Electrokinetics in Microfluidics, Volume 2 (Interface Science and Technology)

Electrokinetics in Microfluidics INTERFACE SCIENCE AND TECHNOLOGY Series Editor: ARTHUR HUBBARD In this series: Vol. ...

Author: Dongqing Li

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Electrokinetics in Microfluidics

INTERFACE SCIENCE AND TECHNOLOGY Series Editor: ARTHUR HUBBARD In this series: Vol. 1: Clay Surfaces: Fundamentals and Applications Edited by F. Wypych and K.G. Satyanarayana Vol. 2: Electrokinetics in Microfluidics By Dongqing Li

INTERFACE SCIENCE AND TECHNOLOGY - VOLUME 2


Dongqing Li Department of Mechanical & Industrial Engineering University of Toronto Toronto, Ontario, Canada

2004

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© 2004 Elsevier Ltd. All rights reserved. This work is protected under copyright by Elsevier Ltd, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier's Rights Department in Oxford, UK: phone (+44) 1865 843830, fax (+44) 1865 853333, e-mail: [email protected]. Requests may also be completed on-line via the Elsevier homepage (http://www.elsevier.com/locate/permissions). In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 20 7631 5555; fax: (+44) 207631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of the Publisher is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier's Rights Department, at the fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. First edition 2004 Library of Congress Cataloging in Publication Data A catalog record is available from the Library of Congress. British Library Cataloguing in Publication Data A catalogue record is available from the British Library. ISBN: 0-12-088444-5 ISSN: 1573-4285 (series) The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands.

Preface A lab-on-a-chip device is a microscale laboratory on a credit-card sized glass or plastic chip with a network of microchannels, electrodes, sensors and electronic circuits. Electrodes are placed at strategic locations on the chip. Applying electrical fields along microchannels controls the liquid flow and other operations in the chip. These labs on a chip can duplicate the specialized functions as performed by their room-sized counterparts, such as clinical diagnoses, PCR and electrophoretic separation. The advantages of these labs on a chip include significant reduction in the amounts of samples and reagents, very short reaction and analysis time, high throughput and portability. Generally, a lab-on-a-chip device must perform a number of microfiuidic functions: pumping, mixing, thermal cycling/incubating, dispensing, and separating. Precise manipulation of these microfiuidic processes is key to the operation and performance of labs on a chip. The recent, rapid development of the emerging microfiuidics and lab-on-achip technology brings a strong demand of understanding the interfacial electrokinetic phenomena in microfiuidic processes. This increasing demand results from two aspects. First, essentially all microfiuidic processes in lab-on-achip applications are electrokinetic processes, which are critical to the operation and performance of the lab-chip devices. Secondly, due to the micron scale of the fluidic channels, and the applications of the advanced microfabrication and the micro surface modification technology, the electrokinetic phenomena in microfiuidics become more complicated and have many unique features, such as the synergetic effects of the flow field and the electrical double layer field, and the coupled effects of surface roughness, surface heterogeneity and electrokinetics. The interfacial electrokinetic phenomena in microfiuidics have not been systematically discussed in any published books. Without sufficient knowledge of these phenomena, one cannot systematically design the microfiuidic and labchip devices and cannot control their operations. The objective of this book is to provide a fundamental understanding of the interfacial electrokinetic phenomena in several key microfiuidic processes, and to show how these phenomena can be utilised to control the microfiuidic processes. For this purpose, this book emphasises the theoretical modelling and the numerically simulation of these

vi

Preface

electrokinetic phenomena in micro fluidics. However, experimental studies of the electrokinetic microfluidic processes are also highlighted with sufficient detail. Chapter 1 gives an overview of the correlation between microfluidics and interfacial electrokinetics. Chapter 2 introduces the basic theory of the electrical double layer field that is required to understand the electrokinetic phenomena in the later chapters. Electrokinetic phenomena such as the streaming potential and the electro-viscous effect in pressure-driven flows in microchannels are discussed in Chapter 3. Since most on-chip microfluidic transport processes are driven by applied electrical fields, Chapters 4, 5, 6, 7, 8 and 9 are devoted to electrokinetic transport processes in microchannels under applied electrical fields. Chapter 4 introduces the basics of electroosmotic flows, transient electroosmotic flows, solution displacing processes, and the Joule heating effects in electroosmotic flow. Chapter 5 discusses the effects of surface heterogeneity on the flow structure and solution mixing. Both homogeneous and heterogeneous three-dimensional surface roughness elements in microchannels have a great influence on electrokinetic flow and mass transport, which is covered in Chapter 6. Chapter 7 introduces several experimental techniques for measuring the electroosmotic flow velocity, visualizing the flow field and temperature field in microchannels. Chapter 8 focuses on the on-chip electrokinetic sample dispensing processes. Chapter 9 discusses electrokinetic motion of micron-sized particles in microchannels. Since zeta potential is a key parameter in the studies of electrokinetic phenomena, Chapter 10 reviews two commonly-used techniques for measuring zeta potential. This book is not intended to provide a comprehensive review of all aspects of electrokinetic processes in microfluidics. The purpose is to introduce a fundamental understanding of the interfacial electrokinetic phenomena in microfluidics by studying some key processes. Considering the researchers and the graduate students who are new to this field, this book tries to provide many details of theoretical modelling, numerical simulations, and experimental set-up and procedures. This book reviews many recent research works in my laboratory. I greatly appreciate the contributions of my graduate students in these research works. Here I would like to name a few: David Erickson, David Sinton (now assistant professor at the University of Victoria), Liqing Ren (now assistant professor at the University of Waterloo), Chunzhen Ye, Yandong Hu and Xiangchun Xuan. Together we learn new things and make progress every day. I would also like to thank Professor Charles Chun Yang, Singapore Technological University, for providing me with his papers on transient electroosmotic flow. Finally, I am indebted to my wife, Liping Wang, and my daughter, Daphne, without their love and constant support this book would not have been written. Dongqing Li

Table of Contents Preface

V

Chapter 1 Lab-on-a-chip, microfluidics and interfacial electrokinetics 1 References 5 Chapter 2 Basics of electrical double layer 2-1 2-2

7 8

Introduction to electrical double layer (EDL) Basic electrokinetic phenomena in microfluidics References

28 29

Chapter 3 Pressure-driven flows in microchannels

30

3-1 3-2 3-3 3-4

Chapter 4 4-1 4-2 4-3 4-4 4-5 4-6 4-7 4-8

Pressure-driven electrokinetic flows in slit microchannels Pressure-driven electrokinetic flows in rectangular microchannels Measured electro-viscous effects New understanding of electro-viscous effects References

32 44 63 74 91

Electroosmotic flows in microchannels Electroosmotic flow in a slit microchannel Electroosmotic flow in a cylindrical microchannel Electroosmotic flow in rectangular microchannels Transient electroosmotic flow in cylindrical microchannels AC electroosmotic flows in a rectangular microchannel Electroosmotic flow with one solution displacing another solution Analysis of the displacing process between two electrolyte solutions Joule heating and thermal end effects on electroosmotic flow References

92 94 98 104 119 131 151 167 184 202

Chapter 5 Effects of surface heterogeneity on electrokinetic flow 5-1 5-2 5-3 5-4 5-5 5-6 5-7

Pressure-driven flow in microchannels with stream-wise heterogeneous strips Pressure-driven flow in microchannels with heterogeneous patches Electroosmotic flow in microchannels with continuous variation of zeta potential Electroosmotic flow in microchannels with heterogeneous patches Solution mixing in T-shaped microchannels with heterogeneous patches Heterogeneous surface charge enhanced micro-mixer Analysis of electrokinetic flow in microchannel networks References

204 206 215 238 251 268 288 298 318

viii

Table of Contents

Chapter 6 Effects of surface roughness on electrokinetic flow 6-1 6-2

Electroosmotic transport in a slit microchannel with 3D rough elements Effects of 3D heterogeneous rough elements References

Chapter 7 Experimental studies of electroosmotic flow 7-1 Measurement of the average electroosmotic velocity by a current method 7-2 Measurement of the average electroosmotic velocity by a slope method 7-3 Microfluidic visualization by a laser-induced dye method 7-4 Velocity profiles of electroosmotic flow in microchannels 7-5 Comparison of the current method and the visualization technique 7-6 Flow visualization by a micro-bubble lensing induced photobleaching method 7-7 Joule heating and heat transfer in chips with T-shaped microchannels 7-8 Joule heating effects on electroosmotic flow References

321 323 344 353 354 356 369 376 390 403 411 427 446 460

Chapter 8 Electrokinetic sample dispensing in crossing microchannels 463 8-1 8-2 8-3 8-4 8-5

Analysis of electrokinetic sample dispensing in crossing microchannels Experimental studies of on-chip microfluidic dispensing Dispensing using dynamic loading Effects of spatial gradients of electrical conductivity Controlled on-chip sample injection, pumping and stacking with liquid conductivity differences References

466 484 496 510 526 541

Chapter 9 Electrophoretic motion of particles in microchannels 9-1 Single spherical particle with gravity effects 9-2 Single cylindrical particle without gravity effects 9-3 Spherical particle in a T-shaped microchannel 9-4 Two particles in a rectangular microchannel References

542 546 563 579 599 616

Chapter 10 Microfluidic methods for measuring zeta potential 10-1 Streaming potential method 10-2 Electroosmotic flow method References

617 618 627 640

Subject Index

641

Lab-on-a-chip, Microfluidics and Interfacial Electrokinetics

1

Chapter 1

Lab-on-a-chip, microfluidics and interfacial electrokinetics The microfabrication technology has advanced microelectronics and computer technologies in an amazing speed, making the modern telecommunication and Internet technology possible, and consequently changed the way we work and the way we live profoundly. It allows rapid new technology development and dramatic cost reduction. Scientists and engineers have realized the tremendous advantages of the microfabrication technology and the enormous potential of applying the microfabrication technology to other fields such as mechanical engineering and biomedical engineering. This leads to the recent rapid development of Micro-Electro-Mechanical Systems (MEMS) and Laboratory-on-a-Chip (LOC) devices. A lab-on-a-chip (LOC) is a microscale chemical or biological laboratory built on a thin glass or plastic plate with a net work of microchannels, electrodes, sensors and electronic circuits. The width or the height of a typical microchannel ranges from 20 to 200 um. Applying electrical fields through the electrodes along microchannels controls the liquid flow and other operations in the chip. These labs on a chip can duplicate the specialized functions as their room-sized counterparts, such as clinical diagnoses of bacteria and viruses, and DNA electrophoretic separation. The advantages of these labs on a chip include dramatic reduction in the amount of the samples and reagents, very short reaction and analysis time, high throughput, automation and portability [1-5]. In conventional chemistry and biology laboratories, an experiment is generally carried out as a series of separate operations (such as measuring samples, mixing solutions, and incubating) using separate tools and techniques. Many different instruments are involved from simple devices such as beakers, pipettes, stirring hot plate, centrifuge and incubator to more sophisticated instruments for PCR amplification, electrophoresis and fluorescent microscopy. Generally, the sample preparation prior to measurements is conducted manually and is labour intensive. Due to the relatively large size of the instruments, a large amount of the reagents or samples is required. This results in higher operating cost and longer time for completing the reaction and analysis. Conducting experiments with different samples or reagents require performing the costly and time-consuming separate experiments. These manual and individual experimental procedures may result in more chances for errors.

2


A LOC device generally consists of a number of integrated microfluidic components such as pump, mixer, reactor, dispenser and separator, as illustrated in Figure 1. Therefore, multiple steps of an experiment can be conducted automatically on a single chip. For example, a sample of an unknown, singlestranded DNA solution and a solution containing a known, single-stranded DNA tagged with fluorescent dye are pumped from the reagent loading wells into a mixer by applying electric fields through the related electrodes. The mixed solution will then flow into a reactor where the unknown DNA fragments will react with the dye tagged DNA probe molecules (i.e., hybridization) at a specified temperature. The matched DNA samples will bind with the DNA probe. Following that, the reaction product will be pumped to the dispenser section. Then, by switching on another electrical field, a plug of DNA molecules will be dispensed into a buffer solution and flow into a separation microchannel where they are separated according to the charge to mass ratio by electrophoresis. Finally, when the separated DNA molecules enter the detection section of the microchannel, a laser beam is applied. The dye causes the DNA fragments to give off light when a laser beam is shone on them. The larger the separated fragment, the stronger the fluorescence. The detected light intensities are fed to a computer which sorts through signals from separated fragments to provide a sample analysis. Because of the size of the microchannels, the amount of the liquids involved in such a LOC is of the order of nanoliters, and hence the required amount of the samples and reagents are significantly less than that required in conventional lab experiments. Furthermore, using the microfabrication technology, we can easily make many parallel microchannel systems on a single chip, so that one chip can perform multiple tasks at the same time.

Figure 1.1.

Illustration of microfluidic components in a lab-on-a-chip device.


3

Currently, improving the technology and reducing the cost in health care is a major driving force for rapid development of LOC technology. The demand to apply LOC technology to genomics and proteomics research, high-throughput screening, drug discovery, point-of-care clinical diagnostic devices has been increasing remarkably over the last decade. There are many examples of the applications of LOC, including micro-total analysis system (u-TAS) [6], microfluidic capillary electrophoretic separation [7,8], electrochromatography [9], PCR amplification [10-14], mixing [15,16], flow cytometry [17], sample injection of proteins for analysis via mass spectrometry [18-20], DNA analysis [21-24], cell manipulation [25], cell separation [26], cell patterning [27,28], fluid handling [29], immunoassay [30-37], enzymatic reactions [38-41], and molecular detection [42]. A recent review of integrated LOC devices can be found elsewhere [43] The most important media in the biomedical analysis and diagnostics are liquids. Common liquids used in LOC devices include whole blood samples, bacterial cell suspensions, protein or antibody solutions and various buffers. Therefore, a key to the functions of the LOC is the quantitatively controlled flow, mass (e.g., sample molecules and particles) transport and heat transfer processes in microchannels. The studies of the transport processes in microchannels are referred to as the microfluidics. Generally, a lab-on-a-chip device must perform the following microfluidic functions: pumping, metering, mixing, flow switching, thermal cycling or incubating, sample dispensing or injection, and separating molecules or particles, etc. Precise manipulation of these microfluidic processes is key to the operation and performance of LOC. Generally, we may classify the transport processes into three categories according to the characteristic dimension, Lc, of the systems: (1) Macroscale systems: Lc > 200 um. (2) Microscale systems: 100 nm < Lc < 200 um. (3) Nanoscale systems: Lc< 100 nm. The characteristics of the transport processes change significantly as the characteristic dimension of the system changes from one category to another category. It should be noted that microchannels have very large surface area to volume ratio. For example, for a microchannel of 100 um in diameter, the surface area-volume ratio is: (2TTRL/7IR2L) = (2/R) = 2x10 (m~ ). Therefore, one can expect significant influence of the liquidchannel wall interface on the microfluidic processes. Because most solid-liquid interfaces have electrostatic charge and consequently an electrical field near the interface, the interfacial electrokinetic phenomena are very important to microfluidic processes. In fact, most of the microfluidic processes on a LOC are electrokinetic processes. For example, electroosmosis is used to generate liquid motion or pump the liquid through microchannels; electrophoresis is used to separate molecules and particles in microchannels. Therefore, interfacial electrokinetic phenomena dominate these microscale transport processes.

4


Because of the complex of the electrokinetic phenomena, the characteristics and controlling parameters of the microfluidic processes vary from system to system and from application to application. Conventional theories of the transport phenomena for macroscopic systems are generally not applicable in microfluidics. At the present, the lack of understanding of the complicated electrokinetic transport phenomena in microchannels makes it difficult to do systematic design and precise operation control of the labs on a chip. It is true that the microfabrication capability is needed to make a LOC device, one must realise that the fundamental understanding of the microfluidic transport processes is essential for the design and for the operation control of LOC devices. Therefore, this book is devoted to provide basic understanding of electrokinetic phenomena in some key microfluidic processes involved in LOC devices.


