Nonlinear Phenomena in Induced Charge Electroosmosis

UNIVERSITY OF CALIFORNIA

Santa Barbara

Nonlinear Phenomena in Induced Charge Electroosmosis

A Dissertation submitted in partial satisfaction of the

requirements for the degree Doctor of Philosophy

in Mechanical Engineering

by

Gaurav Soni

Committee in charge:

Professor Carl D. Meinhart, Chair

Professor George M. Homsy

Professor Todd M. Squires

Professor Hyongsok T. Soh

December 2008

ii

The dissertation of Gaurav Soni is approved.

____________________________________________ George M. Homsy

____________________________________________ Todd M. Squires

____________________________________________ Hyongsok T. Soh

____________________________________________ Carl D. Meinhart, Committee Chair

October 2008

iii


Copyright © 2008

by

Gaurav Soni

iv

To my loving parents, Ramji and Sushila

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ACKNOWLEDGEMENTS

The work contained in this dissertation would have been impossible

without the support of my Ph.D. advisor, Professor Carl Meinhart. The dictionary

does not have enough words to express my gratitude towards Carl. He always

showed confidence in my abilities. He showed me the path and prevented me from

falling in the pitfalls. He gave me the freedom, however, to digress and pick up

knowledge from foreign territories. I am grateful to my co-advisor and committee

member, Professor Todd Squires for pulling me out of scientific deadlocks. He

provided me with a unique insight and theoretical framework for solving complex

problems. His advice often brought a turning point in my research. I am thankful to

my committee member Professor George ‘Bud’ Homsy for evaluating the merit of

my work and offering useful insights through his enlightening comments during

my exams. It was a wonderful experience taking lessons in fluid dynamics from

him and also TA’ing for him. Thanks are also due to my committee member,

Professor Tom Soh, not only for evaluating my work but also for allowing me to

borrow equipment and books from his lab.

This work would have been very difficult to finish without the support,

comraderie and friendship of the past and present members of the Meinhart Lab. It

is a pleasure for me to mention their names here and say thanks to them: Hope

Feldman, Marin Sigurdson, Matthew Pommer, Frederic Bottausci, Caroline

Cardonne, Stephen Bradford, Brian Piorek and Lisan Viel. Special thanks to Dr.

Changsong Ding for teaching me Titanium microfabrication and to Dr. Adam

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Monkowski for making the nanoscale structures used in the chapter six of this

dissertation. I want to take this opportunity to thank my friends Mr. Amitabh

Virmani, Mr. Ankur Saxena and Mr. Amarendra Singh for their friendship and

moral support.

My words are not enough to express my thankfulness to my father Ramji,

and mother Sushila. I simply couldn’t have achieved this milestone without their

love, support and the sacrifices that they made for my education. I am grateful to

my brothers Tej and Vikas for always being there for me and for helping me in all

phases of my life. I am thankful to my sister-in-law Ambika for her kind love.

Finally, I want to say thanks to my wife Ritoo Varma for believing in me and

loving me unconditionally.

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VITA OF GAURAV SONI October 2008

[email protected]

EDUCATION Ph.D. (Mechanical Engineering) University of California Santa Barbara, CA, USA, Nov 2008 Advisor: Carl Meinhart Bachelor of Technology (Mechanical Engineering) Indian Institute of Technology Delhi, India, May 2002 INDUSTRIAL EXPERIENCE Support Engineer Comsol Inc., Los Angeles, CA, July – Sept 2004 Thermal Analysis Engineer Applied Thermal Technologies, Pune, India, Sept 2002 – July 2003 AWARDS Guru Gobind Singh Fellowship, 2008-09 Best paper award at the annual congress of ASME, Seattle, Nov 2007 Merit Fellowship of Department of Mechanical Engineering, UCSB, Sept 2003 Merit Certificate for outstanding GPA, IIT Delhi, 2001 Silver medal for 3rd rank in the Rajasthan Senior Secondary examination, 1997 Silver medal for 8th rank in the Rajasthan Secondary examination, 1995 JOURNAL PUBLICATIONS Impact of Surface Conduction on Induced Charge Electroosmosis (in preparation) Nonlinear Effects in Induced Charge Electroosmosis (in preparation) A Titanium Micro and Nano Structure Based Flat Heat Pipe (in preparation) A new wetting material based on titanium micro and nano structures (in preparation) REFEREED CONFERENCE PROCEEDINGS C. Ding, G. Soni, P. Bozorgi, B. Piorek, C. D. Meinhart, N. C. MacDonald, “A Titanium Based Flat Heat Pipe”, Proceedings of IMECE2008, Paper number: IMECE2008-68967, ASME International Mechanical Engineering Congress and Exposition, October 31-November 6, 2008, Boston, MA, USA G. Soni, T. M. Squires, C. D. Meinhart, “Simulation of highly nonlinear electrokinetics using a weak formulation”, Proceedings of Comsol User Conference, October 9-11, 2008, Boston, MA, USA

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G. Soni, T. M. Squires and C. D. Meinhart, “Nonlinear phenomena in induced charge electroosmosis”, Proceedings of IMECE 2007, American Society of Mechanical Engineers, November 11-15, 2007, Seattle, WA (best paper award). G. Soni, T. M. Squires and C. D. Meinhart, “Nonlinear phenomena in induced charge electroosmosis, A numerical and experimental investigation”, Proceedings of MicroTAS 2007, The 11th International Conference on Miniaturized Systems for Chemistry and Life Sciences, October 7-11, 2007, Paris, France. G. Soni, T. M. Squires and C. D. Meinhart, “Study of nonlinear effects in electrokinetics”, Proceedings of Comsol Users Conference, October 4-6, 2007, Boston, MA M. S. Pommer, A. R. Kiehl, G. Soni, N. S. Dakessian and C. D. Meinhart, “A 3D-3C micro-PIV method”, Proceedings of NEMS 2007, 2nd IEEE International Conference on Nano/Micro Engineered and Molecular Systems, January 16-19, 2007, Bangkok, Thailand. J. Wu, G. Soni, D. Wang and C. D. Meinhart, “AC electrokinetic pumps for micro/nanofluidics”, Proceedings of IMECE 2004, American Society of Mechanical Engineers, November 13-19, 2004, Anaheim, CA.

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ABSTRACT


by

Gaurav Soni

Induced charge electroosmosis (ICEO) refers to production of electroosmotic

slip by way of induced charges. Unlike fixed-charge-zeta potentials, the induced

zeta potentials are proportional to the applied electric field strength which can be

very strong in microfluidic devices. As a result, the induced zeta potentials are

generally much higher than the thermal voltage ( kT zeζ > ). The linear theory of

electrokinetics which is derived under the Debye-Huckel limit ( kT zeζ ) breaks

down for such large induced zeta potentials and predicts unrealistically high

magnitudes of ICEO slip velocities. Moreover, many flow characteristics observed

in experiments can not be explained by the linear theory. A vast discrepancy

between the linear theory and the experiments creates the need for an investigation

of the effects which take place at high zeta potentials (called nonlinear effects).

This dissertation investigates some of these nonlinear effects by the means of

experiments and numerical simulations. An effort has been made to reduce the

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discrepancy between the theory and the experiments and to explain the previously

unexplained experimental flow characteristics.

Induced charge electroosmotic flow was produced on a planar microelectrode

with an AC electric field. The experimental velocity was found to be 2 orders of

magnitude lower than the predictions of the linear theory. It was also found that the

slip velocity saturates at high applied voltages, a feature not predicted by the linear

theory.

A nonlinear electrokinetic model was formulated with the intent of explaining

the experiments. The nonlinear model is more advanced than the linear model. It

solves for the surface conduction of ions through the diffuse layer and also models

the double layer as a nonlinear capacitor which requires exponentially large

amounts of charge to get charged. Surface conduction refers to excess ionic

currents through the diffuse layer. We show that surface conduction through a

nanoscale diffuse layer can cause micron scale gradients in the bulk electric field

and cause severe reduction in the tangential electric field. We show that these

nonlinear effects deteriorate the slip velocity. We are able to reduce the

discrepancy between the theory and the experiments by one order of magnitude.

Finally ICEO flow is produced on a rough surface with nanoscale roughness.

We demonstrate that the roughness of a surface can have dramatic effects on the

flow velocity. These effects are explained with the help of fundamental aspects of

surface conduction.

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TABLE OF CONTENTS

Chapter 1. Introduction ........................................................................................... 1

1.1 Applications of Electrokinetics ................................................... 1

1.2 Motivation ................................................................................... 3

1.3 Outline of Dissertation ................................................................ 7

Chapter 2. Theory of Electrokinetics .................................................................... 10

2.1 Introduction ............................................................................... 10

2.2 Electrokinetic Equations ........................................................... 15

2.3 Theory of a Thin Double Layer: Poisson-Boltzmann Equation 16

2.3.1 The Linear or Debye-Huckel Theory ................................ 19

2.3.2 The Nonlinear or Gouy-Chapman Theory ........................ 20

2.4 Capacitance of the Double Layer .............................................. 22

2.4.1 Linear Capacitance............................................................ 23

2.4.2 Nonlinear Capacitance ...................................................... 23

2.5 Helmholtz Smoluchowski Formula........................................... 24

2.6 Surface conduction and Dukhin Number .................................. 25

2.6.1 Stern Layer Dukhin Number ............................................. 30

Chapter 3. Induced Charge Electroosmosis: Experiments .................................... 33

3.1 Induced Charge Electroosmosis ................................................ 33

3.2 Experimental Evidence for ICEO Flow .................................... 37

3.3 Device Design and Fabrication Process .................................... 38

3.4 Device Packaging...................................................................... 40

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3.5 Experimental Setup ................................................................... 41

3.6 Measurement of ICEO with µPIV............................................. 43

3.7 Dependence on Driving Voltage and Ionic Concentration ....... 47

3.8 Dependence on Driving Frequency........................................... 49

3.9 Effects of Finite Frame Rate ..................................................... 51

3.10 Momentum Diffusion Length................................................ 54

3.11 Numerical Simulations based on Linear Theory................... 55

3.9.1 Boundary Conditions......................................................... 56

3.9.2 Boundary Conditions on the Gate Electrode..................... 57

3.12 Numerical Results ................................................................. 60

3.13 Breaking the Symmetry: Field Effect Flow Control (FEFC) 63

3.14 FEFC in DC Electric Fields .................................................. 65

3.15 FEFC in AC Electric Fields .................................................. 66

3.16 Experimental Evidence of FEFC in AC Electric Fields........ 67

3.17 Linear Simulation of FEFC ................................................... 72

3.18 Uncertainty Analysis ............................................................. 74

3.19 Conclusions ........................................................................... 77

3.20 Appendix to Chapter 3 .......................................................... 79

Microelectrode Fabrication with Image Reversal Lithography .... 79

Chapter 4. Surface Conduction in Induced Charge Electroosmosis ..................... 81

4.1 Effects at High Zeta Potentials.................................................. 83

4.2 A Fundamental Picture.............................................................. 84

4.3 Bulk Equations .......................................................................... 90

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4.3.1 Bulk Boundary Conditions................................................ 90

4.4 Charge Conservation in the Double Layer................................ 91

4.4.1 Double Layer Edge Conditions ......................................... 93

4.4.2 Convergence Issues ........................................................... 93

4.5 Dimensionless Equations .......................................................... 95

4.6 ICEO on a Flat Metal Electrode................................................ 98

4.6.1 Parameters of Study ........................................................ 101

4.6.2 Results and Discussion.................................................... 101

4.6.3 Results of Parametric Study ............................................ 103

4.6.4 Normalized Quantities..................................................... 105

4.6.5 Streamlines ...................................................................... 112

4.7 ICEO on a Metal Cylinder ...................................................... 113

4.7.1 Surface Conduction Model for a 2D Cylinder in Cartesian

Coordinates.................................................................................. 116

4.7.2 Geometry and Boundary Conditions............................... 117

4.7.3 Electric Field Lines around Cylinder .............................. 120

4.7.4 Streamlines around Cylinder........................................... 122

4.7.5 Normalized Quantities..................................................... 124

4.7.6 Parametric Study ............................................................. 128

4.8 Concentration Polarization...................................................... 130

4.9 Conclusions ............................................................................. 133

4.10 Appendix to Chapter 4 ........................................................ 133

4.10.1 Comments on COMSOL Multiphysics Simulations ..... 133

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4.10.2 Weak Form for Finite Element Solution ....................... 136

4.10.3 Weak Form of Double Layer PDE................................ 137

Chapter 5. Simulations vs. Experiments ............................................................. 140

5.1 Numerical Model..................................................................... 140

5.2 Depth Averaging ..................................................................... 143

5.3 Uncertainty in Diffusivity Values ........................................... 146

5.4 Results ..................................................................................... 148

5.5 Simulations vs. Experiments ................................................... 153

5.6 Contribution of Dielectrophoresis........................................... 154

5.7 Contribution of Electrothermal Flow ...................................... 156

5.8 Stern Layer and High Ionic Concentrations............................ 160

5.9 Conclusions ............................................................................. 162

Chapter 6. Induced Charge Electroosmosis on a Rough Surface........................ 165

6.1 Background ............................................................................. 165

6.2 Fundamental Picture................................................................ 168

6.2.1 Thick Double Layers.......................................................... 171

6.3 Experiments............................................................................. 171

6.3.1 Details of the Device .......................................................... 171

6.3.2 Fabrication Process ......................................................... 173

6.3.3 Experimental Results.......................................................... 174

6.4 Asymmetry in Flow................................................................. 178

6.5 Conclusions ............................................................................. 179

Chapter 7. Conclusions and Future Directions ................................................... 181

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7.1 Experimental ........................................................................... 181

7.2 Numerical ................................................................................ 182

7.3 Simulations vs. Experiments ................................................... 182

7.4 ICEO Flow over Rough Surfaces............................................ 183

7.5 Future Directions..................................................................... 184

7.5.1 Bulk Equations ................................................................ 184

7.5.2 Surface Transport Equations ........................................... 185

7.5.3 Boundary Conditions on the Gate Electrode................... 185

Bibliography........................................................................................................ 187

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1

1

Chapter 1. Introduction

This dissertation deals with nonlinear effects in electrokinetics. Electrokinetics

is an important subject because it has applications in numerous areas such as

micro-nano fluidics, lab-on-a-chip, cell separation, bio-detection, molecule/colloid

transport, micro pumping and power generation. Electrokinetics is a very

interesting subject because it falls at the intersection of seemingly unrelated

subjects such as electromagnetism, biology, colloids and hydrodynamics [1-3]. The

motivation behind our work comes from the potential applications of

electrokinetics. Therefore, we will begin this chapter by elaborating a little more

on the applications of electrokinetics. Then we will describe the motivation behind

our work and present the outline of this dissertation.

1.1 Applications of Electrokinetics

Electrokinetics deals with transport of ionic liquids, charged species and

polarizable particles in presence of electric fields. It has roots in 19th century with

the pioneering electrophoresis experiments of Reuss in 1809 [4]. Since then,

electrophoresis has been of great importance to the biochemists who frequently use

gel electrophoresis for DNA fractionation and sequencing [5].

Electrokinetics has been studied in the context of colloids for a long time [1, 6-

14]. In the last two decades, however, it has found applications in micro and

nanofluidic devices popularly known as lab-on-a-chip. Recent emergence of micro

and nano technology has given birth to a renewed interest in the study of

2

electrokinetic physics at nano liter scale [15]. A lot of practical applications have

been developed with electrokinetics. For example, dielectrophoresis has been

established as an effective method of manipulating micron and submicron size

particles [16]. Dielectrophoretic cell sorters have been widely adapted by

biochemists for separating diseased cells from healthy ones [17-19]. The AC

electrothermal stirring method has been used for enhancing the speeds of

biochemical reactions in diffusion-limited micro channels [20, 21].

Another electrokinetic phenomenon, namely electroosmosis has captured a lot

of attention in recent times. Electroosmosis refers to the flow of an ionic liquid on

a charged (fixed or induced) surface in presence of electric fields. Since the flow is

generated on the surface, the flow speeds do not decrease as the size of the channel

is decreased. This offers a great alternative to pressure driven pumping which

suffers from decreasing flow rates and increasing pressure losses as the channel

size is decreased. Various types of micropumps have been developed based on DC

electroosmosis [22, 23]. DC electroosmotic pumps have been employed in

applications such as chromatography [24] and cooling of VLSI chips [25].

Micropumps have been developed for AC electric fields too, for example

micropumping can be achieved by using asymmetric electrode designs in AC

electroosmosis [26-38]. In addition to pumping, AC electroosmosis has also been

used as a method of concentrating DNA in a micro-chamber [39, 40].

More recently, a general theory of Induced Charge Electroosmosis (ICEO) has

been presented [41, 42]. It shows that charges induced by an electric field can lead

to electroosmotic flows on neutral but polarizable surfaces. ICEO flow has been

3

observed experimentally on surfaces of various shapes [43, 44]. Various

applications such as pumps and mixers can be developed by breaking the

symmetry of ICEO flows [45, 46].

Another electrokinetic phenomenon called ‘field effect’ has been used to

develop various fluidic applications such as field effect flow control (FEFC) [47-

53] and nanofluidic field effect transistors [54-57]. FEFC modifies electroosmotic

slip velocity on a surface by modifying its zeta potential. FEFC has been used for

enhancing the velocity of dc electroosmotic flow and for controlling band

dispersion in capillary electrophoretic devices [51-53]. Nanofluidic transistors

have been used for controlling transport of ions and proteins in nanochannels [56,

57].

Recent effort has been put into making nanofluidic batteries which can

generate electricity from pressure driven flow of ions in nanochannels [58-60].

1.2 Motivation

We are motivated to make microfluidic pumps and mixers which can transport

fluids at substantial flow rates. Electroosmosis is a surface driven method of fluid

transport in micro and nano channels and appears to be an ideal candidate for

microscale fluid manipulation. According to Helmholtz-Smoluchowski formula,

the electroosmotic slip velocity, su , is proportional to the zeta potential of the

surface, ζ , and to the applied tangential electric field, E ,

su Eε ζη

= − , (1.1)

4

where ε is the absolute permittivity of the fluid and η is the viscosity. The natural

zeta potential of a glass surface is -0.1 Volt. A simple micropump can be realized

by applying a tangential DC electric field to the walls of a glass capillary.

However, there are a few problems with this approach: firstly, when the capillary

is long, a large voltage difference (100-1000 Volt) is required to create a

substantial electric field; secondly, the zeta potential of glass is fixed at a low value

and therefore a large electric field is required for producing substantial flow.

Thirdly, high DC voltages create electrolysis of water and pose an engineering

challenge to the pump designer.

AC electric fields can solve the problem of electrolysis. The problem of low

zeta potentials can be solved by inducing zeta potentials which are proportional to

the applied electric fields (see chapter 3). The problem of large voltages can be

solved by moving the electrodes close to each other within the channel and by

applying an AC field to them to avoid electrolysis. Linear theory of electrokinetics

(explained in the next chapter) predicts that such devices can generate flow

velocities of the order of 1-10 mm/s for very low applied potentials such as 10

Volts and 100-1000 Hz frequencies. Encouraged by these calculations, we

designed a device which had microelectrodes inside a channel. We applied an AC

electric field to the electrodes and generated induced charge electroosmotic flow

on a metal surface. To our dismay, we found that the measured flow velocities

were 2-3 orders of magnitude lower than what the linear theory predicts. In other

words, we could not produce as high flow velocities as we expected them to be.

5

While this creates a roadblock in the development of high flow rate pumps, our

understanding of the theory of electrokinetics is also challenged. Linear theory has

been derived for very low zeta potentials ( 25 mVoltkT zeζ = , where ζ is the

zeta potential, k is the Boltzmann constant, T is the absolute temperature, z is the

valence of the ions and e is the elementary charge) and works fine in that regime.

For large zeta potentials, however, it fails and predicts unrealistic results. In

induced charge electroosmosis, the zeta potentials are proportional to the applied

field and therefore the zeta potentials can be 10-100 times higher than the thermal

voltage, kT ze . We do expect the linear theory to break down at such large zeta

potentials.

The discrepancy between the linear theory of electrokinetics and experimental

measurements is not new. Many experimentalists have indicated that the linear

theory predicts much higher flow velocities than can be observed in the

experiments [31, 44]. Apart from the magnitude discrepancy, various experimental

features can not be explained by the linear theory either. For example, linear theory

predicts that the slip velocity scales with the square of the applied voltage, 20φ . In

the experiments, however, velocity tends to saturate at high voltages [36, 44]. As

another example, experimentalists have observed that the direction of asymmetric

AC electroosmotic pumping mysteriously reverses direction at high frequencies

[35] but linear theory does not predict so. Yet another example pertains to the

observed dependence of AC electroosmotic velocity on the salt concentration.

Experiments show that the velocity decreases as the salt concentration increases

6

[28] but the linear theory predicts no such effect. Linear theory fails to predict all

these mysterious features observed in the experiments.

All these problems with the linear theory of electrokinetics motivate us to

develop an understanding of the phenomena which may take place at large zeta

potentials and which are not incorporated in the linear theory. There are several

effects which may take place at large zeta potentials. Following are a few of them:

nonlinear capacitance of the double layer, surface conduction [1], faradaic

reactions [2, 38], chemi-osmosis [61, 62], steric interactions between ions in the

crowded environment of the double layer [63], increase in the viscosity of the

double layer due to ion condensation [64], etc. In fact, study of these effects has

yielded some success in predicting the experimental features. For example, the

study of steric effects has been shown to explain the flow reversal in asymmetric

pumping [64]. Similarly, the increase in the viscosity of the double layer has been

shown to explain the concentration dependence of the velocity [64].

In this dissertation, we measure the velocity of induced charge electroosmotic

flow at large zeta potentials. We show some surprising features in the experiments

which can not be explained by the linear theory. We then try to explain those

surprising features with the help of nonlinear simulations. We have devoted our

attention to two nonlinear effects, namely surface conduction and nonlinear

capacitance of the double layer. We have developed a fundamental picture (both

qualitative and quantitative) of surface conduction in ICEO flows. We show some

really surprising features of surface conduction which occur only in nonuniform

ICEO flows. We use this fundamental picture to explain some of our experimental

7

features. We also show that these nonlinear effects indeed help in reducing the

magnitude discrepancy between the theory and the experiments.

1.3 Outline of Dissertation

Chapter 2 presents the basic theory of electrokinetics. It introduces the

equations which govern the transport of ions and fluids in presence of electric

fields. These equations are then simplified to derive the Boltzmann distribution of

ionic species and distributions of potential and velocity inside a thin double layer.

Important concepts such as nonlinear capacitance of the double layer and surface

conductance are also introduced.

Chapter 3 presents the results of our experiments on induced charge

electroosmosis (ICEO). We demonstrate the dependence of velocity on various

experimental parameters such as salt concentration, driving voltage and frequency.

On a practical note, we use field effect to break the symmetry of ICEO flows and

produce micro pumping. We also perform linear simulations and show that the

numerical velocities are 2-3 orders of magnitude higher than the experimental

velocities.

Chapter 4 deals with nonlinear electrokinetic simulations. We develop a model

to simulate surface conduction in steady and time dependents cases. We also

incorporate nonlinear capacitance in our model. Via our simulations, we develop a

fundamental picture of surface conduction in ICEO flows. We show some

surprising consequences of surface conduction in ICEO flows.

8

In chapter 5, we compare the results of our simulations with the experiments.

We show that the nonlinear effects reduce the discrepancy between the theory and

the experiments.

In chapter 6, we present the results of our experiments with rough surfaces. We

show that the roughness of a surface has dramatic impact on the electroosmotic

flow around the surface. We present experimental data to support our claim and

give arguments to explain the results of the experiments.

Chapter 7 concludes the dissertation.

Finally, the Bibliography contains the references cited in the dissertation.

10

Chapter 2. Theory of Electrokinetics

In this chapter, we present the theory of electrokinetics. Most attention has

been paid to electroosmosis and the theory of double layer. The electric double

layer is an essential part of any electrokinetic phenomenon, be it electroosmosis or

dielectrophoresis. We, therefore, present the equations (also called Nernst-Planck

Equations) which govern the transport of ionic species in the presence of electric

fields. These equations have been simplified to derive the Boltzmann distribution

of ionic species. We also apply the Debye-Huckel approximation to derive the

potential distribution and the flow profile within the double layer. An expression

for the Helmholtz Smoluchowski slip velocity has been derived. Some other useful

concepts such as the capacitance of the double layer and the surface conductance

have also been introduced. These concepts are of great importance while studying

the charging dynamics of the double layer and will be used intensively in all the

succeeding chapters.

2.1 Introduction

Electrokinetics deals with the motion of suspended particles and suspending

fluids in presence of electric fields. An example of electrokinetic transport is

electrophoresis [3]. Electrophoresis refers to the motion of a charged particle

suspended in an ionic liquid and subjected to an external electric field. The electric

field causes the particle to move with an electrophoretic velocity u proportional to

its electrophoretic mobility, eµ , and to the electric field strength, E,

11

eu Eµ= . (2.1)

The electrophoretic mobility eµ is related to the zeta potential of the particle

surface, ζ , and can be expressed as the following in a thin electric double layer

limit,

eεµ ζη

= , (2.2)

where ε is the absolute permittivity of the suspending fluid and η is the viscosity.

The definition of ζ will be clear to the reader when the concept of an electric

double layer is introduced in a following section. Electrophoresis is of great

importance to the biochemist who frequently uses gel electrophoresis for DNA

fractionation and sequencing [5]. An example of DNA fractionation by gel

electrophoresis is shown in Fig. 2.1.

Fig. 2.1: Gel electrophoretic separation of DNA fragments based on their electrophoretic mobility which is inversely proportional to the length of the fragment (in base pair, shown on the vertical axis). This image is reprinted from http://www.mun.ca/biology/scarr/Lab_5_gel_1999.gif for demonstration purposes and is not part of the original work contained in this dissertation.

12

Another example of electrokinetics deals with the creation of the ‘electric

double layer’ over a charged surface. When a charged surface (such as the surface

of the particle discussed above) comes in contact with an ionic liquid, counter ions

of the ionic liquid are attracted towards the surface and a diffuse charge cloud is

formed on the surface. This charge cloud is called ‘electric double layer’ [1, 2].

When an external tangential field is applied, the charge in the double layer feels a

net Coulomb force. This force acts only on the double layer and not on the bulk

fluid because the bulk is electroneutral. As a result of the Coulomb force, the

double layer moves relative to the charged surface and creates a fluid motion in the

bulk fluid due to viscous diffusion of momentum. This phenomenon is referred to

as electroosmosis. The famous Helmholtz Smoluchowski equation for

electroosmotic slip velocity, su , is given as

su Eε ζη

= − . (2.3)

This expression is very similar to the expression for the electrophoretic

velocity (2.1) of the suspended particle discussed above. When the particle is

freely suspended, the electric field causes the particle to move with the

electrophoretic velocity. If the particle is fixed in position, an electroosmotic flow

is developed around it with the same magnitude of slip velocity as electrophoresis

but in the opposite direction.

Electroosmosis can be produced not only on surfaces with fixed charge but also

on surfaces which do not have any charge. The surprising phenomenon of

electroosmosis over uncharged surfaces has been termed as Induced Charge

13

Electroosmosis (ICEO) [41-44]. In ICEO, the charge on the surface is induced due

to polarization. Since the charge is induced and moved by the same electric field, it

is a nonlinear phenomenon unlike the linear phenomenon of fixed charge

electroosmosis. The nonlinearity of this phenomenon produces steady flows in AC

fields unlike fixed charge electroosmosis which produces zero time averaged flow

in AC fields. Infact, a steady ICEO flow in AC fields was first discovered on a pair

of closely spaced coplanar electrodes and was labeled as “AC Electroosmosis”. It

was studied in detail by [26, 28, 29, 31]. AC electroosmosis has been used as a

method of concentrating DNA in a micro-concentrator [39, 40]. AC electroosmosis

has also been shown as a mechanism for micro pumping by introducing asymmetry

in the electrode design [32-38]. Later on a general theory of ICEO on any

polarizable surface was given by [41].

