flow, the mass transport equation is

∂c ∇ ⋅ nv = − i (7.7) i ∂t* 249 which can be easily derived. Combining equation (2.41) and (7.7), we have

r ∂X − ∇ 2 X + z ∇ • ()X ⋅ E + Re⋅ Sc ⋅ ∇ • ()X ur + χ i = 0 (7.8) i i i i i ∂t

t* D h2 = χ = 1 = st where t , i , and t0 is the diffusive time scale. For example, if the 1 t0 Di D1 ion species is Na, we know that the diffusion coefficient of Na+ is 1.334×10-9m2/s [30], so

-7 for h=20nm, t0=3.0×10 s. For 1D cases and a small Reynolds number, equation (7.8) becomes

∂ ∂X ∂X  i − ⋅  = χ i  zi X i E y  i (7.9) ∂y  ∂y  ∂t

∂ur * For varying velocity field of flow, the Navier-Stokes equation has a part on ∂t * its left side. Thus the nondimensionalized equations (2.47) should be replaced by

2  ∂v  Fch E0E zi X i Re⋅τ + vr • ∇vr = −∇p + + ∇2vr (7.10) ∂ µ  t  U0 and equation (2.48) should be replaced by

 ∂u ∂u ∂u ∂u  ∂p Fch2 E E Reτ + ε u + v + ε w  = −ε − 0 x z X + ∇2u (7.11)  ∂ 1 ∂ ∂ 2 ∂  1 ∂ µ ∑ i i  t x y z  x U0 i

U h where τ = 0 is a dimensionless factor. D1

Because the Reynolds number is very small, and the pressure driven flow is negligible, equation (7.11) can be simplified to equation (2.51):

ε 2 ∇ 2 = −β u ∑ zi X i (2.51) i which becomes

250 2 2 ∂ u ε = −β z X (2.54) 2 ∑ i i ∂y i for one dimensional cases.

Equations (7.6), (7.9) and (2.54) are the governing equations for the unsteady flow in one dimensional channel, which can be numerically solved by Thomas algorithm or SOR method as we have done in previous chapters.

251 CHAPTER 8

SUMMARY AND FUTURE WORK

8.1 Introduction

In this work, the governing equations for electrokinetic flow are solved for not only monovalent binary electrolytes, but also multi-component electrolytic system containing multivalent ion species. For channel-reservoir systems, the wall concentrations of different ion species are obtained using electrochemical equilibrium consideration. For channel materials with known surface charge density, the results are true predictive, i.e. there are no assumable constants in the model. The results agree very well with the experimental data from three separate sources. This implies that the model is effective for channel height varying from less than ten nanometers to several micrometers. Several parameters for the governing equations are explored, which indicates that the EOF in small channels can be mimicked by the EOF of less concentrated solution in larger channels. The 2D governing equations which are appropriate to study synthetic ion channels or rectangular nanochannels are also solved in this work. For the ion-channel case, we have produces results for both long and short channels relative to the channel height. For the rectangular-nanochannel case, we have shown results for both symmetric and asymmetric channels.

The sizes of macromolecules such as protein and DNA are comparable with the height of nanochannels, and the transport of macromolecules in nanochannels or 252 nanocapillaries is different from the transport of small inorganic ions such as Na+. Thus modification of the model is necessary for macromolecules. Keh et. al. have done much work in related areas. [64], [90]-[92], [93].

Because of the good agreement between the model and the experimental results, this work shows no evidence that the no-slip condition is invalid, or the surface roughness has to be considered. However, slip may occur in gas flow more readily [44], and calculation in more complex geometries than described here have also been presented

[94] [95] [96]. Properties of the fluid such as pH, viscosity and ionicstrength, and the slip conditions also affect the flow and fluxes. [97] [98] [99].

8.2 Transport of Macromolecules in Micro- and Nano- capillaries

Biomolecules such as proteins, ribonucleic acids (RNA) and deoxyribonucleic acids (DNA) are macromolecules, whose molecular weight is tens of thousand Daltons or higher [100]. Figure 8.1 shows a spacefill map of the human serum albumin, a kind of protein existing in human blood serum, whose functions include retention of fluid in the blood. There are about 9000 atoms in each albumin molecule; however albumin is a relatively small protein. Most proteins are not neutral. For instance, at pH=7, an albumin molecule carries net negative charges, and the effective valence of the albumin molecule is about -19. [101]. Structurally proteins are chains consisting of different amino acid residues that have different side chains. At pH 7, the side chains of residues such as aspartic acid and glutamic acid have negative charges, and the side chains of arginie, histidine and lysine have positive charges. The net charges of a protein molecule are determined by the numbers of these residues.