5

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J. Knight, Nature, 418 (2002) 474-5. M. Freemantle, Chemical & Engineering News, 77 (1999) 27-36. I. Wickelgren, Popular Science, Nov. 1998, 57-61. G. Sinha, Popular Science, Aug. 1999, 48-52. S. Borman, Chemical & Engineering News, Feb. 1999, 30-31. S.C. Jakeway, A.J. de Mello and E.L. Russell, Fresenius J. Anal. Chem., 366 (2000) 525-39. D.J. Harrison, K. Fluri, K. Seiler, K. Seiler, Z. Fan, C. Effenhauser and A. Manz, Science, 261 (1993) 895-7. S.B. Cheng, C. D. Skinner, J. Taylor, S. Attiya, W. E. Lee, G. Picelli, and D. J. Harrison, Anal. Chem., 73 (2001) 1472-1479. R.D. Oelschuk, L.L. Shultz-Lockyear, Y. Ning and D. J. Harrison, Anal. Chem., 72 (2000) 585-90. M.U. Kopp, A.J. de Mello and A. Manz, Science, 280 (1998) 1046-8. D. Erickson and D. Li, Int. J. Heat Mass Transfer, 45 (2002) 3759-3770. P. Belgrader, M. Okuzumi, F. Pourahmadi, D.A. Borkholder and M. Northrup, Biosensors & Bioelectronics, 14 (2000) 849-852. J. Khandurina, et al., Anal. Chem., 72 (2000) 2995-3000. E.T. Lagally, I. Medintz and R.A. Mathies, Anal. Chem., 73 (2001) 565-570. A.D. Stroock, S.K.W. Dertinger, A. Ajdari, I. Mezic, H.A. Stone and G.M. Whitesides, Science, 295 (2002) 647-651. D. Erickson and D. Li, Langmuir, 18 (2002) 1883-1892. L.L. Sohn, et al., Proceedings of the National Academy of Sciences of the United States of America, 97 (2000) 10687-10690. D. Figeys, S.P. Gygi, G. McKinnon and R. Aebersold, Anal. Chem., 70 (1998) 37283734. Y. Jiang, P.C. Wang, L.E. Locascio and C.S. Lee, Anal. Chem., 73 (2001) 2048-2053. J. Gao, J.D. Xu, L.E. Locascio and C.S. Lee, Anal. Chem., 73 (2001) 2648-2655. B.A. Buchholz, et al., Anal. Chem., 73 (2001) 157-164. Z.H. Fan, et al., Analy. Chem, 71 (1999) 4851-4859. L. Koutny, et al. Anal. Chem, 72 (2000) 3388-3391. G.B. Lee, S.H. Chen, G.R. Huang, W.C. Sung and Y.H. Lin, Sensors and Actuators BChemical, 75 (2001) 142-148. I.K. Glasgow, et al, IEEE Transactions On Biomedical Engineering, 48 (2001) 570-578. J. Yang, Y. Huang, X.B. Wang, F.F. Becker and P.R. Gascoyne, Anal. Chem, 71 (1999) 911-918. D.T. Chiu, et al. Proceedings of the National Academy of Sciences of the United States of America, 97 (2000) 2408-2413. A. Folch, B.H. Jo, O. Hurtado, D.J. Beebe and M. Toner, J. Biomedical Materials Research, 52 (2000) 346-353. D.D. Cunningham, Anal. Chim. Acta, 429 (2001) 1-18. A. Hatch, et al. Nature Biotechnology, 19 (2001) 461-465. E. Eteshola and D. Leckband, Sensors and Actuators B-Chemical, 72 (2001) 129-133. S.B. Cheng, et al. Anal. Chem, 73 (2001) 1472-1479. T.L. Yang, S.Y. Jung, H.B. Mao and P.S. Cremer, Anal. Chem, 73 (2001) 165-169.

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[34] D.L. Stokes, G.D. Griffin and T. Vo-Dinh, Fresenius J. Anal. Chem., 369 (2001) 295301. [35] K. Sato, M. Tokeshi, T. Odake, H. Kimura, T. Ooi, M. Nakao and T. Kitamori, Anal. Chem., 72 (2000) 1144-7. [36] A. Dodge, K. Fluri, E. Verpoorte and N.F. de Rooij, Anal. Chem., 73 (2001) 3400-9. [37] M. Sharma, A. Saxena, S. Ghosh, J.C. Samantaray and G.P. Talwar, Indian J. Med. Res., 88(1998)409-15. [38] A.G. Hadd, D.E. Raymond, J.W. Halliwell, S.C. Jacobson and J.M. Ramsey, Anal. Chem., 69(1997)3407-3412. [39] D.C. Duffy, H. L. Gillis, J. Lin, N.F. Sheppard and G J . Kellogg, Anal. Chem., 71 (1999) 4669-4678. [40] A.G. Hadd, S.C. Jacobson and J.M. Ramsey, Anal. Chem, 71 (1999) 5206-5212. [41] S.C. Jacobson and J.M. Ramsey, Anal. Chem, 68 (1996) 720-3. [42] B.H. Weigl and P. Yager, Science, 283 (1999) 346-7. [43] D. Erickson and D. Li, Anal. Chem. Acta, 507 (2004) 11-26.

Basics of Electrical Double Layer

7

Chapter 2

Basics of electrical double layer There are many excellent books dealing with electrokinetics and interfaces associated with colloidal particles, such as the books by Hunter [1] and Lyklema [2]. This chapter does not intend to duplicate these references, instead, the objective of this chapter is to provide the basic understanding of the electrical double layer field required in the later chapters dealing with different microfluidic processes. More in-depth discussions of various electrokinetic phenomena will be provided in these chapters whenever appropriate.

8

2-1


INTRODUCTION TO ELECTRICAL DOUBLE LAYER (EDL)

2-1.1 Electrical field in a dielectric medium In this chapter, we will discuss the electrical filed near solid-liquid and liquid-fluid interfaces. Since we will deal with dielectric media in most applications, it is worthwhile to mention some features of dielectric materials. Dielectric materials include plastics, organic liquids, water (including aqueous electrolyte solutions) and gases. Generally, molecules of many dielectric materials are permanently polarized due to their asymmetrical molecular structure. A simple example is HC1. For dielectric materials containing symmetrical molecules or atoms they can also become polarized when exposed to an electrical field, resulting from the relative displacement of orbital electrons, such as the case of He. Therefore, in the presence of an electrical field, molecules of all dielectric materials have a dipole that comprises two equal and opposite charges separated by a distance. The Poisson equation describes the electrical potential in a dielectric medium.

(i) where y/ is the electrical field potential, p is the free charge density and s and s0 are the dielectric constants in the medium and in the vacuum, respectively. The assumption for this form of the Poisson equation is that the dielectric constant s is a constant, independent of position. The Poisson equation is one of the fundamental equations used to evaluate the electrical potential in an electrolyte solution such as NaCl in water. In situations where there is no free charge, for example, in pure organic liquids such as pure oils, p « 0, or in an electrically neutral aqueous solution (i.e., net charge density is zero), the Poisson equation becomes the well-known Laplace equation: (2)

Examples of typical values of dielectric constant are given in the Table 1. 2-1.2 Origin of surface charge Most materials obtain surface electric charge when they are brought into contact with an aqueous solution. The origin of the surface charge includes: (1) Different affinities for ions of different signs to two phases. This may include the following situations: (a) The distribution of anions and cations

9


between two immiscible phases such as oil and water, (b) The preferential adsorption of certain type ions from an electrolyte solution on to a solid surface. For example, surfactant ions specifically adsorbed on a surface result in a positively charged surface if the surfactant is a cationic surfactant, and a negatively charged surface if the surfactant is an anionic surfactant, (c) The differential solution of one type ion over the other from a crystal lattice. For example, when a silver iodide crystal (Agl) is placed in water, dissolution takes place until the product of ionic concentration equals the solubility product [Ag+][I~] = 10~16 (mol/L)2. If equal amounts of Ag+ and F ions were to dissolve, then [Ag+] = [F] =10~8 and the surface would be uncharged. However, silver ions dissolve preferentially, leaving a negatively charged surface. (2) Ionization of surface groups. If a surface has acidic groups (e.g., COOH on the surface), the acidic groups dissociation (e.g., COO" on the surface and H+ in the aqueous solution) will result in a negatively charged surface. If a surface contains basic groups (e.g., OH on the surface), the dissociation of the basic groups (e.g., release OFF in the aqueous solution, and leave a positive charge on the surface) will generate a positively charged surface. In both cases, the magnitude of the surface charge depends on the acidic or basic strength of the surface groups and on the pH of the solution. Decreasing pH will reduce the surface charge for a surface containing acidic groups. Increasing pH will reduce the surface charge for a surface containing basic

Table 1 Examples of dielectric constants Liquid or Solid

Dielectric constant

Water

H2O

E = 80.1 at T = 20°C

Water

H2O

8 = 78.5 at T = 25°C

Ethylene glycol

C2H4(OH)2

£ = 40.7 at T = 25°C

Methanol

CH3OH

E = 32.6 at T = 25°C

Dodecane

Cj2H26

s = 2.0 a t T = 25°C

Sodium chloride

NaCl

s = 6.0 atT = 25°C

Quartz

SiO2

£ = 4.5

atT = 25°C

An empirical relationship between water's dielectric constant and temperature is 0.391T + 0.000741T2. For gases at atmospheric pressure, e « 1 and s increases with pressure. For example, = 1.0006 at 1 atm, and = 1.05 at 100 atm.

10


groups. The surface charge can thus be reduced to zero. Most metal oxides can have either positive or negative surface charge depending on the bulk pH. (3) Charged crystal surfaces. When a crystal is broken, different surfaces may have different properties including different surface charges. 2-1.3 Electrical double layer (EDL) When in contact with an aqueous solution, most solid surfaces carry electrostatic charges or an electrical surface potential. Generally, the solution is electrically neutral (having an equal number of positively charged ions and negatively charged ions). However, the electrostatic charges on the solid surface will attract the counterions in the liquid (i.e., an electrolyte solution or a liquid with impurities). Because of the electrostatic attraction, the counterion concentration near the solid surface is higher than that in the bulk liquid far away from the solid surface. The coion concentration near the surface, however, is lower than that in the bulk liquid far away from the solid surface, due to the electrical repulsion. Therefore there is a net charge (excess counterions) in the region close to the surface. This net charge should balance the charge at the solid surface. The charged surface and the layer of liquid containing the balancing charges is called the electrical double layer (EDL) [1,2], as illustrated in Figure 2.1. Immediately next to the charged solid surface, there is a layer of ions that are strongly attracted to the solid surface and are immobile. This layer is called the compact layer, normally about several Angstroms thick. The charge and potential distributions in the compact layer are mainly determined by the geometrical restrictions of ion and molecule size and the short-range interactions between ions, the wall and the adjoining dipoles. From the compact layer to the electrically neutral bulk liquid, the net charge density gradually reduces to zero. Ions in this region are affected less strongly by the electrostatic interaction and are mobile. This region is called the diffuse layer of the EDL. As will be discussed below, an equation called the Poisson-Boltzmann equation is used to describe the ion and potential distributions in the diffuse layer. The thickness of the diffuse layer is dependent on the bulk ionic concentration and electrical properties of the liquid, usually ranging from several nanometers for high ionic concentration solutions up one or two microns for pure water and pure organic liquids (with extremely low ionic concentration). The boundary between the compact layer and the diffuse layer is usually referred to as the shear plane. In conventional fluid mechanics, the liquid velocity at the shear plane is usually set to be zero, and used as a boundary condition. From the above discussion of the compact layer, one can easily understand why the velocity at the shear plane is zero. Generally, The electrical potential at the solid-liquid interface is difficult to


11

Figure 2.1a. Illustration of the ionic concentration field in an electrical double layer for a flat surface in contact with an aqueous solution.

Figure 2.1b Illustration of an electrical double layer potential field for a flat surface in contact with an aqueous solution, K = [ 2 z2 e2 nx I er so fa T ]m.

12


measure directly. The electrical potential at the shear plane, however, can be measured experimentally [1,2], this potential is called the zeta potential, g, and is considered as an approximation of the surface potential in most electrokinetic models. 2-1-4 Boltzmann distribution In thermodynamics, Boltzmann's distribution refers to Boltzmann's probability law. It states that the probability of an isolated system taking a thermodynamic equilibrium state with an energy W is proportional to exp(-W/kbT), where T is the absolute temperature (K) and kb is the Boltzmann constant (J/K). Applying the Boltzmann probability to the ionic number concentration (number per unit volume) distribution near a charged surface in a symmetric (i.e., valence ratio z:z) electrolyte solution, we have

Here n+ and n are the ionic number concentrations for the cations and anions at a given location in the liquid, y/ is the electrical field potential (V) at a given position in the liquid, z is the absolute value of the ionic valence, e is the fundamental charge of an electron, « J = n^ = n^, nx is the ionic number concentration in the bulk solution (infinite away from the charged surface). In fact, the Boltzmann distribution can be derived more rigorously from equilibrium thermodynamics. Consider a charged surface and the surrounding electrolyte solution in an equilibrium state. At equilibrium the electrochemical potential of the ions must be constant everywhere, i.e.,

In other words, the electrical force and the diffusion force on the ion must balance each other.

For a one-dimensional flat surface system, the above equation can be rewritten as:


13

As the chemical potential (per ion) is given by:

where nx is the number of ions of type i (i.e., n+ or n ) per unit volume, we have

If we integrate this equation from a point in the bulk solution to a point in the EDL, i.e., using the following boundary conditions: In the bulk solution:

where nf is the bulk ionic number density per unit volume of type / ion. This means that at positions infinitely far away from the charged solid surface the bulk solution is electrically neutral or has zero net charge. In the EDL region:

we will obtain the Boltzmann equation:

The above equations give the so-called Boltzman distribution of the ions near a charged surface. In the above derivation, the implied assumptions are: (l)The system is in equilibrium, ions have no macroscopic motion (i.e., no convection and diffusion). (2)The system is subject only to a homogeneous surface electrical double layer field as characterized by the potential i//. For a microscopically heterogeneous surface, the electrical double layer field varies from point to point, Eq. (9) may not be valid. See Chapter 5 for more general treatments.

14


(3) The charged surface is in contact with an infinitely large liquid medium. At positions infinitely away from the charged surface, the potential y/ is zero and the ionic concentration is nx (i.e., the bulk solution boundary condition). Although the majority of the analyses of electrokinetic processes is based on the Boltzman distribution, it should be aware that the conditions listed above may not be satisfied in many situations. For example, in the cases of liquid flow, the system is not in equilibrium. The equilibrium condition, Eq. (4) is not valid in principle. Therefore the applicability of the Boltzman distribution, Eq. (9), is questionable. However, the ion distribution near the solid-liquid interface will not significantly deviate from the Boltzman distribution unless high speed flows are involved (e.g., Re > 10), as will be discussed later (Chapter 5). In the cases of very low speed flows, we can still use the Boltzmann distribution as a good approximation, as proved below. Consider a differential volume element in a flowing electrolyte solution. At a steady state without chemical reaction, the conservation of ion species requires the divergence of the mass fluxes to be zero.

where the jj is the flux of the ith species. Using the Nernst-Planck equation, we have

where the right hand side terms are the flux due to bulk convection, the flux due to the concentration difference (the diffusion process) and the flux due to the migration in an electrical field. Combining Eq. (10) and Eq. (11) yields:

If the flow of the solution is very slow (as it is in most microfluidic applications), the first term can be neglected. If we consider Dt is a constant, Eq. (12) reduces to the following:


15

Under the same boundary conditions, the Boltzmann distribution can be easily derived from Eq. (13). Therefore, if the Boltzmann distribution is valid in lowspeed flow situations, we can use the Poisson-Boltzmann equation (see the next section) to describe the electrical double layer field in these situations. This implies that the electrical double layer field is independent of the flow field. For systems under an externally applied electrical field, one may question the validity of the Boltzmann distribution as well. Consider a typical zeta potential value of lOOmV. If the EDL thickness is of the order of lOnm, the EDL field strength can be estimated as 100 mV/lOnm or 1 x 107 V/m. If the applied electrical field strength is not extremely high, for example, less than lx 105 V/m (or 1 x 103 V/cm), its influence on the EDL field and hence on the ionic distribution is negligible in comparison with the EDL field strength. We can therefore use the Boltzmann distribution and hence the Poisson-Boltzmann equation to describe the EDL field in the studies of flow systems with an applied electrical field such as electroosmotic flow. As another caution, if a liquid is confined in a submicron-sized capillary, the electrical double layer fields of the opposite solid surfaces may overlap. The potential and the net charge density in the middle plane are not zero. Obviously the boundary condition in the bulk solution will be different from what was described above. Since the boundary condition is different, the solution to the differential equation, Eq. (8) will be different, in turn, Eq. (9) is no longer valid in this case [3].