There are two other phenomena which take place only in nonuniform electric

fields, namely 1) Dielectrophoresis, and 2) Electrothermal flow. Dielectrophoresis

(DEP) is a force which acts on the suspended particles in presence of non-uniform

electric fields [26]. DEP arises due to the difference in the polarizability of the

particle and the fluid medium. DEP can be used to move very small particles in a

controlled manner. Recent work in this field has established DEP as an effective

method of manipulating micron and submicron size particles [16].

Dielectrophoresis has been used widely for microfluidic cell separation,

concentration and bio-analysis [18, 19].

Electrothermal flow defines the motion of the suspending medium in the

presence of non-uniform electric fields [26]. Electrothermal flow arises due to

14

gradients in the fluid’s electric properties (namely, the permittivity and the

conductivity). Non-uniform electric fields lead to non-uniform Joule heating of the

fluid and generate temperature gradients. Electric properties like permittivity and

conductivity depend on the temperature and therefore temperature gradients cause

gradients in these properties too. These gradients combined with the electric field

lead to an electrothermal force on the fluid and cause the electrothermal flow. One

important observation of the electrothermal flow is that it creates viscous drag on

the suspended particles. At the micro scale, this viscous drag can be of equal or

even more importance than the DEP force. Therefore, the study of electrothermal

flow is quite important. Recently, electrothermal flow has been used to enhance the

binding reaction between an immobilized ligand and an antigen in a microfluidic

immuno-assay [20, 21]. This opens a lot of scope for the use of electrokinetics in

biological sciences where high yield and throughput are highly desired.

In this thesis we will not pursue the subjects of dielectrophoresis or

electrothermal phenomena. Instead, electroosmosis will be the main subject. Study

of electroosmosis starts with certain electrokinetic equations which govern the

transport of ionic species in a liquid medium in presence of electric fields. In a

very thin double layer limit, these equations can be used to derive distributions of

concentration, electrostatic potential and flow velocity within the double layer. The

canonical case of fixed charge electroosmosis (also called DC electroosmosis) will

be discussed. Important concepts (such as double layer capacitance and surface

conduction) will be established so that they can be used for the detailed study of

induced-charge electroosmosis in the succeeding chapters.

15

2.2 Electrokinetic Equations

We describe the relevant equations for transport of various ionic species in a

dilute electrolyte solution in the presence of electric fields. These equations are

also called the Nernst-Planck equations. For simplicity, we present the equations

for a symmetric electrolyte (z:z), made of ions of valence z± (e.g. KCl).

The electrostatic potential φ obeys Poisson’s equation:

2 ( )n n zeφε

+ −−∇ = − , (2.4)

where n+ and n− are the number densities of positive and negative ions

respectively, e is the elementary charge and ε is the absolute permittivity of

water which is approximately equal to 080.1ε at 200 C, where 0ε is the permittivity

of vacuum.

For some symmetric electrolytes, the positive and negative ions have equal

diffusivities and ionic mobilities. For example, the difference in the diffusivities of

K+ and Cl- ions is less than 4% with 91.96 10k

D +−= × and 92.03 10

ClD −

−= × m2/s

[65]. With these simplifications, the number densities of ions obey the following

conservation equations

. 0n Jt±

±

∂+∇ =

∂, (2.5)

where the ion current densities, J± , are given by

uJ n D n nµ φ± ± ± ±= ∇ − ∇ +∓ , (2.6)

16

where u is the local fluid velocity vector, and µ and D are the mobility and

diffusivity of the ions, respectively. These two quantities are related to each other

as /zeD kTµ = [1]. The three terms on the right hand side of (2.6) represent the

flux of ionic species due to a Coulomb force (same as Ohm’s law,

/ohmJ zeσ φ± ±= − ∇ , with the conductivity 2( ) /n ze D kTσ± ±= ), diffusion due to

concentration gradient and advection by the fluid flow respectively.

The fluid flow is governed by Navier-Stokes equations:

2 ( )u u up ze n nρ η φ+ −⋅∇ = −∇ + ∇ − − ∇ , (2.7)

0u∇⋅ = , (2.8)

where the third term on right hand side of (2.7) represents the Coulomb body force

on the fluid.

2.3 Theory of a Thin Double Layer: Poisson-Boltzmann Equation

Here we present a theoretical framework for a thin double layer ( D aλ ,

where a is the radius of curvature of the surface). Consider an ionic solution which

has equal densities of positive and negative charges. The bulk of this solution will

be electrically neutral due to equal ion densities. When a charged surface comes in

contact with such ionic solution, it attracts counter ions of the solution and repels

co-ions. This creates a narrow region of non-zero charge density close to the

surface. This narrow charged region is referred to as the electric double layer or

Debye layer (see Fig. 2.2). The thickness of the double layer is a function of the

ionic strength of the solution and is generally very small (1-100 nm). The electric

17

potential caused by the surface charge is the highest at the surface and zero in the

bulk. The drop in the potential is caused by the double layer charge. Therefore, the

double layer can be said to act as a capacitor which screens the surface charge

from the bulk of the solution.

++

+

+ + + + + ++ + + + + ++ + + + + +

++

+

++

++

++

++

+

++

+

++

+

++

+

++

+

++

++

++

++

+

+ ++

+-

-

-

- -

-

-

-

-

-

+ + + + + + + + +

- - - - - - - - - - - - - - - - - - - - - - -

us

-q

ζ

x

y

λDDiffuse Layer

Stern Layer

Plane of Slip

E

Fig. 2.2: Electrical double layer (EDL) formation on a charged surface. EDL is much thinner than the bulk. The potential drops exponentially through the charge cloud. The potential difference across the diffuse part of the EDL is called the zeta potential of the surface. A tangential electric field can drive the charge cloud with a velocity that grows exponentially in the double layer. Far field velocity is called the electroosmotic slip velocity.

The double layer is viewed as consisting of two layers, an inner or Stern layer

and an outer or diffuse layer. The Stern layer consists of those ions which are

closest to the charged surface and the fluid in this region experiences a no-slip

condition. As a result, the Stern layer does not contribute to the electroosmotic

flow. The diffuse layer resides on top of the Stern layer and ions in this layer are

mobile. The interface between the two layers is called the plane of shear. The net

Plane of shear

18

potential drop across the diffuse layer is defined as the zeta ζ potential of the

surface.

Let’s consider an infinite flat surface with a fixed charge density - q (per unit

area) (Fig. 2.2). Let’s assume that the double layer is much thinner than the

dimensions of the surface ( D aλ ). If we assume that the fluid is static (i.e. u=0),

then combining (2.5) and (2.6), we obtain a balance between the electromigration

and diffusive fluxes of the ions close to the surface,

dn dD ndy dy

φµ±±= ∓ . (2.9)

The boundary conditions away from the surface, i.e. y →∞ , can be written as,

( ) 0yφ →∞ = , (2.10)

and

0( )n y n± →∞ = , (2.11)

where 0n is the number density of ions in the bulk.

With the help of boundary conditions (2.10) and (2.11), (2.9) can be integrated

to give the distribution of ionic concentration in the double layer, also known as

the Boltzmann distribution,

0 exp( )zen nkTφ

± = ∓ . (2.12)

Now combining (2.4) and (2.12) yields the famous Poisson-Boltzmann

equation,

19

2

02

2 sinh( )n zed zedy kTφ φ

ε= . (2.13)

which can be integrated to determine the potential distribution in the double layer.

The results of the integration are shown in the next two subsections.

2.3.1 The Linear or Debye-Huckel Theory

In the electrokinetic literature, a Debye-Huckel approximation is used for

linearizing the Poisson-Boltzmann equation. This approximation states that when

the zeta potential is much smaller than the thermal voltage ( kT zeφ ), the

nonlinear terms in the electrokinetic equations can be linearized, i.e.,

sinh( ) for ze ze kT zekT kTφ φ φ≈ . (2.14)

Under the Debye-Huckel assumption, (2.13) can be linearlized to,

2

2 2D

ddyφ φ

λ= , (2.15)

where

202 ( )DkT

n zeελ = (2.16)

is the approximate double layer thickness. Note that Dλ decreases with the salt

concentration, 0n . The double layer charge counterbalances the charge on the

surface. At higher ionic concentrations, a thinner region of ionic charge is

sufficient to counterbalance the charge on the surface.

Following are the boundary conditions for the integration of (2.15):

20

( 0) syφ φ= = , (2.17)

and

( ) 0yφ →∞ = , (2.18)

where sφ is the potential at the surface. A simple integration of (2.15) then yields

the potential distribution in the double layer

exp( )sD

yφ φλ

= − . (2.19)

If we assume that the potential at the surface, i.e. sφ , is not much different from

the potential at the plane of slip, i.e. ζ , then

exp( )D

yφ ζλ

≈ − . (2.20)

The preceding analysis shows that, in a thin double layer limit and under the

Debye Huckel assumption, the potential drops exponentially across the double

layer (also shown in Fig. 2.2).

2.3.2 The Nonlinear or Gouy-Chapman Theory

Under the Debye-Huckel approximation, we obtained an exponential decay of

potential in the double layer. This is a canonical result and gives us a great insight

into the double layer potential distribution. This approximation works well for

many surfaces of interests. For example, the zeta potential of a glass surface (in

contact with water of pH<4.0) is about -100 mVolt which is only four times higher

than the thermal voltage at room temperature (~25 mVolt).

21

In many induced charge electroosmotic situations, however, the zeta potential

is caused by the electric field and not by a chemical change. In such cases, the

induced zeta potentials can be much higher than the thermal voltage and thus the

Debye-Huckel approximation does not hold true. Infact, linear theory predicts

much higher slip velocities than can be observed in ICEO experiments [44].

Gouy-Chapman offered the integration of Poisson-Boltzmann equation without

linearization. The nonlinear integration was done by noting that,

22

2

12

d d ddy d dyφ φ

φ⎛ ⎞

= ⎜ ⎟⎝ ⎠

, (2.21)

and hence (2.13) becomes,

2

04 sinh( )n zed zed ddy kTφ φ φ

ε⎛ ⎞

=⎜ ⎟⎝ ⎠

. (2.22)

Integrating the preceding equation yields,

2

04 cosh( ) constantn kTd zedy kTφ φ

ε⎛ ⎞

= +⎜ ⎟⎝ ⎠

. (2.23)

The constant of integration can be evaluated by noting that at distances far

from the surface, 0φ = and ( )0d dyφ = . As a result,

2

04 cosh( ) 1n kTd zedy kTφ φ

ε⎛ ⎞ ⎡ ⎤= −⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠

. (2.24)

The preceding equation can be further simplified to

1 2

08 sinh( )2

n kTd zedy kTφ φ

ε⎛ ⎞= −⎜ ⎟⎝ ⎠

, (2.25)

22

which can then be integrated in the following manner to find the potential

distribution in the double layer,

1 2

0

0

8sinh( 2 )

y

w

n kTd dyze kT

φ

ζ

φφ ε

⎛ ⎞= −⎜ ⎟

⎝ ⎠∫ ∫ , (2.26)

where ζ is taken to be the potential at 0y = (approximately). The result of

integration is the following:

1 2

082 tanh( 4 )lntanh( 4 )

kTnkT ze kT yze ze kT

φζ ε

⎡ ⎤ ⎛ ⎞= −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

, (2.27)

or

14 tanh tanh4

DykT ze eze kT

λζφ −− ⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠. (2.28)

2.4 Capacitance of the Double Layer

According to Helmholtz model of double layer [1], the electrode surface and

the double layer can be seen together as two plates of a parallel plate capacitor.

The charge residing in the double layer counterbalances the charge on the

electrode surface. We can define a capacitance for the double layer. This

capacitance gives us a way to relate the amount of charge in the double layer to the

voltage drop across it. In the following section we present expressions for the

capacitance of the diffuse layer. The capacitance of the Stern layer is unknown

because the ionic properties such as diffusivity and mobility in the Stern layer may

not be the same as the bulk.

23

2.4.1 Linear Capacitance

Under the Debye Huckel approximation, a capacitive relation for the diffuse

layer can be expressed as follows

dq C ζ= − , (2.29)

where q is the double layer charge per unit area and dC is the diffuse layer

capacitance per unit area.

According to Gauss's law, the charge density per unit area of an infinite sheet

of charge is related to the normal electric field in the following manner,

0

ny

dq Edyφε ε

=

= − = . (2.30)

Under the linear assumption, 0 Dy

d dyφ ζ λ== − (see (2.20)). As a result,

D

q ε ζλ

= − . (2.31)

Comparing the two expressions for q , i.e. (2.29) and (2.31), we get an

expression for the linear capacitance of the diffuse layer

dD

C ελ

= . (2.32)

This result is analogous to the parallel plate capacitance with a plate separation

of Dλ .

2.4.2 Nonlinear Capacitance

When the zeta potential is higher than the thermal voltage, the preceding linear

capacitance expression becomes inadequate. In such circumstances, we need to use

24

the results of the Gouy-Chapman theory. We can get the following nonlinear

expression easily by combining (2.25) and (2.30),

1 20(8 ) sinh( )

2zeq kTnkTζε= − . (2.33)

which can be simplified to

2 sinh 2D

zekTq ze kTζε

λ⎛ ⎞⎜ ⎟⎝ ⎠

= − (2.34)

A nonlinear differential capacitance of the diffuse layer can then be defined in

the following manner [2]

cosh( )2d

D

dq zeCd kT

ζζ

ελ

= − = . (2.35)

Note that, for 2kT zeζ , the preceding equation yields d DC ε λ= which is

equivalent to the linear capacitance formula given by (2.32).

2.5 Helmholtz Smoluchowski Formula

Let’s consider a surface with a fixed charge density i.e. a fixed zeta potential.

Let’s subject the surface to a constant tangential electric field E . If there is no

pressure gradient in the system, the fluid flow equation for a parallel flow can be

simplified to,

2 2

2 2

d u d Edy dy

φη ε= . (2.36)

Then using (0) 0, (0) , 0 and 0u u y yφ ζ φ∞ ∞

= = ∂ ∂ = ∂ ∂ = for a thin double

layer, the preceding equation can be integrated to yield

25

1u Eεζ φη ζ

⎛ ⎞= − −⎜ ⎟

⎝ ⎠. (2.37)

Under Debye-Huckel approximation, φ can be substituted from (2.20) to yield

[1 exp( )]D

yu Eεζη λ

= − − − . (2.38)

Then, the electroosmotic slip velocity, su , at the edge of the double layer (i.e.,

y →∞ ) can be written as,

su Eεζη

= − . (2.39)

This is the well known Helmholtz-Smoluchowski equation for electroosmotic

slip velocity. This equation was derived for a thin double layer under Debye-

Huckel approximation ( / 1ze kTφ ), however, it works very well up to

/ 2ze kTφ ≤ according to [66]. According to Squires & Bazant 2004 [41], this

equation remains valid in nonlinear regime as long as

exp( ) 12

D zea kTλ ζ , (2.40)

where a is the radius of curvature of the surface.

2.6 Surface conduction and Dukhin Number

In the previous sections, we looked at the mechanism by which a tangential

flow of ions (and the fluid) is generated (i.e. electroosmosis). A solution was also

obtained which satisfied all the equations and boundary conditions. However, the

flow of ions tangential to the electrode surface creates a current which has not been

26

accounted for yet in the mathematical model. The ions can move along the

electrode surface not only by convection but also by conduction due to the

tangential component of the electric field (see Fig. 2.3).

- - - - -+ + + + ++ - + +

++

+-- -

+ -

λDjs=σsEσs

σ j∞=σEBulk

E

Fig. 2.3: Surface conduction

Usually, the net ionic current in the body of the fluid can be approximated by

j Eσ= . However, when the zeta potentials are large, a lot of charge can reside in

the double layer and the conductivity in the double layer might be much higher

than the bulk. Therefore the approximation ( j Eσ= ) yields a lower value for the

current. The excess current caused by the excess conductivity of the double layer is

referred to as surface conduction (see Fig. 2.3). Surface conduction has been

studied in the colloidal science for a long time [6-8].

Surface current is very important to estimate because excessive amount of

surface current can affect the electric field in the bulk and deteriorate the

27

electroosmotic slip velocity (refer to chapter 4). Surface conduction becomes

important in cases where zeta potentials are large (such as induced charge

electroosmosis in which the zeta potential is induced by an applied electric field).

The excess current (or surface current) can be characterized by a excess surface

conductivity, sσ ,

s sj σ= E , (2.41)

where sj is the excess surface current density per unit width in C s-1 m-1 = A m-1,

sσ is the excess surface conductivity in A V-1 = S and E is the electric field

causing the current. If y is the direction normal to the electrode surface, then it has

been shown that [1]

0

[ ( ) ( )]sj j y j dy∞

= − ∞∫ , (2.42)

where ( )j y is the bulk current density in C s-1 m-2 = A m-2 .

Bikerman, 1940 [6] realized that besides conduction, the charge also gets

advected with the electroosmotic flow. Including the electroosmotic advection in

(2.42) and expressing ( )j y and ( )j ∞ in terms of ionic concentrations and

mobilities, we get

( ) ( ) [ ( ) ( )] ( ) [ ( )]E Ei i i i i ii ij y j n y n z e z en y yεµ ζ φ

η− ∞ = − ∞ − −∑ ∑ , (2.43)

where, the first term is the current density due to conduction of charge relative to

the fluid and the second term is the current density due to the electroosmotic flow.

Here in is the number density of the ith ionic species, iz is the valence, e is the

28

charge of an electron, iµ is the ionic mobility, ζ is the zeta potential and ( )yφ is

the potential distribution in the double layer. The ionic mobility, iµ , can also be

expressed in terms of diffusivity using the Nernst-Einstein Equation,

i ii

z eDkT

µ = . (2.44)

Substituting for iµ in (2.43) from (2.44) yields,

2

2( ) ( ) [ ( ) ( )] [ ( )] ( )E Ei i i i i ii i

e ej y j n y n z D y z n ykT

ε ζ φη

− ∞ = − ∞ − −∑ ∑ . (2.45)

Substituting this expression in (2.42) and taking into account that all

concentrations follow from Gouy-Chapman theory, Bikerman derived for a

symmetric electrolyte,

( )( ) ( )( )2 2

2 22 1 1 1 1ze kT ze kTDs

z e n D e m D e mkT

ζ ζλσ −+ + − −

⎡ ⎤= − + + − +⎣ ⎦ , (2.46)

where,

2 2kTm

ze Dε

η±±

⎛ ⎞= ⎜ ⎟⎝ ⎠

(2.47)

is a dimensionless parameter indicating the relative contribution of electroosmosis

to surface conduction. When the cations and the anions have similar diffusivities

(i.e. D D+ −= ), (2.46) can be simplified to

( )2 24 1 cosh 1

2D

sz e nD zem

kT kTλ ζσ ⎡ ⎤⎛ ⎞= + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

. (2.48)

29

For aqueous solutions of a binary electrolyte (such as KCl), 0.45m ≈ at room

temperature and therefore the contribution of electroosmosis is lower than

conduction.

Usually, the surface conductivity is expressed in terms of a dimensionless

parameter called Dukhin number, Du , defined as.

sDua

σσ

= , (2.49)

where σ is the bulk conductivity and a is the characteristic length scale. The

Dukhin number is the dimensionless ratio of the surface and the bulk

conductivities. This dimensionless ratio has been used in the electrokinetic

literature for a long time and has been used for discriminating between various

regimes of electrokinetics [1].

The bulk conductivity σ can also be expressed in terms of ionic mobility and

concentration as

2zenσ µ= , (2.50)

which, using zeD kTµ = , can be further simplified to

2 22z e nDkT

σ = . (2.51)

Now combining (2.48), (2.49) and (2.51) yields

( )2 1 cosh 12

D zeDu ma kTλ ζ⎡ ⎤⎛ ⎞= + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

, (2.52)

which can also be written as

30

( ) 24 1 sinh4

D zeDu ma kTλ ζ⎛ ⎞= + ⎜ ⎟

⎝ ⎠. (2.53)

2.6.1 Stern Layer Dukhin Number

The previous discussions of surface conduction and the Dukhin number were

solely about excess surface currents in the diffuse part of the double layer. It was

easy to estimate the current density and Dukhin number in the diffuse part of the

double layer because the ionic mobility and diffusivity in the diffuse layer can be

considered to be the same as the bulk. However, the current density in the Stern

layer is hard to estimate because the diffusivity and mobility in the Stern layer are

unknown. The Stern layer surface conductivity, isσ is generally expressed as

i i is i ii

σ σ µ=∑ , (2.54)

where i in the superscript represents inner or Stern layer whereas the i in the

subscript represents the ith ionic species. Here iiσ is the surface charge density in

the Stern layer. Often there is only one ionic species (denoted by i) in the Stern

layer and therefore,

i i

i i i i i is i i

z eDkTσσ σ µ= = . (2.55)

Then using the definition of the bulk conductivity 2 22i i i iz e n D kTσ = , we can

express the Stern layer Dukhin number as follows,

2

i ii i i

i i i

DDuaz en Dσ

= . (2.56)

31

The methods of estimation of Stern layer Dukhin number have been described

in [1]. Assuming an adsorption model, iiσ can be related to in and the specific

Gibbs binding energy ads miG∆ . Making miG∆ more negative increases iiσ but does

not necessarily enhance isσ because tighter bound ions may have a lower lateral

mobility. In some cases, the Stern layer surface current can be stronger than the

diffuse current. This is expected for porous surfaces, containing thick

hydrodynamically stagnant layers with mobile ions in them, as is the case with

porous glass and bacterial cells.

33

Chapter 3. Induced Charge Electroosmosis: Experiments

We present an experimental evidence for the existence of induced charge

electroosmosis on a flat microelectrode. The experimental data contains basic

features of symmetric ICEO flow over a metallic surface. We also present an

experimental method called field effect flow control (FEFC) to break the symmetry

of the symmetric ICEO flows. FEFC can be used for making micropumps whose

pumping velocity as well as the direction of pumping can be modified by applying

a radial voltage to the flow surface. The existence of both ICEO and FEFC has

been demonstrated through experiments. Finite element simulations based on

linear theory (i.e. Debye Huckel approximation) have been performed to verify the

results of the experiments. Not surprisingly, the results of the simulations do not

match with the experiments; the simulations predict almost 2-3 orders of

magnitude higher slip velocities. The applied voltages in the experiments are high

and they induce high zeta potentials ( 10 40 kT zeζ ≈ − ). We expect the linear

theory (and the Debye Huckel approximation) to fail at such high zeta potentials.

In chapter 4 and 5, we will introduce a nonlinear model which reduces the

numerical velocity and reduces the discrepancy between the numerical simulations

and the experiments.

3.1 Induced Charge Electroosmosis

ICEO refers to a phenomenon in which a DC or AC electric field induces

charge on a polarizable surface (metal or dielectric), and produces an

34

electroosmotic slip by applying a body force on the electric double layer [41, 42].

Since the double layer is created and moved by the same electric field, this

phenomenon gives rise to steady flows in both DC and AC electric fields.

Flat electrodes are of much interest in microfluidic devices. Consider a finite

flat conductor surface in contact with an electrolytic solution. When it is subjected

to an external electric field, 0 ˆE E x= , at 0t = , the electric field lines intersect the

surface at right angles and a charge density is induced on the surface because of

charge separation (Fig. 3.1a). However, the field lines start changing their

configuration as a current J Eσ= drives positive ions towards one half of the

surface ( 0x < ) and negative ions to the other half ( 0x > ). This process develops a

double layer on the surface which grows as long as the normal electric field drives

ions into it. In the steady state, assuming that there is no surface conduction or

Faradaic injection, the double layer insulates the surface completely and no electric

field lines can penetrate into it. In this state, all the electric field lines are tangential

to the surface (Fig. 3.1b) and cause an electroosmotic slip directed from the edges

toward the center giving rise to two symmetric rolls above the surface (Fig. 3.1c).

An AC field will drive an identical flow as the change in the direction of the field

changes the polarity of the induced charge as well.

35

Conductor

(a) Electric field at t=0,

(b) Steady State electric field

(a) Steady ICEO Flow

E0

- - - - - - - - - - - +++++++++++

E0

- - - - - - - - - - - ++++++++++++ + + + + + + + + + + - - - - - - - - - - -

x

x

Conductor

Double layer

x

Fig. 3.1: Induced charge electroosmosis on a conducting surface. (a) At time t=0, there is no double layer, hence all the field lines are perpendicular to the surface (shown by electric field lines), (b) In the steady state, the double layer gets completely charged and all the field lines become tangential to the surface, (c) The tangential field causes the double layer to move and produces symmetric ICEO flow (shown by streamlines).

36

The slip velocity su is given by the Helmholtz Smoluchowski equation,

su Eε ζη

= − , (3.1)

where ε and η are respectively the absolute permittivity and viscosity of the ionic

solution, ζ is zeta potential of the surface and E is the electric field component

tangential to the surface. ζ is defined as the potential drop across the diffuse part

of the double layer. In an ideal case when there is no nonlinear effect present (e.g.

Stern layer, surface conduction or faradaic reactions), 0E E= and 0E xζ = . By

substituting these in (3.1), we get

20

2, s

aEu x xεη

≤= − . (3.2)

where a is the length of the conducting plate. This shows that the flow is

symmetric about 0x = (i.e. the center of the surface) and the maximum slip

velocity occurs at the left and right edges. The velocity depends on the square of

the electric field which implies that an AC field also drives a similar flow as long

as the frequency, ω , is low enough that the double layer has time to form,

1cω τ −< , (3.3)

where cτ is the characteristic double layer charging time defined as

cD

aετσ λ

= , (3.4)

where Dλ is the double layer thickness, and σ is the electrical conductivity of the

solution. Existence of cτ can be readily seen by considering an RC circuit with a

37

bulk resistor, /R a σ= , and a double layer capacitor, / DC ε λ= , [27]. The

expression for cτ can also be obtained by a scaling analysis of (3.25).

For a symmetric electrolyte for which the cations and anions have the same

magnitude of valence and diffusivity,

2 2

02z e n DkT

σ = , (3.5)

and

1 2

2 202D

kTz e nελ

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= , (3.6)

where D is the diffusivity of ions, 0n is the numeric concentration of ions in the

bulk, k is the Boltzmann constant, e is the elementary charge, and z is the

valence of the ions.

Combining the preceding definitions of cτ , σ and Dλ , we get

Dc

aDλτ = . (3.7)

3.2 Experimental Evidence for ICEO Flow

Squires and Bazant, 2004 [41] theoretically predicted ICEO flows around

polarizable surfaces when such surfaces are subject to an external electric field in

the presence of an electrolytic solution. Levitan et al. 2005 [43] verified the

existence of ICEO around a cylindrical wire in an ionic liquid. We present

experimental evidence of ICEO flow on a flat metal electrode in the following

sections.

38

3.3 Device Design and Fabrication Process

ICEO experiments were performed in a microfluidic device which simply

consisted of three parallel, equally spaced, coplanar electrodes laid on a glass

substrate (see Fig. 3.2). Image reversal photolithography was used to transfer the

electrode pattern in a photoresist film (AZ5214). Electrodes were made by

depositing 10 nm of titanium and 200 nm of platinum using electron beam

evaporation. Residual photoresist was removed by lift-off in acetone. The

fabrication process is depicted in Fig. 3.3. A step-wise fabrication procedure is

given in an appendix to this chapter.

Gate Electrode

Driving Electrodes

10 mm

20 mm

Fig. 3.2: The ICEO device. Three 200 µm wide electrodes are deposited on a glass substrate. The middle electrode is called ‘gate electrode’; the outer two electrodes are called ‘driving electrodes’.

39

Glass

Photoresist

Metal

PDMS

1)

2)

3)

4)

5)

6)

Glass

Photoresist

Metal

PDMS

1)

2)

3)

4)

5)

6)

Fig. 3.3: Fabrication process for the ICEO device: (1) A glass wafer was cleaned respectively in Acetone, Iso propanol, de-ionized water (DI) and finally in O2 plasma. (2) AZ 5214 photoresist was spin coated on the wafer. (3) The device design was transferred from the mask into the photoresist by image reversal photolithography. (4) Thin layers of metals (Ti/Pt, respectively 10/200 nm) were evaporated on the wafer and (5) then lifted off in acetone. (6) A PDMS chamber was placed on the glass substrate. PDMS readily adheres to the clean glass surface. The chamber was closed by placing another glass piece on top.