253 DNA molecules are the major carriers of genetic information. The well-known double helix structure of DNA molecules was discovered by Watson and Crick [102].

There are four types of nucleotides of which the DNA molecules consist. Each type of nucleotide has one phosphate group, one deoxyribose, and one special nitrogenous base

(Adenine, Guanine, Cytosine or Thymine). The nitrogenous bases on the two chains are paired with each other, and hydrogen bonds are formed between Adenine and Thymine, and between Guanine and Cytosine. A DNA molecule may contain millions of nucleotides, so its molecular weight can be very large. The DNA molecules are highly negatively charged due to the phosphate groups which exist in every nucleotide. A segment of DNA molecule with 12 base pairs is shown in Figure 8.2.

Figure 8.1 A spacefill model of the human serum albumin, from the protein data bank (database code: 1AO6). oxygen, nitrogen, carbon and sulfur atoms are shown as red, blue, charcoal and yellow balls, hydrogen atoms are not shown. 254

Figure 8.2 A Wireframe model of a duplex DNA segment CGCAAATTGGCG, from the protein data bank, (database code:111D). Carbon, oxygen, nitrogen and phosphorus atoms are represented by charcoal, red, blue and yellow wires.

The model of electrophoresis and mass fluxes we described in Chapter 5 can be applied to inorganic and small organic ions and molecules without difficulty. For macromolecules such as proteins and DNAs, things are more complicated. The scale of these molecules is in nanometers, and the model may still work for the transport of macromolecules in microchannels. For nanochannels, the channel height is comparable with the dimension of the molecules. Therefore two assumptions of the model are violated. Electrically, these molecules do not satisfy the point charge assumption, and charge distribution inside the molecules must be considered. Mechanically, these molecules do not satisfy the continuum flow assumption, so the velocity of these

255 molecules must be separate from the velocity of the solvent and small ions. As a result, another model is required for studying the transport of macromolecules in nanochannels.

The easiest way to represent globular proteins in solution is a sphere with charged surface. An EDL is built up on the surface of proteins. Thus an appropriate model should include the effects of this EDL and the EDL on channel wall and their interaction. The study of charged colloidal spheres with EDL has been done by Keh et. al. [64] [90]-[92]

[93] and other researchers.

A segment of the DNA molecule can be theoretically simplified to a cylinder with negatively charged surface. Therefore, a long DNA chain in a nanocapillary can be represented by a cylinder inside a capillary, which has concentric axis. EDL is built up on the inner wall of the capillary and outer wall of the cylinder. Assume the DNA chain is long enough and the surface charges on the DNA molecules are distributed equally. Then the movement of the fluid between the DNA and the capillary wall can be solved by using polar coordinates to rewrite the governing equations for 2D channels. The velocity of DNA relative to the capillary wall can be calculated next as electric migration.

256 Appendix A

For a static EDL system, there is no flow, u* = 0 , and there are no fluxes of ions

r = or other solutes, thus ni 0 From the flux equation (2.41), we have

1 ∗ ∇X + z FX ∇φ = 0 (A1.1) i RT i i

After nondimensionalization, equation (A1.1) becomes

∇ + ∇φ = X i zi X i 0 (A1.2) which can be solved as

− φ = 0 zi X i X i e (A1.3)

0 where Xi is the mole fraction of species i at φ=0. If the channel wall is defined as the

0 zero potential, Xi is the mole fraction of species i at the wall. For asymptotic cases where

0 ε<<1, we can also choose the bulk as zero potential, then Xi is the mole fraction of species i in the bulk. Equation (A1.3) is the dimensionless form of the Boltzmann distribution (2.2). Combining equation (A1.3) with Poisson equation (2.36), we get

β − φ ∇2φ = − 0 zi 2 ∑ zi X i e (A1.4) ε i

257 Equation (A1.4) is the dimensionless form of the Poisson-Boltzmann equation (2.4).

Therefore, the Poisson-Boltzmann equation is a special case, where there is no flow, no net fluxes, and the electric field is static. For cases where the flow and fluxes are little enough, the Poisson-Boltzmann equation can also be applied, being an approximation of the governing equations discussed in this work.