2-1.5 Theoretical model and analysis of EDL According to the theory of electrostatics, the relationship between the electrical potential y/ and the local net charge density per unit volume pe at any point in the solution is described by the Poisson equation: (i) where s is the dielectric constant of the solution. Assuming the equilibrium Boltzmann distribution equation is applicable, the number concentration of the type-i ion in a symmetric electrolyte solution is of the form (9)

16


where niao and z7- are the bulk ionic concentration and the valence of type-i ions, respectively, e is the charge of a proton, Kb is the Boltzmann constant, and T is the absolute temperature. The net volume charge density pe is proportional to the concentration difference between symmetric cations and anions, via

For arbitrary electrolyte solutions that may contain asymmetric ions, the net volume charge density pe is

Obviously, when z, = z = constant, i.e., symmetrical ions, Eq. (15) reduces to Eq. (14). Substituting Eq. (14) into the Poisson equation, Eq. (1), leads to the wellknown Poisson-Boltzmann (P-B) equation. (16) Apparently, the combination of Eq. (1) and Eq. (15) will give a different (more general) form of the P-B equation.

By defining the Debye-Huckel parameter dimensional electrical potential

and the non-

the Poisson-Boltzmann equation, Eq.

(16), can be re-written as:

Generally, solving this equation with appropriate boundary conditions, the electrical potential distribution y/ of the EDL can be obtained, and the local charge density distribution pe can then be determined from Eq. (14).


It

should

be

17

noted that the Debye-Huckel parameter is independent of the solid surface properties and is determined by the liquid properties (such as the electrolyte's valence and the bulk ionic concentration) only. \lk is normally referred to as the characteristic thickness of EDL and is a function of the electrolyte concentration. Values of \lk range, for example, from 9.6 nm at 10~3 M to 304.0 nm at 10"6 M for a KC1 solution. When the ionic concentration is 10~6 M, the solution is considered practically the pure water. The thickness of the diffuse layer usually is about three to five times of Ilk, and hence may be larger than one micron for pure water and pure organic liquids. Let's see an example of how to calculate 1/k. Consider pure water at T = 298K and use the following parameters: a = 78.5, so = 8.85 x 10~12 C2/Nm2, e = 1.602 x 10~19 C, kb = 1.381 x 10~23 J/K, and Na = 6.022 x 1023 /mol. Note that ttoo is the bulk ionic number concentration and is expressed in terms of the molarity M (mole/liter) by:

Put all the above values into

and we have

here Mis the molarity of a symmetrical (z:z) electrolyte. For example, z =1, we have

18


As seen from the above table, when the bulk ionic concentration increases, more counterions are attracted to the region close to the charged solid surface to neutralize the surface charge. Consequently, the double layer thickness is reduced, and it seems that the EDL is "compressed". 2-1.6 EDL field near a flat surface Consider two flat surfaces separated by a distance 2a in an aqueous solution, as illustrated in the Figure 2.2 below. The EDL filed is onedimensional. If we set the origin of the X-axis in the middle plane between the two plats, the P-B-equation can be re-written as: (19)

If the distance a is much larger than the EDL thickness, the boundary conditions are: (i.e., infinitely away from the surface) (i.e., at the surface or shear plane)

Figure 2.2.

Illustration of a system that consists of two identical flat surfaces.


19

If the electrical potential is small compared to the thermal energy of the ions, i.e., (zey/\< A^r|) so that the right-hand side term in Eq. (19) can be approximated by the first terms in a Taylor series. This transforms Eq. (19) to (20) In literature, this treatment is called the Debye-Huckle linear approximation. At 25°C, this linear assumption requires that y/ < 25 mV. The solution of the above equation can be easily obtained as: sinh(£X)

(21)

For a flat surface in contact with an infinitely large aqueous solution, if we set the origin of the coordinate system on the wall, as illustrated in Figure 2.3 below, the P-B equation in dimensional form is given by (22)

with the boundary conditions: (i.e., at the surface or shear plane) (i.e., infinitely away from the surface) Eq. (22) has the following solution, if the Debye-Huckle linear approximation is used, (23) It should be noted that distance EDL thickness

(24)

is often refereed to as the electrokinetic distance, a measure of the distance relative to the EDL thickness.

20


Figure 2.3. The electrical double layer field near a flat surface in an infinitely large liquid.

For the flat surface system, it is possible to obtain an exact analytical solution to the P-B equation without using the linear assumption (see Hunter's book[l]): (25)

(26)

In this equation, k is the Debye-Huckle parameter. The boundary conditions are the following: At the surface or the shear p l a n e , ; and at points in the bulk liquid infinitely away from the surface, 0. A comparison of the EDL potential predicted by the exact solution (Eq.(26)) and the linear solution (Eq. (23)) is given in Figure 2.4. Once is known, the ionic concentration distribution and the net charge density distribution p (x) can be determined by using Eq. (9) and Eq.(14), respectively. Figure 2.5 illustrates the ionic concentration distributions predicted by using the exact solution (Eq.(26)) and the linear solution (Eq. (23)).


21

Figure 2.4. Comparison of EDL potential fields predicted by the linear solution and the non-linear (exact) solution.

Figure 2.5. Ion distribution in diffuse layer: Accumulation of positive ions and expulsion of negative ions in the EDL region of a negatively charged surface.

22


As seen from the above figure, the ions in the diffuse layer are not all of the same sign. At a given position from the charged surface, the concentration difference between the positive ions and the negative ions determines the local net charge density. The charge on the wall surface is balanced by (1) an accumulation of charges of opposite sign (i.e., counterions) and (2) a deficit of charges of the same sign (co-ions) compared to their concentrations in the bulk liquid. The charge per unit area on the surface is called the surface charge density, denoted by a 0 (C/m2). a0 must be balanced by the charge in the adjacent solution: (27)

From the Poisson equation,

we have

Integrating the P-B equation

yields (translate the result in dimensional form):

the surface charge density is related to the surface potential via the following equation:


23

(28)

2-1.7 EDL field around a spherical surface For the ID EDL field around a spherical particle of radius a, the P-B equation can be written as: (29) The boundary conditions are: (At the sphere surface or the shear plane) (Infinitely away from the sphere surface) Using the Debye-Huckle linear approximation, it can be shown that (30) The charge on the particle surface must balance the charge in the EDL so that

Use the Poisson equation

and the linearized P-B equation

we have

24


(31)

(32) Using the above equations the potential on the surface of the sphere can be written as (33)

The first term represents the potential on the surface due to the charge on the spherical particle itself. The second term is the potential due to the atmosphere of charge of opposite sign, i.e., it is the potential due to a spherical shell of charge -Q and radius {a + 1/k). If we can measure the zeta potential, £, experimentally, the electrokinetic charge on the particle is given by

2-1.8 EDL field around a cylinder If we consider cylindrical symmetry (i.e., neglecting the end effects), the P-B equation for a cylinder of radius a in a symmetrical electrolyte solution is given by, in cylindrical coordinates,


25

where R = kr. Under the linear assumption, (zet//) < kt,T, the solution is: (36)

(37)

(38)

(39)

2-1.9 Dependence of surface charge and zeta potential on ion concentration and pH The effects of the bulk ionic concentration on the EDL potential and particularly the surface potential (or approximately the zeta potential) can be analyzed qualitatively as follows. When we derived the Boltzmann distribution, we showed the following equilibrium condition:

This equation can also be re-arranged into (40)

26


Figure 2.6. Illustration of the surface charge density dependence on pH and the bulk ionic concentration.

Figure 2.7. Illustration of the zeta potential dependence on pH and the bulk ionic concentration.


Consider Kb = 1.381x10 1, we have

23

27

J/K, e = 1.602 xlO" 19 C, T = 298K, and assume z,•.=

This equation implies that a ten-fold increase in the ion concentration, nh will have a relatively small change in the potential. Note that if we choose z, = - 1 , the sign in the above equation will change, indicating that the effect of positive and negative ions on the potential change is opposite. Generally, for a given solid surface, the surface charge and zeta potential are functions of the bulk ionic concentration and valence as well as pH, as shown in Figures 2.6 and 2.7. Clearly, for a given electrolyte solution, the surface charge and zeta potential can be changed from positive to negative by varying the pH value of the solution.

28

2-2


BASIC ELECTROKINETIC PHENOMENA IN MICROFLUIDICS

Electroosmosis Consider a stationary solid surface in contact with an aqueous solution. When an electric field is applied, the excess counterions in the diffuse layer of the EDL will move under the applied electrical force. This is called the electroosmosis. As the ions move, they drag the surrounding liquid molecules to move with them due to the viscous effect, resulting in a bulk liquid motion. Such a liquid motion is called the electroosmotic flow. For example, by applying an electric field along a microchannel, we can electroosmotically "pump" liquids to flow through the microchannel. Electrophoresis Consider a (solid, liquid or gas) particle in a bulk liquid phase. When an electric field is applied to the bulk liquid, because the particle surface has electrostatic charge, the particle can be induced to move (relative to the stationary or moving liquid) under the applied electrical field. Such a particle motion is called the electrophoresis. Streaming potential In absence of an applied electric field, when a liquid is forced to flow through a capillary or microchannel under an applied hydrostatic pressure difference, the counterions in the diffuse layer (mobile part) of the EDL are carried towards the downstream end, resulting in an electrical current in the pressure-driven flow direction. This current is called the streaming current. Corresponding to this streaming current, there is an electrokinetic potential called the streaming potential. This flow induced streaming potential is a potential difference that builds up along a capillary or microchannel. This streaming potential acts to drive the counterions in the diffuse layer of the EDL to move in the direction opposite to the streaming current, i.e., opposite to the pressure-driven flow direction. The action of the streaming potential will generate an electrical current called the conduction current. At a steady state, the streaming current will be balanced by the conduction current, and hence the net current in the capillary/microchannel is zero.


29

REFERENCES [1] [2] [3]

R.J. Hunter, "Zeta Potential in Colloid Science: Principles and Applications", Academic Press, London, 1988. J. Lyklema, "Fundamentals of Interface and Colloid Science", Vol. I and II, Academic Press, 1995. W. Qu and D. Li, J. Colloid Interface Sci., 224 (2000) 397-407.

30


Chapter 3

Electro-viscous effects on pressure-driven liquid flow in microchannels Just as rapid advances in microelectronics have revolutionized computers, appliances, communication systems and many other devices, microfluidic technologies will revolutionize many aspects of applied sciences and engineering, such as heat exchangers, pumps, gas absorbers, solvent extractors, fuel processors, on-chip biomedical and biochemical analysis instruments. These lightweight, compact and high-performance micro-systems will have many important applications in transportation, buildings, military, environmental restoration, space exploration, environmental management, biochemical and other industrial chemical processing. Fundamental understanding of liquid flow in microchannels is critical to the design and process control of various microfluidic and lab-on-chip devices used in chemical analysis and biomedical diagnostics (e.g., miniaturized total chemical analysis system). However, many phenomena of liquid flow in microchannels, such as unusually high flow resistance, cannot be explained by the conventional theories of fluid mechanics. These are largely due to the significant influences of interfacial electrokinetic phenomena such as electroviscous effects at the micron scale. Although interfacial electrokinetic phenomena such as electro-osmosis, electrophoresis and electro-viscous effects are well known to colloidal and interfacial sciences for many years, the effects of these phenomena on transport phenomena (such as liquid flow and mixing in fine capillaries) generally are less well understood. Partially this is because in colloidal sciences the electrokinetic phenomena usually are studied for closed systems, while in the studies of micro transport phenomena the systems are open systems and involve complicated boundary conditions. In various microfluidic processes, a desired amount of a liquid is forced to flow through microchannels from one location to another. Depending on the specific structures of the microfluidic devices, the shape of the cross-section of the microchannels varies. Two typical cases are the slit microchannels (formed between two parallel plates) and the trapezoidal (may be approximated as rectangular) microchannels made by microfabrication processes. This chapter

Electro-viscous Effects on Pressure-driven Liquid Flow in Microchannels

31

will show how to model and evaluate the interfacial electrokinetic effects on pressure-driven liquid flow through these microchannels. The concept of streaming potential, streaming current, conduction current, electro-viscous effects will be introduced. Some existing experimental evidences of the electroviscous effects will also be presented.

32

3-1


PRESSURE-DRIVEN ELECTROKINETIC FLOW IN SLIT MICROCHANNELS

When a liquid is forced through a microchannel under an applied hydrostatic pressure, the counterions in the diffuse layer (mobile part) of the EDL are carried towards the downstream end, resulting in an electrical current in the pressure-driven flow direction. This current is called the streaming current. Corresponding to this streaming current, there is an electrokinetic potential called the streaming potential. This flow induced streaming potential is a potential difference that builds up along a microchannel. This streaming potential acts to drive the counterions in the diffuse layer of the EDL to move in the direction opposite to the streaming current, i.e., opposite to the pressuredriven flow direction. The action of the streaming potential will generate an electrical current called the conduction current, as illustrated in Figure 3.1. It is obvious that when ions move in a liquid, they will pull the liquid molecules to move with them. Therefore, the conduction current will produce a liquid flow in the opposite direction to the pressure driven flow. The overall result is a reduced flow rate in the pressure drop direction. Under the same conditions, if the reduced flow rate is compared with the flow rate predicted by the conventional fluid mechanics theory without considering the presence of the EDL, it seems that the liquid would have a higher viscosity. This is usually referred to as the electro-viscous effect [1,2]. In this section, we consider liquid flow through a microchannel with a slitshaped cross-section [3,4], such as a channel formed between two parallel plates, as illustrated in Figure 2. We assume the length, L, and the width, W, of the slit channel are much larger than the height, H = 2a, of the channel, so that both the electrical double layer (EDL) field and the flow field can be considered as one dimensional (i.e., with variation in the channel height direction only).

Figure 3.1.

Illustration of the flow-induced electrokinetic field in a microchannel.


Figure 3.2. A slit microchannel. a « W, and a « smaller than the channel's width and length.

33

L, i.e., the channel's height is much

3-1.1 The electrical double layer field Essentially all liquid flows in microchannels are low speed flow. As discussed in Chapter 2,we will assume that the flow field has no appreciable effects on the ion concentration field. The ion distribution is given by the Boltzmann equation. The EDL field is therefore independent of the flow field. Consider a liquid containing simple symmetric ions, i.e., the valences of the ions are the same, z = z+ = z~. The bulk ionic concentration is nx. For the slitmicrochannel, we consider only the EDL fields near the top and the bottom channel surfaces. The one-dimensional EDL field for a flat surface is described by the following form of the Poisson-Boltzmann equation:

(i) The local net charge density in the liquid is given by (2)

Non-dimensionalizing Eqs. (1) and (2) via (3)

34


we obtain the non-dimensional form of the Poisson-Boltzmann equation as:

(4) (5) where

a

n

d

i

s

the Debye-Huckel parameter

and 1/k is the characteristic thickness of the EDL.

is the

electrokinetic separation distance or the ratio of the half channel's height to the EDL thickness. Therefore, the parameter K can be understood as the relative channel's height with respect to the EDL thickness. For example, K = 10 means that the half channel height a is 10 times of the EDL thickness 1/k. If the electrical potential is small as compared to the thermal energy of the ions, i.e., so that the sinh function in Eq. (4) can be approximated by This transforms Eq. (4) to

(6) This treatment is usually called the Debye-Huckel linear approximation [1,3,4]. The solution of Eq. (6) can be obtained easily. As illustrated in Figure 3.2, if the separation distance between the two plates is sufficiently larger so that the EDL fields near the two plates are not overlapped, the appropriate boundary conditions for the EDL fields are: At the centre of the slit channel: At the solid surfaces: With these boundary conditions Eq. (6) can be solved and the solution is given by: (?)


35

Eq.(7) allows us to plot the non-dimensional EDL potential field in the slit channel. Due to the symmetry, we plot only the non-dimensional EDL potential field from one surface to the centre of the channel, as shown in Figure 3.3. In this figure, the zeta potential is assumed to be 50 mV. For a given electrolyte, a large K implies either a large separation distance between the two plates or a small EDL thickness. If the separation distance {2a) is given, increase in the bulk ionic concentration nx will increase the value of Debye-Huckel parameter k (recall k = (2naoz2e21ssokbT)112), the double layer thickness (1/k) is reduced (or the double layer is "compressed"), and the electrokinetic separation distance is increased. It can be seen from this figure that as K increases, the double layer field (the region) exists only in the region close to the channel wall. For example, appreciable EDL potential exists only in a region less than a few percent of the channel cross-section area for K = 80. However, for dilute solutions such as pure water (infinite dilute solution), the value of the electrokinetic separation distance K = a * k is small, and hence the EDL filed (the region) may affect significant portion of the flow channel, as shown in Figure 3.3.