The electrodes were placed in contact with a microchamber filled with an ionic

solution. An AC signal was applied between the two outer electrodes (also called

driving electrodes); the middle electrode was kept floating i.e. it was not connected

to any power source. The floating (i.e. the middle) electrode is analogous to the

polarizable surface shown in Fig. 3.1. The AC electric field produced by the two

driving electrodes causes the floating electrode to get polarized and leads to

formation of double layer on the floating electrode. The external field also causes

the double layer to move, resulting in a symmetric ICEO flow.

40

The floating electrode is also denoted as the gate electrode. This name for the

electrode is chosen because of its analogy with a solid state field effect transistor

(FET). In a solid state FET, a gate voltage controls the electronic current between a

source and a drain terminal. In our device, the driving electrodes are used for

producing an electric field and set up a symmetric ICEO flow on the gate

electrode. The symmetry of the flow can be broken by applying an external

potential on the gate electrode (equivalent to field effect). A simple micropump

based on AC electric fields with directional control of the flow can thus be made.

We have discussed field effect in ICEO in a later section of this chapter.

3.4 Device Packaging

The overall dimensions of the bottom glass substrate were

20mm×10mm×500µm (Fig. 3.4). The gap between the two driving electrodes

was 800 µm (Fig. 3.5). The gate electrode was located symmetrically between the

two driving electrodes. A 2mm×4mm×125µm flow chamber was cut into a 125

µm thick PDMS sheet and placed on the electrodes. The net volume of the flow

chamber was 1 µLiter. A 1 µLiter volume of a KCl solution was dropped into the

chamber using a micro pipette. The ionic solution was seeded with 700 nm

florescent polystyrene beads for flow tracing. A 500µm thick blank glass chip was

placed atop the chamber to close it. A cross section of the ICEO chamber is shown

in Fig. 3.5.

41

ICEO flow chamber Top glass

coverPDMS sheetglass

substrate

Connection pads

10 mm

20 mm

Driving electrodesGate electrode

Fig. 3.4: Picture of a fully packaged device. A cut PDMS sheet is sandwiched between the bottom glass substrate and a top glass cover to form a closed flow chamber.

Driving electrode 1 Driving electrode 2Flow chamber Gate electrode

x

y

2 mm0.8 mm

0.2 mm0.2 mm 0.2 mm

0.125 mm

Fig. 3.5: Cross section of the flow chamber. The gap between the driving electrodes is 800 µm. The flow chamber is 125 µm deep.

3.5 Experimental Setup

An AC function generator produces a signal which is applied to the driving

electrodes (Fig. 3.6). In order to visualize the ICEO flow, 700 nm red-fluorescent

polystyrene particles (Duke scientific, Fremont, CA) are suspended in the ionic

fluid. The final concentration of fluorescent particles in the working fluid is 0.02%

42

by volume. A 100 W mercury arc lamp (Optiquip, Highland Mills, NY), an epi-

fluorescence microscope (Nikon Eclipse E600FN), an optical filter cube

(excitation 532 nm, emission 612nm, Chroma, Rockingham, VT), a 10x objective

lens (NA 0.25) and a CCD camera (Hamamatsu, 1280 1024 12× × -bit) are the main

components of the imaging system. The fluorescent particles are excited using

green light (532 nm wavelength) and upon excitation, they emit red light (612 nm)

which is recorded with the CCD camera.

glassPDMSglass

~AC function generator

CCD

Objective lensFluorescent particles in KCl

Floating electrode

Excitation filter

Hg lamp

Focusing lensFilter cube

Emitted lightExcitation light

White light

µPIV

ICEO device

Fig. 3.6: The experimental setup for flow measurement.

43

3.6 Measurement of ICEO with µPIV

The velocity of the fluid was determined by measuring the velocity of the

suspended fluorescent particles which were 0.7 µm in diameter. The particles close

to the electrodes have high velocity (due to ICEO slip) whereas the particles close

to the top-cover are slow (due to no-slip).

Since the particle diameter is much larger than the double layer thickness (~10-

30 nm), it’s impossible to measure the velocity within the double layer with such

large particles. Apart from this, the excitation light illuminates the entire flow field

and the lens collects light from all the particles in the illuminated flow field.

Therefore, what we measure is not the velocity in the focal plane; instead it is a

weighted depth average of the entire flow field. However, the highest relative

contribution to the depth average comes from the focal plane and the relative

contribution decays sharply as the distance from the focal plane increases. For

these reasons, the measured velocity (i.e. the weighted depth average) is a good

approximation for the velocity in the focal plane and we will call it the ‘slip

velocity’. The reader must keep in mind that it’s not the true value of the slip

velocity but a very good approximation. For a discussion of the weighted depth

averaging, the reader is referred to section 5.2.

For a typical experiment, approximately 50 consecutive image frames were

recorded using a µPIV system [67]. A µPIV program calculated the cross

correlations from the sequential image pairs and averaged the correlations before

producing a final velocity vector field [68]. The uncertainty in the experimental

44

data is about 3.6% (with respect to full scale) and has been discussed in section

3.18.

Fig. 3.7: A typical image of a µPIV experiment. The gate electrode is visible in the center. The inner edges of the two driving electrodes are also shown. Fluorescent particles are also clearly visible.

For the symmetric ICEO flow, the velocity has a symmetric vector field. A

typical vector field is shown in Fig. 3.8. The fluid moves from the edges towards

the center on all three electrodes. Flow is observed to be symmetric on the gate

electrode. The vector field on the gate electrode alone is also shown in Fig. 3.9 for

clarity. The fluid flows symmetrically from the two edges ( 100x mµ= ± ) towards

the center of the gate ( 0x = ). The velocity is highest close to the edges and zero at

the center. At the center, the fluid moves out of the plane (not shown in this vector

field). The highest velocity is observed to be 45 µm/s (Fig. 3.10) for a driving

voltage of 0 pp20Vφ = , and a frequency of 100Hzf = in purified water

( 17σ = µS/cm). The frequency of 100 Hz was chosen such that there is enough

45

time for the double layer to charge. Taking 17σ = µS/cm and

1280.1 8.854 10ε −= × × F/m for the purified water and 91.995 10D −= × m2/s for K+

and Cl- ions, we find 28.85D Dλ ε σ= = nm (refer to (3.5) and (3.6)). Then

taking 200a = µm, we find 0.0029c Da Dτ λ= = s (refer to (3.4)). We can now

calculate a dimensionless frequency, 0.29 1cfτ = < . In a linear regime (i.e.

kT zeζ < ), a dimensionless frequency less than 1 will ensure that the double layer

has enough time to charge. In a nonlinear regime, however, the double layer

requires an exponentially large amount of charge and the double layer might not

charge completely even at very low frequencies (refer to (2.34)).

The electrical conductivity of the purified water was measured to be 17 µS/cm

(Oakton Ectest microprocessor based conductivity meter). The purified water was

actually a 50:1 mixture of deionized (DI) water and Duke Scientific fluorescent

microsphere polymer suspension (Fremont, CA). DI water has very low

conductivity (0.055 µS/cm) and a tiny fraction (1/50th) of microsphere suspension

raises its conductivity to 17 µS/cm. Microspheres were added essentially for

visualizing the flow. The content of the microsphere suspension is not disclosed by

Duke Scientific due to proprietary issues. Therefore, we don’t know what ions are

present in the particle solution.

46

Fig. 3.8: A typical vector field for symmetric ICEO flow on a planar electrode.

Fig. 3.9: Velocity vectors on the gate electrode, for 0 pp20Vφ = and 100Hzf = in purified water with 17σ = µS/cm.

47

-1 -0.5 0 0.5 1x 10-4

-5

-4

-3

-2

-1

0

1

2

3

4

5x 10-5

Slip

Vel

ocity

, us (m

/s)

Distance from the center x (m)

Fig. 3.10: Variation of velocity along the gate electrode for 0 pp20Vφ = and 100Hzf = in purified water ( 17σ = µS/cm).

3.7 Dependence on Driving Voltage and Ionic Concentration

Fig. 3.11 shows the maximum slip velocity, maxu , as a function of the driving

voltage and the ionic concentration. The experiments were conducted for a range

of driving voltage at a constant frequency of 100 Hz and in two different solutions:

purified water ( 17σ = µS/cm) and 1 mM KCl (165 µS/cm). The velocity in the

purified water is significantly higher than that in 1 mM KCl solution. Many

workers have also observed that the electroosmotic slip velocity decreases as the

ionic concentration increases [28]. Such a reduction in slip velocity has been

attributed to Stern layer. At high ionic concentrations, the double layer becomes

very thin and its capacitance becomes comparable to that of the Stern layer. As a

48

result, a large part of the surface potential is dropped across the Stern layer,

leaving a small drop across the diffuse layer and reducing the slip velocity [30].

The velocity scales with 20φ for low values of 0φ confirming the trend

predicted by the linear theory. However, it becomes almost constant at high values

of 0φ . The saturation of velocity at high voltages can be attributed to the nonlinear

effects (such as surface conduction and nonlinear surface capacitance) which

become strong at high voltages and do not let the velocity grow. It’s important to

note that the saturation takes place at a voltage of around 15 Vpp in the purified

water and at about 20 Vpp in 1 mM KCl. In other words, the flow in the purified

water saturates sooner than in 1 mM KCl. This behavior can be explained by

comparing the double layer thicknesses in the two solutions. The double layer is

much thicker in the purified water than in 1 mM KCl. The surface currents are

stronger for thicker double layers because there is more diffuse charge (see chapter

4). In other words, surface conduction is much stronger in the purified water than

in 1 mM KCl. Therefore, the saturation in the purified water takes place at a lower

voltage than it does in 1 mM KCl.

49

0 5 10 15 20 25 30 350

0.5

1

1.5

2

2.5

3

3.5x 10-5

Driving Voltage, φ0 (Vpp)

Max

imum

Slip

Vel

ocity

, um

ax (m

/s)

PurifiedWater

1 mMKCl

us=4E-7φ02

Fig. 3.11: Variation of maximum slip velocity with the driving voltage. Initially the velocity scales quadratically with the voltage but eventually saturates. This saturation behavior is attributed to surface conduction. Surface conduction is stronger in the purified water than in 1 mM KCl; therefore the velocity saturates at a lower voltage in purified water than in 1 mM KCl. The measurements were performed at 100 Hz.

3.8 Dependence on Driving Frequency

Fig. 3.12 exhibits the effect of frequency on the maximum slip velocity. The

velocity stays almost a constant for f<300 Hz but decreases significantly for higher

frequencies. This indicates that the characteristic frequency of the double layer

charging is less than 300 Hz. The experiments were performed in 1 mM KCl at 18

Vpp.

50

101 102 103 104 10510-7

10-6

10-5

10-4

Driving Signal Frequency, f (Hz)

Max

imum

Slip

Vel

ocity

, um

ax (m

/s)

Fig. 3.12: Variation of maximum slip velocity with driving frequency. As the frequency increases, the velocity decreases. This is because at high frequencies, the double layer does not get enough time to get charged. The experiments were performed in 1 mM KCl at 18 Vpp.

We chose to perform our ICEO experiments in presence of AC electric fields

because AC fields yield the advantage of reduced electrolysis. In presence of DC

electric fields, a lot of gas bubbles are generated due to electrolysis. Apart from

this, DC electric field will cause the driving electrodes to get shielded by double

layer completely. In such a case, there will be no electric field in the bulk because

entire potential will be dropped across the double layers on the driving electrodes.

These problems motivate us to run our experiments in presence of AC electric

fields. However, the frequency of the driving AC field affects the velocity. As the

frequency increases, the available time for the double layer formation decreases.

51

As a result, the double layer does not charge completely and the zeta potential is

reduced.

3.9 Effects of Finite Frame Rate

The frequency of frame acquisition is generally much smaller than the driving

AC frequencies. For example, in our experiments, the PIV frames (i.e. images)

were captured at a frame rate of 13.2 Hz, whereas the driving frequency was 100

Hz. As a result, PIV images yield a time averaged velocity field. However, since

the frame acquisition is started at a random time, the frame acquisition process

might have a phase lag with respect to the driving ac cycle. If the phase lag is non-

zero, the time averaged slip velocity might have a contribution from the time

dependent component. Let’s quantify how much effect a finite frame rate has on

PIV velocity calculations.

In a linear regime, both ζ and E can be assumed to vary in synch with the

driving voltage. In other words, sin(2 )ftζ π∝ and sin(2 )E ftπ∝ . As a result, the

ICEO flow velocity can be represented as

202 sin (2 )u u ftπ= (3.8)

where 0u is the time averaged flow velocity. Note that, when integrated from 0t =

to 1/f, (3.8) yields a time averaged velocity equal to 0u . The preceding expression

can now be separated into a steady and a time dependent component as follows,

0 0 cos(4 )u u u ftπ= − (3.9)

52

The time variation of the velocity is shown in Fig. 3.13. Let’s say a frame is

captured at an arbitrary time, t1, and the second frame is captured at a time t1+∆t,

where ∆t the time gap between the two frames. Since, the frame capturing was

started at an arbitrary time, a phase lag, say 1φ , occurs between the velocity

variation and the frame acquisition. Since the frequency of frame acquisition is

different from the ac frequency, a different phase lag, , say 2φ , when the second

frame is captured. The phase lag keeps changing as more and more frames are

acquired.

When the first two frames are cross-correlated, a time averaged velocity is

obtained. The time average can be represented as follows,

( )0 0 10

1 cos(4 )t

u u u ft dtt

π φ∆

= − +∆ ∫ , (3.10)

which can be simplified to

( ) ( )( )00 1 1cos(4 )sin sin(4 )cos

4uu u f t f tf t

π φ π φπ

= − ∆ + ∆∆

. (3.11)

Note that the time dependent component of the velocity no longer produces a

zero average and leaves a non-zero residual which depends upon the phase 1φ . The

phase dependent residual can be viewed as a source of error because the exact

value of 1φ is not known. The phase dependent residual can be made negligible by

averaging over several pairs of images. Let’s say n pairs of images are captured

and the velocity is obtained by averaging n cross correlations. As a result, the

average velocity can be roughly estimated as,

53

( ) ( )00

1 1

cos(4 ) sin sin(4 ) cos4

n n

i ii i

uu u f t f tnf t

π φ π φπ = =

⎛ ⎞= − ∆ + ∆⎜ ⎟∆ ⎝ ⎠∑ ∑ . (3.12)

If n is large, the phase dependent residual in (3.12) will become very small

because the phase iφ keeps changing its value from one frame to another.

Although the values of iφ have a particular order because the images were

captured continuously but this order depends on f and ∆t. If we assume that that the

ac cycle and the image acquisition start in synch and then digress from each other,

the phase iφ can be expressed as

( )2 2 intL 2 , n=0,1,2,...,ni if t if tφ π= ∆ − ∆⎡ ⎤⎣ ⎦ , (3.13)

where ( )intL is the highest lower integer. In general, f and ∆t might not have an

integer product (i.e. 1f t∆ ≠ ) and therefore iφ may attain various non-integer

values. Averaging over a large number of phases will then yield a negligible phase

dependent residual in (3.12). In our experiments, we have 100f = Hz,

1/13.2 0.0757t∆ = = and 50n = . We find that for these parameters, the phase

dependent residual is approximately 4 orders smaller than the steady component,

0u . Hence, we can say that the measurement error due to a finite frame rate and

due to the phase lag is negligible.

54

0 1/f 2/f 3/f 4/f 5/f

0

u0

2u0

Time, t

Vel

ocity

, u

φ1φ2

Frame time, ∆t

Frame 1 t1

Frame 2 t1+∆t

Phase lag forfirst PIV pair

Phase lag forsecond PIV pair

Fig. 3.13: Effect of finite frame rate. The finite frame rate yields a time averaged velocity. But the phase lag between ac cycle and frame acquisition causes a phase dependent component in the average velocity. When averaged over a large number of frames, the phase dependent component becomes negligible.

3.10 Momentum Diffusion Length

Since the experiments are performed under AC electric fields, we must check if

the time dependent component of the velocity has enough time to diffuse to a

significant distance from the electrode. The momentum diffusion length can be

derived from a scaling analysis of the following equation,

2

2

u ut y

ρ η∂ ∂=

∂ ∂. (3.14)

55

A quick scaling analysis yield a momentum diffusion length, y fδ ν≈ ,

where ν is the kinematic viscosity of water. Taking 610ν −= m2/s and f=100 Hz,

we find 100yδ = µm. The chamber height is approximately 125 µm. This shows

that the time dependent component of the velocity can diffuse upto a significant

portion of the chamber, under the experimental ac frequencies. The PIV

measurements yield only a time averaged velocity (as was discussed in the

previous section) and therefore a momentum diffusion length smaller than the

chamber height does not have much impact on our PIV measurements.

3.11 Numerical Simulations based on Linear Theory

ICEO flow can be simulated numerically by solving the electrokinetic

equations described in chapter 2. The electrokinetic equations govern the

electrostatic potential and the concentrations of various ionic species. However,

the double layer, which is only 1-100 nm thick, is orders of magnitude smaller than

the depth of the chamber (125 µm). Therefore, one needs a really high resolution

mesh in order to discretize the electrokinetic equations in the double layer. Since

enormous computer resources will be needed for simulating the double layers in a

microfluidic environment, workers tend to model the double layer as a set of

effective boundary conditions. One of the effective boundary conditions is derived

by writing a charge conservation equation for the double layer. The other effective

boundary condition pertains to the mean concentration of the salt [61, 62]. The

double layer charging process not only affects the electric field in the bulk but also

causes gradients in the bulk salt concentration. The salt gradients, however, can be

56

ignored when the zeta potentials are low and the salt concentration can be assumed

to be uniformly equal to the far field concentration. This leaves us with the

problem of solving only for the electrostatic potential in the bulk and the double

layer charge density. Once the electrostatic potential and the double layer charge

density are obtained, we can find the slip velocity from the Helmholtz

Smoluchowski equation and use that as a boundary condition in the Navier Stokes

flow simulation.

The electrostatic potential in the bulk obeys Laplace’s equation,

2 0φ∇ = . (3.15)

The time averaged flow velocity in the bulk is obtained by solving steady state

Navier-Stokes equations,

2u u upρ η⋅∇ = −∇ + ∇ , (3.16)

0u∇⋅ = . (3.17)

Fluid transport in microfluidic devices is mostly dominated by diffusion and

therefore the nonlinear inertial terms can be ignored in (3.16).

3.9.1 Boundary Conditions

The walls of the chamber are insulated, i.e.

ˆ 0n φ⋅∇ = , (3.18)

0u = . (3.19)

57

On the left driving electrode we can apply a sinusoidal potential to simulate

AC fields,

0 sin2

tφφ ω= , (3.20)

and similarly, on the right driving electrode,

0 sin2

tφφ ω= − , (3.21)

where 0φ is the amplitude of the driving voltage. The voltage in peak-to-peak units

will be reported as 02φ .

In our simulations, we assume that there is no slip on any boundary except the

gate. In that case, the side walls, the top wall and the entire bottom wall except the

gate electrode will have the following b.c.

0u = . (3.22)

3.9.2 Boundary Conditions on the Gate Electrode

A double layer is considered on the gate electrode. If we ignore nonlinear

effects such as Faradaic reactions, surface conduction and concentration

polarization [61, 62], the charge conservation equation for the double layer charge

is given as

ˆq n Et

σ∂= − ⋅

∂. (3.23)

where q is the double layer charge per unit area. The left hand side (lhs) of the

preceding equation represents the rate of accumulation of double layer charge per

58

unit area. The right hand side (rhs) represents the normal flux of the charge into

the double layer. Note that n is the outward normal unit vector.

Under linear assumption, q can be related to the zeta potential, ζ , by treating

the double layer as a linear capacitor, i.e.

D

q ε ζλ

= − . (3.24)

Combining (3.23) and (3.24) and using E φ= −∇ yields

ˆD

nt

ε ζ σ φλ

∂= − ⋅ ∇

∂. (3.25)

The zeta potential of the gate electrode is given as

els

qC

ζ φ φ= − + , (3.26)

where elφ is potential of the equipotential gate electrode and sC is the capacitance

of the Stern layer. The third term on rhs represents the drop across the Stern layer.

Note that the Stern layer has been modeled as a linear capacitor which is a good

assumption according to [2]. Combining (3.24) and (3.26) yields

1

el

D sC

φ φζε

λ

−=⎛ ⎞+⎜ ⎟

⎝ ⎠

, (3.27)

which shows that the zeta potential ζ is reduced by a factor ( )1 δ+ where

D sCδ ε λ= is a parameter representing the ratio of diffuse to Stern layer

capacitance. This parameter accounts for the Stern layer effect. The higher the

value of δ , the lower the zeta potential ζ .

59

Combining (3.25) and (3.27) yields the following form of the double layer

charge conservation equation,

( )

( ) ˆ1

el

D

nt

φ φε σ φλ δ

∂ −= − ⋅ ∇

+ ∂. (3.28)

Since the preceding equations are linear, all field quantities can be represented

as phasors when using AC fields, e.g. ( )ˆIm i te ωφ φ= where φ is the complex

amplitude of φ . Note that phasors can only be used with linear equations. Using

phasors and rearranging the terms of (3.28), we get a simple boundary condition

for the bulk potential,

( ) ( )ˆ ˆ ˆˆ1 el

D

in ωεφ φ φλ σ δ

⋅∇ = − −+

. (3.29)

The time averaged Helmholtz Smoluchowski slip velocity can also be

represented in terms of phasors

( ) ( )2

0ˆ ˆIm Im

2i t i t

s su e E e dtπ ω ω ωω ε ζ

π η= − ∫ . (3.30)

After some algebra, the preceding equation can be simplified to

( )1 ˆ ˆˆ ˆconj( ) conj( )4s s su E Eε ζ ζη

= − + , (3.31)

where conj represents the complex conjugate and ˆsE is the phasor amplitude of the

tangential component of the electric field. This equation can be used as the slip

boundary conditions in the fluid flow solution.

60

3.12 Numerical Results

The linear model was simulated using a commercial finite element package

COMSOL (Comsol Inc., Stockholm, Se). The geometry used in the simulations is

the same as that shown in Fig. 3.5. It’s not clear what ions are present in the

purified water because purified water is a mixture of deionized (DI) water and

fluorescent particle suspension. For numerical purpose, we have assumed that K+

and Cl- are the main ions present. An average diffusivity value of 91.995 10D −= ×

m2/s at room T=200C was used in the numerical simulations. The gate potential elφ

was taken to be zero. Following are the constants used in simulations.

Table 3.1: Constants for Numerical Simulations

Constant

Value Description

a 200e-6 m Width of the gate electrode D 1.995e-9 m2/s Average diffusivity of K+ and Cl- k 1.381e-23 J/K Boltzmann constant T 293.15 K Room temperature η 1.002e-3 Pa-s Viscosity of water at 200 C

rε 80.1 Relative permittivity of water at 200 C

oε 8.854e-12 F/m Absolute permittivity of vacuum e 1.602e-19

Coulomb Elementary charge

Z 1 Valence of K+ and Cl- ions

Fig. 3.14 shows the numerical value of the slip velocity for purified water

( 17σ = µS/cm) when 20 Vpp (i.e. 0 10φ = Volt) at 100 Hz is applied to the driving

electrodes. The magnitude of the velocity is also shown in Fig. 3.15 along with the

61

experimental data. The numerical value of the slip velocity is almost two orders of

magnitude higher than the experiments. This indicates that the linear theory is very

inadequate in predicting the slip velocities for high zeta potentials. Please note that

in this particular experiment, the zeta potential is of the order 1 Volt which is 40

times higher than the thermal voltage ( 25.9kT ze = mVolt) at room temperature.

The zeta potential can be estimated as follows. The applied voltage of 20 Vpp

means that the maximum instantaneous potential difference between the driving

electrodes is 0 10φ = Volt. The driving electrodes have a gap of 1 mm

approximately. This yields a bulk electric field of 410sE ≈ V/m. The maximum

zeta potential is then / 2 1sE aζ ≈ = Volt. Here a (the width of the gate electrode)

is taken to be 200 µm.

62

-1 0 1x 10-4

-5

-4

-3

-2

-1

0

1

2

3

4

5x 10-3

Distance from the center, x (m)

Slip

Vel

ocity

, us (m

/s)

Linear simulation

Fig. 3.14: The slip velocity on the gate electrode as predicted by a linear simulation. The simulations were performed for purified DI water ( 17σ = µS/cm) and driving conditions of 20 Vpp at 100 Hz.

-1 0 1x 10-4

10-6

10-5

10-4

10-3

10-2


Slip

Vel

ocity

, us (m

/s)

Linear simulation

Experimental data

Fig. 3.15: Comparison of numerical and experimental slip velocity magnitudes for purified DI water ( 17σ = µS/cm) and driving conditions of 20 Vpp at 100 Hz. The numerical values are almost 2 orders of magnitude higher than the

63

experiments. It shows that the linear theory fails to predict the correct magnitudes of the slip velocity.

3.13 Breaking the Symmetry: Field Effect Flow Control (FEFC)

The symmetry of the flow can be useful for several purposes such as mixing

and enhanced transport of suspended species; but it becomes a roadblock when it

comes to pumping. The symmetric flows do not pump the fluid in any one

direction; instead the fluid moves in fixed vortices. In order to pump, the symmetry

has to be broken. The symmetry of ICEO flow can be broken by breaking the

symmetry of the induced charge. ICEO flows are symmetric when the charge

residing on the metal surface is symmetric. This condition is automatically broken

when the surface has a native charge (such as a pre-charged surface). In such

cases, the native charge gives rise to an electroosmotic flow which moves in one

direction in presence of DC electric field. In presence of AC fields, however, such

a surface again yields zero time averaged flow. Apart from this, we don’t have

native charge on many surfaces of interest, such as the electrodes in a microfluidic

system. In such cases, there is another way of breaking the symmetry of ICEO

flow, that is, by applying a radial voltage to the metal surface. The radial voltage

essentially produces a uniform charge density in addition to the symmetric induced

charge. This method yields an asymmetric ICEO flow leading to a scheme for

micro pumping. This method can also be understood in terms of zeta potentials. In

case of symmetric ICEO, the zeta potential of the surface has a symmetric nature.

64

By applying an external radial voltage to this surface, we can modify its zeta

potential and create asymmetries. This method is called ‘field effect flow control’

(FEFC) and is portrayed in Fig. 3.16.

+ + + + - - - - -+

E

+ + + + + + + - -+

E

VgFloating gate

surface

Symmetric flow Net pumping

+ + + + - - - - -+

E

+ + + + + + + - -+

E

VgFloating gate

surface

Symmetric flow Net pumping

Fig. 3.16: Concept of field effect flow control. When no external voltage is applied to the gate electrode, it developed a symmetric bipolar charge density and produces a symmetric ICEO flow. However, when an external voltage is applied to the gate, its charge density no longer remains symmetric and breaks the symmetry of the flow. As a result, a net pumping is achieved on the gate surface. The method of modifying the flow field by applying a gate voltage is called field effect flow control (FEFC).

The name ‘field effect’ is derived from semiconductor physics. Our FEFC

pump has a strong analogy with a semiconductor field effect transistor (FET). A

field effect transistor has three electrodes: a source, a gate and a drain. When an

electric field is applied between the source and the drain, electrons flow from the

source to the drain. However, the flux of the electrons depends strongly on the gate

voltage. By modifying the potential of the gate, one can modify the amount of

current flowing between the source and the drain. In a similar fashion, the gate

potential of an FEFC pump determines the direction and the magnitude of the

pumping velocity.

65

3.14 FEFC in DC Electric Fields

Field effect has been used in the past for enhancing the flow velocities of DC

electroosmotic pumps. In these works, an electroosmotic flow was established on a

glass surface by applying a tangential DC field. A gate electrode was embedded in

the surface of the glass. When the potential of the gate is fixed by connecting it to a

power supply, the zeta potential of the glass surface is modified. As a result, the

slip velocity is modified. This method has been utilized for enhancing the velocity

of DC electroosmotic flows [48, 53] for controlling band dispersion in capillary

electrophoretic devices [51] and recently for developing nanofluidic field effect

transistors [54-56].