258 Appendix B

+ - + - 2+ 2- Na Cl K H2PO4 Ca HPO4 Solution 1 1.0623 0.0261 - - - - Solution 2 1.0106 0.0261 0.0517 - - - Solution 3 1.0623 0.0248 - 0.0013 - - Solution 4 0.6886 0.0327 - - 0.1898 - Solution 5 1.0579 0.0213 - - - 0.0002 PBS 1.0302 0.0211 0.0277 0.0002 - 0.0002

Table B.1 Average molarity of ions in the nanochannel. The channel height is 4nm.

For the channel-reservoir system shown in Figure 3.35 and Table 3.1, Table 3.2 and Table 3.3 present the channel average molarity and wall molarity for 20nm high channel. For h-4nm, the channel average molarity of ion species and corresponding wall molarity is shown in Table B.1 and Table B.2, respectively. Solution 4 which contains multivalent counterion species: Ca2+ also shows less flow rate than other solutions.

259 Flow rate + - + - 2+ 2- Na Cl K H2PO4 Ca HPO4 λ (nm) ε (µl/min) Solution 11.6144 0.0008 - - - - 0.126 0.0316 0.44 1 Solution 11.0404 0.0008 0.5494 - - - 0.127 0.0317 0.44 2 Solution 11.6123 0.0008 - 0.0000 - - 0.126 0.0316 0.44 3 Solution 6.1202 0.0017 - - 5.1848 - 0.0832 0.0208 0.38 4 Solution 11.5727 0.0006 - - - 0.0000 0.127 0.0317 0.44 5 0.294 PBS 11.2856 0.0006 0.0000 - 0.0000 0.127 0.0317 0.44 6

Table B.2 Molarity of different ions at the walls of the channel. The channel height is 4nm. The thickness of the EDL is given in this table, and the volume flow rate is shown in the last column, for an imposed voltage of 0.05V over a channel length of 3.5 µm.

260 BIBLIOGRAPHY

[1] Kulinsky, L., “Study of the fluid flow in microfabricated microchannels”, UMI Dissertation Service, 1998.

[2] Maluf, N., “An introduction to Microelectromechanical system engineering”, Artech House, Inc., 2000.

[3] Senturia, S. D., “Microsystem design”, Kluwer Academic Publishers, 2001.

[4] Khoo, M., Liu, C., “A novel micromachined magnetic membrane microfluid pump”, 0-7803-6465-1/00 2000 IEEE, 2394-2397

[5] Starkey, D. E., Han, A., Bao, J. J., Ahn, C. H., Wehmeyer, K. R., Prenger, M. C., Halsall, H. B., Heineman, W. R., “Fluorogenic assay for β-glucuronidase using microchip-based capillary electrophoresis”, Journal of Chromatography B, 2001, 762, 33-41.

[6] Gass, V., van der Schoot, B. H., de Rooij, N. F., “Nanofluid handling by micro-flow- sensor based on drag force measurements”, IEEE, 1993, 167-172.

[7] Nonner, W., Gillespie, D., Henderson, D., Eisenberg, R., “Ion accumulation in a biological calcium channel: effects of solvent and confining”, Journal of Physical Chemistry, 2001, 105, 6427-6436.

[8] Fujita, H., “Studies of micro actuators in Japan”, IEEE, 1989, 1559-1564.

[9] Anderson, R. C., Bogdan, G. J., Barniv, Z., Dawes, T. D., Winkler, J., Roy, K., “Microfluidic biochemical analysis system”, IEEE Transducers, 1997, 477-480.

[10] Choi, J.-W., Oh, K. W., Thomas, J. H., Heineman, W. R., Halsall H. B., Nevin, J. H., Helmicki, A. J., Henderson, H. T., Ahn, C. H., “An integrated microfluidic biochemical detection system with magnetic bead-based sampling and analysis capabilities”, 0-7803-5998-4/01 2001 IEEE, 447-450.