Figure 3.3. Non-dimensional electrical double layer potential distribution near the channel wall (£= -50 mV). Center of the channel: X= 0; channel wall: X= 1.0.

36


It should be pointed out that Figure 3.3 is the results of using the DebyeHuckel linear approximation, Eq.(7), and that the linear approximation is valid for small surface potential situations (i.e., y/

2

1

-

1

1

1

1

4

6

8

10

\

V 0

2

Electrokinetic Separation Distance K

Figure 3.5. Variation of the ratio of apparent viscosity to the bulk viscosity with the electrokinetic separation distance Kat £= -50 mV.


43

distance K in Figure 3.5. It is observed that for C, = -50mV, the ratio \ij\i is approximately 2.75 when K = 2, and then decreases as K increases, approaching a constant value equal to one for very large values of K. For lower values of zeta potential, the trend is the same except the value of the ratio, [ij\x, is lower. Generally, the higher the zeta potential C,, the higher the ratio nJn.

44

3-2


PRESSURE-DRIVEN ELECTROKINETIC FLOWS IN RECTANGULAR MICROCHANNELS

For most lab-on-a-chip devices, the cross-section of the microchannels made by micromachining technology is close to a rectangular (trapezoidal, more exactly) shape. As the EDL field depends on the geometry of the solid-liquid interfaces, the EDL field will be two-dimensional in a rectangular microchannel. In such a situation, the two-dimensional Poisson-Boltzmann (P-B) equation is required to describe the electrical potential distribution in the rectangular channel; and the corner of the channel may have particular contribution to the EDL field, subsequently to the fluid flow field. [5,6] 3-2.1 EDL field in a rectangular microchannel In order to consider the electro-viscous effects on liquid flow in rectangular microchannels, we must evaluate the body force generated by the flow induced electrokinetic field in the equation of motion. To do so, we must evaluate the distributions of the electrical potential and the net charge density of the EDL. Consider a rectangular microchannel of width 2W, height 2H, and length L as illustrated in Figure 6.

Figure 3.6.

A rectangular microchannel (height 2H, width 2W).

In this case, the two-dimensional Poisson equation provides the relationship between the electrical potential y/ and the net charge density per unit volume pe at any point in the solution,

(30)


45

where e is the dielectric constant of the solution. Assuming the equilibrium Boltzmann distribution equation is applicable, the number concentration of the type-i ion in a symmetric electrolyte solution is of the form (31) where «7CO and z7- are the bulk concentration and the valence of type-i ions, respectively, e is the charge of a proton, Kb is the Boltzmann constant, and T is the absolute temperature. The net volume charge density pe is proportional to the concentration difference between symmetric cations and anions, via. (32)

Substituting Eq.(32) into the Poisson equation, Eq. (30), leads to the well-known Poisson-Boltzmann equation in a two-dimensional form. (33)

By defining the Debye-Huckel parameter

and the hydraulic

diameter of the rectangular microchannel Df, = /TTLL and introducing the dimensionless groups: Y = -£-, Z = -^-, K = KD/,, and vF = :f^-, the above equation can be non-dimensionlized as (34)

Due to symmetry, Eq. (34) is subjected to the following boundary conditions in a quarter of the rectangular cross section (35a)

46


(35b)

where g, defined byg = f^,

is the non-dimensional zeta potential at the channel

wall (here g is the zeta potential). For small values of y/ (Debye-Huckel approximation, which physically means that the electrical potential is small in comparison with the thermal energy of ions, i.e. zey/ | < K^T), the Poisson-Boltzmann equation can be linearized as

(36) By using the separation of variable method, the solution to the linearized P-B equation, Eq. (36), can be obtained. Therefore, the electrical potential distribution in the rectangular microchannel is of the form

(37)

For large values of \\i, the linear approximation is no longer valid. The EDL field has to be determined by solving Eq. (33) or (34). In order to solve this non-linear, two-dimensional, differential equation, a numerical finite-difference scheme may be introduced to derive this differential equation into the discrete, algebraic equations by integrating the governing differential equation over a control volume surrounding a typical grid point. The non-linear source term of Eq.(34) is linearized as (38)


47

where the subscript (n+1) and n represent the (n+l)th and the nth iterative value, respectively. The derived discrete, algebraic equations can be solved by using the Gauss-Seidel iterative procedure. The solution of the linearized P-B equation with the same boundary conditions may be chosen as the first guess value for the iterative calculation. The under-relaxation technique can be employed to make this iterative process converge fast. The details of how to obtain the numerical solutions of Eq. (34) can be found elsewhere [6,7]. After the electrical potential distribution inside the rectangular microchannel is computed, the local net charge density can be obtained from Eq.(32) as (39) Figure 3.7 shows a comparison of the EDL field in a rectangular microchannel predicted by the linear solution and by the complete numerical solution. In the calculation, the liquid is a dilute aqueous 1:1 electrolyte solution (concentration is lxlO"6 M) at 18°C. The rectangular microchannel has a crosssection 30 x 20 urn with a zeta potential of-75 mV. Because of the symmetry, the EDL field is plotted only in one quarter of the microchannel. With such a relatively high zeta potential, the linear approximation is obviously not good. It can be seen that there is a very steep decrease in the potential in the case of the complete solution, while the linear solution predicts a more gradual decay of the potential. The linear solution predicts a thicker layer from the wall that has an appreciable non-zero EDL field. 3-2.2 Flow field in a rectangular microchannel Consider the case of a forced, two-dimensional, laminar flow through a rectangular microchannel as illustrated in Figure 6. The equation of motion for an incompressible liquid is given by (40)

In this equation, pt and Hf are the density and viscosity of the liquid, respectively. For a steady-state, fully-developed flow, the components of velocity V satisfy u = u(y,z) and v = w = 0 in terms of Cartesian coordinates. dV

- -

-

Thus both the time term -r- and the inertia term (F-V)F vanish. Also, the

48


hydraulic pressure P is a function of xonly and the pressure gradient ~

is

constant. If the gravity effect is negligible, the body force F is only caused by the action of an induced electrical field Ex (see the explanation of the electrokinetic potential) on the net charge density p e (Y, Z) in the electrical double layer region, i.e. Fx = Expe. With these considerations, the Eq. (40) is reduced to

(41)

Defining the reference Reynolds number

PfDhU

Re 0 =—

and non-

dimensionalizing the Eq. (41) via the following dimensionless parameters (42a)

(where U is a reference velocity, Po is a reference pressure, and go is a reference electrical potential), one can obtain the non-dimensionlized equation of motion

by (149) Since the capillary wall is non-conducting, the conservation of electric current gives (150) where <J(T) = X{T)C is the temperature dependent electric conductivity with C the molar concentration and A(r) the molar conductivity of the solution, respectively. The molar concentration is dependent on the bulk ionic concentration n^'mEq. (148)by nx -NaC with Na the Avogadro'snumber. While the flow field in Eq. (145), EDL potential field in Eq. (147), and applied electric potential field in Eq. (150) are all restricted in the liquid region, the temperature field must be extended to cover the whole computational domain (see Figure 4.38). Within the liquid region, the energy equation in the presence of electroosmotic flow effect is given by (151) where C is the specific heat of the liquid and is assumed to be constant in this work, k(l) the temperature dependent thermal conductivity of the liquid. Here, we have neglected the term associated with viscous dissipation (i.e., the thermal energy converted from the mechanical energy) because it is small compared to the Joule heat. Within the two solid regions, the energy equations become (152)

(153)

188


where the subscripts w and p denote the capillary wall and polyimide coating, respectively. Modeling simplifications It is rather difficult to solve the set of Eqs. (145), (147), (150)-(153) without simplifications. The main problem lies in the simultaneous presence of three separate length scales: the capillary length of millimeters, the capillary radius of micrometers and the EDL thickness of nanometers. A complete solution on all the three length scales would require a prohibitive amount of memory and computational time. Therefore, two methods have been proposed to solve this problem. One is to artificially increase the order of magnitude of the EDL thickness, and a qualitative nature of the flow field can thus be obtained [7,9]. The other way is to apply a slip boundary condition at the wall, and avoid the solution of the EDL field [67,41]. In this section, the second approach is chosen, where the liquid at the capillary internal wall is assumed to slip at a velocity Vwall =neoEz with jj.eo =-S(T)SO£(T)/II(T) the electroosmotic mobility and Ez the local electric field in the axial direction (the axial coordinate z is indicated in Figure 4.38). Another simplification is made with respect to the momentum equations. Since electroosmotic flows are generally limited to small Reynolds numbers, the convection term (i.e., the second term on the left-hand side) in Eq. (145) can be neglected. In addition, it has been shown that the characteristic time tsteady for an electroosmotic flow to reach the steady state is on the order of milliseconds [7,67], which is far less than the characteristic time of thermal diffusion in this system (on the order of seconds) [68]. Hence, we can reasonably omit the transient term in the momentum equations as long as the time step sizes selected in the numerical simulation are greater than tsteady [67]. Finally, the set of non-dimensional equations determining the applied electric potential field, flow field, and temperature field is summarized as

Electroosmotic Flows in Microchannels

189

where the length is scaled by the capillary internal diameter (i.e., 2R in Figure 4.38), O = /(p with (p the potential applied at the capillary inlet, V = v/Frey with

the

reference

velocity

Vref = srefe0 10~5 M, in the case where pressure driven flow is directed parallel with the surface plane. In more general cases we can infer from Figure 5.15 that the velocity normal to the surface in the double layer must be on the order of 10~3 mm/s before significant a non-Poisson-Boltzmann characteristics will be observed. Though in principal the mechanisms are the same, it is not clear based on the above results if these limitations are also applicable to modeling electroosmotically-driven flows over heterogeneous surfaces.

Effects of Surface Heterogeneity on Electrokinetic Flow

235

Figure 5.16 Diagram showing net charge density within the double layer region for the case of a homogeneous surface ^-potential of -60mV and an oppositely charged patch of +60mV, where the flow is directed into the paper. The detailed image shows the applied body force on the local charges due to the induced electric potential gradient between the positive and negative surfaces.

5-2.4 Oppositely charged heterogeneous patches A special case of theoretical interest occurs when the heterogeneous patches have an equal but opposite charge to that of the homogeneous surface (i.e. the surface has a ^-potential of -60mV while the heterogeneous patches have a (^-potential of +60mV). In such a case there is an excess of positive ions over the homogeneous surface, as before, but negative ions dominate over the heterogeneous surface to balance out the positive surface charge as is shown in Figure 5.16. The difference in the surface potential results in an electrostatic potential gradient within the double layer region near the transition zone between the two regions similar to that shown in Figure 5.9b. The presence of this potential gradient results in a body force on the ions in the double layer, however in this case an opposite body force is applied to the negative ions over the positively charged patch, as shown in the detailed image in Figure 5.16.

236


Consequently the net body force over the region is zero and the circulation velocity perpendicular to the main flow, as shown in Figure 5.8, becomes negligible. Figure 5.17 shows a contour plot of the double layer potential field along the F5 (A) and F6 (B) planes, as defined in Figure 5.7, for the ^-potential case described above using the close packed surface pattern with a bulk ionic concentration of lxlO"5 M and a Reynolds number of 10. Note that in this figure the darkest region corresponds to the -60 mV surface, as before, while the lightest is the +60mV heterogeneous patch. As can be seen the convective influence on the double layer region is identical for both cases unlike in previous cases where an asymmetry existed depending on the direction of the direction of the streaming potential induced y-directional velocity (Figures 5.13 and 5.14). In this case however the net streaming potential is negligible (since the average C,potential is zero) and this effect is thus much smaller. The variations in the surface potential observed here are induced as a result of oppositely charged ions being transported (through both advection and diffusion) from one region to the other. In the first transition region shown in Figure 5.17a for example positive ions are being transported from the homogeneous region to the heterogeneous patch. As a result this decreases the local net negative charge density and a disturbance of the local electric potential field is observed. Similarly for the second transition region negative ions are transported from the region above the heterogeneous patch to the homogeneous region, resulting in a decrease in the positive net charge density and an opposite disturbance of the local double layer field. 5-2.5 Summary Surface electrokinetic heterogeneity occurs in a variety of electrokinetic characterization and microfluidic applications. A theoretical model for pressuredriven electrokinetic flow over a patch-wise heterogeneous surfaces is discussed in this section. This model based on a simultaneous solution to the coupled Nernst-Planck, Poisson and Navier-Stokes Equations reveals the synergetic effects of the heterogeneous EDL fields and the flow field on the resulting 3D flow structure. The presence of a heterogeneous patch was shown to induce a circulating flow pattern perpendicular to the main flow axis, which was caused by an electrostatic potential difference between the double layer fields over the homogeneous surface and heterogeneous patch. The strength of this circulation was found to be proportional to both Reynolds number and double layer thickness. At higher Reynolds number {Re =10) convective effects were found to be significant to the double layer distribution and led to a deviation from the classical Poisson-Boltzmann distribution observed at lower Reynolds numbers.


237

Figure 5.17. Electrostatic potential contours in the double layer region parallel with the flow axis on (A) T5 (z = 0|J.m) and (B) T6 (z = 25 um) surfaces with Re = 10.0 for the close packed heterogeneous surface pattern. In each case the ^-potential of the homogeneous region and heterogeneous patch are C,o = -60mV and t,p = +60mV respectively and the bulk ionic concentration is no, = lxl0~5 M.

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5-3


ELECTROOSMOTIC FLOW IN MICROCHANNELS WITH CONTINUOUS VARIATION OF ZETA POTENTIAL

The driving force for electroosmotic flow in microchannels depends on the local net charge density and the strength of the externally applied electrical field. Since the net charge density is dependent on the EDL field and hence on the zeta potential, C, , the electroosmotic flow behavior in turn is dependent on the zeta potential. Generally, the zeta potential is a function of the ionic valence, the ionic concentration of the electrolyte solution and the surface properties of the microchannel wall. For a system with a simple electrolyte solution and a homogeneous channel wall, the zeta potential is considered constant. However, in practice, the liquids involved in various lab-chips are solutions containing biological particles (e.g. DNA and protein) and bio-polymers. The adhesion of these particles to the channel wall will cause the non-uniform zeta potential along the channel, depending on the distribution and the extent of the adhered particles on the wall. Therefore, the understanding of the flow behavior in such a situation is important for manipulating the flow in biochip devices. Several researchers [18,19,38,39,40] have investigated the effects of variable zeta potential on electroosmotic flow. Anderson and Idol [38] developed an infinite series solution for flow in a cylindrical microchannel with a zeta potential varying as a cosine or sine function in the flow direction. Long [39] considered the similar type of zeta potential variation in planar and cylindrical capillaries, and developed approximate solutions by considering the heterogeneity as the small perturbations to the velocity. Herr [19] experimentally investigated the flow field of the electroosmotic flow and the sample dispersion rate in open capillaries with a step change in zeta potential (i.e. £j = 0 and 0 ) and presented a simple model for the fluid velocity and the dispersion rate. Stroock, et al [18] studied two types of surface charge change in rectangular channels. One is the surface charge variation along a direction perpendicular to the applied electrical field, which generates a two-directional electroosmotic flow. Another is the surface charge variation along a direction parallel to the electrical field, which generates a re-circulating flow. Potocek, et al [40] investigated the influence of the discrete step change in zeta potential on the velocity profile of the electroosmotic flow by considering the zeta potential change between two steps. The above mentioned works all assumed thin electrical double layer and used the electroosmotic velocity as the slip boundary condition. Because the electrical body force responsible for electroosmotic flow depends on the local net charge density, therefore, a more general treatment to the effects of zeta potential (or EDL) variation on electroosmotic flow should consider the local net charge density change along the channel. This requires solving for the Poisson-Boltzmann equation which governing the EDL field.