Let’s derive an expression for FEFC pumping velocity based on linear theory.

If we ignore the Stern layer, then the zeta potential of the gate surface can be

expressed as

gζ φ φ= − , (3.32)

where gφ is the gate potential. If E is the external electric field, then under the

linear assumption, the slip velocity can be given as

( )s gu Eε φ φη

= − − , (3.33)

If we assume that the bulk potential decays linearly, then

E xφ ≈ − , (3.34)

and therefore,

66

( )s gu E x Eε φη

= − + . (3.35)

Note that the slip velocity will be symmetric about the center 0x = if 0gφ = .

However, for any other applied gate potential i.e. 0gφ ≠ , the slip velocity is not

symmetric. The net pumping velocity can be expressed as,

/ 2

/ 2

1 a

pump sau u dx

a −= ∫ . (3.36)

Substituting for su from (3.35) and integrating we get

pump gu Eε φη

= − . (3.37)

Note that if 0gφ < , the net pumping velocity is from left to right and vice

versa. The pumping velocity varies linearly with the gate potential gφ in the linear

theory limit.

3.15 FEFC in AC Electric Fields

In order to enable field effect flow control in AC electric fields, the gate

potential also has to change polarity in unison with the driving voltage. Then only

a non-zero time average flow can be achieved. The phase gap between the gate and

the driving voltages can either be zero or 1800. These two phase gaps will produce

pumping in opposite directions. Evidence of FEFC in AC electric fields was first

presented by Mutlu et al. in 2004 [69]. They embedded a metal electrode in an

insulating material and applied very high potentials to the embedded gate

electrode. We develop the idea of AC FEFC further and present a detailed

67

experimental study. We do not insulate our gate electrode with any insulating

material. Instead, our gate electrode makes direct contact with the ionic liquid and

therefore we don’t have to apply high potentials to the gate.

3.16 Experimental Evidence of FEFC in AC Electric Fields

AC FEFC experiments were performed with the device also used for

symmetric ICEO flow described in the previous sections. The experiments were

performed in three different KCl solutions: 1 mM ( 165σ = µS/cm), 10 mM

( 1400σ = µS/cm) and 50 mM ( 2950σ = µS/cm). A driving voltage of

0 pp60Vφ = at 500Hzf = was applied between the driving electrodes in all the

experiments. Let’s say that the left driving electrode is at 60 Vpp while the right

driving electrode is grounded. Since the gate electrode is symmetrically placed

between the two driving electrodes, the floating potential of the gate can be

assumed to be 0 pp/ 2 30VgV φ= = at 500Hzf = . Therefore, if we apply a potential

of pp30 VgV = at 500 Hz to the gate, a symmetric flow will be produced. However,

if the applied gate potential is different from 30 Vpp, the flow will no longer remain

symmetric. For pp30VgV < , the net flow will be from left to right and for

pp30VgV > , the net flow will be from right to left. This behavior has been

demonstrated experimentally (see Fig. 3.17).

68

Fig. 3.17: Field effect flow control. Shown are the experimentally measured velocity vectors on the gate electrode for various values of gate voltage. The large arrow shows the direction of net pumping. For pp30VgV < , the fluid flows from left to right whereas for pp30VgV > , the flow goes from right to left. The experiments were performed in 1 mM KCl under the driving conditions of 60 Vpp at 500 Hz.

Velocity profiles for several gate voltages are shown in Fig. 3.18. The velocity

profile is not symmetric for any pp30VgV ≠ . Infact, the velocity is mostly positive

for pp30VgV < and negative for pp30 VgV > .

69

-1 -0.5 0 0.5 1x 10-4

-8

-6

-4

-2

0

2

4

6

8x 10-5

x (m)

Slip

Vel

ocity

, us (m

/s)

Vg=18.8 Vpp

22.2 Vpp

28.1 Vpp

31.4 Vpp

35.5 Vpp

Vg=39.2 Vpp

25.2 Vpp

Fig. 3.18: Experimentally measured slip velocity profile on gate electrode for various values of gate voltage. The experiments were performed in 1 mM KCl under the driving conditions of 60 Vpp at 500 Hz.

These velocity profiles can be integrated over the width of the gate electrode in

order to find the net pumping velocity,

/ 2

/ 2

1 a

pump sau u dx

a −= ∫ . (3.38)

Naturally, 0pumpu = when the flow is symmetric (i.e. pp30VgV = ). The net

pumping velocity pumpu has been plotted for three different ionic concentrations as

a function of gV in Fig. 3.19. As the ionic concentration increases, the net pumping

velocity decreases. This behavior is again attributed to an increased Stern layer

effect at high ionic concentrations. The net pumping velocity is symmetric about

70

pp30VgV = . The direction of pumping can be easily reversed by modifying the

potential on the gate. Note that, 0pumpu > for pp30VgV < and 0pumpu < for

pp30VgV > . FEFC thus provides us with a great tool to not only modify the

pumping velocity but also the direction of pumping with the ease of modifying the

gate voltage.

10 15 20 25 30 35 40 45 5050-5

-2.5

0

2.5

5x 10-5

Gate Voltage, Vg (Vpp)

Net

pum

ping

vel

ocity

, upu

mp (m

/s) 1 mM

10 mM

50 mM

Fig. 3.19: Net pumping velocity as a function gate voltage. For pp30VgV < , the net pumping velocity is positive whereas it’s negative for pp30VgV > . The experiments were performed in 1 mM KCl under the driving conditions of 60 Vpp at 500 Hz.

Net pumping velocities of 50 µm/s can be achieved while working with 1 mM

KCl solution (see Fig. 3.19). The pumping velocity does not increase further even

if the gate voltage is biased further. We define the bias as 30gV − i.e. the

difference between the applied and the floating gate potentials. The saturation

71

behavior at high bias is attributed to surface conduction. The saturation effect is

more prominent for low ionic concentration (1 mM) than for high ionic

concentrations (10 mM and 50 mM). This behavior can be justified by comparing

the double layer thicknesses for these solutions. The double layer thickness

decreases as the ionic concentration increases. When the double layer is very thin,

the surface conduction currents are negligible because there is not enough diffuse

charge to create the surface current. In other words, surface conduction is almost

absent in the 10 mM and 50 mM experiments and therefore the velocity does not

saturate at high bias in the gate voltage.

The pumping velocity is also a function of the driving frequency. As the

frequency increases, the net pumping velocity decreases (see Fig. 3.20).

103 104 105 10610-6

10-5

10-4

Driving Frequency, f (Hz)

Net

Pum

ping

Vel

ocity

, upu

mp (m

/s)

Fig. 3.20: The net pumping velocity as a function of the driving frequency. The velocity decreases as the frequency increases. The experiments were performed in 1 mM KCl under the driving conditions of 60 Vpp at 500 Hz. The gate voltage was fixed at 15 Vpp.

72

3.17 Linear Simulation of FEFC

The net pumping velocity for the preceding experimental conditions was also

calculated through linear numerical simulations. The gate voltage was varied to

obtain the pumping velocity as a function of the gate voltage. The numerical

pumping velocity as a function of the gate voltage is shown in Fig. 3.21. The

numerical pumping velocity varies linearly with Vg, unlike the experiments. The

linear theory does not take into account surface conduction, the reason to which

the saturation behavior of experimental velocity (Fig. 3.19) was attributed. The net

pumping is zero for Vg=30 Vpp also observed in the experiments.

The numerical magnitudes of pumpu have also been compared with the

experimental data (see Fig. 3.22). The numerical values are almost three orders of

magnitude higher than the experiments. This time the discrepancy (3 orders) is

higher than before (see Fig. 3.15). This is because the experimental data in Fig.

3.22 was obtained in 1 mM KCl whereas the data in Fig. 3.15 was obtained in

purified water. The Stern layer effect is more prominent at higher ionic

concentration and reduces the experimental velocity. We have not taken the Stern

layer into account in our simulations. Therefore, the discrepancy in 1 mM KCl is

higher.

The numerical simulations do not predict the saturation behavior of velocity at

high voltages which was observed in the experiments (Fig. 3.19). This indicates

that the linear theory is inadequate for simulating high zeta potential situations

such as those encountered in our experiments.

73

10 20 30 40 50-0.03

-0.02

-0.01

0

0.01

0.02

0.03


Net

Pum

ping

Vel

ocity

, upu

mp (m

/s)

Linear Simulations

Fig. 3.21: Numerical values of the net pumping velocity as a function of gate voltage. The simulations were performed in 1 mM KCl under the driving conditions of 60 Vpp and 500 Hz.

10 20 30 40 5010-5

10-4

10-3

10-2

10-1


Mag

nitu

de o

f Net

Pum

ping

Vel

ocity

u pu

mp (m

/s)

Experimental Data

Linear Simulations

Fig. 3.22: Comparison of numerical and experimental values of the net pumping velocity. Numerical values are 3 orders of magnitude higher than the experiments. The simulations and experiments were both performed in 1 mM KCl under the driving conditions of 60 Vpp and 500 Hz.

74

3.18 Uncertainty Analysis

The uncertainty of a µPIV measurement originates from the uncertainty in the

measurement of two quantities: the displacement of the particles, x∆ , and the time

interval between the two frames of correlation, t∆ . The velocity can be expressed

as

xut

∆=∆

(3.39)

If ( )xδ ∆ and ( )tδ ∆ are the uncertainties in x∆ and t∆ , then the standard

error analysis yields the following expression for the uncertainty in the velocity,

uδ ,

( ) ( )2 2x tu

u x tδ δδ ∆ ∆⎛ ⎞ ⎛ ⎞

= +⎜ ⎟ ⎜ ⎟∆ ∆⎝ ⎠ ⎝ ⎠ (3.40)

In our experiments, the frames were obtained at a frame rate of 13.2 Hz, which

yields 0.075 st∆ = . The time interval, t∆ , is controlled by an electronic camera

controller and the uncertainty ( )tδ ∆ is generally very small. The uncertainty in

the displacement, ( )xδ ∆ , on the other hand, depends on many factors such as the

Brownian motion, numerical aperture of the microscope, wavelength of the light,

size of the particles, pixel resolution of the CCD camera, choice of the

interrogation method and the choice of the correlation-peak finding method.

Therefore, the relative uncertainty in displacement, i.e. ( ) /x xδ ∆ ∆ , is generally

much larger than the relative uncertainty in the time interval, ( ) /t tδ ∆ ∆ . In other

75

words, the uncertainty in the µPIV measurement is mainly due to the uncertainty in

determining the displacement of the particles,

( )xuu x

δδ ∆⎛ ⎞≈ ⎜ ⎟∆⎝ ⎠

(3.41)

Since submicron size particles (0.7 µm) were used for tracing the flow, the

uncertainty due to Brownian motion must be taken into account. A first order

estimate of this error relative to the displacement in the x-direction is given as [67],

1/ 22

1 2B

s Dx u t

ε = =∆ ∆

(3.42)

where 2s is the random mean square particle displacement associated with

Brownian motion and D is the Brownian diffusion coefficient of the particles. Note

that Bε is equivalent to an inverse Peclet number, i.e. 1B Peε −= where

BPe u Dδ= and 2B D tδ = ∆ . The Brownian error is small when the flow is fast

or the particles are large. We can make an estimate of the Brownian error in our

experiments by estimating the diffusion coefficient of the particles. According to

the Einstein equation, the diffusion coefficient of a particle of radius a is given as

6

kTDaπη

= (3.43)

For 0.7 µm particles we estimate 133 10D −≈ × m2/s. The characteristic velocity

and the time interval are respectively 6~ 100 10u −× m/s and 0.075t∆ = s.

Substituting these values in (3.42) yields 2%Bε = .

76

We also have to consider presence of a random error in our measurements. The

random error is caused by the imperfections in the particle images. If a particle

image diameter (when projected back into the flow field) is big enough to cover 2-

3 pixel of the CCD array, its displacement can be determined within 1/10th of the

its image diameter (projected back into the flow field). The diffraction limited

diameter of a particle image is given by [70]

1/ 22

2 1.22e pd M d

NAλ⎡ ⎤⎛ ⎞= +⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦ (3.44)

where pd is the diameter of the particle (0.7 µm), λ is the wavelength of the

emitted light (612 nm), NA is the numerical aperture of the lens (0.25) and M is the

magnification of the lens (10x). When projected back into the flow, the image

diameter becomes /ed M which is approximately equal to 3 micron in our case

and covers 4-5 pixels of our CCD array. As mentioned before, we estimate that the

random errors is approximately 10% of the diameter of the particle image when

projected back into the flow field [71], i.e. ( ) 0.1 3 0.3xδ ∆ = × = µm. In our PIV

analysis, a 20 micron wide interrogation spot was used with a 50% overlap. As a

result, the full scale displacement that we can measure, FSx∆ , is approximately 10

micron (half of the interrogation region width). Referencing the error to full scale

displacement, we get a relative uncertainty of 3%

( ( )/ / 0.3 /10 0.03FS FSu u x xδ δ= ∆ ∆ = = ). This relative uncertainty can be reduced

77

by increasing the time gap between the two frames so that the particle displaces by

a longer distance between the frames.

In a nutshell, the relative uncertainty due to Brownian motion is approximately

2% and the random error is approximately 3% (when referenced to the full scale

displacement). Since different types of uncertainties are independent of each other,

a total uncertainty can be found by taking the root of the sum of the squares, i.e.

total uncertainty = 2 22 3 3.6%+ = .

3.19 Conclusions

We have presented extensive experimental data which verifies the existence of

ICEO flows in microfluidic devices. We observed symmetric ICEO flows on a

planar microelectrode and measured the flow velocities using micro particle image

velocimetry. Slip velocity of 40 µm/s was observed in purified water under driving

conditions of 20 Vpp at 100 Hz. From our experiments, we learned that the slip

velocity decreases as the ionic concentration increases. This behavior was

attributed to the Stern layer effects. We also learned that the slip velocity is a

function of the driving voltage and the frequency. As the driving voltage increases,

the slip velocity increases quadratically. However, at very high voltages, the slip

velocity stops growing and either decreases or stays constant as the voltage is

increased further. This saturation behavior has been attributed to surface

conduction. Our claim about surface conduction is bolstered by the observation

that the saturation behavior occurs at a lower voltage in purified water than in 1

mM KCl. Surface conduction is higher when the ionic concentration is lower and

78

thus causes a saturation of slip velocity at a lower voltage. The velocity decreases

as the frequency increases.

A pumping strategy based on field effect flow control (FEFC) was discussed

and demonstrated experimentally. Field effect proves to be a good way of pumping

at microscale. It gives the user flexibility of modifying the magnitude and the

direction of pumping by simply modifying the voltage applied to the gate

electrode. Pumping velocity of 50 µm/s was obtained in 1 mM KCl under driving

condition of 60 Vpp at 500 Hz and an applied gate voltage of 15 Vpp. The relative

uncertainty in the experiments was estimated to be approximately 20%.

We also performed numerical simulations based on linear Debye Huckel

theory. Our simulations predict results which are 2-3 orders of magnitude higher

than the experimental data. Such a high discrepancy between the linear simulations

and the experiments imply that the linear theory is inadequate for simulating our

experiments. In our experiments, the induced zeta potentials are much higher than

the thermal voltage (10-40 times higher). Under such condition, we do expect the

linear theory to fail.

In the next chapter, we will introduce a nonlinear model of double layer which

takes into account nonlinear effects such as nonlinear surface capacitance and

surface conduction. Surface conduction in particular will be explored in detail. We

will show that high amounts of surface currents can flow when the induced zeta

potentials are high. These currents cause significant reduction in the tangential

component of the electric field and therefore deteriorate the slip velocity. In

chapter 5, a full AC nonlinear model will be simulated to make better predictions

79

about our experiments. It will be shown that the combined effects of nonlinear

surface capacitance and surface conduction indeed help in bridging the

discrepancy observed in the current chapter.

3.20 Appendix to Chapter 3

Microelectrode Fabrication with Image Reversal Lithography

Following are the steps for the fabrication of microelectrodes in ICEO device 1. Clean a glass wafer in acetone (3 minutes in ultrasound bath) 2. Clean in isopropanol (3 minutes in ultrasound bath) 3. Dry by blowing N2 gas 4. Stir clean in a piranha solution (H2SO4:H2O2 = 2:1) at 800 C for 10

minutes 5. Dry by blowing N2 gas 6. Dehydrate on a hotplate at 1100 C for 5 min 7. Spin coat HMDS at 4000 rpm for 45s 8. Spin coat AZ5214 photoresist at 4000 rpm for 45s 9. Soft bake on a hotplate at 950 C for 90s 10. Mask-expose with a contact aligner for 8s 11. Post exposure bake on a hotplate at 1100 C for 1 min 12. Flood expose on a contact aligner for 1 min 13. Develop in AZ400:H20 (1:4) for 25s 14. Rinse with deionized water 15. Dehydrate on a hotplate at 950 C for 5 min 16. Deposit metal with electron beam evaporator, 10 nm Ti, 200 nm Pt 17. Lift off residual metal in acetone with ultrasound bath

81

Chapter 4. Surface Conduction in Induced Charge

Electroosmosis

We have simulated highly nonlinear electrokinetic phenomena which occur at

high zeta potentials. Special attention has been paid to the understanding of surface

conduction in ICEO flows. We show that surface conduction through nanoscale

diffuse layer can cause micron scale gradients in the bulk electric field and reduce

the slip velocity thereby. Our work reveals some dramatic and surprising

consequences of surface conduction in ICEO flows. In ICEO flows, the zeta

potentials are induced by the applied electric field. As a result, the zeta potentials

are generally high and very nonuniform on the electrode surface. Nonuniform zeta

potentials give rise to nonuniform surface conduction currents which lead to

unique results not observed in phenomena with uniform zeta potentials. Khair and

Squires 2008 [72] had analyzed the case of a surface with a uniform zeta potential

and demonstrated that surface conduction pulls the electric field lines into the

double layer close to the stagnation points (or corners) of the surface. The electric

field lines ‘heal’ from this pulling after a short region (called healing region) close

to the corners and become parallel to the surface again. We find these corner

healing regions in our analysis of nonuniform ICEO flows too; however, we find

another healing region in the middle of the surface where the lines are expelled out

of the double layer. This middle healing region is caused by the nonuniform

surface currents inherent to ICEO flows. We will solve two canonical cases of

ICEO on a flat metal electrode and on a metallic cylinder. In cases where surface

82

currents are strong, the length of the healing regions can be comparable to the

surface dimensions and the tangential component of the electric field is

significantly reduced in the healing zones. Surprisingly the tangential field is

increased in the center of the surface. Overall, surface conduction reduces the

tangential field and the zeta potential and therefore reduces the slip velocity

significantly. Our analysis also shows that surface conduction grows not only with

the induced zeta potential but also with the double layer thickness. In cases where

double layer is infinitesimally thin, surface conduction becomes negligible and has

minor effect on the slip velocity.

We develop a fundamental picture (both qualitative and quantitative) of surface

conduction in ICEO flows, showing its unique features. We also introduce a

numerical model to simulate surface conduction both in steady and time dependent

cases. In time dependent cases (such as AC), nonlinear capacitance will also play a

role. In steady cases, however, surface capacitance does not play any role because

enough time is provided for the double layer (capacitor) to charge. In the presented

numerical model, the electric double layer is realized by solving a partial

differential equation (PDE) on the surface. The PDE is derived by conserving ionic

charge in the double layer. A normal ohmic current and a tangential surface

conduction current are balanced with the rate of charge accumulation. This model

is more advanced than the linear model based on Debye Huckel approximation

because the linear model ignores the presence of surface conduction altogether and

uses a linearized surface capacitance. These simplifications are not valid for

situations in which the zeta potential is much higher than the thermal voltage and

83

therefore our nonlinear model serves as a tool for simulating high zeta potential

situations.

4.1 Effects at High Zeta Potentials

When a surface acquires charge because of a spontaneous chemical change, we

can expect its zeta potential to be of the same order as the thermal voltage ( kT ze )

because chemical changes are facilitated by thermal energy. For example, when a

glass surface comes in contact with water, it acquires negative charge and develops

a zeta potential of about -0.1 Volt (only four times higher than the thermal voltage

at room temperature). This is, however, not the case in ICEO where the zeta

potential is induced by the electric field. The zeta potential in such cases is

proportional to the electric field strength. In many microfluidic experiments,

electric fields are very strong and can induce several times higher zeta potentials

than the thermal voltage.

There are several effects which can take place at high zeta potentials, e.g.,

nonlinear double layer capacitance, surface conduction, chemi-osmosis [61, 62]

and faradaic reactions [2, 38].

At high zeta potentials, the double layer behaves as a nonlinear capacitor

requiring exponentially large amount of ionic charge, q, which according to the

Gouy-Chapman model is expressed as

2 sinh 2D

zekTq ze kTζε

λ⎛ ⎞⎜ ⎟⎝ ⎠

= − . (4.1)

84

This means that an exponentially long time is required for charging the double

layer completely. This results in very poor charging when using AC electric fields.

The double layer can not acquire enough charge even at very low frequencies,

yielding a low zeta potential, a low tangential electric field and therefore a low slip

velocity.

The surface conductivity, sσ , also becomes very high at high zeta potentials.

According to the Gouy-Chapman model for surface conductivity [1], sσ is an

exponentially growing function of zeta potential, ζ ,

( ) 24 1 sinh 4s Dzem kTζσ λ σ ⎛ ⎞

⎜ ⎟⎝ ⎠

= + , (4.2)

where m is a dimensionless parameter indicating the relative contribution of

electroconvection to surface conduction. In presence of large zeta potentials, a

significant amount of current flows parallel to the surface through the double layer.

When a lot of charge flows through the double layer, a similarly large amount of

charge has to be supplied from the bulk in form of a normal flux [72]. This causes

the electric field to have a large normal component and a low tangential

component. This creates a deteriorating impact on the slip velocity. Next we

expand this argument by drawing a fundamental picture of surface conduction.

4.2 A Fundamental Picture

Before discussing ICEO, let’s consider a steady electroosmotic flow on a finite

size plate with a fixed negative charge. In a case when the zeta potential of the

surface is low ( kT zeζ ), the electric field outside the double layer can be

85

assumed to be equal to the applied uniform field of E∞ [73]. This is, however, not

true when the surface is highly charged, i.e. kT zeζ > . In this case, large surface

current flows through the diffuse layer for which the ions have to be supplied from

the bulk. Khair and Squires, 2008 [72] showed that the electric field lines are

drawn into the double layer close to the edges of plate (Figs. 4.1 and 4.2) bringing

ions into the double layer, conserving charge thereby. The field lines again become

parallel to the surface after a healing length HL and the electric field becomes

equal to E∞ . By a simple current counting argument, Khair and Squires, 2008 [72]

showed that the healing length is equal to the ratio of surface-to-bulk electrolyte

conductivities,

( ) 24 1 sinh4

sH D

zeL mkT

σ ζλσ

⎛ ⎞≡ = + ⎜ ⎟⎝ ⎠

. (4.3)

The healing length can be derived easily by applying charge conservation on a

volume V extending a distance Dλ in normal direction and a tangential distance

HL from the left edge of the plate. A surface current s xEσ leaves the right

boundary of this volume; however no surface current enters on the left boundary

because there is no surface charge upstream of the plate. To maintain the current

balance, a normal current y HE Lσ enters from the bulk. Note that the surface

conductivity, sσ , has units of S, whereas the bulk conductivity, σ , has units of

S/m. By equating the two currents, we reach the aforementioned expression for

HL . The same argument works for the right edge of the electrode. For highly

86

charged surfaces, the healing length can be much larger than the debye length Dλ .

For example, taking D250 mV and 10 nmζ λ= = gives 1.5 µmHL ≈ . In other

words, nanoscale surface conduction can cause micro scale gradients in the bulk

electric field (see Fig. 4.2). Surface conduction has been studied in colloidal

science for a long time and has serious consequences for the electrophoretic

mobility of charged particle [7]. In fact, at sufficiently large zeta potentials, the

electrophoretic mobility of a charged particle decreases with increasing ζ because

the tangential field around the particle is reduced due to surface conduction [8-11].

In context of electrophoresis, the ratio of healing length to the particle radius is

also known as Dukhin number, Du, and is given as [1, 13]

( ) 24 1 sinh 4sH DL zeDu ma a a kT

λσ ζσ

⎛ ⎞⎜ ⎟⎝ ⎠

= = = + (4.4)

The preceding equation accounts only for diffuse layer surface conduction. We

have not accounted for the Stern layer surface conduction because the ionic

mobility and diffusivity in the Stern layer are unknown. Strategies to estimate the

Stern layer Dukhin number are enumerated in [1].

87

- - - - -0sj = Dλs s xj Eσ≈

+

- - - - -- -0sj =

y yj Eσ≈ y yj Eσ≈

+ +

++

++

+ +

+ + +

E∞

constant sσconstant ζ0sj = 0sj =LH LH

0ζ = 0ζ =

Fig. 4.1: The fundamental picture of surface conduction on a highly charged plate. Consider charge conservation in a control volume close to the left edge of the plate. A surface current, s xEσ , leaves the inner boundary of the volume but no surface current enters the outer boundary because there is no surface charge outside the plate. As a result, a normal current, y HE Lσ , enters from the bulk causing the electric field lines to be pulled into the diffuse layer and maintaining the charge conservation. This picture is adapted from Khair and Squires 2008 [72].

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -LH LH

E∞

Fig. 4.2: Electric field lines on a highly charged plate. The field lines are pulled into the double layer as a result of surface current flowing through the diffuse layer. The field ‘heals’ from the discontinuity in the surface current in a distance LH and becomes parallel to the surface again.

Now let’s discuss the main subject of this chapter, i.e. surface conduction

effects in ICEO flows. Consider ICEO flow on a floating metal electrode subjected

to a uniform applied field E∞ . Since the electrode is not charged, it must acquire a

bipolar charge distribution in order to maintain its charge neutrality. This yields a

88

nonuniform zeta potential distribution along the width of the electrode, with the

maximum zeta potential, maxζ± , on the two edges and a zero zeta potential at the

center. In a linear case, zeta potential will vary linearly, E xζ ∞= , with 0x = at the

center of the electrode. However, when ζ is large, surface current will flow from

the two edges (composed of opposite ions, see Fig. 4.3). A current counting

analysis similar to the one above will yield healing regions close to the two edges.

However, at the center of the electrode, the surface conductivity is zero (because

0ζ = ). As a result, the ions of the surface current are forced to escape into the

bulk before they reach the center. In turn, the electric field lines are expelled out

from the double layer (focus on left to the center), yielding another region in which

the field has a large normal component (see Fig. 4.4). A similar but reverse

argument works on the other half of the electrode (right to the center).

-

+ + + + +0sj = Dλ

+

- - - - +- -0sj =


+

++

E∞

maxζ−maxζ+0ζ =

0sj =0sj = 0sj =,maxsj ,maxsj

---

+ + - -++

--

LH LH

0ζ = 0ζ =

-

+ + + + +0sj = Dλ

+

- - - - +- -0sj =


+

++

E∞

maxζ−maxζ+0ζ =

0sj =0sj = 0sj =,maxsj ,maxsj

---

+ + - -++

--

LH LH

0ζ = 0ζ =

Fig. 4.3: Fundamental picture of surface conduction on a floating electrode subjected to an applied field. The electrode acquires a bipolar charge and a nonuniform zeta potential distribution. Surface currents yield healing regions at the edges. However, in the center, where zeta potential is zero, no surface current can flow and therefore the ions are expelled out of the double layer. This yields another healing region in the middle of the electrode.

89

- - - - - - - - - - - - - - - -LH LH

E∞

+ + + + + + + + + + + + + + + +- - - - - - - - - - - - - - - -LH LH

E∞

+ + + + + + + + + + + + + + + +

Fig. 4.4: Electric field lines on a floating electrode subject to an applied electric field. The field lines are first pulled into the double layer and then expelled out because of nonuniform surface currents. This feature is unique to ICEO flows where nonuniform zeta potentials are induced by the applied electric field.