261 [11] Desai, A., Bökenkamp, D., Yang, X., Tai, Y.-C., Marzluff, E., Mayo, S., “Microfluidic sub-millisecond mixers for the study of chemical reaction kinetics”, IEEE Transducers, 1997, 167-170.

[12] Harrison, D. J., Fluri, K., Chiem, N., Tang, T., Fan, Z., “Micromachining chemical and biochemical analysis and reaction systems on glass substrates”, IEEE Transducers ’95 Eurosensors IX, 1995, 752-755.

[13] Ikuta, K., Hasegawa, T., Adachi, T., Maruo, S., “Fluid drive chips containing multiple pumps and switching valves for biochemical IC family”, 0-7803-5273-4/00 2000 IEEE, 739-744.

[14] Rasmusson, M., Wall, S., “Electrostatic characterization of Al-modified, nanosized silica particles”, Colloids and Surfaces A: Physicochemical and Engineering Aspects, 1997, 122, 169-181.

[15] Oberholzer, M. R., Lenhoff, A. M., “Protein adsorption isotherms through colloidal energetics”, Langmuir, 1999, 15, 3905-3914.

[16] Cohen, C. B., Chin-Dixon, E., Jeong, S., Nikiforov, T. T., “A microchip-based enzyme assay for protein kinase A”, Analytical Biochemistry, 1999, 273, 89-97.

[17] Israelachvili, J., “Intermolecular & surface forces”, 2nd Edition, 1994, Academic Press Inc.

[18] Bird, R. B., Stewart, W. E., Lightfoot, E. N. Transport Phenomena, John Wiley and Sons, 1960.

[19] Conlisk, A. T., “Falling film absorption on a cylindrical tube”, AIChE J., 1992, 38, 1716-1728.

[20] Conlisk, A. T., “Analytical solutions for the heat and mass transfer in a falling film absorber”, Chemical Engineering Science, 1995, 50, 651-660.

[21] Conlisk, A. T., “Analytical solutions for falling film absorption of ternary mixtures, part 1: theory”, Chemical Engineering Science, 1996, 51, 1157-1168.

[22] Conlisk, A. T., Mao, Jie, “Non-isothermal absorption on a horizontal tube part 1: film flow”, Chemical Engineering Science, 1996, 51, 1275-1285.

262 [23] Helmholtz, H., “Ueber einige gezetze der verteilung elektrischer ströme in körperliche leitern mit anwendung auf die thierisch-elektrischen versuche”, Pogg. Ann. Physik und Chemie, 1853, 89, 211-233, 353-377.

[24] Debye, P., Hückel, E., “The interionic attraction theory of deviations from ideal behavior in solution”, Zeitschrift für Physik, 1923, 24 (185).

[25] Gouy, G., “About the electric charge on the surface of an electrolyte”, Journal of Physics A, 1910, 9 (457).

[26] Verwey, E. J. W., Overbeek, J. Th. G, Theory of Stability of Lyophobic Colloids, Elsevier: Amsterdam, 1948.

[27] Burgeen, D., Nakache, F. R., “Electrokinetic flow in ultrafine capillary slits”, The Journal of Physical Chemistry, 1964, 68, 1084-1091.

[28] Rice, C.L., Whitehead, R., “Electrokinetic flow in a narrow capillary”, Journal of Physical Chemistry, 1965, 69, 4017-4024.

[29] Levine, S., Marriott, J.R., Robinson, K., “Theory of electrokinetic flow in a narrow parallel-plate channel”, Faraday Transaction II, 1975, 71, 1-11.

[30] Newman, J. S. Electrochemical Systems, Prentiss-Hall: Englewood Cliffs, NJ, 1973, 138-240.

[31] Hunter, R. J. Zeta Potential in Colloid Science, Academic Press: London, 1981, 59- 124.

[32] Probstein, R. F., Physicochemical Hydrodynamics, Butterworths: Boston, 1989, 161- 200.

[33] Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, Academic Press: New York, 1959, 284-335.

[34] Barcilon, V., Chen, D.-P., Eisenberg, R. S., Jerome, J. W., "Qualitative properties of steady-state Poisson-Nernst-Planck systems: perturbation and simulation study", SIAM Journal on Applied Mathematics, 1997, 57 (3), 631-648.