239

In this section, we consider that the zeta potential changes over a range of values along a cylindrical microchannel [41]. Without assuming a thin EDL field, the non-linear Poisson-Boltzmann equation and the momentum equation were solved numerically. The influences of the heterogeneous section size and the direction of the zeta potential change on the flow rate, the velocity profile and the induced pressure distribution are discussed. In the following, two cases of electroosmotic flow in a 10-cm long heterogeneous microchannel will be analyzed. In the first case, the microchannel has 100 equal-size sections and the zeta potential linearly changes from 10 mV to 200 mV over the 100 sections. In the second case, the microchannel has 100 unequal-size sections and the zeta potential linearly changes from 10 mV to 200 mV over the 100 sections. In this case, the section size (i.e. the length of the section) is determined by the following equation: (17) where Lj is the length of the i -th section and / is a constant chosen as 1.1. Since the objective here is to investigate the qualitative effects of the heterogeneous section size on the electroosmotic flow behavior, the specific choice of the function in Eq. (17) will not affect the generality of the results. For a cylindrical microchannel, the relationship between the electrical potential, y/(r), and the net charge density per unit volume, pe, at any point in the liquid is described by the Poisson equation, ,,8) where £ is the dielectric constant of the solution and £o is the permittivity of vacuum. Assuming that the equilibrium Boltzmann distribution is applicable, the net volumetric charge density, pe, is proportional to the concentration difference between cations and anions, for symmetric electrolyte solution such as KC1 (z: z = 1:1) solution, given by: (19) where nica and z,are the bulk ionic concentration and the valence of type / ion, respectively, e is the charge of a proton, kj, is Boltzmann constant and T is the temperature.

240


Substituting the net charge density in the Poission equation by Eq.(19), and introducing the dimensionless variables

where a is the radius of the microchannel, the non-dimensional PossionBoltzmann equation can be written as follows: (20)

where K: , the Debye-Huckle parameter. Because of the symmetry of the EDL field in the cylindrical microchannel, Eq. (20) is subjected to the following nondimensional boundary conditions:

(21) Eq.(20) describing the electrical double layer potential is a nonlinear partial differential equation that must be solved numerically. Taylor series expansion is used to linearize the nonlinear source term on the right-hand side of Eq. (20) as follows: (22) where y/ is the value for y/ obtained in the previous iteration [42]. A numerical finite difference scheme was used to discretize the governing differential equation and the resulting system of algebraic equations were solved using the Gauss-Seidel iterative technique, with successive over-relaxation employed to improve the convergence time [43]. Since the electrical potential field varies dramatically within a small distance of the channel walls, variable grid spacing [44] was employed to ensure that as the surface was approached, the grid spacing was refined enough to capture the sharp gradients. Once the electrical potential field i// (r ) has been found, the net charge density can be determined by Equation (19). For a steady and fully developed flow, the equation of motion is given by


241

(23)

where u z is the velocity component in the channel length direction, viscosity, p is the density of the fluid, VP is the induced pressure which will be discussed shortly, pe is the local net charge density and electrical field strength applied to the microchannel. Substituting the Eq. (19) for the net charge density into the Eq. introducing the following non-dimensional variables,

/u is the gradient E is the (23) and

where U is an arbitrary reference velocity, the non-dimensional equation of motion can be obtained, (24)

Eq. (24) is subjected to the no-slip and symmetric boundary conditions:

For the electroosmotic flow of a homogeneous liquid through different sections of a microchannel, certain conditions must be satisfied. Because of the uniform bulk liquid properties along the flow path, and assuming negligible effects of the surface conductance, the electrical current and the applied field strength are constant axially. The remaining key condition is the constant volume flow rate or the continuity condition. Consider a heterogeneous channel with two equal-size sections, each has a different zeta potential, and the same external pressure at the both ends of the channel. It can be understood easily that, if the zeta potential is higher, the EDL field will be stronger and hence the electroosmotic flow rate will be larger. If, for example, the high zeta potential section is in the downstream and the low zeta potential section is in the upstream, the liquid in the downstream section would have a higher electroosmotic

242


velocity, and the liquid in the upstream section would have a lower velocity. This would cause different flow rate in different sections and hence violate the continuity condition. In reality, a vacuum tends to form between these two sections since the liquid in the downstream section is moving faster than that in the upstream section. Because of the same pressure (i.e. atmospheric pressure) at the both ends of the channel, a negative pressure is induced between these two sections. This induced negative pressure will generate a positive pressure gradient for the downstream section to slow the flow and a negative pressure gradient for the upstream section to increase the flow rate. In this way, the continuity condition will be satisfied. This is also why Eq.(23) or (24) has an induced pressure gradient term. The induced pressure gradient will introduce more complexity in the numerical solution. Firstly, the Possion-Boltzmann equation has to be solved to find the electrical potential distribution in each section along the channel, y/(r,z). Once the electrical potential is found, the net charge density can be determined by Eq. (19). Secondly, a guess value of the induced pressure gradient profile is chosen for the first 99 sections (from left to right). With this guess value, the equation of motion can be solved for these sections and the volumetric flow rate in these sections can be determined by the following equation: (25) where Qj is the volumetric flow rate and Uj (r) is the local velocity for the i -th section, respectively. During this process, the pressure gradient for the first section is fixed and the pressure gradients for the other sections are adjusted to satisfy the continuity condition: (26) Finally, because of the same pressure at the both ends of the channel, the pressure gradient in the last section has to be determined by the following condition: (27)


243

Figure 5.18 Flow rates versus the microchannel diameter for a 1x10 5 M KC1 solution under electrical field strength of 350 V/cm.

Figure 5.19 Flow rates versus the electrical field strength for a 1x10 5 M KC1 solution in a microchannel of 100 |am in diameter.

244


where VP;- is the pressure gradient of the i-th section. With this pressure gradient, the equation of motion can be solved for the last section and the volumetric flow rate for this section can be determined by Eq. (25). The continuity condition, Q\ =Q\§§, has to be checked again at this moment. If the continuity condition is not satisfied, the pressure gradient for the first section has to be adjusted and repeat the above procedure until the continuity condition is satisfied. The above equations and the matching boundary conditions for the EDL field and the flow field were solved numerically. KCl electrolyte solution is used as the testing liquid and the following physical properties of KCl electrolyte solution were used: s = 80, sQ= 8.854 x 1(T12 CV 'm ', and /u = 0.90 x 10"3 kg

Figure 5.20 The velocity field (upper), the induced pressure field and the zeta potential distribution (lower) versus the distance in flow direction for a lxl(T 5 M KCl solution in a microchannel of 100 nm in diameter and under an electrical field strength of 350 V/cm. In this case, the section size is axially constant.


245

m 's l [45]. An arbitrary reference velocity of U= 1 mm/s was used to nondimensionalize the velocity. Figure 5.18 shows the dependence of the electroosmotic flow rate on the diameter of the microchannel. As the diameter increases, the flow rate increases. The variations of the section size and the trend of the zeta potential change in the flow direction will not affect this result. As discussed previously, the velocity of the electroosmotic flow and hence the flow rate depend on the zeta potential. In this case, the zeta potential changes from 10 mV to 200 mV along the channel. As clearly seen from Figure 5.18, for the system with equal-size sections, the flow rate generated with the average zeta potential (the average value of the maximum and the minimum zeta potentials) is the same as the flow rate for the case with a linearly increasing zeta potential distribution and the case with a

Figure 5.21 The velocity field (upper), the induced pressure field and the zeta potential distribution (lower) versus the distance in flow direction for a lxl0~ 5 M KC1 solution in a microchannel of 100 (im in diameter and under an electrical field strength of 350 V/cm. In this case, the section size is axially constant.

246


linearly decreasing zeta potential distribution in the flow direction. Therefore, for the case of equal section size, the average zeta potential can be used to evaluate the electroosmotic flow rate in a microchannel with a non-uniform zeta potential distribution. For comparison, the flow rates for homogeneous microchannels with a zeta potential of 200 mV and 10 mV, respectively, are also plotted in Figure 5.18. However, for the case of unequal section size, the flow rates are different when the trend (i.e. increase or decrease) of the zeta potential change along the flow direction changes. When the zeta potential linearly decreases in the flow direction, the flow rate is smaller than that in the system with a linearly increasing zeta potential distribution in the flow direction. This is because when the zeta potential decreases in the flow direction, the lower zeta potential takes a large portion of the microchannel as the section size increases in the flow direction (i.e. I ] = 7.26x10" 4 mm, £j=200mV and L 1 0 0 =9.1mm, ClOO =10mV). This means a weaker EDL field in this portion of the channel. On the other hand, in the system with an increasing zeta potential distribution in the flow direction, the higher zeta potential sections occupy a large portion of the channel (i.e. L\ =7.26x10" mm, £]=10mV and Z]oo=9.1mm, £l00 =200mV). This implies a stronger EDL field in this portion of the channel. The flow rate depends on the EDL field and hence will be smaller in the system with a decreasing zeta potential distribution in comparison with that in the system with an increasing zeta potential distribution. The influence of the electrical field strength on the flow rate exhibits a similar behavior as shown in Figure 5.19 and can be understood similarly. Figure 5.20 shows the velocity field (upper) and the induced pressure field (lower) in a microchannel with a linearly decreasing zeta potential distribution and the equal-size sections. For a simple electroosmotic flow in a microchannel with uniform zeta potential, the plug-like velocity profile is expected. However, if the zeta potential is non-uniform such as specified in this work, the net charge density within the liquid varies axially. Meanwhile, the applied electrical field strength (V/m), which depends on the electrical current and the conductivity of the electrolyte solution, is constant axially. Consequently, the electrical body force generating the electroosmotic flow, which depends on the net charge density and the electrical field strength, is different from section to section. This would imply different volumetric flow rates for different sections. For example, if the zeta potential is lower in a section, the electrical force exerted on the fluid would be smaller and hence the generated flow rate is lower. For an incompressible liquid, the continuity condition requires the same flow rate throughout the microchannel. In order to achieve the same flow rate as that in the next section with a higher zeta potential, a negative pressure gradient (the


247

pressure decreases in the flow direction) is introduced to increase the flow rate. As seen in Figure 5.20, the pressure starts decreasing from a maximum pressure to the atmospheric pressure at the exit when the zeta potential is lower than 105 mV, the average value of the maximum and the minimum zeta potential of the microchannel. The nearly parabolic velocity profile was found for the downstream sections with lower zeta potentials (e.g. £=10mV), since the dominant driving force in these sections is the induced pressure gradient. For the upstream sections with higher zeta potentials, because of the presence of the negative pressure gradient in the downstream sections and the same atmospheric pressure at the both ends of the microchannel, a positive pressure gradient (the pressure increases in the flow direction) must exist to slow down the flow in these upstream sections. As shown in the figure, for the sections with higher zeta

Figure 5.22 The velocity field (upper), the induced pressure field and the zeta potential distribution (lower) versus the distance in flow direction for a 1CT5 M KC1 solution in a microchannel of 100 |^m in diameter and under an electrical field strength of 350 V/cm. In this case, the section size increases axially.

248


Figure 5.23 The induced pressure distribution versus the distance in flow direction for a 1CT5 M KC1 solution in a microchannel of 50 u.m in diameter and under an electrical field strength of 350 V/cm.

potentials (e.g. C, = 200 mV), the electroosmotic velocity profile is distorted by the positive pressure gradient. If the zeta potential increases in the flow direction, the distorted electroosmotic velocity profiles in the high zeta potential sections and the nearly parabolic velocity profiles in the low zeta potential sections were also found. As shown in Figure 5.21, the pressure distribution in the flow direction is different from that shown in Figure 5.20, since the directions of the zeta potential variation in these two cases are opposite. The variation of the induced pressure in this case can be understood in a similar way. However, for a microchannel with unequal-size sections (the section size increases in the flow direction in this paper), when the zeta potential decreases from 200 mV to 10 mV in the flow direction, the electroosmotic velocity profile in the higher zeta potential sections (e.g., the C, = 200 mV section) is distorted more significantly as shown in Figure5 in comparison with that in the system with equal-size sections. This is because a major portion of the channel has lower


249

zeta potentials in this case and hence the flow rate is smaller than that of the equal section size system, as seen in Figure 5.18. Therefore, for the sections with the same high zeta potentials, in order to reach this small flow rate, a bigger induced pressure gradient is required resulting in the more distorted velocity profile. Figure 5.23 shows that the induced pressure field depends on the direction of the zeta potential variation and the section size distribution. As discussed earlier, for the sections with higher zeta potentials, the induced pressure gradient is positive (the pressure increases) and for the sections with lower zeta potentials, the induced pressure gradient is negative (the pressure decreases). Therefore, when the zeta potential decreases from 200 mV to 10 mV in the flow direction, the pressure increases first from the atmospheric pressure until the zeta potential decreases to a specific value, and then the pressure decreases from a maximum value to the atmospheric pressure at the exit. When the zeta potential increases from 10 mV to 200 mV, the pressure decreases first from the atmospheric pressure until the zeta potential reaches a specific value, and then the pressure increases from a minimum value to the atmospheric pressure at the exit. This specific zeta potential depends on the section size distribution. For the case of equal section size, this specific zeta potential is the average value of the maximum and the minimum zeta potentials; however, for the case of unequal section size, this value is between the maximum and the minimum zeta potential and can be determined by the numerical results. Also in Figure 5.23, we can see that the induced pressure in the case of equal section size is bigger than that in the case of unequal section size. This is because for unequal section size system, the zeta potential changes more quickly over a very small distance. For example, in the case of decreasing zeta potential, the zeta potential decreases from 200 mV to 105 mV over 0.845% of the total length of the microchannel. Therefore, the variation of the zeta potential over almost the total length of the microchannel (i.e. 99.155% of the total length) is smaller (i.e. from 105 mV to 10 mV) in comparison with that in the case of equal section size, where the variation of the zeta potential over the same distance is approximately 190 mV. Recall that the induced pressure gradient is introduced because of the non-uniform zeta potential distribution, it is easy to understand that the smaller the variation of the zeta potential, the smaller the induced pressure gradient. From the above analysis, we see that the different types of velocity profiles in the upstream and the downstream sections are caused by the nonuniform zeta potential distribution. The effects of the unequal section size and the direction of the zeta potential change in the flow direction on the flow rate are demonstrated. For the equal section size system, the trend of the zeta potential change has no effect on the flow rate and the average zeta potential can be used to evaluate the electroosmotic flow rate. However, for the unequal

250


section size system, the flow rate in the case of increasing zeta potential is bigger than that in the case of decreasing zeta potential. This is because the higher zeta potential takes a larger portion in the former case and hence the EDL field is stronger in this case. The effects of the unequal section size and the direction of the zeta potential change in the flow direction on the induced pressure distribution are also demonstrated. The induced pressure distribution is different when the trend of the zeta potential variation in the flow direction changes. For the unequal section size system, the induced pressure is smaller than that in the system of equal section size, because the variation of the zeta potential over almost the total length of the channel in this case is smaller than that in the system with equalsize sections.


5-4

251

ELECTROOSMOTIC FLOW IN MICROCHANNELS WITH HETEROGENEOUS PATCHES

Patches-wise surface heterogeneity is the configuration most close to the reality. In order to fully understand and potentially exploit the effects of patcheswise surface heterogeneity on electroosmotic flow, this section will discuss the complex three-dimensional electroosmotic flow structure in the presence of patches-wise surface heterogeneity [23]. The model used here is based on a simultaneous solution to the Nernst-Planck, Poisson and Navier-Stokes equations, which also allows us to avoid the assumption of Boltzmann double layer distribution and to adopt a more general approach. Let's consider the electroosmotically driven flow through a slit microchannel (i.e. a channel formed between two parallel plates) exhibiting one of the three periodically repeating heterogeneous surface patterns shown in Figure 5.24. The first two of these patterns, Figures 5.24(a) and 5.24(b), are in essence one-dimensional surface patterns and are considered here in order to provide a validation of the results via a qualitative comparison with other published studies [14,18]. Since the pattern is repeating, the computational domain is reduced to that over a single periodic cell, demonstrated by the dashed lines in Figure 5.24. To further minimize the size of the solution domain it has

Figure 5.24 Periodic surface patterns (a) lengthwise strips, (b) crosswise strips, (c) patchwise pattern. Dark regions represent heterogeneous patches, light regions are the homogeneous surface. In each case the percent heterogeneous coverage is 50%.