Note that the surface current in ICEO is very nonuniform; it’s zero not only at

the two edges but also at the center of the electrode. The surface current has a

maximum somewhere between the edge and the center. The location of the

maximum is also the location where the electric field lines start being expelled out

(left to the center). Note that, nonuniform surface currents virtually leave no width

where the electric field is completely tangential to the surface. On almost all the

points, the field has a normal component, either being pulled in or expelled out.

Only in the very center of the electrode, the electric field is completely tangential

and can attain a very large value. A reduced tangential field renders a low zeta

potential and low slip velocity.

90

4.3 Bulk Equations

Under the assumption that the bulk is electroneutral and the salt concentration

in the bulk is uniform, the electrostatic potential in the bulk, φ , satisfies Laplace’s

equation,

2 0φ∇ = . (4.5)

The time averaged flow velocity in the bulk is obtained by solving steady state

Navier-Stokes equations,

2u u upρ η⋅∇ = −∇ + ∇ , (4.6)

0u∇⋅ = . (4.7)

4.3.1 Bulk Boundary Conditions

At all the walls, electric insulation and no-slip are applied as boundary

conditions, i.e.,

ˆ 0n φ⋅∇ = , (4.8)

and

0u = . (4.9)

At the electrodes where electroosmotic slip occurs, following are the boundary

conditions: Assuming that the potential drop across the Stern layer is negligible,

the bulk potential outside the double layer can be expressed as

elφ φ ζ= − , (4.10)

91

where elφ is the potential of the equipotential surface formed by the electrode. We

will show a strategy to obtain ζ in the next section. The slip velocity on the

electrode is given by Helmholtz Smoluchowski equation

ˆ ˆu s sεζ φη

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= ∇ ⋅ , (4.11)

where s is a vector tangential to the electrode surface.

4.4 Charge Conservation in the Double Layer

Assuming that the double layer is infinitely thin ( 1D aλ ) and that the double

layer charging does not cause any gradients in the bulk salt concentration, we can

derive a conservation-law for the double layer surface charge density q. Consider a

small patch of a thin double layer (Fig. 4.5). The charge is brought into it by a

normal ohmic flux. This charge can accumulate or leak tangentially from the

edges.

( )ˆ. dsEn σ−

ds dl ( )ˆ . dlEl s sn σq

Ohmic flux in

Surface conduction flux out

Fig. 4.5: Conservation of surface charge. A tangential surface conduction flux and a normal ohmic flux constitute the conservation law for the double layer surface charge density q.

92

A conservation law can then be simply written as

Rate of accumulation = Flux in – Flux out, (4.12)

which in mathematical form can be expressed as

( ) ( )ˆ ˆE Es sls s l

q ds n ds n dlt

σ σ∂ = − ⋅ − ⋅∂∫ ∫ ∫ , (4.13)

where n is the outward normal vector and ˆln is a vector tangential to the surface

but perpendicular to the edge. The second term on right hand side (rhs) represents

a linear integral on the boundary of the surface. We can convert it into an area

integral by applying divergence theorem,

( ) ( )ˆ E Es s ss s s

q ds n ds dst

σ σ∂ = − ⋅ − ∇ ⋅∂∫ ∫ ∫ . (4.14)

Here, s∇ is the tangential gradient operator on the outer surface of the double

layer. If the electrode surface lies in the x-y plane, the tangential gradient operator

will simply be ˆ ˆs x x y y∇ = ∂ ∂ + ∂ ∂ .

Since (4.14) holds true for any arbitrarily small surface, the integrals can be

removed and we get

( ) ( )ˆ E Es s sq nt

σ σ∂ = − ⋅ −∇ ⋅∂

. (4.15)

Using the electrostatic relations E φ= −∇ and Es sφ= −∇ (where φ is bulk

electrostatic potential right outside the double layer), we get

( ) ( )ˆ s s sq nt

σ φ σ φ∂ = ⋅ ∇ +∇ ⋅ ∇∂

, (4.16)

93

4.4.1 Double Layer Edge Conditions

Since the surface PDE (4.16) contains spatial derivatives, we need some spatial

boundary conditions (called edge conditions here, to distinguish them from the

bulk boundary conditions) to solve it. We have previously argued that the surface

conduction flux is zero at the edge of the electrode and therefore the following can

be used as an edge condition (also see Fig. 4.6)

ˆ ( ) 0s sln σ φ⋅ ∇ = , (4.17)

Electrode Surfaceˆln

Edgeˆ ( ) ( )s s s

q nt

σ φ σ φ∂= ⋅ ∇ +∇ ⋅ ∇

∂ˆ ( ) 0l s sn σ φ⋅ ∇ =

ˆ ( ) 0l s sn σ φ⋅ ∇ =

ˆ ( ) 0l s sn σ φ⋅ ∇ =

ˆ ( ) 0l s sn σ φ⋅ ∇ =

n

s

ˆln

ˆln

ˆln

Fig. 4.6: Conditions on the edges of the electrode. The surface conduction flux is zero on the edges. Here n is the outward surface normal vector, s is an arbitrarily oriented tangential vector, and ˆln is a vector tangential to the surface but perpendicular (outward) to the edge.

4.4.2 Convergence Issues

We can solve (4.16) for q but several convergence problems may arise because

this PDE does not contain any spatial diffusion of q. The spatial derivatives of φ ,

which occur in this PDE, are supplied from the bulk and therefore they behave as

94

source terms. Essentially, there is no diffusive term available for q in this equation.

The absence of diffusive terms combined with the presence of source terms leads

to an infinite Péclet number and can create problems in convergence. Moreover,

magnitude of q is supposed to grow exponentially as predicted by (4.1); as a result

the computer will solve for a variable which is very large, leading to large

numerical errors. One way to counter these problems is to change the dependent

variable from q to ζ . This is explained below.

By differentiating (4.1) with respect to time, we can obtain an expression for

q dt∂ in terms of dtζ∂ ,

cosh 2D

q zet kT t

ζ ζελ

⎛ ⎞⎜ ⎟⎝ ⎠

∂ ∂= −∂ ∂

. (4.18)

Now, by using the relation elφ φ ζ= − , where elφ is the potential of the

electrode surface and by ignoring the Stern layer, we can obtain an expression for

the tangential divergence of φ in terms of ζ

( ) ( )s s s s s sσ φ σ ζ∇ ⋅ ∇ = −∇ ⋅ ∇ . (4.19)

Please note that elφ is a constant because the metal electrode forms an

equipotential surface. Combining the two preceding equations with (4.16), we get

( )

ˆ ( ) ( )cosh 2

s s sD nt ze kT

λ σ φ σ ζζε ζ

⎡ ⎤⎣ ⎦⋅ ∇ −∇ ⋅ ∇∂ = −∂

. (4.20)

Now, using the definitions c Daτ ε σλ= and sDu aσ σ= , the preceding

equation reduces to

95

( )

2ˆ ( )

cosh 2s s

c

an a Dut ze kT

φ ζζτζ

⎡ ⎤⎣ ⎦⋅∇ − ∇ ⋅ ∇∂ = −

∂, (4.21)

which is a well behaved equation in ζ . The first term in the numerator of the rhs

( ˆan φ⋅∇ ) is supplied from the bulk and behaves as a source. The second term, on

the other hand, behaves as a diffusion term in ζ .

Another advantage of changing the dependent variable from q to ζ is that it

produces a factor of ( )1/ cosh 2ze kTζ on the rhs; in other words, the rhs is

divided by a large number making the rhs small; this makes the problem easier to

converge. Magnitudes of ζ are generally much smaller than q and therefore we

prefer to solve for ζ instead of q.

When one wants to solve for a steady state, (4.21) is reduced to

ˆ ( ) 0s sn a Duφ ζ⋅∇ − ∇ ⋅ ∇ = . (4.22)

4.5 Dimensionless Equations

Following are the scales for making the equations dimensionless. Asterisk in

the superscript represents a dimensionless quantity.

*x xa= ,

*THφ φ φ= ,

*TH Dq q εφ λ= ,

*ct tτ= ,

*u uHSu= ,

(4.23)

96

*HSp p u aη= ,

where, TH kT zeφ = is the thermal voltage and HSu is an appropriate scale for the

Helmholtz Smoluchowski velocity; HSu will be chosen later based on geometry

and application.

The dimensionless equations for the bulk can then be written as

*2 * 0φ∇ = , (4.24)

* * * * * *2 *Reu u up⋅∇ = −∇ +∇ , (4.25)

* * 0u∇ ⋅ = , (4.26)

where Re HSu aρ η= is the Reynolds numbers. In most of microfluidic situations,

Re 1 and therefore the convective terms in (4.25) can be neglected.

The boundary conditions for the walls are

* *ˆ 0n φ⋅∇ = , (4.27)

and

* 0u = . (4.28)

At the electrodes where electroosmotic slip occurs

* * *elφ φ ζ= − , (4.29)

and

( )2

* * * * ˆ ˆu TH

HS

a s su

εφ η ζ φ= ∇ ⋅ . (4.30)

97

The dimensionless form for the double layer charge conservation (see (4.21)) is

expressed as

( )

* * * * **

* *

ˆ ( )

cosh 2s sn Du

tφ ζζ

ζ

⎡ ⎤⎣ ⎦⋅∇ −∇ ⋅ ∇∂ = −

∂, (4.31)

which, for a steady state analysis, simplifies to

* * * * *ˆ ( ) 0s sn Duφ ζ⋅∇ −∇ ⋅ ∇ = . (4.32)

The edge conditions in form of dimensionless variables are given as

* *ˆ ( ) 0sln Du ζ⋅ ∇ = . (4.33)

Dukhin number can also be expressed in terms of dimensionless variables,

( ) ( )* 2 *4 1 sinh 4Du mλ ζ= + , (4.34)

where *λ is the dimensionless double layer thickness, *D aλ λ= . Similarly, the

nonlinear capacitive relation reduces to

( )* *2sinh 2q ζ= − . (4.35)

We will now use the preceding nonlinear numerical model for solving two

canonical cases: (1) ICEO flow on a flat metal electrode in presence of a steady

DC electric field, and (2) ICEO flow on a metal cylinder in presence of a steady

DC electric field.

98

4.6 ICEO on a Flat Metal Electrode

Consider a horizontal electrode of width a placed at the bottom wall of a

chamber of side L. The chamber side is much larger than the width of the

electrode, L*=L/a=15 (see Fig. 4.7). The chamber is filled with a symmetric

( z z z+ −= = and D D D+ −= = ) electrolyte solution such as KCl+H20. We will refer

to the horizontal electrode as a gate electrode. We will carry out a steady state

ICEO flow analysis in presence of a DC electric field.

0=*u

** *

2 *

2( )u xxφζ

α∂

=∂

**

2Lφ α=

*2 * 0φ∇ =

* *ˆ 0,n φ⋅∇ =

* *

* * * 0Duy x xφ ζ⎛ ⎞∂ ∂ ∂

− =⎜ ⎟∂ ∂ ∂⎝ ⎠

**

2Lφ α= −

* *φ ζ= −

*

* 0Duxζ∂

=∂

*

* 0Duxζ∂

=∂

* *ˆ 0n φ⋅∇ = * *ˆ 0n φ⋅∇ =

*2 * * * 0u p∇ −∇ =

* * 0u∇ ⋅ =

L*

1

0=*u 0=*u

0=*u 0=*u

p*=0

yx

Fig. 4.7: Steady state simulation model for ICEO flow on a flat electrode with L/a=15. In dimensionless form, the width of the electrode is 1. Geometry is not shown to scale.

99

The two vertical sides of the chamber serve as a pair of driving electrodes. A

DC electric field is produced in the chamber by applying a potential difference

between the two driving electrodes. Assume that the driving electrodes do not

saturate (i.e. no double layer form on them) and can maintain a constant DC

electric field in the chamber. Dimensionless potentials of * * 2Lφ α= ± are applied

on the left and right driving electrodes respectively (see Fig. 4.8). If none of

surface conduction, Stern layer and faradaic injections is present, in the steady

state a DC electric field of *0E α= is produced as a result of the applied potential

difference. This electric field produces a zeta potential of * *xζ α= on the gate

surface (see Fig. 4.8).

x*

Bulk

pot

entia

l, φ*

α/2

-α/2

αL*/2

-αL*/2

0

-L*/2 L*/2-1/2 1/2

Fig. 4.8: Steady state bulk potential in the ICEO chamber when surface conduction and faradaic injections are absent. The net potential drop across the width of the electrode is equal to α . In absence of Stern layer, this yields

* *xζ α= . Note that the maximum zeta potential is equal to / 2α .

100

In dimensional form the linear estimates can be written as 0 THE aαφ≡ and

2THζ αφ≡ . Therefore, a natural scale for normalizing the slip velocity would be

2 2

02

THHS

Eua

εζ εα φη η

= = . (4.36)

Using this scale, we can define the normalized slip velocity on the gate as

*

* *2 *

1 2ˆ ˆuHS

x xu x xφ φε ζ ζ

η α∂ ∂= =∂ ∂

. (4.37)

The gate electrode is kept floating, i.e., not connected to voltage source. Since

the gate electrode is placed at equal distances from the two driving electrodes, its

floating potential vanishes i.e. * 0elφ = . One consequence of zero floating potential

is that the bulk boundary condition becomes

* *φ ζ= − . (4.38)

Assume that the depth of the chamber into the paper is infinite. Therefore, only

a 2D model needs to be simulated. The charge conservation PDE for this system is

simply (see (4.32))

* *

* * *( ) 0Duy x xφ ζ

−∂ ∂∂ =∂ ∂ ∂

, (4.39)

and the condition on the two edges can be expressed as

*

* 0Duxζ∂ =∂

(4.40)

101

4.6.1 Parameters of Study

The parameters of study are α and the dimensionless double layer

thickness *λ . As described in Fig. 4.8, / 2α is the maximum ‘linear’ value of *ζ .

Naturally, as α increases, *ζ increases. As a result, both Dukhin number and the

surface conduction flux increase. Similarly, from (4.4), Dukhin number (and

surface conduction flux) is proportional to *λ . Surface conduction effects grow

with *λ . We will vary these two parameters to quantity the effect of surface

conduction on ICEO flow.

4.6.2 Results and Discussion

The problem was solved with COMSOL Multiphysics (Comsol Inc.,

Stockholm, Se), a commercial finite element software package. The details of the

weak formulation, the grid size and the solution procedure are given in an

appendix to this chapter.

Let’s first consider the case of very thin double layer and small induced zeta

potential, namely * 410λ −= and 0.1α = . In this case, surface conduction is

negligible. In the steady state, the double layer gets completely charged and the

electric field lines become completely tangential to the electrode, showing no

existence of a healing region (Fig. 4.9a). In other words, the field heals

instantaneously. The normal component of the electric field is uniformly zero on

the electrode. This picture is similar to what the linear theory predicts.

102

However, when α and *λ are large ( 25α = , * 0.01λ = ), the steady state

configuration of electric field lines is quite different (Fig. 4.9b). In this case, the

electric field lines have a large normal component on the entire electrode surface,

especially close to the edges. This leads to loss of tangential electric field and

consequently to loss of slip velocity (Fig. 4.10). Fig. 4.10 shows the normalized

slip velocity, *su , on the gate electrode. *

su is linear in the former case but has a

surprising profile in the latter case. In the latter case, *su is zero on the edges and

peaks close to the center at a point on either side of the center. This is because the

tangential field is zero on the edges and maximum in the center of the gate.

E

(b) α=25, λ*=0.01

x*

y*

E

(a) α=0.1, λ*=10-4

x*

y*

Fig. 4.9: Steady state electric field lines in the vicinity of a flat electrode subjected to a DC electric field directed from left to write. (a) For very thin double layer and a very low induced zeta potential (i.e. * 410λ −= and 0.1α = ), the electric field is completely tangential to the electrode. (b) For a thick double layer and a large induced zeta potential ( * 0.01λ = , 25α = ), surface conduction become

103

important. Healing regions are created and the normal component of the electric field becomes large so that the nonuniform surface currents can be maintained.

Fig. 4.10: Steady-state slip velocity on the gate electrode. Two cases are shown, (1) * 410λ −= , 0.1α = and (2) * 0.01λ = , 25α = . In the first case, the normalized slip velocity is much higher than the second case. The loss in normalized velocity in the second case is due to surface conduction which becomes prominent at high zeta potentials (i.e. large α ) and in thick double layers (i.e. large *λ ).

4.6.3 Results of Parametric Study

Fig. 4.11 shows the maximum normalized slip velocity, *maxu , on the gate

electrode as a function of α and *λ . For *λ = 0 , i.e. an infinitely thin double

α=0.1, λ*=10-4

α=25, λ*=10-2

104

layer, *maxu is independent of α , indicating that surface conduction becomes

unimportant for infinitely thin double layers. For non-zero *λ , however, *maxu is a

strong function of α . For * 0λ > , *maxu decreases with increasing α , implying that

the importance of surface conduction increases as induced zeta potentials increase.

Similarly, as *λ increases at a fixed α , *maxu decreases. This indicates that surface

conduction effects grow as *λ increases. The double layer thickness is inversely

proportional to ( )1/ 2σ . Therefore, thin double layers take place in low conductivity

solutions. When the bulk conductivity is low, the ratio of excess surface

conductivity to bulk conductivity is high, yielding a high Dukhin number.

10-1 100 101 102

0.2

0.4

0.6

0.8

1

α

u* max

λ*=0.1λ*=0.0316λ*=0.01λ*=3.16E-3λ*=1E-3λ*=3.16E-4λ*=1E-4

λ*=0λ*

Fig. 4.11: Maximum normalized velocity *maxu as a function of the parameters

α and *λ . *maxu decreased with both α and *λ , because surface conduction

increases as either of α or *λ increases

105

4.6.4 Normalized Quantities

In the last section, we examined the normalized slip velocity (see (4.37)) as *λ

and α were varied. Now let’s examine some other normalized quantities defined

in table 4.1.

Table 4.1: The normalized quantities for ICEO on a flat electrode

Quantity Normalized Expression Zeta potential *2ζ α

Tangential electric field *

*1

xφ

α∂−∂

Normal electric field *

*1

yφ

α∂−∂

Slip velocity * *

2 *2

xζ φα

∂∂

Surface conduction flux *

*1Du

xφ

α∂−∂

106

The following results were obtained for a constant *λ ( 0.01= ) while α was

varied. Fig. 4.12 shows the variation of Dukhin number on the gate electrode for

several values of α . Dukhin number increases with increasing α , indicating the

increasing importance of surface conduction.

Fig. 4.12: Variation of Dukhin Number on the gate electrode for 25, 10, 7.5, 5, 2.5, 1 α = ⋅ ⋅ ⋅ and a constant value of *λ ( 0.01= ). The Dukhin

number increases with α indicating the increasing importance of surface conduction. The Dukhin number is zero at the center of the gate because the zeta potential is zero at the center

α

α=25

α=10

α=1

107

Fig. 4.13 shows the normalized zeta potential, *2ζ α , on the gate electrode for

several values of α . For 1α < , *2ζ α has a linear variation, with a maximum

value of 1, matching the predictions of the Debye-Huckel theory. As α increases,

*2ζ α decreases. This indicates that surface conduction creates a loss in zeta

potential.

Fig. 4.13: Variation of normalized zeta potential, *2ζ α , on the gate electrode for 25, 10, 7.5, 5, 2.5, 1 α = ⋅ ⋅ ⋅ and a constant value of *λ ( 0.01= ). The normalized

zeta potential decreases as α increases. The loss in zeta potential is due to increased surface conduction at high values of α .

α

α=25α=10

α=1

108

Fig. 4.14 shows the normalized tangential field, *xE α , on the gate electrode.

For 1α < , *xE α has a uniform value of 1 over the entire gate electrode, indicating

that, in the absence of surface current, the field outside the double layer is

completely tangential and equal to the applied field. At higher values of α , *xE α

goes to zero at the edges, indicating that the field lines are drawn into the double

layer at the edges. As a result, *xE α is reduced significantly in the healing regions

close to the edges. In the center, however, *xE α attains a large value.

Fig. 4.14: Variation of normalized tangential electric field, *xE α , on the gate

electrode for 25, 10, 7.5, 5, 2.5, 1 α = ⋅ ⋅ ⋅ and a constant value of *λ ( 0.01= ). The normalized tangential field decreases as α increases. The loss in the tangential field is due to increased surface conduction at high values of α .

α α=25

α=10

α=1

109

Fig. 4.15 shows the normalized normal electric field, *yE α , on the gate

electrode. For 1α < , *yE α is uniformly zero on the entire gate electrode

indicating absence of any substantial bending of field lines into the double layer.

As α grows, *yE α grows too, indicating that a lot of lines are drawn into (or

expelled out) of the double layer.

Fig. 4.15: Variation of normalized normal electric field, *yE α , on the gate

electrode 25, 10, 7.5, 5, 2.5, 1 α = ⋅ ⋅ ⋅ and a constant value of *λ ( 0.01= ). The normalized normal field increase as α increases. For large values of α , surface currents become strong and force the electric field to acquire a large normal component. Also notice that the normal field flips sign of either side of the center. The points at which the flip takes place mark the locations where the ions start escaping out of the double layer.

α α=25

α=10

α=1

110

Fig. 4.16 shows the normalized slip velocity, *u , on the gate. For 1α < , the

slip velocity varies linearly on the gate, with the maximum value of 1, matching

with the Debye Huckel theory. As α increases, *u goes to zero at the edges and is

decreased at other points too (especially close to the edges). As α increases, the

location of the maximum velocity shifts towards the center. Interestingly, *u is

large in a small region on either side of the center of the gate. This is because the

tangential field is large in those regions.

Fig. 4.16: Variation of normalized slip velocity, *u , on the gate electrode for 25, 10, 7.5, 5, 2.5, 1 α = ⋅ ⋅ ⋅ and a constant value of *λ ( 0.01= ). The normalized

slip velocity decreases as α increases. The loss in the velocity is due to increased surface conduction at high values of α .

α α=25

α=10

α=1

111

The normalized surface conduction flux, *xDuE α , is shown in Fig. 4.17.

*xDuE α increases with α . *

xDuE α is zero not only on the edges but also at the

center of the gate. It’s zero at the center because *ζ is zero there. Note that Du is

also zero at the center. On either side of the center, *xDuE α first increases from

zero at the edge, attains a maximum and then falls back to zero at the center. This

results in the surprising pattern of electric field lines shown in Fig. 4.9b. The

location of the maxima corresponds to the point where the normal electric field

flips sign. This is the point where the ions start leaking out of the double layer.

Fig. 4.17: Variation of normalized surface conduction flux, *xDuE α , on the gate

electrode for 25, 10, 7.5, 5, 2.5, 1 α = ⋅ ⋅ ⋅ and a constant value of *λ ( 0.01= ). The normalized surface conduction flux increases as α increases. The flux is zero not only at the two edges but also at the center of the gate. The flux attains maxima at a point between the center and the edge. The location of the maxima also marks

α=1

α

α=25

α=10

α=1

Nor

mal

ized

Sur

face

Con

duct

ion

Flux

112

the location at which the ions start escaping out of the double layer. The non-uniformity of the flux is the main reason behind the surprising electric field lines shown in Fig. 4.9b.

4.6.5 Streamlines

Fig. 4.18 shows fluid flow streamlines for 0.1 and 25α = at a constant value of

*λ ( 0.01= ). For 0.1α = , the streamline density is highest at the two edges because

the slip velocity is highest at the edges. For 25α = , the stream line density shifts

towards the center because the slip is highest at a point close to the center on either

side of the center (see Fig. 4.16); at the edges the velocity is zero due to surface

conduction.

(a) 0.1α =

113

\ (b) 25α =

Fig. 4.18: Flow stream lines for * 0.01λ = and (a) 0.1α = and (b) 25α = . For 0.1α = , surface conduction is negligible and the streamlines show a linear flow

profile. At 25α = , the streamline density shifts towards the center of the gate indicating that the region of slip shrinks as surface conduction grows.

4.7 ICEO on a Metal Cylinder

ICEO flow over a cylinder is of interest in many practical applications.

Consider a metal cylinder of radius a with its center at the origin (Fig. 4.19). When

a horizontal DC electric field of magnitude E0 is applied, double layer forms on the

cylinder surface and ICEO flow is produced (Figs. 4.20a-c). In an ideal case when

none of surface conduction and faradaic injections is present, the double layer gets

completely charged. In the steady state and ignoring the Stern layer, the zeta

114

potential, the tangential electric field and the slip velocity on the cylinder surface

are given by [41]

0( ) 2 cosE aζ θ θ= , (4.41)

0ˆ 2 sinEs E θ⋅ = − , (4.42)

20 ˆ2 sin 2su E a sε θη

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= , (4.43)

where s is a unit vector tangential to the cylinder surface.

x

y ns

θa

E0

x

y ns

θa

E0

Fig. 4.19: A metal cylinder subjected to an external electric field.

115

E0

--

-

--

-

--

-

--

-

--------- --

+++++++++

++

-

-

-

++

+

++

+

++

+

++

+

++

+

E0

++++++++++++++

- - - -----------

- - - -----------

--------- --

+++++++++

++

-

-

-

++

+

(a) (b)

(c)

Fig. 4.20: ICEO on a metal cylinder. (a) When a horizontal DC field of magnitude E0 is applied at time t=0, cylinder surface forms an equipotential surface leading to induction of charge on its surface (also called polarization). However, since the cylinder is kept in an ionic solution, the ions travel towards the surface of the cylinder. (b) In the steady state, the ionic cloud shields the cylinder’s surface charge completely and insulates it from the electric field. As a result, all the electric field lines become tangential (shown by field lines). (c) The double layer is moved by the tangential electric field and thus a left-right, top-bottom symmetric ICEO flow is produced around the cylinder (shown by streamlines).

116

4.7.1 Surface Conduction Model for a 2D Cylinder in Cartesian Coordinates

The double layer model described in section 4.5 can be implemented on a

cylinder surface as well. We need to transform the model in Cartesian coordinates

so that it can be easily implemented in commercial software packages such as

COMSOL.

Let’s derive the dimensionless steady state double layer PDE for a 2D cylinder

in Cartesian coordinates. The double layer PDE for a cylindrical surface can be

simply written as

* *

* * * 0Dun s sφ ζ⎛ ⎞

⎜ ⎟⎜ ⎟⎝ ⎠

∂ ∂∂− =∂ ∂ ∂

, (4.44)

where *n∂ ∂ and *s∂ ∂ represent the normal and tangential derivatives on the

surface respectively. Now consider a 2D cylinder of dimensionless radius 1 with

its center at the origin. The normal and tangential vectors for the surface of the

cylinder are defined as

ˆˆ cos sinˆˆ sin cosxnys

θ θθ θ

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠

=−

. (4.45)

However, for this particular geometry, *cos xθ = and *sin yθ = and therefore,

**

* *ˆˆˆˆ

x y xnys y x

⎛ ⎞⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

=−

. (4.46)

Using the preceding definition of the normal vector, the normal derivative of a

quantity on a 2D cylindrical surface can be obtained as

117

* * *

* * * ** * *n x y

n x yφ φ φφ∂ ∂ ∂= ⋅∇ = +∂ ∂ ∂

. (4.47)

Similarly, the tangential derivative can be obtained as

* * *

* * * ** * *s y x

s x yζ

ζ ζ ζ∂ ∂ ∂= ⋅∇ = − +∂ ∂ ∂

. (4.48)

More complicated expressions can also be calculated in similar fashion; for

example the tangential divergence is obtained as,

* * *

* * * ** * * * * *Du y x Du y x

s s x y x yζ ζ ζ⎡ ⎤⎛ ⎞ ⎛ ⎞⎛ ⎞

⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

∂ ∂ ∂∂ ∂ ∂= − + − +∂ ∂ ∂ ∂ ∂ ∂

. (4.49)

Combining the preceding three equations with (4.44) and doing some algebra

yield the double layer PDE in Cartesian coordinates, which is simply written as,

( )* * *ˆ 0Jsn Duφ⋅∇ −∇ ⋅ = , (4.50)

where

* *ˆ ˆ ˆn x x y y= + , (4.51)

and

* * * *

*2 * * *2 * ** * * *ˆ ˆJs y x y x x x y y

x y y xζ ζ ζ ζ⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∂ ∂ ∂ ∂= − + −∂ ∂ ∂ ∂

. (4.52)

4.7.2 Geometry and Boundary Conditions

We have solved the problem only on a half cylinder because of symmetry (See

Fig. 4.21). Consider a half cylinder of radius a placed at the bottom wall of a

118

chamber of side L. The chamber side is much larger than the radius of the cylinder,

L*=L/a=20.