[35] O’Malley, R., E., Jr., “Singular perturbation methods for ordinary differential equations”, Applied Mathematical Sciences (Springer-Verlag New York Inc), 1991, 89.

263 [36] Qu, W., Li, D., “A model for overlapped EDL fields”, Journal of Colloid and Interface Science, 2000, 224, 397-407.

[37] Conlisk, A. T., Mcferran, J., Zheng, Z., Hansford D. J, “Mass transfer and flow in electrically charged micro- and nano-channels”, Analytical Chemistry, 2002, 74, 2139-2150.

[38] Friedl, W., Reijenga, J. C., Kenndler, E., “Ionic strength and charge number correction for mobilities of multivalent organic anions in capillary electrophoresis”, Journal of Chromatography A, 1995, 709, 163-170.

[39] Gavryushov, S. and Zielenkiewicz, P., “Multivalent ion distribution around a cylindrical polyion: contribution of polarization effects due to difference between dielectric properties of the macromolecule’s interior and the aqueous solvent”, Journal of Physical Chemistry B, 1997, 101, 792-797.

[40] Vlachy, V., “Ion-partitioning between charged capillaries and bulk electrolyte solution: an example of negative rejection”, Langmuir, 2001, 17, 399-402.

[41] Adamczyk, Z., Belouschek, P., Lorenz, D., “Electrostatic interactions of bodies bearing thin double-layers, I. General formulation”, Ber. Bunsenges. Phys. Chem., 1990, 94, 1483-1492.

[42] Adamczyk, Z., Belouschek, P., Lorenz, D., “Electrostatic interactions of bodies bearing thin double-layers, II. Exact numerical solutions”, Ber. Bunsenges. Phys. Chem., 1990, 94, 1492-1499.

[43] Cuvillier, N., Rondelez, F., "Breakdown of the Poisson-Boltzmann description for electrical double layers involving large multivalent ions", Thin Solid Films, 1998, 327-329, 19-23.

[44] Beskok, A., Karniadakis, G. E., “Simulation of heat and momentum transfer in complex microgeometries”, Journal of Thermophysics and Heat Transfer, 1994, 8 (4), 647-655.

[45] Zhu, W., Singer, S. J., Zheng, Z., Conlisk, A. T. “Study of electroosmotic flow in a model electrolyte”, submitted

[46] http://www.dur.ac.uk/sharon.cooper/lectures/colloids/interfacesweb1.html#_Toc449 417608 (Oct 19, 2003)

264 [47] Chung., S. H., “Invalidity of the continuum theories of electrolytes in nanopores”, Chemical Physics Letters, 2000, 320 (1-2), 35-41.

[48] Conlisk, A. T., “On the Debye-Hückel approximation in electroosmotic flow in micro- and nano-channels”, AIAA 2002-2869, 3rd AIAA theoretical fluid mechanics meeting, June 2002.

[49] Alberty, R. A., “Physical chemistry”, 6th Edition, John Wiley & Sons, Inc., 1983.

[50] Bockris, J. O’M., Reddy, A. K. N., “Modern electrochemistry”, 2nd Edition, Plenum Press, New York and London, 1998.

[51] Kohonen, M. M., Karaman, M. E., Pahsley, R. M., “Debye Length in Multivalent Electrolyte Solutions”, Langmuir, 2000, 16, 5749-5753.

[52] Zheng, Z., Hansford, D. J., Conlisk, A. T., “Effect of multivalent ions on electroosmotic flow in micro and nanochannels”, Electrophoresis, 2003, 24 (17), 3006-3017.

[53] Smith G.D., “Numerical solution of partial differential equations: Finite difference methods”, 3rd Edition, Clarendon Press, London,1985.

[54] Hansford, D., Desai, T., Ferrari, M., “Nano-scale size-based biomolecular separation technology”, Biochip Technology, Harwood Academic Publishers, 2001, 341-362, eds, Cheng, J., Kricka, L.J.

[55] Walczak, R., private communication.

[56] Coe, J., private communication.

[57] Alarie, J. P., Ramsey, M., private communication.