252


been assumed that the heterogeneous surface pattern is symmetric about the channel mid-plane, resulting in the computational domain shown in Figure 5.25. As a result of these two simplifications, the inflow and outflow boundaries, surfaces 2 and 4 (as labelled in Figure 5.25 and referred by the symbol F in the following discussion), represent periodic boundaries on the computational domain, while surface 3 at the channel mid-plane represents a symmetry boundary. From Figure 5.24 it can be seen that in all cases the surface pattern is symmetric about F5 and F6 and thus these surfaces also represent symmetry boundaries. Further details regarding periodicity related to this study are provided throughout this section. For a general discussion of periodic boundary conditions, however, the reader is referred to a paper by Patankar, Liu and Sparrow [33]. As mentioned above, electroosmotic flow occurs when an applied driving voltage interacts with the net charge in the electrical double layer near the surface resulting in a local net body force that causes fluid motion. In order to model the flow through this periodic unit, it requires a description of the ionic species distribution, the double layer potential, the flow field and the applied potential. To begin the divergence of the ion species flux (here we consider a monovalent, symmetric electrolyte as our model species), often referred to as the Nernst-Planck conservation equations, is used to describe the positive and negative ion densities (given below in non-dimensional form), (28a) (28b) where A^ and N~ are the non-dimensional positive and negative species concentrations (N* = n+/nm N~ = rT/n^, where n^ is the bulk ionic concentration), O is the non-dimensional electric field strength (& = e(j)/kbT where e the elemental charge, kb is the Boltzmann constant and T is the temperature in Kelvin), V is the non-dimensional velocity (V = v/v0 where v0 is a reference velocity) and the ~ sign signifies that the gradient operator has been non-dimensionalised with respect to the channel half height (ly in Figure 5.25). The Peclet number for the two species are given by Pe+ = vol/D+ and Pe~ voly/D~ where D+ and D~ are the diffusion coefficients for the positively charged and negatively charged species respectively. Along the heterogeneous surface (Fj in Figure 5.25) and symmetry boundaries of the computational domain (T 3 , F5 and F6), zero flux boundary conditions are applied to both Eq. 28(a) and Eq. 28(b), see reference [12,22] for explicit equations. As mentioned earlier, along faces F2 and F4, periodic


Figure 5.25 boundaries.

253

The periodical computational cell showing location of the computational

conditions are applied which take the form shown below, (28c) (28d) The velocity field is described by the Navier-Stokes equations for momentum, modified to account for the electrokinetic body force, and the continuity equation as shown below, (29a) (29b) where F is a non-dimensional constant (F = nx ly kb T / JJ. v0) which accounts for the electrokinetic body force, Re is the Reynolds Number (Re = p v0 ly/ /u, where

254


p is the fluid density and n is the viscosity) and P is the non-dimensional pressure. Along the heterogeneous surface, Fj, a no slip boundary condition is applied. At the upper symmetry surface, F3, we enforce a zero penetration condition for the y-component of velocity and zero gradient conditions for the x and z terms respectively. Similarly a zero penetration condition for the z-component of velocity is applied along F5 and F6 while zero gradient conditions are enforced for the x and y components. As with the Nernst-Planck equations, periodic boundary conditions are applied along F2 and F4. For the potential field, we choose to separate, without loss of generality, the total non-dimensional electric field strength, Q, into two components (30) where the first component, *F(X,Y,Z), describes the electrical double layer field and the second component, E(X), represents the applied electric field. In pressure driven flows, E(X) is the flow induced streaming potential [12,22] and is generally quite weak. In this case however the gradient of E(X) is of similar order of magnitude to the gradient of f(^7,Z) and thus cannot be decoupled to simplify the solution to the Poisson equation. As a result for this case the Poisson equation has the form shown below, (31a) where K is the non-dimensional double layer thickness (K2 = JC2 ly2/2 where K = (2 rioo e2lswkb T)m is the Debye-Huckel parameter). Two boundary conditions commonly applied along the solid surface for the above equation are either an enforced potential gradient proportional to the surface charge density, or a fixed potential condition equivalent to the surface ^-potential. In this case we choose the former resulting in the following boundary condition applied along the heterogeneous surface (31b) where © is the non-dimensional surface charge density ( 0 = a lye / Sy, kb T, where a is the surface charge density). Along the upper and side boundaries of the computational domain, F3, F5 and F6, symmetry conditions are applied while periodic conditions are again are applied at the inlet and outlet, F2 and F4. In order to determine the applied electric field, E(X), we enforce a constant current condition at each cross section along the x-axis as shown in Eq. 32(a):


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Figure 5.26 Electroosmotic flow over a surface with lengthwise heterogeneous strips with a/,omD = -4xl(T 4 C/m2 and c/,etero = +4xl(T4 C/m2 in a lxlO"5 M (1/K = 0.1 urn) monovalent electrolyte solution, (a) top-view of the velocity vectors, (b) velocity variation in the channel width direction, (c) velocity profile in channel height direction.

256


(32a) where nx in the above is a normal vector in the x direction and J* and J~ are the non-dimensional positive and negative current densities. Assuming monovalent ions for both positive and negative species (as was discussed earlier) and using the ionic species flux terms from Eq. 28(a) and Eq. 28(b), condition 32(a) reduces to the following,

(32b)

which is a balance between the conduction (term 2) and convection (term 3) currents with an additional term to account for any induced current due to concentration gradients. Solving Eq. 32(b) for the applied potential gradient yields:

(32c)

The above system of equations was solved by the finite element method [36] using the BLOCS (Bio-Lab-on-a-Chip Simulation) software developed by the Laboratory of Microfluidics and Lab-on-a-Chip at the University of Toronto. In the numerical calculations, the computational domain was discretized using 27-noded three-dimensional elements, which were refined within the double layer region near the surface and further refined in locations where discontinuities in the surface charge density were present. Triquadratic basis


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Figure 5.27 Electroosmotic flow over a surface with crosswise heterogeneous strips with <ShOmo = -4xl(T 4 C/m2 and ahelero = +4xl(T 4 C/m2 in a lxlO"5 M monovalent electrolyte solution (1/K = 0.1 |j.m) (a) side view of velocity vectors in x-y plane (b) Magnitude of x direction velocity at various locations along the channel height.

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functions were used for all the unknowns ht', N~, Vx, Vy, Vz, and f. In general the use of these higher order elements and basis functions (as opposed to 8-noded trilinear elements) were found to be optimal as they could better approximate the sharp gradients in velocity, species concentration and electric potential within the double layer region. In all cases periodic conditions were imposed using the technique described by Saez and Carbonell [37]. The solution algorithm begins by forming the computational mesh and performing the elemental matrix integration. All solution variables were then set to zero with the exception of N+, N~ and W which were initially approximated with a classical Poisson-Boltzmann distribution [34,35]. Using these as initial guesses a semi-implicit, Newtonian iteration technique was used to solve Eqs. 28(a), 28(b) and 31 (a) simultaneously. In general it was found that the convergence could only be achieved when the double layer potential, W, in the non-linear terms of Eqs. 28(a) and 28(b) were fully implicit to the solution and the N^ and N~ terms explicit. Once convergence of all 3 variables was obtained, the forcing term in the Navier-Stokes Equations could be evaluated and Eqs. 29(a) and 29(b) were solved concurrently using a penalty method [36]. Following convergence of the flow solution, Eq. 32(c) was evaluated at each cross section to yield updated values of E(X). The above process was repeated until convergence of E(X) was obtained. First, let's apply the above model to the cases of length-wise and crosswise strip patterns as shown in Figures 5.24(a) and 5.24(b). [14] considered a case with a sinusoidally varying surface charge density as opposed to the step changes examined here, however, as will be shown, a similar flow structure exists. To do this we consider a computational domain with dimensions lx = 50îm, ly = 4 = 25 urn (the x-y-z co-ordinate system is the same as in Figure 5.25) containing a 10~5 M KC1 solution (thus a double layer thickness, 1/K, of 0.1 urn) and an applied driving voltage of 500 V cm"1. Chemical properties of K+ and Cl" ions were taken from Vanysek [46]. We consider a homogeneous surface charge density of o/,omo = -4x10~4 C m~2 and the heterogeneous strips with Ghetero = +4x10"4 C nf2 (corresponding to ^-potentials of approximately -50 mV and +50 mV, respectively). This implies an excess of positive ions over the homogeneous region and negative ions over the heterogeneous region. The results for the length-wise strips are shown in Figure 5.26 and Figure 5.27 for the crosswise strips. Figure 5.26(a) shows a top view of the velocity vectors at the symmetry plane (F3, ly = 25 urn). As was observed by Stroock et. al. [18] the flow is directed along the negative x-axis over the heterogeneous patch and the positive x-axis over the homogeneous patch, in both cases becoming weaker and approaching zero near the transition point. Also of interest is the change in the


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Figure 5.28 Electroosmotic flow streamlines over a patchwise heterogeneous surface pattern with ohomo - -4x10~ 4 C/m2 and (a) a/,etera = -2x1 (T4 C/m2 (b) ahetem = +2x1 CT4 C/m2 (c) <Shetem = +4x10"4 C/m2. The grey scale represents magnitude of velocity perpendicular to the applied potential field, scaled by the maximum velocity in a homogeneous channel. Arrow represents direction of applied electric field.

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magnitude of the induced flow velocity with increasing distance from the surface. As can be seen in Figures 5.26(b) and 5.26(c) the velocity reaches a maximum at a distance of approximately 3/K from the surface (which is approximately equivalent to the edge of the double layer where the net charge density is essentially zero) and then becomes lower and lower as the distance from the double layer increases. From Figure 5.26(b) it is also apparent that velocity profile changes from a near step change within the double layer, where the electroosmotic force is strongest, to a significantly more curved and flatter profile as the distance from the surface becomes greater and viscous forces begin to dominate. Extrapolating from these results, one would expect the flow, in a very large channel, to eventually reach a near stagnant flow at the mid-plane, which was an observation originally made by Ajdari [14] for the sinusoidal varying surface pattern. As is shown in by the velocity vectors in Figure 5.27(a), the presence of crosswise heterogeneous strips results in a series of alternating clockwise and counter-clockwise tumbling regions, or regions of flow circulation whose centre axis is perpendicular to the direction of the applied electric field. An analogous effect was predicted by Ajdari [14] for the sinusoidal surface pattern and observed experimentally by Stroock et. al. [18] for a similar strip pattern, lending further credibility to the proposed model. Figure 5.27(b) shows the magnitude of the x-direction velocity at increasing distances from the surface. Similar to the previous case a near step change in vx is observed at the discontinuity in surface charge density within the double layer and the maximum velocity is obtained at a distance of 3/K from the surface. At further distances from the surface, viscous forces begin to dominate and momentum diffusion tends to broaden this step change while reducing the magnitude of the velocity eventually resulting in the sinusoidal type pattern above the circulation axis at the symmetry plane (y = 25um). Next, let's consider the two-dimensional patchwise surface pattern shown in Figure 5.24(c) and look into the resulting three-dimensional flow structure. We consider a lx = 50um, ly = l7 = 25 urn computational domain, as shown in Figure 5.25, containing a 10~5 M KC1 solution and with an applied driving voltage of 500 V cm"1 and a ahomo = -4xlO~4 C/m2. The simulations revealed three distinct flow patterns, depending on the degree of surface heterogeneity, as shown in Figure 5.28. The grey scales in these figures represent the magnitude of the velocity perpendicular to the direction of the applied electric field (v2 = vy2 + v/) scaled by the maximum velocity in a homogeneous channel, which in this case is 1.7 mm/s. At low degrees of surface heterogeneity (o),e/ero < -2x10"4 C/m2) the streamline pattern shown in Figure 5.28(a) was obtained. As can be seen a net


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Figure 5.29 Velocity vectors in the x-z plane over a patchwise heterogeneous surface with Ghomo = -4x10" 4 C/m2 and ahelero = +2x10"4 C/m2 at distances of (a) 0.05 um, (b) 1.5 um and (c) 10.5 Um above the surface.

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counter-clockwise flow perpendicular to the applied electric field is present at the first transition plane {i.e. at the initial discontinuity in the heterogeneous surface pattern) and a clockwise flow at the second transition plane. This flow circulation is a pressure-induced effect. It results from the transition from the higher local fluid velocity (particularly in the double layer) region over the homogeneous surface on the right hand side at the entrance to the left hand side after the first transition plane (and vice versa at the second transition plane). To satisfy continuity then there must be a net flow from right to left at this point, which in this case takes the form of the circulation discussed above. The relatively straight streamlines parallel with the applied electric field indicate that at this level the heterogeneity is too weak to significantly disrupt the main flow. While a similar circulation pattern at the transition planes was observed as the degree of surface heterogeneity was increased into the intermediate range (-2x1 (T4 C/m2 < Ohetero < +2x10~4 C/m2), it is apparent from the darker contours shown in Figure 5.28(b) that the strength of the flow perpendicular to the applied electric field is significantly stronger reaching nearly 50% of the velocity in the homogeneous channel. Unlike in the previous case it is now apparent that the streamlines parallel with the applied electric field are significantly distorted due to the much slower or even oppositely directed velocity over the heterogeneous patch. Figure 5.29 shows the evolution of the flow pattern for this case with increasing distance from the surface. As was noted above, with respect to the length-wise strip pattern, the flow patterns parallel with the surface tend to degrade as the distance from the double layer region increases. This effect is again visible here, as a clear circulation pattern can be observed very near the surface with the flow in the opposite direction to the applied electric field over the heterogeneous patch, Figure 5.29(a). As the distance from the surface is increased the circulation becomes stronger reaching a maximum near the edge of the double layer region, Figure 5.29(b). Farther out from the surface, Figure 5.29(c), the circulation pattern is lost leaving a more uniform flow field with only a slightly disturbed velocity over the heterogeneous regions. At even higher degrees of heterogeneity (+2x1 CT4 C/m2< uhetew< +4x10"4 2 C/m ) a third flow structure is observed in which a dominant circulatory flow pattern exists along all three co-ordinate axes, as shown in Figure 5.28(c). This results in a negligible, or even non-existent, bulk flow in the direction of the applied electric field (which is to be expected since the average surface charge density for these cases is very near zero). As indicated by the grey scales, the velocity perpendicular to the flow axis has again increased in magnitude, reaching a maximum at the edge of the double layer near the symmetry planes at the location where a step change in the surface charge density has been imposed. As electroosmotically driven flow is an inherently surface driven effect, it is of significant practical interest (in mixing applications for example) to


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Figure 5.30 Influence of heterogeneous patch size on velocity vectors in the F(, plane over a patchwise heterogeneous surface with a/,omo = -4x10" 4 C/m2 and ,omo and Ohetem, increased the magnitude of the flow circulation, but had no effect on the overall flow structure. As a result the flow structures shown in the preceding figures are quite general and are not significantly affected by either of these quantities. Further simulations revealed that, for the cases examined here where the thickness of the double layer is significantly less than the channel height and the heterogeneous patch size, the double layer thickness (or equivalently the bulk ionic concentration, n^) had only a minor effect on the flow field. For thicker double layers (for example n^ = 10~6 M, 1/K = 1 um) a slight reduction in the magnitude of the flow field was observed (at equivalent surface potentials), however the overall flow structure was not significantly altered. The full Nernst-Planck-Poisson formulation used here (as opposed to the traditional Poisson-Boltzmann equation) allows us to examine how the electrophoretic influence of the applied electric field and the convective effects


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Figure 5.31 Influence of applied electric field on net charge density distribution in the electric double layer with c t e m o = -4xlO~4 C/m2, <jhetero = +lxlO~4 C/m2 in a l x l t r 5 M (1/K = 0.1 u.m) electrolyte solution, (a) y = 0 |im (b) y = 0.05 u.m and (c) y = 0.2 u.m.

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of the irregular flow structures will disturb the double layer structure. Figure 5.31 shows the scaled net charge density (n+ - n~)lnx distribution along the F6 surface at different y positions for the computational domain discussed above with Ghomo = -4xl(T 4 C/m2 and ohetero = +4xlO"4 C/m2. From Figure 5.3l(a) it is apparent that the near surface net charge density field becomes slightly distorted compared with the Poisson-Boltzmann distribution (which is represented by the Vapp = 0 case) as the applied electric field is increased. In an earlier work concerning pressure driven flow over heterogeneous surfaces [12,22] it was observed that at Reynolds numbers greater than 10 and a bulk ionic strength less than 10~5 M, convective effects can significantly distort the double layer field from the traditional Poisson-Boltzmann distribution. For the electroosmotic case examined here, the highest Reynolds number obtained was Re = 0.09 and thus the convective terms in Eqs. 28(a) and 28(b) could not have significantly influenced the result. The cause of the disturbance observed here is the electrophoretic influence of the applied electric field (which is orders of magnitude stronger than the induced streaming potential in the pressure driven case). In this case the applied field induces a net flux of positive ions from left to right and negative ions from right to left (see Eqs. 28(a) and 28(b)). As a result an increase in the magnitude of the net charge density is observed near the first step change in surface potential as the ion flux from the left (where positive ions are in excess) meets the ion flux from the right (where negative ions are in excess). In contrast at the second transition point a decrease in the net charge density is observed since in both cases the net ion flux is away from this point. As is apparent from Figures 5.31(b) and 5.31(c) however this effect is limited to the region very near the surface as double layer field is shown to resemble very closely a Poisson-Boltzmann distribution at distances of 0.5/K and 2/K from the surface. In general this rearrangement of the net charge density field had only a small affect on the local fluid velocity, in that the calculated velocity was slightly higher in the regions where the net charge density was increased and vice versa. As was noted above, however this did not have a significant influence on the overall flow structure. In summary, the presence of periodically heterogeneous patches is shown to induce three distinct flow structures depending on relative difference between the surface charge density of the homogeneous and the heterogeneous regions. Small differences in the charge density are shown to induce fluid motion perpendicular to the applied electric field however the bulk flow remains largely unaffected. As the degree of heterogeneity is increased the streamlines in the direction of the applied electric field become significantly distorted, however the effect is minimized as the distance from the surface is increased. When oppositely charged surfaces are encountered a strong circulatory flow regime is observed in which flow perpendicular to the applied electric field is of the same


267

order as that parallel with it and the net volume flow rate becomes negligible. It is also shown that the induced flow patterns are limited to a layer near the surface with a thickness equivalent to the length scale of the heterogeneous patch and that as the average size of the heterogeneous region decreases the effect becomes smaller in magnitude and more localized. In addition it was demonstrated that while convective effects are small, the electrophoretic influence of the applied electric field could distort the net charge density field near the surface, resulting in a significant deviation from the traditional Poisson-Boltzmann double layer distribution.