0=*u

( )* * * *2

2 ˆ ˆ( )u θ ζ θ φ θα

= ⋅∇

**

4Lφ α=

*2 * 0φ∇ =

* *ˆ 0,n φ⋅∇ =

**

4Lφ α= −

* *φ ζ= −

0J s = 0J s =

* *ˆ 0n φ⋅∇ = * *ˆ 0n φ⋅∇ =

*2 * * * 0u p∇ −∇ =

* * 0u∇ ⋅ =

a

0=*u 0=*u

p*=0

yx

( )* * *ˆ 0Jsn Duφ⋅∇ −∇ ⋅ =

1

L*

θ

Flow Symmetry

Flow Symmetry

Fig. 4.21: Surface conduction model for a 2D metal cylinder. Only half the cylinder is simulated due to top-bottom symmetry. The chamber is much larger than the cylinder, i.e. L*=L/a>>1. In nondimensional units, the radius of the cylinder is 1.

When dimensionless potentials of * * 4Lφ α= ± are applied on the left and right

walls respectively, a DC electric field *0 2E α= is produced away from the

cylinder. In the linear regime, this electric field causes a zeta potential of

* cosζ α θ= on the cylinder surface (refer to (4.41)). In dimensional form, these

119

estimates can be written as 0 2THE aαφ≡ and THζ αφ≡ . Using these estimates, a

natural scale for normalizing the slip velocity can be obtained as

2 2

02

THHS

Eua

εζ εα φη η

= = , (4.53)

using which we can define the normalized slip velocity on the gate as

( ) ( )* * *2

1 2ˆ ˆ ˆ ˆ*u (x)HS

s s s suε ζ φ ζ φη α

= ∇ ⋅ = ∇ ⋅ . (4.54)

Here we would like to introduce a new variable *su for the magnitude of the

slip velocity, given as

( )* * * *2

2 ˆsu sζ φα

= ∇ ⋅ . (4.55)

Using the definition of s from (4.45) and combining (4.54) and (4.55), the

velocity on the cylinder surface can be written as

( )* * * *ˆ ˆ ˆ*u (x) s su s u y x x y= = − + . (4.56)

As mentioned above equal and opposite potentials are applied to the two

driving electrodes ( * * 4Lφ α= ± ); the cylinder surface is kept floating; the floating

potential on the cylinder surface vanishes i.e. * 0elφ = because it is placed at equal

distances from the two driving electrodes. One consequence of zero floating

potential is that the bulk boundary condition becomes

* *φ ζ= − . (4.57)

The condition on the two poles (i.e. 0 and θ π= ) of the half cylinder is

( )* *ˆ 0s Du ζ⋅ ∇ = , (4.58)

120

which is equivalent to

0sJ = . (4.59)

4.7.3 Electric Field Lines around Cylinder

Fig. 4.22 shows the steady state electric field lines around the cylinder

respectively for 1, 20 and 100α = at a constant *λ of 0.001. For low induced zeta

potential (i.e. 1α = ), the electric field lines are completely tangential to the

cylinder surface which is equivalent to the linear regime; For 20α = and 100α = ,

a lot of electric field lines terminate perpendicular to the cylinder showing large

healing regions due to surface conduction. The extent of healing region grows as

α increases. For 100α = , the healing region extends up to a short angle from the

equator (i.e the top or / 2θ π= ) of the cylinder. This shows that in presence of

strong zeta potentials, the healing regions can be as big as the dimension of the

cylinder itself.

121

(a) Electric field lines around cylinder for 1α =

(b) Electric field lines around cylinder for 20α =

122

(c) Electric field lines around cylinder for 100α =

Fig. 4.22: Electric field lines around the cylinder for * 0.001λ = and 1, 20 and 100 respectivelyα = . For 1α = , the lines are perfectly tangential to the

cylinder, indicating that surface currents are negligible. For 20α = , the field lines acquire a large normal component and give rise to a healing zone between the poles and the equator of the cylinder. For 100α = , the healing zone is very large and extends upto a short angle from the equator of the cylinder. At the equator, the lines are expelled out because of nonuniform surface currents.

4.7.4 Streamlines around Cylinder

Fig. 4.23 shows flow streamlines respectively for 0.01α = and 20α = at a

constant *λ of 0.001. For 1α = , fluid slip at the entire cylinder surface is apparent.

For 20α = , the stream line density shifts towards the equator (i.e. / 2θ π= ). This

shows that the region, in which the slip is substantial, shrinks with increasing

surface conduction.

123

Fig. 4.23(a): Flow streamlines around the cylinder for 1α =

Fig. 4.23(b): Flow streamlines around the cylinder for 20α =

Figure 4.23: Flow streamlines around the cylinder for (a) 1α = and (b) 20α = . For 1α = , the fluid slip all around the cylinder. For 20α = , the region of

slips shrinks significantly and the streamline density shifts towards the equator of the cylinder..

124

4.7.5 Normalized Quantities

Similar to the flat electrode case, let’s examine various normalized quantities

on the surface of the cylinder as a function of α . Various normalized quantities

are defined in table 4.2.

Table 4.2: Normalized quantities on a metal cylinder

Quantity Normalized Expression Zeta potential *ζ α

Tangential electric field

* *1 s φα

− ⋅∇

Normal electric field * *1 n φα

− ⋅∇

Slip velocity *

* *2

2 sζ φα

⋅∇

Surface conduction flux

* *1 ˆDu s φα

− ⋅∇

Fig. 4.24a shows Dukhin number for several values of α and a constant *λ of

0.001. Dukhin number increases as α increases. It’s always zero at 2θ π= due to

asymmetric nature of induced zeta potential.

Figs. 4.24b-f show that, at very low values of α (up to 1α = ), all the

quantities correspond to the linear estimates (refer to (4.41), (4.42), (4.43)). With

the increase of α , the normalized tangential electric field decreases close to the

poles but increases at the equator; the normalized normal electric field increases

everywhere but flips sign a point between the pole and the equator on either side of

the equator; the normalized zeta potential decreases; the normalized surface

125

conduction flux increases and finally the normalized slip velocity decreases close

to the poles but increases close to the equator. The surface conduction flux is zero

the two poles and also at the equator. It attains a maximum between the equator

and the pole, on either side of the equator.

These plots indicate that at high induced zeta potentials, surface conduction

becomes important and deteriorates electroosmotic slip velocity. Similar to the flat

electrode case, the surface conductivity is zero at the equator of the cylinder and

therefore, the ionic current leaks out of the double layer on either side of the

equator causing the surprising electric field lines shown in Fig. 4.22.

Fig. 4.24 (a): Dukhin number on the cylinder for * 0.001λ = and 1, 10, 15, 20 and 30α = . The Dukhin number increases with α indicating the

increasing importance of surface conduction. The Dukhin number is zero at the equator (i.e. / 2θ π= ) of the cylinder because the zeta potential is zero at the equator.

α

126

Fig. 4.24 (b): Normalized zeta potential on the cylinder for * 0.001λ = and 1, 10, 15, 20 and 30α = . The normalized zeta potential decreases with α . The

loss in zeta potential is caused by the increasing surface conduction effect.

Fig. 4.24 (c): Normalized tangential electric field on the cylinder for * 0.001λ = and 1, 10, 15, 20 and 30α = . The normalized tangential field

decreased close to the poles ( 0 and θ π= ) but increases around the equator ( / 2θ π= ) as the values of α increases. The loss in the field is caused by the increasing surface conduction effect.

α

α

127

Fig. 4.24 (d): Normalized normal electric field on the cylinder for * 0.001λ = and 1, 10, 15, 20 and 30α = . The normalized normal electric field increases as α increases. Increased surface conduction causes the field to acquire a large normal component. The field flips sign between a pole and the equator due to nonuniform surface currents around the cylinder surface.

Fig. 4.24 (e): Normalized slip velocity on the cylinder for * 0.001λ = and 1, 10, 15, 20 and 30α = . The slip velocity decreases as α increases. The region of

α

α

128

significant velocity shrinks to a minimum as α increases. The location of maximum slip velocity shifts towards the equator indicating the flow region shrinks towards the equator. The loss in velocity is caused by the increased surface conduction.

Fig. 4.24 (f): Normalized surface conduction flux on the cylinder for * 0.001λ = and 1, 10, 15, 20 and 30α = . The flux increases with α indicating the

increasing importance of surface conduction. The flux is zero not only at the poles but also at the equator of the cylinder. The flux acquires a maximum between a poles and the equator. The nonuniformity of the flux causes the surprising electric field gradients shown in Fig. 4.22.

4.7.6 Parametric Study

The parameters of study are *λ and α . Fig. 4.25 shows the normalized slip

velocity *su at / 4θ π= as a function of the parameters *λ and α . For * 0λ = , *

su

is independent of α indicating that surface conduction is unimportant for

infinitesimally thin double layers. As *λ increases, *su decreases. Similarly, as α

α

129

increases, *su decreases. Increasing surface conduction causes the loss of slip

velocity. At very high induced zeta potentials i.e. 50α > , *su decays almost

quadratically with α i.e * 2su α−∝ (focus on the straight line behavior on the log-

log plot for 50α > ). Note that the normalizing scale for slip velocity grows

quadratically with α ( 2 2 2THHSu aεα φ η= ). Multiplying the dimensionless velocity

( * 2su α−∝ ) with its scale ( 2 2 2THHSu aεα φ η= ) will yield the slip velocity which is

independent of α . In other words, for 50α > , the velocity stops growing with α

and becomes saturated. Increased surface conduction arrests any further increase in

the slip velocity. This behavior was also observed in the experiments of chapter 3.

Fig. 4.25: Normalized slip velocity *su at / 4θ π= as a function of the

parameters *λ and α . The values of *λ increase in the direction of the arrow, in the following order: 0, 0.0001, 0.001, 0.01, and 0.1. *

su decreases with both *λ and α because of increasing surface conduction. For 50α > , *

su scales as

λ*

130

* 2su α−∝ , shown as a straight line on the log-log plot. This indicates that at high

values of α , the real slip velocity saturates and does not grow with increasing voltages. The further growth of the velocity is stopped by surface conduction.

4.8 Concentration Polarization

In our preceding analysis, we have ignored the concentration gradients in the

bulk. We assumed that the salt concentration is uniform in the entire bulk.

However, when zeta potential is high, surface transport of ions can cause gradients

in the bulk salt concentration. The length scale of these gradients can be as big as

the healing length of the bulk electric field. This phenomenon is generally known

as ‘concentration polarization’ [61, 62]. Fig. 4.26 shows a fundamental picture of

how surface conduction can lead to concentration polarization. At the two edges of

the electrode, the bulk electric field not only drives counter-ions into the diffuse

layer but also drives away the co-ions, leading to a ‘salt sink’. In the middle,

however, the counter-ions are reunited with co-ions leading to ‘salt source’. As a

result, a high concentration zone is set up in the middle of the electrode whereas a

salt depletion zone at the two edges. The difference in the concentration drives a

‘chemi-osmotic’ flow. This flow is in a direction opposite to the electroosmotic

flow and can slow the flow down.

131

-

+ + + + +Dλ

+

- - - - +- -

+

++

E∞

---

+ + - -++

--

- - + +- - + +

Salt Sink Salt SinkSalt Source

-

+ + + + +Dλ

+

- - - - +- -

+

++

E∞

---

+ + - -++

--

- - + +- - + +

Salt Sink Salt SinkSalt Source

Fig. 4.26: Surface conduction leads to ‘concentration polarization’. The electric field drives counter-ions into the double layer, but at the same time the co-ions are repelled away, creating a salt sink at the two edges of the electrode. In the middle of the electrode, however, the counter-ions are reunited with the co-ions leading a salt source. This leads to gradients in salt concentration in the bulk. For large ζ , the gradients can extend into the bulk, giving rise to a ‘chemi-osmotic’ slip velocity which opposes electroosmotic slip velocity.

To account for concentration polarization, we need to solve the convection-

diffusion equations for the salt concentration.

2c u c D ct∂ + ⋅∇ = ∇∂

. (4.60)

The electrostatic potential follows the following current conservation equation:

( ) 0c φ∇⋅ ∇ = . (4.61)

The modified Stokes equations for fluid flow and the effective boundary

conditions for the excess double layer charge and excess double layer salt

concentrations are given in chapter 7. The net slip velocity is the sum of

electroosmosis and ‘chemi-osmosis’

24 1ln coshslip

kT ze cux ze kT c x

εζ φ ε ζη η

∂ ∂⎛ ⎞ ⎛ ⎞= − ⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠. (4.62)

132

For large ζ , chemi-osmosis can be of the same order as electroosmosis.

However, we have not accounted for chemi-osmosis in our model. Directions to

incorporate chemi-osmosis are given in chapter 7.

An approximate dimensionless solution for the salt concentration was obtained

by neglecting the convection of the salt and by neglecting the effect of salt

gradients on the bulk electric field. Fig. 4.27 shows the salt concentration in the

bulk for 20α = . A high concentration zone is established above center of the gate

whereas salt depletion zones are established near the edges. The initial

concentration in the bulk was 1.

Fig. 4.27: Concentration polarization due to surface transport. The color shows

the concentration whereas the streamlines show the electric field. A high concentration zone is established above center of the gate whereas salt depletion zones are established near the edges. The initial concentration in the bulk was 1 and 20α = . The concentration solution was obtained by neglecting convection and the effect of concentration gradient on the electric field.

133

4.9 Conclusions

A qualitative and quantitative picture of surface conduction effects in ICEO

has been developed via numerical simulations. It has been found that the

nonuniform induced zeta potentials, which are inherent in ICEO, lead to

nonuniform surface currents and cause surprising gradients in the bulk electric

field. Healing regions, in which the electric field has large normal component,

were found at the corners as well as in the middle of the inducing surface. We

predict that surface conduction creates large normal electric fields and thus reduces

the tangential field and the slip velocity.

A numerical model for simulating surface conduction in steady and time

dependent cases has been presented. In this model, the electric double layer is

handled by solving charge conservation PDE on the electrode surface. A variable

substitution has been presented for improving the convergence of the model. The

new model has excellent convergence properties and can be used for solving

electrokinetic problems for very high induced zeta potentials (100-1000 times

higher than the thermal voltage). The model also allows for nonlinear surface

capacitance.

4.10 Appendix to Chapter 4

4.10.1 Comments on COMSOL Multiphysics Simulations

A commercial finite element package COMSOL Multiphysics (Comsol Inc.,

Stockholm, Sweden) was used for solving the mathematical model described

134

above. Lagrange quadratic elements were used for all the dependent variables.

Since the problem deals with DC electric fields, a stationery solver was used with

zero initial conditions for all the variables. A Direct (UMFPACK) solver was

chosen for solving the system of linear algebraic equations. First the electrostatics

and the double layer PDE were solved simultaneously. Then their solution was

used as a linearization point for the bulk fluid flow problem.

The maximum element size in the bulk was chosen to be 0.1. The mesh

element size on the ICEO electrode boundary was chosen to be 10-4. This ensures

that a mesh independent solution is achieved. Mesh independence is demonstrated

in Fig. 4.28. The normalized kinetic energy has been plotted as function of *λ and

the size of the mesh element on the flat gate electrode, h. The kinetic energy, KE,

is defined as the integral of the square of the slip velocity on the gate electrode.

KE(h) is divided by KE(hmin) to obtain the normalized kinetic energy. The

minimum element sized was hmin=5x10-5. Plot shows that the solution is almost

mesh independent at h=10-4.

135

10-5

10-4

10-3

10-2

10-1

0.8

0.85

0.9

0.95

1

Mesh Element Size on the Gate Electrode, h

Nor

mal

ized

Kin

etic

Ene

rgy

KE

(h)/K

E(h

min

)

λ*=10-4

λ*=10-3

λ*=10-2

λ*=10-1

λ*=1

Fig. 4.28: The normalized kinetic energy as a function of *λ and the element size, h. Normalized KE becomes mesh almost mesh independent for h<10-4.

Weak non-ideal constraints were used for handling the electrostatic dirichlet

boundary condition ( * *φ ζ= − ). The weak non-ideal constraints remove

bidirectional coupling between the bulk and the surface electrostatic fields and

therefore give the most accurate solution for this situation. If one uses weak ideal

constraints instead, a slight quantitative difference arises in the results, but the

results do not change qualitatively and all the conclusions stay exactly the same.

Weak ideal constraints cause this slight error in the results because of bidirectional

coupling between the two fields. However, the weak non-ideal constraints make

the problem harder to solve because of unidirectional coupling. To mitigate this

136

problem, one should first solve the problem with weak ideal constraints and then

use this solution to as an initial condition to find the final solution with weak non-

ideal constraints. It’s easier to achieve convergence with weak ideal constraints

because of its bidirectional nature. For a description of various weak constraints in

COMSOL, please refer to the COMSOL manuals available from Comsol Inc.

(Stockholm, Sweden).

For a mathematical introduction to weak solutions, the reader can refer to [74,

75]. For the convenience of the reader, however, we give a brief introduction to the

weak forms and weak solutions in this appendix. We have also derived the weak

form of the double layer PDE so that it can be directly implemented in any finite

element code.

4.10.2 Weak Form for Finite Element Solution

Finite element methods generally seek a weak solution to the PDE in question.

As an example consider Poisson equation,

2 ( ) ( ),x x xu f∇ = ∈Ω (4.63)

Suppose H is an infinite-dimensional function space that is rich enough to

include in its closure all functions that may be of interest as candidates for the

solution u. For all v H∈ , we define the defect 2(v) vd f=∇ − . Then we say that

w is a weak solution to the problem equation (4.63) provided that the defect d(w) is

orthogonal to the entire function space, that is, if

v ( )d 0 vxd w HΩ

= ∀ ∈∫ . (4.64)

137

v is generally referred to as a test function. Equation (4.64) is also referred as

the weak form of (4.63). For more information, the reader is referred to [74, 75].

We can further simplify (4.64) as

2v d v d 0x xw fΩ Ω

∇ − =∫ ∫ . (4.65)

The first term on left hand side can be simplified to

( )2v d v d v dx x xw w wΩ Ω Ω

∇ = ∇⋅ ∇ − ∇ ⋅∇∫ ∫ ∫ . (4.66)

Applying divergence theorem on the first term on the right hand side yields

2 ˆv d v d v dx s xw n w wΩ ∂Ω Ω

∇ = ⋅∇ − ∇ ⋅∇∫ ∫ ∫ , (4.67)

where ∂Ω is the boundary of Ω . The boundary integrals can be handled by

invoking the boundary conditions. For example, if a Neumann boundary condition

is specified on the boundary, i.e., ˆ ( )n w b x⋅∇ = , then we can simply substitute it in

the boundary integral. Dirichlet conditions are handled by implementing

constraints [75].

4.10.3 Weak Form of Double Layer PDE

To implement the charge conservation equation in COMSOL, it’s necessary to

derive its weak form. When doing a steady state analysis, weak form is simply

given as (refer to (4.32))

* * * * *ˆv d v d 0s ss ss sn Duφ ζ⋅∇ + ∇ ⋅∇ =∫ ∫ (4.68)

where s represents the surface and v is a test function for *ζ . Note that the

edge condition ( )* *ˆ 0sln Du ζ⋅ ∇ = was invoked in deriving (4.68).

138

For a time dependent analysis, however, the weak form is given as (refer to

(4.31)),

( )

* * * * **

* *

ˆ ( )v d v dcosh 2

s ss ss s

n Dut

φ ζζζ

⋅∇ −∇ ⋅ ∇∂ = −∂∫ ∫ (4.69)

The second term on the right hand side of the preceding equation can be

expanded as follows

( ) ( )( )( )

* * * * * * * ** * *

* * *

* * *

vv tanh 2

( ) v 2v d d dcosh 2 cosh 2 cosh 2

s s ss s s s

s st ss s s

DuDu Du

ζ ζ ζ ζζ

ζζ ζ ζ

∇ ⋅∇ − ∇ ⋅∇∇ ⋅ ∇

= ∇ ⋅ ∇ −

⎡ ⎤⎢ ⎥⎛ ⎞ ⎣ ⎦⎜ ⎟⎜ ⎟

⎝ ⎠∫ ∫ ∫

(4.70)

The first term on the right hand side of the preceding equation can be

simplified by applying the divergence theorem

( ) ( )

* * * * ** *

v v ˆd dcosh 2 cosh 2

s ls s sls l

DuDu nζ ζζ ζ

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

∇ ⋅ ∇ = ⋅∇∫ ∫ (4.71)

However, we have * *ˆ 0ss ζ⋅∇ = on the edges and therefore

( )

* * **

v d 0cosh 2

ss ssDu ζ

ζ

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

∇ ⋅ ∇ =∫ (4.72)

Finally combining (4.69), (4.70) and (4.72) yields the desired weak form,

* * * * * * * *

* * *

* * *

vv tanh( 2)ˆ 2vv d d d

cosh( 2) cosh( 2)s s s

s s s s

s s s

Dun

t

ζ ζ ζ ζζ φ

ζ ζ

⎡ ⎤⎢ ⎥⎣ ⎦∇ ⋅∇ − ∇ ⋅∇

∂ ⋅∇= − −∂∫ ∫ ∫ .

(4.73)

139

5

140

Chapter 5. Simulations vs. Experiments

In this chapter we report the results of our nonlinear numerical simulations

which were intended to simulate the conditions of our experiments. We show that

the nonlinear effects indeed help us reduce the discrepancy between the

simulations and the experiments. In chapter three, we had reported that the linear

simulations predict two to three orders of magnitude higher velocities than the

experiments. In this chapter, we reduce that discrepancy by one order by

employing the nonlinear model incorporating the nonlinear surface capacitance

and surface conduction.

5.1 Numerical Model

Since the experiments were performed under AC electric fields, the time

dependent equations from section 4.5 were used. For the convenience of the

reader, the time dependent numerical model is being repeated below and is

portrayed in Fig. 5.1. The simulation geometry represents a two dimensional cross

section of the ICEO chamber used in our experiments. The geometry has been

nondimensionalized with the width of the gate electrode ( 42 10 ma −= × ).

In the bulk, we solve the following equations,

*2 * 0φ∇ = , (5.1)

* * *2 * 0up =−∇ +∇ , (5.2)

* * 0u∇ ⋅ = . (5.3)

141

ˆ 0n φ⋅∇ =* * * *00.5 sin(2 )f tφ φ π=

ˆ 0n φ⋅∇ = ˆ 0n φ⋅∇ =ˆ 0n φ⋅∇ =

Driving electrode 1 Driving electrode 2Gate

x

y

*2 * 0φ∇ =*2 * * * 0u p∇ −∇ =

* * 0u∇ ⋅ =

* * * *00.5 sin(2 )f tφ φ π= −* *φ ζ= −

( )* * * *

** * * **

1 ( )cosh 2

Dut y y yζ φ φ φ ζ

ζ

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

∂ ∂ ∂ ∂= − −∂ ∂ ∂ ∂

* 0p =

** *

2 *

*0

1

where 10

suxφζ

αα φ

∂=

∂=

0 on all boundaries except the gate*u =

*

* 0Duxζ∂

=∂

*

* 0Duxζ∂

=∂

Fig. 5.1: The dimensionless geometry and the time-dependent numerical model incorporating surface conduction and nonlinear surface capacitance. All dimensions shown to the scale

On the chamber walls, we apply insulation and no slip boundary conditions.

* *ˆ 0n φ⋅∇ = , (5.4)

* 0u = . (5.5)

On the two driving electrodes we specify sinusoidal potentials and a no slip

boundary condition,

( )

( )

** * *0

** * *0

sin 2 On left driving electrode2

sin 2 On right driving electrode2

f t

f t

φφ π

φφ π

= +

= −, (5.6)

* 0u = , (5.7)

where *0φ is the dimensionless amplitude of the driving voltage and *

Cf fτ= is

the dimensionless driving frequency.

142

On the gate electrode, the bulk electrostatic potential is related to its zeta

potential and is given as

* * *elφ φ ζ= − , (5.8)

In case of symmetric ICEO flow, we set * 0elφ = . We solve the following time

dependent PDE on the gate electrode

( )

* **

* * * **1 ( )

cosh 2Du

t y x xζ φ ζ

ζ

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

∂ ∂ ∂ ∂= − −∂ ∂ ∂ ∂

, (5.9)

with the following conditions on the edges of the gate electrode

*

* 0Duxζ∂ =∂

, (5.10)

where

( ) ( )* 2 *4 1 sinh 4Du mλ ζ= + , (5.11)

where m is a dimensionless parameter indicating the relative contribution of

electroconvection to surface conduction. For aqueous solutions of KCl at room

temperature, 0.45m ≈ .

Finally, the time dependent normalized slip velocity on the gate electrode is

given by

*

* *2 *

1( )su txφζ

α∂=∂

(5.12)

where *0 10α φ= . This value of α was chosen because in this particular

geometry, the linear scale for maximum instantaneous *ζ is equal to *0 10φ (=α ).

The rms (root mean square) *ζ is approximately *0 10 2φ ( 2α= ) and the

143

dimensionless rms electric field is approximately *0 5 2φ ( 2α= ). Product of

these two quantities yields ( )2*0 10φ which is equal to 2α . Using the rms zeta

potential and the rms electric field, we can get the following scale for the velocity

2 2

THHSu

aα φε

η= . (5.13)

Scaling the slip velocity with the preceding scale yields (5.12).

The time averaged slip velocity, *su can be obtained by averaging the slip

velocity over the duration of the solution,

*

* * ** 0

1 sT

s ss

u u dtT

= ∫ , (5.14)

where *sT is the dimensionless time up to which the solution is obtained. The

solution will be periodic after a short transient period. During the transient period,

the solution develops and achieves periodicity. The transient period is generally

shorter than one period of the AC cycle.

5.2 Depth Averaging

Before we compare our numerical results with the experiments, we would like

to point out a particular feature of µPIV experiments. In µPIV experiments, entire

volume of the fluid is illuminated uniformly. The particles in the object plane

(focal plane) are in perfect focus and contribute highest to the PIV image but the

particles which are away from the object plane are also visible, of course out of

focus. They might not be perfectly focused but they do contribute to the PIV

144

image. Hence, µPIV does not yield the velocity field at a single plane; instead it

yields a certain weighted average of the velocity over the depth of the interrogation

region. As a result, the measured velocity is slightly different from the real velocity

in the focal plane. The relative contribution of the light from a particle away from

the focal plane decreases as its distance from the focal plane increases. In other

words, the highest contribution to the velocity measurement comes from the focal

plane.

Let’s define a parameter PIVε indicating the relative contribution of the

particles which are a distance y away from the focal plane to the velocity

measurement. The weighted depth-averaged velocity, 0u , can then be represented

as,

0

( , , ) ( )( , )

( )PIV

PIV

u x y z y dyu x z

y dy

ε

ε= ∫

∫. (5.15)

If we arbitrarily set ( 0) 1PIV yε = = , then according to [76],

22

1

31

PIV

corr

yy

ε =⎡ ⎤⎛ ⎞⎢ ⎥+ ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

, (5.16)

where corry is an experimental parameter (called the depth of correlation) and

depends upon the properties of the imaging system (the numerical aperture, NA,

and the magnification, M) and the diameter of the particles, dp [76]. Depth of

correlation is defined as the axial distance from the object plane in which a particle

becomes sufficiently out of focus so that it no longer contributes significantly to

145

the signal peak in the particle-image-correlation function. From (5.16), we can see

that indeed ( ) 0.01PIV corryε = (i.e. the contribution decreases to 1/100th at y=ycorr).

We want corry to be as small as possible so that the maximum contribution comes

from the focal plane (y=0) and the contribution from other planes is minimized.

corry can be made small by using a lens with high NA and by using small particles.