[58] Sadr, R., Yoda, M., Zheng, Z., Conlisk, A. T., “An experimental and analytical study of electroosmotic bulk and near wall flows in two dimensional micro and nanochannels”, IMECE2003-42917.

[59] Sadr, R., Yoda, M., Zheng, Z., Conlisk, A. T., “An experimental and analytical study of electroosmotic flow in microchannels”, Submitted.

[60] Heally, T. W., and White, L. R., “Ionizable surface group models of aqueous interfaces”, Advances in Colloid and Interface Science, 9 (1978) 303-345.

265 [61] Baker, D., R., “Capillary electrophoresis”, New York: Wiley, 1995.

[62] Wehr, T., Rodriguez-Diaz, R., Zhu, M., “Capillary electrophoresis of proteins”, New York, M. Dekker, 1999.

[63] Fan, Z. H., Harrison, D. J., “Micromachining of capillary electrophoresis injectors and separators on glass chips and evaluation of flow at capillary intersections”, Anaytical Chemistry, 1994, 66, 177-184.

[64] Keh, H. J., Wu, J. H., “Electrokinetic flow in fine capillaries caused by gradients of electrolyte concentration”, Langmuir, 2001, 17, 4216-4222.

[65] Kutter, J. P., “Current developments in electrophoretic and chromatographic separation methods on microfabricated devices”, Trends in Analytical Chemistry, 2000, 19 (6), 352-363.

[66] Paul, P. H., Garguilo, M. G., Rakestraw, D. J., “Imaging of pressure- and electrokinetically driven flows through open capillaries”, Analytical Chemistry, 1998, 70, 2459-2467.

[67] Sounart, T. L., Baygents, J. C., “Simulation of electrophoretic separations by the flux-corrected transport method”, Journal of Chromatography A, 2000, 890, 321- 326.

[68] Cárdenas, A. E., Coalson, R. D., Kurnikova, M. G., “Three-dimensional Poisson- Nernst-Planck theory studies: influence of membrane electrostatics on Gramicidin A channel conductance”, Biophysical Journal, 2000, 79 (1), 80-93.

[69] Gillespie, D., Nonner, W., Eisenberg, R. S., “Coupling Poisson-Nernst-Planck and density functional theory to calculate ion flux”, Journal of Physics: Condensed Matter, 2002, 14, 12129-12145.

[70] Yang, J., Kwok, D. Y., “Microfluid flow in circular microchannel with electrokinetic effect and Navier’s slip condition”, Langmuir, 2003, 19, 1047-1053.

[71] Barcilon, V., Chen D., Eisenberg, R. S., Ratner, M. A., “Barrier crossing with concentration boundary conditions in biological channels and chemical reactions”, The Journal of Chemical Physics, 1993, 98 (2), 1193-1207.

[72] Chen, D. P., Eisenberg, R. S., Jerome, J. W., Shu, C.-W., “Hydrodynamic model of temperature change in open ionic channels”, Biophysical Journal, 1995, 69, 2304- 2322. 266 [73] Chen, D., Lear, J., Eisenberg, R., “Permeation through an open channel: Poisson- Nernst-Planck theory of a synthetic ionic channel”, Biophysical Journal, 1997, 72, 97-116.

[74] Levitt, D. G., “Kinetics of diffusion and convection in 3.2-Å pores: Exact solution by computer simulation”, Biophysical Journal, 1973, 13, 186-206.

[75] Levitt, D. G., “The mechanism of the sodium pump”, Biochimica et Biophysica Acta, 1980, 604, 321-345.

[76] Levitt, D. G., “Exact continuum solution for a channel that can be occupied by two ions”, 1987, 52, 455-466.

[77] Levitt, D. G., “Continuum model of voltage-dependent gating: Macroscopic conductance, gating current, and single-channel behavior”, Biophysical Journal, 1989, 55, 489-498.

[78] Levitt, D. G., “General continuum theory for multiion channel”, Biophysical Journal, 1991, 59, 271-277.

[79] Levitt, D. G., “General continuum theory for multiion channel II. Application to acetylcholine channel”, Biophysical Journal, 1991, 59, 278-288.

[80] Levitt, D. G., “Modeling of ion channels”, the Journal of General Physiology, 1999, 113, 789-794.