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5-5


SOLUTION MIXING IN T-SHAPED MICROCHANNELS WITH HETEROGENEOUS PATCHES

As demonstrated in the previous sections, the presence of heterogeneous patches on a surface can result in local circulation within the bulk flow near these heterogeneous patches. Creating these bulk flow circulation regions can be explained by considering a simple model illustrated in Figure 5.32. In this figure, electroosmotically induced flow over a homogeneous surface with a zeta potential C, = -\q o\ is compared with that for a surface with an oppositely charged heterogeneous patch characterized by C, = + \q 0\. For the homogeneous case, Figure 5.32a, the electroosmotic body force applied to the liquid continua within the double layer is equivalent at each point along the flow axis and results in a constant bulk liquid velocity at the edge of the diffuse double layer, veo, which can be described by [47]: (33) where neo = (ewg I JJ.) is the electroosmotic mobility, Sy, is the electrical permittivity of the solution, JJ. is the viscosity, q is the zeta potential of the channel wall, and <j> is the applied electric field strength. The same relation holds true for the heterogeneous surface in Figure 5.32b except the electroosmotic body force is now applied to a region with excess negative ions and thus is in the opposite direction to that in the homogeneous regions. The interaction of these local flow fields with the bulk flow results in the regional circulation zones, as is illustrated in Figure 5.32b. In this section, the effect of surface heterogeneous patches on microfluidic mixing [22] will be discussed. Let's consider a simple T-shaped microfluidic mixing system, as shown in Figure 5.33. This simple arrangement has been used for numerous applications including the dilution of a sample in a buffer [48], the development of complex species gradients [49,50], and measurement of the diffusion coefficient [51]. In addition the T-sensor has also been exploited to make diagnostic determinations of analyte concentrations [52] by introducing the analyte of interest in one stream to a receptor molecule in the other (e.g. a pH [53] or fluorescence [54] indicator) producing a measurable signal which can be related to the parameter of interest [51]. Generally most microfluidic mixing systems, whether pressure or electrokinetically driven, are limited to the low Reynolds number regime and thus species mixing is strongly diffusion dominated, as opposed to convection or turbulence dominated at higher Reynolds numbers. Consequently mixing tends to be slow and occur over


269

relatively long distances and time. As an example the concentration gradient generator presented by Dertinger et. al. [49] required a mixing channel length on the order of 9.25mm for a 45um x 45(im cross sectional channel or approximately 200 times the channel width to achieve nearly complete mixing. Enhanced microfluidic mixing over a short flow distance is highly desirable for lab-on-a-chip applications. One possibility of doing so is to utilize the local circulation flow caused by the surface heterogeneous patches. Without loss of the generality, we will consider that two electrolyte solutions of the same flow rate enter a T-junction separately from two horizontal microchannels, and then start mixing while flowing along the vertical microchannel, as illustrated in Figure 5.33. The flow is generated by the applied electrical field via electrodes at the upstream and the downstream positions.

Figure 5.32. Electroosmotic flow near the double layer region for (a) a homogeneous surface (C, = -\C,0\) and (b) a homogeneous surface with a heterogeneous patch (C, = +\C,0\). Over the heterogeneous patch, the excess cations are attracted to the positive electrode resulting in an electroosmotic flow in the opposite direction to that over the homogeneous regions with an excess anion concentration. Arrows represent streamlines and 1/K refers to the characteristic thickness of the electrical double layer.

270


Figure 5.33. T-Shaped micromixer formed by the intersection of two microchannels, showing a schematic of the mixing or dilution process.

Electroosmotic flow results from the interaction of an applied electric field with the unbalanced ions in the electrical double layer (EDL) and is described by the Navier-Stokes Equations subject to an electroosmotic body force and the continuity Equation (given below in non-dimensional form): (34a) (34b) where Fis the non-dimensional velocity (V = v/veo, where veo is calculated using Eq. (33)), P is the non-dimensional pressure, r is the non-dimensional time and Re is the Reynolds number given by Re = pveoL/r\ where L is a length scale taken


271

as the channel width (w from Figure 5.33) in this case. In general the high voltage requirements limit most practical electroosmotically driven flows in microchannels to small Reynolds numbers, therefore to simplify Eq. (34a) we ignore transient and convective terms and limit ourselves to cases where Re < 0.1. Fe represents the non-dimensional electroosmotic body force give by, (34c)

where W represents the non-dimensional EDL field strength (W = zy/ e/kbT, where kb is the Boltzmann constant, T is the absolute temperature, e is the charge of an electron, and z is the ionic valence), is the non-dimensional applied electric field strength (0 = §/§max, where ^>max is the maximum applied voltage) and the ~ symbol over the V operator indicates the gradient with respect to the non-dimensional coordinates (X = x/w, Y = y/w and Z = z/w). Apparently from Eq. (34c), evaluation of the electroosmotic body force requires a description of both f a n d 0, which can be given by Eq. (35) and Eq. (36) respectively, (35) (36) where K is the non-dimensional double layer thickness (K = KW, where K is the Debye-Huckel parameter, given by K: = (2naz2e2/swkbT)1/2 where n^ is the bulk ionic concentration of the aqueous medium. 1/K is usually referred to as the characteristic thickness of the double layer as shown in Figure 5.32.). Inherent in Eqs. (35) and (36) are a few assumption that are worth discussing. The description of the applied electric field by the homogeneous Poisson equation is a simplification valid only in cases where the bulk conductivity of the aqueous solution does not change significantly within the channel. In the majority of cases of fluidic transport in microchip devices the species of interest are transported in a buffer solution with an ionic concentration 100 times greater than that of the species. In these cases the buffer conductivity is assumed to dominate and thus remains constant, independent of the local species concentration. This assumption is also applicable to Eq. (35) that has been derived for a symmetric ionic species with concentration nx . n^ is referred to the buffer concentration and the contribution of the more dilute species of interest have been ignored. By decoupling the equations for the EDL and applied electric field, it has been assumed that the charge distribution near the

272


wall is unaffected by the externally applied field. Along the same lines, the description of the EDL field by the non-linear Poisson-Boltzmann equation assumes that the double layer distribution is also undisturbed by external convective influences. These assumptions is generally valid so long as double layer thickness is not large, or equivalently the ionic concentration of the solution is not very low and the flow remains in the limit of low Re. Numerical simulation of electrokinetic flow and species mixing in a Tshaped mixer is complicated by the simultaneous presence of three separate length scales; the mixing channel length (mm), the channel cross sectional dimensions (um) and the double layer thickness, 1/K (nm), which we will refer to as L\, L2 and L3 respectively. In general the amount of computational time and memory required to fully capture the complete solution on all three length scales would make such a problem nearly intractable. Since the channel length and cross sectional dimensions (Lj, L2) are required to fully define the problem, most previous studies [55-57] have resolved this problem by either eliminating or increasing the length scale associated with the double layer thickness (L3). Both Bianchi et. al. [56] and Patankar et. al. [57] accomplished this by artificially inflating the double layer thickness to bring its length scale nearer that of the channel dimensions. This allowed them to fully solve for the EDL field, calculate the electroosmotic forcing term, and incorporate it into the NavierStokes Equation without any further simplification. It did however not fully eliminate the third length scale and considerable mesh refinement was still required near the channel wall. In these cases this approach was tractable largely because the geometries were selected such that the channel length (Lj) was not as large as that examined here. In a different approach Ermakov et. al. [55] set Fe = 0 in Eq. (34a) and applied a slip boundary condition at the channel, veo from Eq. (33). By this method the double layer length scale (L3) is completely eliminated from the solution domain, and a description of the double layer region is no longer required. A similar slip condition approach was used by Stroock et. al. [18,31] in their simulations of electrokinetically induced circulating flows. In both cases excellent agreement was obtained with experimental results. For the mixing problem interested here, the large mixing channel length and the extension to 3D mixing require that we follow the slip condition approach. As shown in Figure 5.33, we consider the mixing of equal portions of two buffer solutions, one of which contains a species of interest at a concentration, co. Species transport by electrokinetic means is accomplished by 3 mechanisms: convection, diffusion and electrophoresis, and can be described by Eq. (37), (37)


273

Figure 5.34. Electroosmotic streamlines at the midplane of a 50|im T-shaped micromixer for the a) homogeneous case with C, = -42 mV, b) heterogeneous case with six symmetrically distributed heterogeneous patches on the left and right channel walls and c) heterogeneous case with six offset patches on the left and right channel walls. All heterogeneous patches are represented by the crosshatched regions and have a C, = + 42mV. The applied voltage is <j>app = 500 V/cm.

274


where C is the non-dimensional species concentration (C = c/c0, where c0 is original concentration of the interested species in the buffer solution.), Pe is the Peclet number {Pe = veow/D, where D is the diffusion coefficient), and Vep is the non-dimensional electrophoretic velocity equal to vep/veo where vep is given by: (38)

and juep ~(swgp/ri)

is the electrophoretic mobility (£w is the electrical

permittivity of the solution, \i is the viscosity, gp is the zeta potential of the tobe-mixed molecules or particles) [47]. As we are interested in the steady state solution, the transient term in Eq. (37) can be ignored. Before continuing it would be useful to clarify the relationship between the three different velocities discussed above: the electroosmotic velocity Veo (Eq. (33)), the bulk flow velocity V (Eqs. (34a) and (34b)), and the electrophoretic velocity Vep (Eq. (38)). As shown in Figure 5.32, Veo, is the induced velocity at the edge of the double layer caused by the application of the external electric field. As described above this quantity is used as a boundary condition on the Navier-Stokes equations, which are then used to determine the local velocity of the bulk flow at each point, V. While velocity at the boundary, Veo, governs the magnitude V, it is important to distinguish between these two quantities. The velocity Vep, represents the mobility of the species in the applied electric field. For a neutral species the electrophoretic velocity is zero, leaving V as the only velocity in Eq. (37). A charged species however will be attracted to either the positive or negative electrode, depending on the sign of the charge, at a speed described by Vep. In such a case the total species convection is described by the superposition of V and Vep as shown in Eq. (37). For further information on species mobility and its contribution to the convection diffusion equation (or more precisely the Nernst-Planck equation), the reader is referred to the texts by Lyklema [34,35]. In all simulations a square geometry was used with the depth equaling the width of the channel, w, and for consistency the arm length, Larm, was also assigned the value of w. The length of the mixing channel, Lmix, was dictated by that required to obtain a uniform concentration (i.e. fully mixed) at the outflow boundary. Depending on the simulation conditions this required Lmix to be on the order of 200 times the channel width. The applied electric field strength, Eq. (36) was solved subject to max/(Lmix + Lam)), and a mixing channel length of 15mm. These conditions are applied to all simulations discussed below unless otherwise specified. A homogeneous electroosmotic mobility of 4.0x10"8 mVVs is chosen, corresponding to a homogeneous C, potential of -42 mV. In Figure 5.34 we compare the mid-plane flow field near the T-intersection of (a) a homogeneous mixing channel, with that of a mixing channel having (b) a series of 6 symmetrically distributed heterogeneous patches on the left and right channel walls and (c) a series of offset patches also located on the left and right walls respectively. For clarity the heterogeneous regions are marked as the crosshatched regions in this and all subsequent figures. A C,potential of t, = +42mV was assumed for the heterogeneous patches. As expected both Figures 5.34b and 5.34c do exhibit regions of local flow circulation near these heterogeneous patches, however their respective effects on the overall flow fields are dramatically different. In Figure 5.34b it is apparent that the symmetric circulation regions force the bulk flow streamlines to converge into a narrow stream through the middle of the channel. The curved streamlines shown in Figure 5.34c show the more tortuous path through which the bulk flow passes as a result of the offset, non-symmetric circulation regions. Figure 5.35 compares both the 3D and channel midplane concentration profiles for the three heterogeneous arrangements shown in Figure 5.34. In all these figures a neutral mixing species (i.e. jj,ep = 0, thereby ignoring any


277

Figure 5.36. Species concentration contours in the cross section of the T-shaped micromixer at 200p.m downstream (upper image) and at 750u,m downstream (lower image) for the 3 cases shown in Figure 5.35; (a) homogeneous case, (b) heterogeneous case with symmetrically distributed heterogeneous patches, and (c) heterogeneous case with offset patches.

electrophoretic transport for the time being) with a diffusion coefficient D = 3x10~10 m2/s is considered. As expected both symmetric flow fields discussed above have yielded symmetric concentration profiles as shown in Figures 5.35a and 5.35b. While mixing in the homogeneous case is purely diffusive in nature, the presence of the symmetric circulation regions, Figure 5.35b, enables enhanced mixing by two mechanisms, firstly through convective means by circulating a portion of the mixed downstream fluid to the unmixed upstream region, and secondly by forcing the bulk flow through a significantly narrower region, as shown by the convergence of the streamlines in Figure 5.35b. In this second mechanism, the local concentration gradients are increased leading to a greater diffusive flux along the channel width (x-axis in Figure 5.33), N, given byEq.(39),

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(39)

where Ldi/f is the diffusion length scale. In the homogeneous case Ldiff is equivalent to the channel width, w. For the case shown in Figure 5.34b however the heterogeneous regions force the bulk flow to pass through a narrower region, resulting in L^ff < w and thereby increasing JV. This enhancement of the concentration gradients is apparent from Figures 5.36a and 5.36b, which compare the cross sectional concentration contours for these two cases at distances of 200u.ni (top plot) and 750um (bottom plot) downstream from the intersection. At 200um downstream (within the heterogeneous region) the convergence of the bulk flow into a narrow stream has led to the greater concentration gradients, as compared to Figure 5.36a, within this narrow band at the center of the channel. This results in much stronger diffusion and a more uniform species distribution further downstream, and thus better mixing, as shown in the 750um downstream contour plots. In Figure 5.35c the convective effects on the local species concentration is apparent from the concentration contours generated for the non-symmetrical, offset patch arrangement. In Figure 5.36c the influence of these offset patches on the mixing is apparent again leading to a similar enhancements in the concentration gradients at the 200um downstream point and a more uniform distribution at 750um downstream. In this case however the concentration distribution is no longer symmetric about the center axis. Influence of the heterogeneous ^-potential In the above, we have demonstrated qualitatively how the presence of heterogeneous regions can enhance the mixing process in a T-shaped micromixer. To quantify the degree of enhanced mixing, we introduce a mixing efficiency, e, given by Eq. (40),

(40)

where c is the cross sectional concentration profile at a distance y downstream and cx and c0 are the concentration profiles associated with a completely mixed and completely unmixed states respectively. A fully mixed state therefore would


279

have a 100% mixing efficiency while the unmixed state would have a 0% mixing efficiency. Figure 5.37 demonstrates the influence of the (^-potential of the heterogeneous region on the mixing efficiency for a 50um mixing channel with a 500um long heterogeneous region on all four channel faces (beginning at the intersection) for an electrically neutral species with D = 3x10~10 m2/s. The applied electric field strength is <j)app = 500 V/cm and the total mixing channel length is 15mm. As in the previous cases the channel has a (^-potential of 42mV in all regions outside the heterogeneous patches. From Figure 5.37 it is apparent that in all cases the introduction of heterogeneous patches does lead to an increase in the mixing efficiency over the homogeneous case. The largest improvement comes from the oppositely charged patch arrangement (C, = +42mV) where the length to achieve 50% mixing efficiency is decreased by nearly 30% (390|am vs. 550um). Similarly, the length to achieve 70% mixing is also decreased by 30% (820um vs. 1230urn).