In our experiments, we have used a 10x lens with NA=0.25 and particles with

diameter of dp=0.7 µm. For these conditions, following the analysis of [76], we

determine that 18.58corry ≈ µm and thus we can plot PIVε for our experiments (see

Fig. 5.2).

-3 -2 -1 0 1 2 3x 10-5

0

0.2

0.4

0.6

0.8

1

Distance from the focal plane (m)

ε PIV

Fig. 5.2: Relative contribution of particles to the velocity measurement as a function of their distance from the focal plane for NA=0.25, M=10 and dp=0.7 µm. The contribution of a particle decreases as its distance from the focal plane increases. The depth of correlation, corry , for these experimental conditions is approximately 18.58 µm. Note that ( ) 0.01PIV corryε = .

146

Depth averaging in µPIV has significant consequences for our experiments.

First, the depth of correlation is much larger ( 18.58corry ≈ µm) than the double

layer thickness (10-50 nm). Second, the particles used in our experiments are also

much larger (0.7 µm) than the double layer thickness. These two observations

indicate that it’s impossible for us to measure the real slip velocity on the gate

electrode. At best, we can measure a weighted depth-average of the velocity.

Therefore, in order to draw a meaningful comparison between the experiments and

the simulations, we should perform the weighted depth-averaging in our numerical

simulations also.

5.3 Uncertainty in Diffusivity Values

In many of our experiments, we have used an aqueous fluid with a measured

conductivity of 417 10 S/mσ −= × . We call this solution ‘purified water’ because

it’s a 50:1 mixture of DI water ( 40.055 10 S/mσ −= × ) and a proprietary particle

solution obtained from Duke scientific, Fremont, CA. The resultant conductivity of

this solution is clearly due to the content of the particle solution because the

original conductivity of the DI water is very low ( 40.055 10 S/mσ −= × ) and the

conductivity rises to the reported high value ( 417 10 S/mσ −= × ) as soon as the

particle solution is added. We specifically don’t know what ions are present in the

particle solution. The knowledge about ions and specifically their diffusivity is

critical to determining the double layer thickness, D Dλ ε σ= and the charging

time, C Da Dτ λ= . These parameters are important because Dλ is used in

147

determining Dukhin number and cτ determines if the double layer can charge fast

enough in an AC electric field. Since, we don’t have the accurate knowledge of D,

we exercise the freedom to make assumptions about ions present in the solution.

We have considered two different cases in our simulations:

Case 1: Assume that the present ions are K+ and Cl-. We measured the pH of

water before and after adding the particles and found that the two measurements

are very close to each other (5.7 and 5.46 respectively). This implies that particle

solution probably does not contain any acidic or basic material; instead it might

contain a salt which does not alter the pH of water. Based on this observation, we

assume the presence of K+ and Cl- because KCl is a common salt in chemistry. The

diffusivities of K+ and Cl- ions are respectively -91.957 10× and -92.032 10× m2/s

[65]. The average of the two, 91.995 10D −= × m2/s, was used in our simulations.

Or

Case 2: Assume that the present ions are H+ and -13HCO . According to [66],

when DI water is exposed to atmosphere, it absorbs a lot of CO2 and forms

carbonic acid, 2 3H CO . This is evident from the pH values (5.46-5.7). 2 3H CO

solution is generally dominated by H+ and -13HCO ions. The diffusivities of H+ and

-13HCO ions are respectively -99.311 10× and -91.185 10× m2/s [65]. Following the

analysis of ref. [66], we use the average of the two ionic diffusivities (i.e.

-9=5.248 10D × m2/s) in our simulations.

148

Using the preceding two values of average D and the measured conductivity

417 10 S/mσ −= × , we calculate 8D 2.885 10λ −= × and 8

D 4.679 10λ −= × m

respectively. The characteristic time scales are 32.892 10cτ−= × s and

31.783 10cτ−= × respectively. The second case yields a thicker double layer but its

charging time is shorter. In other words, the second case suffers from higher

amount of surface conduction but at the same time has a faster charging dynamics.

These two effects counter act against each other. Therefore, we expect that the

results of the simulations are not so much affected by the exact value of D. Results

of the simulation for these two values of D will be presented next.

5.4 Results

The preceding nonlinear numerical model was solved in COMSOL (Comsol

Inc., Stockholm, Se). A time dependent solver was used for solving the

electrostatic problem. The electrostatic solution was used for calculating the time

averaged slip velocity *su . The flow solution was then obtained with a stationary

solver using *su as a boundary condition. The linear algebraic system was solved

with a direct (UMFPACK) linear system solver. Mesh was created by using a

maximum element size of 5e-4 on the gate boundary, 1e-5 on the two edges of the

gate and 0.05 in the rest of the domain (see Fig. 5.3). Non-ideal weak constraints

were used for handling the dirichlet conditions ( * * *elφ φ ζ= − , and * *

su u= ) on the

gate electrode.

149

Fig. 5.3: Mesh inside the simulation geometry. The mesh was highly refined on

the gate electrode and on the edges of the gate electrode. The mesh was also sufficiently refined in the bulk for obtaining an accurate bulk flow solution.

First we present the results of simulations under the following conditions:

0 10 Vφ = , 100 Hzf = , 417 10 S/mσ −= × , 91.995 10D −= × m2/s at 020T = C. The

values of various other constants were reported in chapter 3. These conditions

yield the following: 82.885 10 mDλ−= × , 32.892 10 scτ

−= × , *0 395.856φ = ,

* 0.289f = , 39.586α = and * 41.442 10λ −= × . Note that the characteristic zeta

potential scale, α , is almost 40 times higher than the thermal voltage. The solution

was obtained for durations longer than the period of the AC signal so that the

solution has time to develop fully and reach periodicity. In this case, the solution

was obtained upto * 14t ≈ which is four times longer than the AC signal period

( *1 3.458f = ). Fig. 5.4 shows the dimensionless zeta potential, *ζ , at the left

edge of the gate electrode (i.e. * *0.5, 0x y= − = ) as a function of time. The

Gate Electrode

150

solution is almost periodic for * 2t > . Note that the maximum value of *ζ is

around 10, meaning that the zeta potential is 10 times higher than the thermal

voltage. We can expect the nonlinear effects to be strong at such high zeta

potentials.

Fig. 5.4: Temporal evolution of dimensionless zeta potential *ζ , at the left edge of the gate electrode (i.e. * *0.5, 0x y= − = ). The solution first goes through a short transient period ( * 2t < ) and then achiever periodicity with the same period as the AC cycle. The periodic solution can be called fully developed. Results are shown for 0 10 Vφ = , 100 Hzf = , 417 10 S/mσ −= × and 91.995 10D −= × m2/s at

020T = C.

Fig. 5.5 shows the streamlines for the time averaged flow. The flow is

symmetric about the center and shows the general features of symmetric ICEO

flow.

ζ*

151

Fig. 5.5: Time averaged streamlines on top of the gate electrode for 0 10 Vφ = , 100 Hzf = , 417 10 S/mσ −= × and 91.995 10D −= × m2/s at 020T = C. The ICEO

flow is symmetric about the center of the gate. Fig. 5.6 shows the time averaged value of the normalized slip velocity, *

su

for -9=1.995 10D × m2/s (i.e. for KCl), while the other parameters are the same as

before. The velocity is zero on the edges which is a feature of surface conduction

dominated flows (as explained in chapter 4). We have also shown the weighted

depth-average of the velocity on the same figure. The weighted depth-average is

lower than the slip velocity and will help in reducing the discrepancy between the

simulations and the experiments.

In Fig. 5.7 we show the slip and the weighted depth-averaged velocity for

-9=5.248 10D × m2/s (i.e. for HCO3), while the other parameters are the same as

before. A quick comparison between Figs. 5.6 and 5.7 reveals that the velocities in

the latter case are slightly lower due to increased surface conduction. However, the

difference between the two is small. The reason behind this was explained in

section 5.3.

152

-0.5 0 0.5

-0.1

-0.05

0

0.05

0.1

0.15

x*

<u* >

Slip VelocityWeighted Depth Average

Fig. 5.6: The dimensionless time averaged slip velocity on the gate electrode for 91.995 10D −= × m2/s and 0 10 Vφ = , 100 Hzf = , 417 10 S/mσ −= × at

020T = C. The weighted depth-averaging produces a lower velocity than the slip velocity.

-0.5 0 0.5

-0.1

-0.05

0

0.05

0.1

0.15

x*

<u* >

Slip VelocityWeighted Depth Average

Fig. 5.7: The dimensionless time averaged slip velocity on the gate electrode for 95.248 10D −= × m2/s and 0 10 Vφ = , 100 Hzf = , 417 10 S/mσ −= × at

153

020T = C. The weighted depth-averaging produces a lower velocity than the slip velocity.

5.5 Simulations vs. Experiments

Fig. 5.8 shows the experimental data along with simulation results for

0 10 Vφ = , 100 Hzf = , 417 10 S/mσ −= × and 91.995 10D −= × m2/s at 020T = .

The linear velocity is more than two orders of magnitude higher than the

experimental velocity. The nonlinear simulation reduces the velocity by one order

of magnitude. Depth averaging of nonlinear velocity reduces the discrepancy even

further and brings the velocity within a factor of 6 of the experiments. Depth-

averaging is important because a direct comparison of the numerical slip velocity

with experimental data is not meaningful. In our experiments, we can not measure

the real slip velocity; instead we measure a weighted depth-average of the velocity.

Therefore, it’s more meaningful to compare the experimental data with the

weighted depth-average of the numerical results.

154

-1 0 1x 10-4

10-6

10-5

10-4

10-3

10-2


Slip

Vel

ocity

, us (m

/s)

Linear simulation

Experimental data

Depth averagednonlinear simulation

Nonlinearsimulation

Fig. 5.8: Comparison of different estimates of ICEO flow velocity for 0 10 Vφ = , 100 Hzf = , 417 10 S/mσ −= × . The linear theory predicts two orders

of magnitude higher slip velocity than the experiments. The nonlinear theory reduces the discrepancy by more than one order of magnitude. Depth averaging of the nonlinear flow field brings the velocity within a factor of 6 of the experimental data. Simulations (both linear and nonlinear) were performed for 91.995 10D −= × m2/s.

5.6 Contribution of Dielectrophoresis

In the experiments, the flow tracing particles may be subjected to

dielectrophoresis (DEP). In order to determine whether DEP is responsible for

discrepancy between the numerical and experimental velocities, DEP force [26]

was calculated from

232 DEP rmsF r K Eπε= ℜ ∇ , (5.17)

155

where r is the particle radius, Kℜ is the real part of the Clausius-Mossotti factor

and E is the magnitude of the electric field. The difference between the particle

velocity pu and the fluid velocity fu was calculated from Stokes’ law,

6r p f DEPu u u F rπη= − = , (5.18)

where ru denotes the difference between the particle velocity and the fluid

velocity. Since the electric field varies significantly within the healing length, a

good estimate for 2rmsE∇ can be found by

2

2 012rms

H

EL L

φ⎛ ⎞∇ ≈ ⎜ ⎟⎝ ⎠

, (5.19)

where a is the length of the gate electrode, L is the gap between the driving

electrodes, HL is the healing length predicted by (4.3) and 0φ is the amplitude of

the driving voltage. Taking *max 10ζ = (from Fig. 5.4) and 28.85 nmDλ = (from

section 5.4), we estimate the healing length to be 6HL ≈ µm. Finally taking

0 10φ = Volt, a=200 µm, L=800 µm, r=350 nm & 1Kℜ ≈ and combining (5.17),

(5.18) and (5.19), we find 0.4ru ≈ µm/s, which is approximately 2 orders or

magnitude smaller than the experimental ICEO velocity (45 µm/s). The

contribution of dielectrophoresis to the particle velocity measurements can

therefore be ignored.

156

5.7 Contribution of Electrothermal Flow

The electrothermal flow is generated when an electric field acts on a fluid with

nonuniform conductivity and permittivity. Nonuniform electric fields cause

nonuniform Joule heating of the fluid. As a result, the temperature of the fluid

varies within the device and leads to gradients in fluid properties such as

conductivity, permittivity, density, viscosity etc. The gradients in these properties

are proportional to the gradients in the temperature. A simplified expression for the

electrothermal force for water under low AC frequencies can be written as [26],

00 20.0121 ( )E

Ef T E εωτ

= − ∇ ⋅+

. (5.20)

We have performed time averaged simulations (using steady state solvers) to

obtain an approximate scale for the electrothermal flow velocity in our

experiments. The simulations were performed for our experimental conditions:

0 10 Vφ = , 100 Hzf = , 417 10 S/mσ −= × , 91.995 10D −= × m2/s at 020ambientT = C.

The electrostatic potential is governed by the Laplace’s equation. Other governing

equations are shown later in this section along with a scaling analysis. The

electrostatic boundary conditions are: insulation on the all non-metallic walls,

constant potentials ( 5 V± ) on the two driving electrodes and φ ζ= − on the gate

electrode. ζ was obtained by solving (4.22) on the gate electrode. The thermal

boundary conditions are: insulation of the left and right side walls and outward

flux of *( )w ambient glassk T T d− on all top and bottom boundaries (including the

three electrodes). Here wk is the thermal conductivity of water and 500glassd =

157

µm. The outward flux boundary condition was derived by assuming that the

temperature outside the device is ambient. The flow boundary conditions are: ‘no-

slip’ at all the boundaries. A reference pressure is defined at an arbitrary point in

the domain.

The time averaged electric field lines inside the device are shown in Fig 5.9.

The electric field causes Joule heating and gives rise to two hot zones, one on the

left of the gate electrode, and the other on the right (Fig. 5.10). The maximum

temperature rise is negligible (0.0027 K). The electrothermal force is highest close

to the edges of the electrodes and has a magnitude of approximately 0.02 N/m3

(Fig. 5.11). The resultant electrothermal flow velocity is approximately 0.6 nm/s

(Fig. 5.12), which is 4 orders of magnitude smaller than the experimentally

measured ICEO slip velocity. Therefore, the electrothermal flow can be ignored in

our electrokinetic simulations.

Now we will show a scaling analysis which predicts magnitudes of various

quantities very reliably, without performing any simulations. The temperature rise

of water as a result of Joule heating is governed by the following equation,

( ) 2 0w rmsk T Eσ∇⋅ ∇ + = , (5.21)

where wk is the thermal conductivity of water. Since the temperature is highest in

the gaps between the driving and the gate electrodes (Fig. 5.10), an estimate for the

temperature rise can made as follows

( )

20

2 22wk T

LGφσ∆ ⎛ ⎞

⎜ ⎟⎝ ⎠

∼ , (5.22)

158

where G is the gap between the driving and the gate electrodes (see Fig. 5.9). Note

that G is the most relevant length scale for calculating the temperature gradients.

Rearranging (5.22), we find

22

0

8 w

GTk L

σφ ⎛ ⎞∆ ⎜ ⎟⎝ ⎠

∼ . (5.23)

Taking 0.6 W/mKwk = (at room temperature), 417 10 S/mσ −= × , 0 10φ = Volt,

G=300 µm and L=800 µm, we find, 0.003T∆ ≈ K, which is very close to the

simulation results. Now using (5.20), the electrothermal force on the fluid can be

roughly estimated to be 2020.012 0.017

/ 2ETf

G Lφε ∆⎛ ⎞≈ =⎜ ⎟

⎝ ⎠ N/m3, which is also very

close to the simulation results. The fluid flow in the absence of any pressure

gradients is governed by

2 0Eu fη∇ + = , (5.24)

The electrothermal flow velocity can then be approximated as

2 106 10E Hu f L η −≈ ×∼ m/s which is again very close to the simulation results.

Here HL is the healing length and has a value of 6 µm for our experimental

conditions. Since the electrothermal force is highest in a very small region close to

the edges, the healing length, HL , is the most relevant length scale for estimating

the electrothermal flow velocity. The electrothermal flow velocity is approximately

4 orders of magnitude smaller than the ICEO slip velocity measurements (~45

µm/s). Therefore, electrothermal flow can be neglected while simulating ICEO.

159

The electrothermal flow is negligible in our experiments because the electric

conductivity of our fluid is very low and the Joule heating is negligible.

Fig. 5.9: Time averaged electric field lines for the experimental conditions.

Fig. 5.10: Temperature rise in water as a result of Joule heating for the experimental conditions.

G GL

160

Fig. 5.11: Electrothermal force magnitude and vectors for the experimental conditions.

Fig. 5.12: Electrothermal flow velocity magnitude and vectors for the experimental conditions.

5.8 Stern Layer and High Ionic Concentrations

For the purified water of our experiments, we obtained a reasonable agreement

between the simulations and the experiments. However, many other experiments

were performed in higher salt concentrations and the agreement at higher

161

concentrations was found to be poorer. We can explain this in the following way:

At higher concentrations, the double layer gets thinner and the surface conduction

effects become smaller. Therefore, in the simulations, one would get higher

velocity in higher ionic concentrations. However, from our experimental

experience, we know that this is not the case in real life. In real life, the velocities

decrease as the ionic concentrations increase. This behavior is generally blamed to

the Stern layer which becomes very important at high ionic concentrations.

The model presented in the previous sections does not take the Stern layer into

account and therefore predicts the unrealistic effect of higher velocities at higher

concentrations. To encounter this problem, we present a way to incorporate the

Stern layer in our nonlinear model. In order to incorporate the Stern layer, the

double layer model (5.9) is modified in the following way.

( ) modified

* **

* * * **1 ( )

cosh 2Du

t y x xζ φ ζ

ζ

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

∂ ∂ ∂ ∂= − −∂ ∂ ∂ ∂

(5.25)

where

modified

*

1 cosh2

Du ζδ= + (5.26)

where D SCδ ε λ= is a parameter determining the importance of the Stern layer.

The Stern layer has been modeled as a linear capacitor with a capacitance SC . The

bulk boundary condition (5.8) is modified to

* * * *el qφ φ ζ δ= − + (5.27)

162

Derivation of (5.26) becomes clear when one makes the substitution,

( )* *2sinh / 2q ζ= − in (5.27), differentiates it with respect to *x .

The parameter, D SCδ ε λ= , is unknown and its value depends solely on the

discretion of the user. In electrokinetic literature, workers have used δ as a fitting

parameter and chosen a value which matches the simulations with the experiments

[43]. In cases, where induced zeta potential is not large, it’s reasonable to blame

the entire discrepancy on the Stern layer. In such cases, reasonably low values of

δ (<1) match the results of the simulations with the experimental data [43].

However, in cases where induced zeta potentials are large, one can not ignore the

possibility of other nonlinear effects such as surface conduction and nonlinear

surface capacitance. In such cases, it’s not reasonable to blame the entire

discrepancy on the Stern layer. In fact, when we try to chose a value of δ while

ignoring other nonlinear effects, we ends up getting an unreasonably high value of

δ ( 1) for matching the results. This again bolsters our claim that at high

induced zeta potentials, nonlinearities play very important role in electrokinetics.

5.9 Conclusions

In this chapter, we solved a time dependent nonlinear model encompassing

nonlinear surface capacitance and surface conduction and compared the numerical

results with our experimental data. We showed that the combined effects of

nonlinear capacitance, surface conduction and depth-averaging bring the numerical

velocity within a factor of 6 of the experimental measurements for the driving

163

conditions of 0 10 Vφ = , 100 Hzf = , 417 10 S/mσ −= × . The results of our

nonlinear simulations still do not match the experimental data exactly. The cause

of the remaining discrepancy is not clear to us but there are several other effects

which can be responsible for this remaining discrepancy. Stern layer is the most

probable reason for it. We have presented a modified model for incorporating the

Stern layer. We also can not ignore the possibilities of chemi-osmosis [61, 62] and

steric effects between the finite size ions [63, 64]. Directions for incorporating

chemi-osmosis are given in chapter 7.

164

6

165

Chapter 6. Induced Charge Electroosmosis on a Rough

Surface

We present the results of our electrokinetic experiments on a rough surface.

We show experimentally that the roughness of a surface can have dramatic impact

on the electroosmotic flow on the surface. In the previous chapters, we had

analyzed the flow around smooth surfaces; but in many practical situations,

surfaces are not smooth and have roughness. We have produced induced charge

electroosmotic flow on a rough surface containing randomly distributed nano

structures of characteristic dimension 200 nm. Experiments show that the flow is

upto 50% slower on the nanoscale roughness as compared to the flow on a smooth

surface. This behavior is attributed to enhanced surface conduction on the

nanoscale features of the rough surface. As described previously, the surface

conductivity is inversely proportional to the characteristic dimension of the

surface. In case of a smooth electrode, the characteristic dimension is equal to the

width of the electrode (200 micron). In case of a rough electrode, however, the

characteristic dimension is on the order of the roughness (200 nm). Due to a small

length scale, surface conduction is greatly amplified on the rough surface and leads

to slower ICEO flows. Here we present an experimental evidence for this effect.

6.1 Background

In order to understand the electroosmotic flow around small features, it is

useful to think about the electrophoretic motion of a small particle in an ionic

166

liquid. In a reference frame attached to the particle, the electrophoretic motion of

the particle relative to the fluid appears as the electroosmotic flow of the fluid

relative to the particle. In other words, the electroosmotic flow of the fluid around

the particle causes the particle to move relative to the fluid with the same relative

speed as the speed of electroosmosis but in the opposite direction.

It has been established that the electrophoretic mobility of a charged particle

depends not only on its zeta potential but also on the ratio of the double layer

thickness to the size of the particle [8-11]. O’Brien and White, 1978 [9] calculated

the electrophoretic mobility of a spherical particle as a function of its

dimensionless zeta potential ( * ze kTζ ζ= ) and the product aκ where κ is the

reciprocal debye length (i.e. 1Dκ λ−= ) and a is the radius of the particle. They

showed that for a sufficiently large particle ( 4aκ > ), the dimensionless mobility

( * 6e e kTµ πη µ ε= ) decreases as aκ decreases at a fixed *ζ (see Fig. 6.1). In other

words, as the dimension of the particle, a, gets smaller for a fixed κ and *ζ , its

mobility also gets lower. According to Squires et al. [77], the reduction in

electrophoretic mobility with decreasing aκ can be understood in terms of surface

conduction. In a previous chapter, we reported that Dukhin number is a function of

aκ ,

( ) 24 1 sinh4zeDu m

a kTζ

κ⎛ ⎞= + ⎜ ⎟⎝ ⎠

(6.1)

For small values of aκ , we can expect surface conduction to be very

prominent. Surface conduction reduces the tangential field around the particle (see

167

Fig. 6.2b) and increases the normal electric field. This reduction in tangential

electric field reduces electrophoretic velocity of the particle.

Fig. 6.1: Dimensionless electrophoretic mobility ( * 6e ee kTµ πη µ ε= ) of a particle as a function of its dimensionless zeta potential * ze kTζ ζ= and aκ in a KCl solution. The mobility increases with aκ for 4aκ > . This picture has been reprinted from O’Brien and White 1978 [9]. © Royal Society of Chemistry 1978.

For a very small particle or a very thick double layer ( 1aκ ), the preceding

analysis does not hold true. In this limit, the particle becomes so small compared to

168

the double layer that it does not feel the forces acting on the double layer and the

electrophoretic mobility is simply derived by equating the Coulomb force on the

particle to the Stokes drag. In other words, it’s not required to consider the

presence of the double layer in the calculation of the electrophoretic mobility when

the particle is too small ( 1aκ ). The reader is referred to references [3] and [9]

for an analysis of this regime of electrophoretic mobility.

6.2 Fundamental Picture

Let’s consider the electric field lines around a positively charged cylinder

subject to a horizontal electric field. When the zeta potential of the cylinder is low,

the surface currents are absent, and the electric field lines become tangential to the

surface of the cylinder (see Fig. 6.2a). However, when the zeta potential is high,

the surface currents are high and the electric field line configuration appears as

shown in Fig. 6.2b. Since the surface conduction current is highest at the equator

of the cylinder (because the tangential electric field, Es, is highest at the equator)

and zero at the poles (because Es=0 on the poles), electric field acquires a large

normal component at between the poles and the equator to maintain the current at

the equator. The normal component of the electric field supplies the ions for the

high surface current at the equator. As a result, the tangential field around the

cylinder is reduced and electroosmotic slip velocity around the cylinder is severely

deteriorated.

169

E0Pole, js=0 Equator, js≈0

Enormal=0

+ + + ++

++

+

++++

++

+

+

+ Low ζ

Fig. 6.2 (a): Electric field lines around a lowly charged cylinder

E0Pole, js=0 Equator, high js

High Enormal

+ + + ++

++

+

++++

++

+

+

+ High ζ

Fig. 6.2(b): Electric field lines around a highly charged cylinder

Fig. 6.2: Electric field lines around (a) a lowly charged cylinder and (b) a highly charged cylinder. In the former case the zeta potential and surface conduction are low and therefore the electric field is completely tangential to the cylinder surface. In the latter case, the zeta potential and surface conduction are high. The surface current is zero at the pole but highest at the equator. To maintain the high surface current at the equator, field acquires a large normal component between the pole and the equator. The tangential field around the cylinder is reduced which reduces the electroosmotic slip velocity.

170

According to Khair and Squires 2008 [72], the length of the region in which

the electric field has large normal component (called ‘healing region’) is roughly

proportional to the ratio of the surface conductivity to the bulk conductivity

( ) 24 1 sinh4H s DzeL mkTζσ σ λ ⎛ ⎞≡ = + ⎜ ⎟

⎝ ⎠ (6.2)

When ζ is large, HL can be several times longer than Dλ . When the

dimension of the cylinder is small, a large portion of the cylinder surface is

covered with the healing zone, yielding a negligible electroosmotic flow rate.

Now let’s consider a positively charged surface with its roughness

approximated by a regular shape such as sinusoid (Fig. 6.3). Suppose that the

dimension of the roughness is small and the zeta potential is high. In such a case,

the healing length can be comparable to the dimension of the roughness and the

tangential field is severely reduced on most of the surface. Consequently, the

electroosmotic flow rate is also reduced.

++++ + +

++++

+++ + +

+++ +

+++ + +

++++

+++ + +

+++ +

+++ + +

++++

+++ + +

+++

E0 high jsLow js

high Enormal

Fig. 6.3: Electric field lines on a charged rough surface. The roughness has been approximated as sinusoids. When the zeta potential and the surface currents are high, the electric field lines acquire a large normal component and the tangential component is reduced. As a result, the electroosmotic flow rate is also reduced.

171

6.2.1 Thick Double Layers

When the roughness is much smaller than the double layer ( 1aκ ), it can’t

affect the charging dynamics of the double layer and the surface current is again

determined by the global dimension of the electrode such as the width of the

electrode.

6.3 Experiments

6.3.1 Details of the Device

We measured induced charge electroosmotic flow in a device similar to the one

described in chapter 3 but the gate electrode had a specially grown nanoscale

roughness. As a brief recapitulation, this device had three coplanar microelectrodes

(two driving electrodes and a gate electrode) laid on a glass substrate. Each

microelectrode was 200 µm in width. A PDMS chamber was carefully placed on

the electrodes. The chamber was filled with a KCl solution seeded with fluorescent

particles. An AC signal was applied between the two driving electrodes and the

flow was measured on the gate electrode using µPIV.

The roughness of the electrode was realized by growing a nano structured

titania (NST) film on a titanium electrode. NST is automatically formed when a Ti

surface is oxidized in hydrogen peroxide. NST was covered with a thin layer of

metal to make its surface conductive for achieving ICEO flow. The characteristic

size of NST was 200 nm (see Fig. 6.4). NST was grown only on the gate electrode

because that’s where the ICEO flow of interest takes place; the driving electrodes

172

were kept smooth i.e. no NST was grown on them. However, it was only a half

length of the gate electrode where NST was grown. The remaining half was kept

smooth. This way we obtained two different portions, rough and smooth, on the

same gate electrode. The interface between the two portions was clearly visible

under the microscope (see Fig. 6.5). This allowed us to measure the slip velocity

on both of the portions, rough and smooth, simultaneously and enabled us to

compare the velocities on the two portions under exactly the same conditions.

200 nm

Fig. 6.4: An SEM image of the Nano-Structured Titania (NST) covered with a thin layer of metal. The characteristic dimension of NST is 200 nm. NST device was fabricated by Dr. Adam Monkowski who, at the time of fabrication, was a graduate student in Prof. Noel MacDonald’s group at UCSB [78].