[81] Nonner, W., Eisenberg, B., “Ion permeation and glutamate residues linked by Poisson-Nernst-Planck theory in L-type calcium channels”, Biophysical Journal, 1998, 75, 1287-1305.

[82] Van der Straaten, T. A., Tang, J., Eisenberg, R. S., “Three-dimensional continuum simulations of ion transport through biological ion channels: effect of charge distribution in the constriction region of porin”, Journal of Computational Electronics 2002, 1, 335-340.

[83] Rao, I. J., Rajagopal, K. R., “The effect of the slip boundary condition on the flow of fluids in a channel”, Acta Mechanica, 1999, 135, 113-126.

[84] Li, D., “Electro-viscous effects on pressure-driven liquid flow in microchannels”, Colloids and Surfaces A:Physicochemical and Engineering Aspects, 2001, 195, 35- 57.

267 [85] Bhattacharyya, S., Zheng, Z., Conlisk, A. T., “Asymptotic solutions for electroosmotic flow in two-dimensional charged micro- and nanochannels”, 33rd AIAA Fluid Dynamics Conference and Exhibit, 2003-3582.

[86] Duck, P. W. “Pulsatile flow through constricted or dilated channels”, Mech. Appl. Math., 1980, XXXIII, 77-92.

[87] Keh, H. J., Tseng, H. C., “Transient electrokinetic flow in fine capillaries”, Journal of Colloid and Interface Science, 2001, 242, 450-459.

[88] Maxey, M. R., Riley, J. J., “Equation of motion for a small rigid sphere in a nonuniform flow”, Physics of Fluids, 1983, 26 (4), 883-889.

[89] Jackson, J. D., “Classical Electrodynamics”, 3rd Edition, John Wiley & Sons, Inc, 1998.

[90] Keh, H. J., Wei, Y. K., “Osmosis through a fibrous medium caused by transverse electrolyte concentration gradients”, Langmuir, 2002, 18, 10475-10485.

[91] Keh, H., J., Wei, Y. K., “Diffusioosmosis and electroosmosis of electrolyte solution in fibrous porous media”, Journal of Colloid and Interface Science, 2002, 252, 354- 364.

[92] Keh, H, J., Hsu, W. T., “Electric conductivity of a suspension of charged colloidal spheres with thin but polarized double layers”, Colloid Polym Sci, 2002, 280, 922- 928.

[93] Wu, J. H., Keh, H. J., “Diffusioosmosis and electroosmosis in a capillary slit with surface charge layers”, Colloid and Surfaces A: Physiochem. Eng. Aspects, 2003, 212, 27-42.

[94] Jacobson, S. C., Hergenröder, R., Koutny, L. B., Warmack, R. J., Ramsey, J. M., “Effects of injection schemes and column geometry on the performance of microchip electrophoresis devices”, 1994, 66 (7), 1107-1113.

[95] Nitsche, L. C., Brenner, H., “Hydrodynamics of particulate motion in sinusoidal pores via a singularity method”, AIChE Journal, 1990, 36 (9), 1403-1419.

[96] Qiao, R. and Aluru, N. R., “A compact model for electroosmotic flow in microfluidic devices”, Journal of Micromechanics and Microengineering, 2002, 12, 625-635.

268 [97] Ariza, M. J., Rodríguez-Castellón, E., Muñoz, M., Benavente, J., “Surface chemical and electrokinetic characterizations of membranes containing different carriers by x- ray photoelectron spectroscopy and streaming potential measurements: study of the effect of pH”, Surface and Interface Analysis, 2002, 34, 637-641.

[98] Craig, V. S. J., Neto, C., Williams, D. R. M., “Shear-dependent boundary slip in an aqueous Newtonian liquid”, Physical Review Letters, 2001, 87 (5), 054504.

[99] Scales, P. J., Grieser, F., Healy, T. W., White, L. R., Chan, D. Y. C., “Electrokinetics of the silica-solution interface: a flat plate streaming potential study”, Langmuir, 1992, 8, 965-974.

[100] Stryer, L. “Biochemistry”, 4th Edition, W. H. Freeman and Company, New York, 1995.

[101] Mcferren, J., Private communication.

[102] Watson, J. D, Crick, F. H. C., “A structure for deoxyribose nucleic acid”, Nature, 1953, 171, 737-738.

269