173

NST portion

Smooth portion

Interface

200 µm

Fig. 6.5: A microscopic picture of the gate electrode. NST is grown on one half (top) of the electrode whereas the other half (bottom) is kept smooth. The interface between the two portions is clearly visible.

6.3.2 Fabrication Process

A glass wafer was cleaned respectively in Acetone, Iso propanol, de-ionized

water (DI) and finally in O2 plasma. AZ 5214 photoresist was spin coated on the

wafer and the device design was transferred from the mask into the photoresist by

image reversal photolithography. Thin layers of metals (Ti/Au/Ti, respectively

50/2000/5000 Å) were evaporated on the wafer and then lifted off in acetone. AZ

4110 positive resist was spin coated and exposed through a mask yielding a pattern

which hid half length of the gate electrode and revealed rest of the pattern. The top

Ti layer of the revealed pattern was etched in a solution of (49%HF): H2O2: DI (5:

5: 100). The gold layer was exposed due to etching of Ti. The resist was removed

with Acetone. Then the Ti layer remaining on one half of the gate electrode was

174

oxidized in 10% H2O2 preheated to 850C on a hotplate. Oxidation was carried out

for 8 minutes. Nano-Structured Titania (NST) is formed on Ti surface as a result of

this oxidation [78-82]. NST was annealed at 3000C for 1 hour in air. Then another

AZ 5214 image reversal lithography, exactly similar as before, was carried out and

layers of Ti/Pt (25/250 Å) were evaporated on all the electrodes. During

evaporation, the wafer was held at an angle of 200-300 and rotated at about 0.5 rpm

to get uniform metal coverage on NST. The residual metal was then lifted off in

acetone.

6.3.3 Experimental Results

ICEO flow velocity on the gate electrode was measured in various KCl

concentrations and at various driving voltages at a constant frequency of 100 Hz.

Figs. 6.6a-d show the velocity profiles for the following ionic concentrations and

driving voltages: (a) purified water ( 17σ = µS/cm), 20 Vpp, (b) 5 mM

( 835σ = µS/cm), 14 Vpp, (c) 10 mM ( 1400σ = µS/cm), 20 Vpp, and (d) 10 mM

( 2950σ = µS/cm), 20 Vpp. In all of the above, the maximum velocity on the rough

portion of electrode is 1.1-1.4 times lower than the maximum velocity on the

smooth portion.

175

-1.5 -1 -0.5 0 0.5 1 1.5x 10-4

-4

-3

-2

-1

0

1

2

3

4x 10-5

x (m)

Slip

Vel

ocity

(m/s

)

Smooth

Rough

(a) Purified Water, 20 Vpp

-1.5 -1 -0.5 0 0.5 1 1.5x 10-4

-1.5

-1

-0.5

0

0.5

1

1.5x 10-5

x (m/s)

Slip

Vel

ocity

(m/s

)

Rough

Smooth

(b) 5 mM KCl, 14 Vpp

176

-1.5 -1 -0.5 0 0.5 1 1.5x 10-4

-1.5

-1

-0.5

0

0.5

1

1.5x 10-5

x (m)

Slip

Vel

ocity

(m/s

)

Smooth

Rough

(c) 10 mM KCl, 20 Vpp

-1.5 -1 -0.5 0 0.5 1 1.5x 10-4

-6

-4

-2

0

2

4

6x 10-6

x (m)

Slip

Vel

ocity

(m/s

)

Rough

Smooth

(d) 20 mM KCl, 20 Vpp

Fig. 6.6: ICEO flow velocity on the gate electrode for various KCl concentrations and driving voltages at 100 Hz. The maximum velocity on rough

177

portion is approximately 1.1-1.4 times lower than the maximum velocity on the smooth portion.

Fig. 6.7 shows the maximum slip velocity as a function of the driving voltage

at 100 Hz in 5 mM KCl. The maximum slip velocity on the rough portion is

consistently lower than velocity on the smooth portion for all driving voltages. The

velocity first increases with the driving voltage and then starts decreasing. The

decrease in the velocity at high voltages is attributed to enhanced surface

conduction due to large zeta potentials.

6 8 10 12 14 16 180

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10-5

Driving Voltage, φ0 (Vpp)

Max

imum

Slip

Vel

ocity

, um

ax (m

/s)

Smooth

Rough

Fig. 6.7: The maximum slip velocity on the gate electrode as a function of driving voltage in 5 mM KCl at 100 Hz. The velocity is always lower on the rough portion of the gate electrode than the velocity on the smooth portion.

178

6.4 Asymmetry in Flow

The observed slip velocity is not symmetric about the center of the gate (note

that the velocity is not zero at the center in Fig. 6.6). The reason behind the

asymmetry is not related to the roughness. It was observed in many other

experiments (with and without NST). There is a stray electroosmotic flow which

superimposes on the symmetric ICEO flow and gives rise to the asymmetry in the

results. The stray electroosmotic flow originates from the capacitive coupling

between the gate electrode and the metallic stage of the microscope. The stage of

the microscope can be considered grounded and therefore it is at a lower potential

than the floating potential of the gate electrode. The glass between the gate and the

stage can be seen as a dielectric material of the capacitor. The potential difference

between the gate and the stage creates a stray zeta potential on the gate electrode

which gives rise to a unidirectional stray electroosmotic flow. This effect can also

bee seen as ‘Field Effect’ (see chapter 3) caused by the grounded stage. This flow

is always in one direction even if the electric field changes direction (as in AC

Field Effect shown in chapter 3). We can easily identify the stray electroosmotic

velocity. It’s equal to the velocity at the center of the gate. We have not subtracted

the stray velocity from our data so that the reader’s attention can be drawn to this

surprising effect. To learn more about the origin of this stray flow, the reader is

referred to Mansuripur, Pascall and Squires 2009 [83].

179

6.5 Conclusions

We experimentally demonstrated that the slip velocity of an ICEO flow is

reduced due to the roughness of the surface. We found that a 200 nm roughness

can reduce the velocity by a factor of 1.4 in high concentrations of KCl (10 mM)

and under high induced zeta potentials (~ 1 Volt). We attribute this reduction of

velocity to surface conduction. Surface conductivity is inversely proportional to

the product aκ . On a smooth surface, ‘a’ represents the global length scale of the

surface (e.g. the width of the electrode). However, when the surface has roughness,

the dimension ‘a’ becomes equal to the dimension of the roughness which is much

smaller than the global length scale. As a result, for small roughness features, the

localized surface conduction becomes very prominent and causes reduction in the

local tangential electric field which reduces the slip velocity. However, when the

roughness is much smaller than the double layer, it can be ignored altogether

because it can’t affect the double layer charging dynamics.

180

7

181

Chapter 7. Conclusions and Future Directions

Following are the conclusions of our work.

7.1 Experimental

1. Symmetric ICEO flow was observed on a planar microelectrode. Slip velocity

of 40 µm/s was observed in purified water under driving conditions of 20 Vpp

at 100 Hz applied over a distance of 1 mm. The linear theory predicts a slip

velocity of ~4 mm/s which is 100 times higher than the experimental data. We

estimate that the induced zeta potentials in our experiments are 10-40 times

higher than the thermal voltage ( 10 40 kT zeζ ≈ − ) and therefore we can

expect the linear theory to break down.

2. The velocity is observed to depend on the salt concentration. The higher the

salt concentration, the lower the velocity. We attribute this to Stern layer

effects.

3. At higher applied voltages, the velocity does not scale as 20φ . Instead it

saturates and does not increase further. The voltage at which saturation starts

depends on salt concentration. The higher the salt concentration, the higher the

voltage at which saturation starts. We attribute this to surface conduction.

4. Field effect flow control has been demonstrated with AC electric fields. Net

pumping velocity of 50 µm/s was obtained. Field effect proves to be a good

pumping method with the flexibility of modifying the flow rate and the

freedom to reverse the flow direction by just varying the gate voltage.

182

7.2 Numerical

5. A novel method to simulate nonlinear electrokinetic effects has been

formulated. This model can simulate surface conduction in steady and time

dependent cases. In the time dependent cases (such as AC), a nonlinear surface

capacitance affects the charging dynamics of the double layer. In the steady

cases, the capacitance does not play any role because enough time is given for

the double layer to charge.

6. The nonuniform induced zeta potentials, which are inherent in ICEO, lead to

nonuniform surface currents and cause surprising gradients in the bulk electric

field. Surface conduction through a nanoscale diffuse layer causes micron scale

gradient in the bulk electric field. Surface current pulls the electric field lines

into the double layer to maintain the surface current. This gives rise to healing

regions at the corners. However, the lines are expelled out of the surface once

again in the middle of the surface due to nonuniform surface current. Overall,

surface conduction creates large normal electric field and reduces the

tangential field causing a reduction in the slip velocity.

7.3 Simulations vs. Experiments

7. The combined effect of nonlinear capacitance, surface conduction and

volumetric averaging brings the numerical velocity within a factor of 7-10 of

the experimental data. The nonlinear model reduces the discrepancy between

the theory and the experiments by atleast one order of magnitude.

183

8. The remaining discrepancy (factor of 7-10) could not be explained. However,

there are several other nonlinear effects which have not been accounted for in

our work. The two major effects which can be incorporated in the classical

theory are Stern layer and chemi-osmosis. Contribution of Stern layer can’t be

predicted and has to be tuned by a user-defined parameter. Chemi-osmosis

refers to concentration gradient driven flow of ions. Surface currents cause

gradients not only in the bulk electric field but also in the bulk salt

concentration. The concentration gradient driven flow counters the

electroosmotic flow and further reduces the slip velocity. Directions for

simulating chemi-osmosis are given in the later part of this chapter.

7.4 ICEO Flow over Rough Surfaces

9. ICEO flow was produced on a nanoscale rough surface. The characteristic

dimension of the roughness was 200 nm. It was found that the flow over

roughness is slower by a factor of 1.4 than the flow over a smooth electrode.

10. The reduction in velocity due to roughness was explained in terms of surface

conduction. On a rough surface, the amount of surface current is determined by

the roughness length scale which is much smaller than the length of the

electrode. As a result surface conduction is much more prominent on a rough

surface than on a smooth surface. This reduces the tangential field and the slip

velocity.

184

7.5 Future Directions

We showed that surface flow of ions through nano scale diffuse layer creates

microscale bulk gradients in the electric field. It can also create bulk gradients in

the salt concentration. Bulk salt concentrations set up a ‘chemi-osmotic’ flow of

ions (and fluid) in a direction opposite to electroosmosis.

The nonlinear formulation can be expanded to include the surface transport of

the salt as well. Following are the ‘simplified’ governing equations and effective

boundary conditions for surface transport of charge and salt both.

7.5.1 Bulk Equations

Following the analysis of [62], we present ‘simplified’ equations for the

concentration field, c, the excess surface charge, q, and the excess surface salt

concentration, w for a planar gate electrode. In the following, electroosmotic

convection of q and w has been neglected. For a complete formulation, the reader

is referred to [62].

1. Conservation of conduction current in the bulk

( ) 0c φ∇ ⋅ ∇ = . (7.1)

2. Advection-diffusion of salt in the bulk (units of c are ‘number per unit

volume’)

2c u c D ct∂ + ⋅∇ = ∇∂

. (7.2)

3. Navier-Stokes equation for bulk fluid flow

185

2 ln cu pc

η ε φ φ∞

⎛ ⎞∇ −∇ = ∇ ⋅∇ ∇⎜ ⎟

⎝ ⎠. (7.3)

The term on the right hand side of the preceding equation represents chemi-

osmotic transport of fluid. The continuity equation is given as

0u∇⋅ = . (7.4)

7.5.2 Surface Transport Equations

4. Conservation of excess charge, q, in the double layer (units of q are ‘number

per unit area’)

1q c Dze DzeqD w ct x c x kT x kT y

φ φ∂ ∂ ∂ ∂ ∂⎛ ⎞= + +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠. (7.5)

5. Conservation of excess salt, w, in the double layer (units of w are ‘number

per unit area’)

1w c Dze cwD q Dt x c x kT x y

φ∂ ∂ ∂ ∂ ∂⎛ ⎞= + +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠. (7.6)

7.5.3 Boundary Conditions on the Gate Electrode

2 sinh2D

c zeq cc kT

ζλ ∞∞

⎛ ⎞= − ⎜ ⎟⎝ ⎠

. (7.7)

24 sinh4D

c zew cc kT

ζλ ∞∞

⎛ ⎞= ⎜ ⎟⎝ ⎠

. (7.8)

The slip velocity is given by,

24 1ln coshslip

kT ze cux ze kT c x

εζ φ ε ζη η

∂ ∂⎛ ⎞ ⎛ ⎞= − ⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠. (7.9)

186

The second term on the right hand side represents the contribution from chemi-

osmosis. At large ζ , chemi-osmotic velocity can be as high as the electroosmotic

velocity.

187

Bibliography

1. Lyklema, J., Fundamentals of interface and colloid science, Volume II:

Solid-liquid interfaces. 2001: Academic press.

2. Bard, A.J. and L.R. Faulkner, Electrochemical methods, Fundamentals and

applications. second ed. 2001: John Wiley & Sons, Inc.

3. Morgan, H. and N.G. Green, AC Electrokinetics: colloids and

nanoparticles. 2003: Research Studies Press Ltd., Baldock, Hertfordshire,

England.

4. Reuss, F., Sur un nouvel effet de le électricité glavanique. Mém. Soc.

Imp. Nat. Mosc., 1809. 2: p. 327.

5. Southern, E.M., Detection of specific sequences among DNA fragments

separated by gel electrophoresis. J Mol Biol., 1975. Nov 5;98(3): p. 503-

17.

6. Bikerman, J.J., Electrokinetic equations and surface conductance. A survey

of the diffuse double layer theory of colloidal solutions. Trans. Faraday

Soc. , 1940. 35: p. 154-160.

7. Deryaguin, B.V. and S.S. Dukhin, Theory of surface conductance. Colloid

J. USSR, 1969. 31: p. 277.

8. Dukhin, S.S. and B.V. Deryaguin, Electrokinetic phenomena. Serface and

Colloid Science, 1974. 7.

188

9. O'Brien, R.W. and L.R. White, Electrophoretic mobility of a spherical

colloidal particle. J. Chem. Soc., Faraday Trans. 2, 1978. 74: p. 1607 -

1626.

10. O'Brien, R.W., The solution of the electrokinetic equations for colloidal

particles with this double layers. Journal of Colloid and Interface Science,

1983. 92: p. 204.

11. Russel, W.B., D.A. Saville, and W.R. Schowalter, Colloidal dispersions.

1989: Cambridge University Press.

12. Gamayunov, N.I., V.A. Murtsovkin, and A.S. Dukhin, Pair interaction of

particles in electric field. 1. Features of hydrodynamic interaction of

polarized particles. Colloid J. USSR, 1986. 48: p. 197.

13. Dukhin, S.S., Non-equilibrium electric surface phenomena. Advances in

Colloid and Interface Science, 1993. 44: p. 1.

14. Murtsovkin, V.A., Nonlinear flows near polarized disperse particles.

Colloid J. Russ. Acad. Sci., 1996. 53: p. 947.

15. Squires, T.M. and S.R. Quake, Microfluidics: Fluid physics at the nanoliter

scale. Reviews of Modern Physics, 2005. 77(3): p. 977-1026.

16. Gascoyne, P.R.C. and J. Vykoukal, Particle separation by

dielectrophoresis. ELECTROPHORESIS, 2002. 23(13): p. 1973-1983.

17. Gascoyne, P.R.C., et al., Dielectrophoretic separation of cancer cells from

blood. Industry Applications, IEEE Transactions on, 1997. 33(3): p. 670-

678.

189

18. Pethig, R. and G.H. Markx, Applications of dielectrophoresis in

biotechnology. Trends in Biotechnology, 1997. 15(10): p. 426-432.

19. Hu, X.Y., et al., Marker-specific sorting of rare cells using

dielectrophoresis. Proceedings of the National Academy of Sciences of the

United States of America, 2005. 102(44): p. 15757-15761.

20. Sigurdson, M., D.Z. Wang, and C.D. Meinhart, Electrothermal stirring for

heterogeneous immunoassays. Lab on a Chip, 2005. 5(12): p. 1366-1373.

21. Feldman, H.C., M. Sigurdson, and C.D. Meinhart, AC electrothermal

enhancement of heterogeneous assays in microfluidics. Lab on a Chip,

2007. 7(11): p. 1553-1559.

22. Yao, S.H., et al., Porous glass electroosmotic pumps: design and

experiments. Journal of Colloid and Interface Science, 2003. 268(1): p.

143-153.

23. Yao, S.H. and J.G. Santiago, Porous glass electroosmotic pumps: theory.

Journal of Colloid and Interface Science, 2003. 268(1): p. 133-142.

24. Jacobson, S.C., et al., Open-Channel Electrochromatography on a

Microchip. Analytical Chemistry, 1994. 66(14): p. 2369-2373.

25. Jiang, L.N., et al., Closed-loop electroosmotic microchannel cooling system

for VLSI circuits. Ieee Transactions on Components and Packaging

Technologies, 2002. 25(3): p. 347-355.

190

26. Ramos, A., et al., Ac electrokinetics: a review of forces in microelectrode

structures. Journal of Physics D-Applied Physics, 1998. 31(18): p. 2338-

2353.

27. Ramos, A., et al., AC electric-field-induced fluid flow in microelectrodes.

Journal of Colloid and Interface Science, 1999. 217(2): p. 420-422.

28. Green, N.G., et al., Fluid flow induced by nonuniform ac electric fields in

electrolytes on microelectrodes. I. Experimental measurements. Physical

Review E, 2000. 61(4): p. 4011-4018.

29. Gonzalez, A., et al., Fluid flow induced by nonuniform ac electric fields in

electrolytes on microelectrodes. II. A linear double-layer analysis. Physical

Review E, 2000. 61(4): p. 4019-4028.

30. Ramos, A., et al., Comment on "Theoretical Model of Electrode

Polarization and AC Electroosmotic Fluid Flow in Planar Electrode

Arrays". Journal of Colloid and Interface Science, 2001. 243(1): p. 265-

266.

31. Green, N.G., et al., Fluid flow induced by nonuniform ac electric fields in

electrolytes on microelectrodes. III. Observation of streamlines and

numerical simulation. Physical Review E, 2002. 66(2): p. 11.

32. Ajdari, A., Pumping liquids using asymmetric electrode arrays. Physical

Review E, 2000. 61(1): p. R45-R48.

191

33. Ramos, A., et al., Pumping of liquids with ac voltages applied to

asymmetric pairs of microelectrodes. Physical Review E, 2003. 67(5): p.

11.

34. Seibel, K., et al. Transport properties of ac electroosmotic micropumps on

labchips. 2003. Cambridge, UK.

35. Ramos, A., et al., Pumping of electrolytes using arrays of asymmetric pairs

of microelectrodes subjected to ac voltages, in Electrostatics 2003. 2004,

Iop Publishing Ltd: Bristol. p. 187-192.

36. Studer, V., et al., An integrated AC electrokinetic pump in a microfluidic

loop for fast and tunable flow control. Analyst, 2004. 129(10): p. 944-949.

37. Garcia-Sanchez, P., et al., Experiments on AC electrokinetic pumping of

liquids using arrays of microelectrodes. Ieee Transactions on Dielectrics

and Electrical Insulation, 2006. 13(3): p. 670-677.

38. Olesen, L.H., H. Bruus, and A. Ajdari, ac electrokinetic micropumps: The

effect of geometrical confinement, Faradaic current injection, and

nonlinear surface capacitance. Physical Review E, 2006. 73(5).

39. Wong, P.K., et al., Electrokinetic bioprocessor for concentrating cells and

molecules. Analytical Chemistry, 2004. 76(23): p. 6908-6914.

40. Bown, M.R. and C.D. Meinhart, AC electroosmotic flow in a DNA

concentrator. Microfluidics and Nanofluidics, 2006. 2(6): p. 513-523.

41. Squires, T.M. and M.Z. Bazant, Induced-charge electro-osmosis. Journal of

Fluid Mechanics, 2004. 509: p. 217-252.

192

42. Bazant, M.Z. and T.M. Squires, Induced-charge electrokinetic phenomena:

Theory and microfluidic applications. Physical Review Letters, 2004.

92(6): p. 4.

43. Levitan, J.A., et al., Experimental observation of induced-charge electro-

osmosis around a metal wire in a microchannel. Colloids and Surfaces a-

Physicochemical and Engineering Aspects, 2005. 267(1-3): p. 122-132.

44. Soni, G., Squires, T.M., Meinhart, C.D., Nonlinear Phenomena in Induced

Charge Electroosmosis. Proceedings of 2007 ASME International

Mechanical Engineering Congress and Exposition, Seattle, Washington,

2007: p. IMECE2007-41468.

45. Squires, T.M. and M.Z. Bazant, Breaking symmetries in induced-charge

electro-osmosis and electrophoresis. Journal of Fluid Mechanics, 2006.

560: p. 65-101.

46. Gangwal, S., et al., Induced-charge electrophoresis of metallodielectric

particles. Phys. Rev. Lett., 2008. 100: p. Art no. 058302.

47. Hayes, M.A., I. Kheterpal, and A.G. Ewing, Effects of buffer pH on

electroosmotic flow-control by an applied radial voltage for capillary zone

electrophoresis. Analytical Chemistry, 1993. 65(1): p. 27-31.

48. Schasfoort, R.B.M., et al., Field-effect flow control for microfabricated

fluidic networks. Science, 1999. 286(5441): p. 942-945.

193

49. Polson, N.A. and M.A. Hayes, Electroosmotic flow control of fluids on a

capillary electrophoresis microdevice using an applied external voltage.

Analytical Chemistry, 2000. 72(5): p. 1088-1092.

50. Lee, C.Y., C.H. Lin, and L.M. Fu, Band spreading control in

electrophoresis microchips by localized zeta-potential variation using

field-effect. Analyst, 2004. 129(10): p. 931-937.

51. Lee, G.B., et al., Dispersion control in microfluidic chips by localized zeta

potential variation using the field effect. Electrophoresis, 2004. 25(12): p.

1879-1887.

52. Sniadecki, N.J., et al., Induced pressure pumping in polymer microchannels

via field-effect flow control. Analytical Chemistry, 2004. 76(7): p. 1942-

1947.

53. van der Wouden, E.J., et al., Field-effect control of electro-osmotic flow in

microfluidic networks. Colloids and Surfaces a-Physicochemical and

Engineering Aspects, 2005. 267(1-3): p. 110-116.

54. Daiguji, H., P.D. Yang, and A. Majumdar, Ion transport in nanofluidic

channels. Nano Letters, 2004. 4(1): p. 137-142.

55. Fan, R., et al., Inorganic nanotube nanofluidic transistors for single

molecule detection. Abstracts of Papers of the American Chemical Society,

2005. 229: p. U141-U142.

56. Karnik, R., et al., Electrostatic control of ions and molecules in nanofluidic

transistors. Nano Letters, 2005. 5(5): p. 943-948.

194

57. Karnik, R., K. Castelino, and A. Majumdar, Field-effect control of protein

transport in a nanofluidic transistor circuit. Applied Physics Letters, 2006.

88(12): p. 3.

58. van der Heyden, F.H.J., et al., Electrokinetic Energy Conversion Efficiency

in Nanofluidic Channels. Nano Lett., 2006. 6(10): p. 2232-2237.

59. van der Heyden, F.H.J., et al., Power Generation by Pressure-Driven

Transport of Ions in Nanofluidic Channels. Nano Lett., 2007. 7(4): p. 1022-

1025.

60. Pennathur, S., J.C. Eijkel, and A. van den Berg, Energy conversion in

microsystems: is there a role for micro/nanofluidics? Lab Chip, 2007. 7: p.

1234-1237.

61. Chu, K.T. and M.Z. Bazant, Nonlinear electrochemical relaxation around

conductors. Physical Review E (Statistical, Nonlinear, and Soft Matter

Physics), 2006. 74(1): p. 011501.

62. Khair, A.S. and T.M. Squires, Fundamental aspects of concentration

polarization arising from nonuniform electrokinetic transport. Physics of

Fluids, 2008. 20: p. 087102.

63. Kilic, M.S., M.Z. Bazant, and A. Ajdari, Steric effects in the dynamics of

electrolytes at large applied voltages. I. Double-layer charging. Physical

Review E, 2007. 75(2).

64. Bazant, M.Z., et al., Nonlinear Electrokinetics at large applied voltages.

arXiv:cond-mat/0703035v2 [cond-mat.other], 2007.

195

65. CRC Handbook of Chemisty and Physics, 88th Edition. 2007-2008.

66. Chen, C.H. and J.G. Santiago, A planar electroosmotic micropump. Journal

of Microelectromechanical Systems, 2002. 11(6): p. 672-683.

67. Santiago, J.G., et al., A particle image velocimetry system for microfluidics.

Experiments in Fluids, 1998. 25(4): p. 316-319.

68. Meinhart, C.D., S.T. Wereley, and J.G. Santiago, A PIV algorithm for

estimating time-averaged velocity fields. Journal of Fluids Engineering-

Transactions of the Asme, 2000. 122(2): p. 285-289.

69. Mutlu, S., et al., Enhanced electro-osmotic pumping with liquid bridge and

field effect flow rectification. Micro Electro Mechanical Systems, 17th

IEEE International Conference on MEMS, 2004: p. 850- 853.

70. Meinhart, C.D., S.T. Wereley, and J.G. Santiago, PIV measurements of a

microchannel flow. Experiments in Fluids, 1999. 27(5): p. 414-419.

71. Prasad, A.K., et al., Effect of resolution on the speed and accuracy of

particle image velocimetry interrogation. Experiments in Fluids, 1992. 13:

p. 105-116.

72. Khair, A.S. and T.M. Squires, Surprising consequences of ion conservation

in electro-osmosis over a surface charge discontinuity. Under consideration

for publication in J. Fluid Mech., 2008.

73. Yariv, E., Electro-osmotic flow near a surface charge discontinuity. J.

Fluid Mech., 2004. 521: p. 181.

196

74. Johnson, C., Numerical solution of partial differential equations by the

finite element method. Studentlitteratur/Cambridge University Press, 1987.

75. Olesen, L.H., Computational Fluid Dynamics in Microfluidic Systems

Technical University of Denmark, 2003(Masters Thesis).

76. Olsen, M.G. and R.J. Adrian, Out-of-focus effects on particle image

visibility and correlation in microscopic particle image velocimetry.

Experiments in Fluids, 2000. 29(7): p. S166-S174.

77. Squires, T.M., A.S. Khair, and R. Messinger. Electrokinetic Flows Over

Rough Surfaces. in 2008 AIChE Annual Meeting. November 16-21, 2008.

Philadelphia, PA.

78. Monkowski, A., PhD Thesis, Microfabricated Structures and Devices

Featuring Nanostructured Titania Thin Films, in Department of

Mechanical Engineering. 2007, University of California: Santa Barbara,

CA.

79. Tengvall, P., et al., Interaction between hydrogen peroxide and titanium: a

possible role in the biocompatibility of titanium. Biomaterials, 1989. 10(2):

p. 118-120.

80. Ichinose, I., H. Senzu, and T. Kunitake, A Surface Sol-Gel Process of TiO2

and Other Metal Oxide Films with Molecular Precision. Chem. Mater.,

1997. 9(6): p. 1296-1298.

197

81. Wu, J.-M., et al., Porous titania films prepared from interactions of

titanium with hydrogen peroxide solution. Scripta Materialia, 2002. 46(1):

p. 101-106.

82. Jane P. Bearinger, C.A.O.J.L.G., Effect of hydrogen peroxide on titanium

surfaces: In situ imaging and step-polarization impedance spectroscopy of

commercially pure titanium and titanium, 6-aluminum, 4-vanadium.

Journal of Biomedical Materials Research Part A, 2003. 67A(3): p. 702-

712.

83. Mansuripur, T.S., A.J. Pascall, and T.M. Squires, Asymmetric induced

charge electro-osmotic flows over symmetric surfaces: the role of

capacitive coupling. Physical Review E, 2009. To be submitted.

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