Figure 5.37. Effect of the heterogeneous patch ^-potential on the mixing efficiency of a 50|o.m T-shaped micromixer with §ipp = 500V/cm. A single 500u,m patch on all four sides of the channel was used and the homogeneous (^-potential is -42m V. Species diffusivity is 3xlO~10 m2/s and zero electrophoretic mobility is assumed.

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In general it was observed that the circulation regions shown in Figures 5.34b and 5.34c were only present when the heterogeneous (^-potential was of opposite sign to that of the homogeneous surface. Additionally the size of the circulation regions was observed to increase as the magnitude of the heterogeneous (^-potential was increased. Referring to Figures 5.34, 5.35, and 5.36, larger circulation regions would force the bulk flow to pass through a narrower region leading to shorter local diffusion length and thus by Eq. (38) enhanced mixing. Despite the lack of circulation zones in the £ = -21mV and C, = OmV cases, a slight convergence of the flow streamlines to a narrower region at the center of the channel, caused by the localized slowing of the velocity field near the channel walls, was observed and proved to be sufficient to provide some degree of enhanced mixing. In Figure 5.37 it can also be seen that all cases exhibit an approximate step change in the mixing efficiency immediately after the heterogeneous region. This occurs as a result of the higher concentration gradients in the center of the channel making the mixing efficiency appear artificially low. Referring to Figure 5.36b it can be seen that within the heterogeneous region the gradients in the concentration profile occur over a relatively thin band in the middle of the channel cross section, leading to the improved mixing, as discussed above. Outside of this narrow band the species appear nearly unmixed resulting in an artificially low mixing efficiency. When the streamlines diverge rapidly at the termination of the heterogeneous patch, see Figure 5.34b, the concentration gradients are expanded to encompass the full channel width, as shown in the lower image in Figure 5.36b. At this point the true mixing efficiency is obtained, demonstrating the overall effect of the heterogeneous patch. This effect is also well demonstrated on the 2D image in Figure 5.35b where the expansion of the concentration gradients after the termination of a heterogeneous patch and the contraction at the beginning of the heterogeneous patch both are well shown. Also in Figure 5.37 it can be observed that the C, = +42mV case has a significantly higher mixing efficiency at the intersection (downstream distance = 0|am) than the other cases. This immediate enhancement is a result stronger circulation zone convecting a greater portion of the mixed downstream solution upstream to the entrance. Interestingly however this effect appears to be localized to the heterogeneous region, as the absolute difference between the mixing efficiencies for the C, = +42 mV and C, = +21mV cases outside the heterogeneous region is no greater than one would expect due to the enhanced concentration gradients (based on the trend from the lower C, potential simulations). Therefore, while the re-circulation mechanism does provide significant improvement of the initial mixing efficiency (at the beginning of the heterogeneous region), it appears as though the enhanced concentration gradients are the dominant mechanism for overall enhanced mixing.


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Influence of heterogeneous patch size and arrangement Figures 5.38a and 5.38b show the influence of the heterogeneous patch size on the mixing efficiency for two different applied voltages, (a) <j)app = 500 V/cm and (b) tj>app = 200 V/cm. As before we consider the case of a single heterogeneous patch with £ = +42mV on all four sides of a homogeneous surface with C, = -42mV, a 50u.m by 50um cross section channel and D = 3xlO~10 m2/s. Apparent in both cases is that the larger patches, exposing the solution to the enhanced concentration gradients for a longer time, will increase the overall mixing efficiency. It is not surprising that as the app is decreased, resulting in slower flow in the lengthwise direction and allowing greater time for diffusion, the mixing efficiency is improved. Additionally the longer retention time in the heterogeneous region for the lower voltage case also serves to improve the effectiveness of the heterogeneous region, reducing the required mixing length to attain both 50% and 70% mixing efficiency by about 70%, compared with 30% for the larger applied voltage case. To investigate the effects of patch distribution we consider the two general arrangements shown in Figures 5.34b (symmetric) and 5.34c (offset) and the same transport conditions as those detailed above with app = 500V/cm. We consider both four patch and six patch arrangements (Figures 5.34b and 5.34c represent the 6 patch arrangement). The patch sizes are adjusted so that in each case a total of 250um of heterogeneous surface is spread out over the first 500um distance downstream of the intersection on both the left and right sides (e.g. in Figure 5.34c there are three 83.3 um patches on the right surface for a total of 250um of heterogeneous surface). The resulting mixing efficiencies for these cases are compared in Figure 5.39. In Figure 5.39 it is apparent that, in terms of the mixing efficiency outside the heterogeneous region, for both cases there is almost no difference between the 4 and 6 patch arrangements. In addition, very little difference between the symmetric and offset distributions is observed, suggesting that mixing efficiency is more a function of patch size, rather than the arrangement. The most striking feature of the results displayed in Figure 5.39 is the large oscillations in the mixing efficiency within the heterogeneous region, most significantly for the symmetric patch case. As discussed above, near step changes in the mixing efficiency are observed at the termination of a heterogeneous patch due to the expansion of the concentration gradients. Referring to Figure 5.35b it is apparent that as the flow passes through multiple heterogeneous regions the concentration gradients experience numerous expansion and contraction zones resulting in the observed oscillations in the mixing efficiency. Since the concentration gradients are compressed into a significantly smaller region than is observed in the homogenous case, as is

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Figure 5.38. Effect of the heterogeneous patch size on the mixing efficiency of a 50u.m T-shaped micromixer with (a) (j>app = 500V/cm and (b) §app = 200V/cm. A single patch on all four sides of the channel with C, = +42mV was used and the homogeneous (^-potential is -42mV. Species diffusivity is 3xlO~10 m2/s and zero electrophoretic mobility is assumed.


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shown in the upper images of Figures 5.36a and 5.36b, it is not surprising that in some cases the mixing efficiency in the heterogeneous region drops below that of the homogenous channel. Influence of channel size Figure 5.40 compares the mixing efficiency in a 50 um x 50 urn channel with that in a 25 urn x 25 um channel for both the homogeneous and heterogeneous cases. As expected the smaller diffusion length, Ldiff in Eq. (39), in the 25 um channel leads to an increased mixing efficiency over that exhibited by the 50um homogeneous channel. Also in Figure 5.40, it is apparent that the influence of the heterogeneous patches is significantly greater for the 25 um channel, with 50%

Figure 5.39. Influence of the heterogeneous patch arrangement on the mixing efficiency of a 50um T-shaped micromixer with app=500V/cm. The symmetric and offset patch arrangements are as shown in Figures 5.34b and 5.34c respectively (6 patch arrangement shown in both cases). The homogeneous ^-potential is -42mV and all heterogeneous patches have C, = +42mV. Species diffusivity is 3xlO~'° m2/s and zero electrophoretic mobility is assumed.

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Figure 5.40. Effect of channel size on the mixing efficiency for both homogeneous and heterogeneous cases with <j>apP = 500V/cm. For the heterogeneous cases a single 500um patch on all four sides of the channel was assumed with C, = +42mV and a homogeneous ^-potential of -42mV was used. Species diffusivity is 3x1 (T10 m2/s and zero electrophoretic mobility is assumed.

reductions in Lmix at both the 50% and 70% mixing efficiency points (compared to 30% for the larger channel) and resulting in nearly complete mixing at only 500um distance downstream. Influence of species diffusivity and electrophoretic mobility In the above discussions, we have confined the analysis to electrically neutral molecules, where the electrophoretic transport component in Eq. (37) has been neglected, with constant diffusion coefficient of D = 3xl0~10 m2/s. Now we seek to understand how the enhanced mixing efficiency is affected by the diffusivity and electrophoretic mobility of the mixing species. In Figures 5.41a and 5.41b we reexamine the case of a 50um x 50u.m channel and a single heterogeneous region with C, = +42mV on all four sides of the first 500u.m of the mixing channel. Figure 5.41a shows the effect of species diffusivity by


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Figure 5.41. Influence of (a) species diffusivity and (b) electrophoretic mobility on the mixing efficiency in a 50nm channel for both homogeneous and heterogeneous cases with (j>app = 500V/cm. For the heterogeneous cases a single 500um patch on all four sides of the channel was assumed with t, = +42mV and a homogeneous ^-potential of -42m V was used.

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examining diffusion coefficients ranging from 1x10 10 m2/s to3xl0 10 m2/s. As in the previous cases an almost step change in the mixing efficiency is observed at the termination of the heterogeneous region (both Figures 5.41a and 5.41b) due to the expansion of the channel concentration gradients as discussed in detail previously. From Eq. (39) it is apparent that for lower D values the species flux will decrease and thus the observed lower mixing efficiency should result. This is clearly observed in Figure 10a where significantly lower mixing efficiencies are observed for the D = lxlO"10 m2/s case for both homogenous and heterogeneous channels. Also apparent in Figure 5.41a is that while in all cases the introduction of surface heterogeneity does improve the mixing efficiency, the absolute reduction in the mixing length is approximately equal for all pairs. This suggests that the effects of a higher diffusion coefficient on the mixing efficiency is neither enhanced nor degraded due to the presence surface heterogeneity. Figure 5.41b illustrates the influence of the electrophoretic mobility on the mixing in both homogeneous and heterogeneous channels, for the identical simulation conditions listed above and D = 3xl0~10 m2/s. Since a negative electrophoretic mobility increases the magnitude of the convection term in Eq. (37) (since Fand Vep will be in the same direction) whereas a positive mobility tends to reduce its magnitude, it would be expected that at a fixed downstream distance the mixing efficiency of the later of these would be considerably higher than that of the former. This is well demonstrated in Figure 5.41b where both heterogeneous and homogenous surfaces exhibit significant increases in the mixing efficiency as fiep is increased from -1.5xl0~8 m2/Vs to +1.5xl0~8 m2/Vs. Of interest in this plot is the significant increase in the initial mixing efficiency exhibited by the positive mobility case with the heterogeneous region. This significant initial enhancement is a result of the superposition of the electrophoretic and electroosmotic transport in the heterogeneous region near the wall (where the local flow direction is opposite that of the bulk flow) allowing a greater circulation of downstream ions into the unmixed initial solution. In summary, for low Reynolds number electroosmotically driven flows in microfluidic devices, species mixing is inherently diffusion dominated, resulting in poor mixing efficiency or requiring long transport distances and retention time. This section discussed the effects of surface electrokinetic heterogeneity on the electroosmotic flow mixing efficiency of a T-shaped micromixer. While all cases of surface heterogeneity were shown to enhance species mixing, the greatest improvements were found when the ^-potential of the heterogeneous surface was of opposite sign to that of the homogeneous surface, resulting in localized circulation zones within the bulk flow field. The local flow circulation enhances the mixing by firstly convection means by circulating a portion of the mixed downstream fluid to the unmixed upstream region, and secondly forcing the bulk flow through a significantly narrower region to increase the local


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concentration gradients. For a fixed ^-potential, the length of the heterogeneous patch was found the most important factor while the patch distribution had only marginal effects on the mixing efficiency. In general the mixing efficiency improvement by decreasing the applied voltage and the channel size will be enhanced through the introduction of surface heterogeneity, in some cases resulting in a 70% reduction in the required mixing length. The combined effect of the heterogeneous patch and a positive electrophoretic mobility was shown to be significant in enhancing the initial mixing efficiency at the T-intersection.

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5-6


HETEROGENEOUS SURFACE CHARGE ENHANCED MICRO-MIXER

Enhancing the diffusion-dominated microscale mixing processes is key to numerous lab-on-a-chip applications. For applications where the use of external power supplies, actuators and auxiliary equipment are permitted, mixing schemes based on a variety of technologies such piezoelectrics [58,59], pneumatics [60], acoustic radiation [61], and oscillating voltage [62], have been successfully developed for microfluidic devices. In many applications, however, where ease of manufacturability, simplicity of design and the portability of the devices take precedence, passive mixing schemes, which do not rely on external stimulation or intricate components, are preferable. Two approaches to flow manipulation are prevalent in passive micromixers, the first relying on channel geometry to generate chaotic advection and increased circulation, and the second on channel surface properties. Notable initiatives, amenable to intermediate Reynolds number flows (i.e. >1) of the geometric category, have included helical or zigzag microchannels, in which mixing occurs as a result of eddies present at the channel bends [63-65]. Alternatively, oblique ridges formed on the bottom of the channel and designed to introduce a transverse component of flow have also proven effective for pressure-driven flows with Reynolds numbers between 0 and 100 and channel lengths on the order of centimetres [66] This technique has also demonstrated a negligible increase in bulk flow resistance, a constraint limiting the effectiveness of mixers based on parallel lamination for electrokinetic applications [67]. Recently a passive electrokinetic micro-mixer based on the use of surface charge heterogeneity was developed [68]. The surface charge patterns are designed to create localized flow disturbances thereby inducing an advective component in the flow to enhance the mixing. This section will discuss the experimentally observed effects of surface charge heterogeneity on microscale species transport, and the performance of the developed micro-mixer in terms of mixing efficiencies and required channel lengths. The micro-mixer is a T-shaped microchannel structure made from Polydimethylsiloxane (PDMS) and glass. Microchannels were fabricated using a rapid prototyping/soft-lithography technique [69]. Specifically, photomasks were designed in AutoCAD, exported as PDF files, and printed on a 3500 dpi image setter. Glass slides were soaked overnight in acetone, dried on a hot plate at 200°C, exposed to oxygen plasma (Harrick Plasma Cleaner model PDC-32G) for 5 minutes and again heated to 200°C subsequently, in order to prepare the surface for coating. 1.5 mL of SU-8 25 photoresist was then distributed onto each slide and degassed in a high vacuum. The photoresist was spin-coated at 1800 rpm for 10s, and at 4000 rpm for 40s with a ramping phase of 5 s between stages in order to obtain a smooth film with a thickness of 8 microns (Special


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Coating System Spin Coat model G3P-8). Films were baked at 65°C for 3 minutes and at 95 °C for 7 minutes to harden. The photomasks were positioned on the photoresist films and exposed to UV light for 1 min. A two-stage postexposure bake at 65°C for 1 minute and 95°C for 2 minutes was then conducted. Masters were developed in 4-Hydroxy-4-methyl-2-pentanone for approximately 2 minutes or until the photoresist rinsed cleanly off. Subsequent to developing, masters were placed under a heat lamp for several hours. To form the microchannels, PDMS was poured over the masters and cured at 65 °C for approximately 2 hours at a pressure of-34 kPa (gauge). The rapid prototyping/soft lithography technique as described above allowed for control of and flexibility in surface charge patterning configurations, examples of which are detailed in Figure 5.42. To selectively pattern the surface charge heterogeneity, the following procedure was followed (Figure 5.43). The PDMS master, featuring a channel configuration corresponding with the pattern of heterogeneities to be examined, was reversibly sealed to a glass slide. Using suction forces with an approximate flow rate of 2.5mL/min, the PDMS master was flushed sequentially with 0.1M sodium hydroxide for 2 minutes, deionized water for 4 minutes, and 5% Polybrene solution for 2 minutes, resulting in selective regions of positive surface charge while leaving the majority of the glass slide with its native negative charge. This polyelectrolyte coating

Figure 5.42 Surface charge patterning configurations with a mixing region length L consistent for all configurations, a patch length 1, a patch width w, and patch spacing s for (A) In-line pattern (B) Staggered pattern (C) Serpentine pattern (D) Herringbone pattern and (E) Diagonal pattern.

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procedure is based on that developed by Liu et al. [70] for capillary electrophoresis microchips. All fluid was then removed from the channel and the system was left undisturbed for 40 minutes before flushing again with water for 20 minutes. To promote bonding of the polymer to the surface, the slide was aged in air for 24 hours prior to use. Before removing the PDMS master, the location of the heterogeneities was landmarked. A T-shaped microchannel, 200um in width and approximately 8 um in depth, was then permanently sealed to the glass slide such that the patterned surface heterogeneities were appropriately positioned within the mixing channel. Using the well-established current monitoring technique [71], the zeta potentials of the native-oxidized PDMS and the polybrene-coated surface were determined using a sodium carbonate/bicarbonate buffer of pH 9.0. Briefly, a dilute buffer was introduced into a uniaxial PDMS channel with the surface treatment of interest. The current was monitored and allowed to stabilize under a constant applied voltage potential. A concentrated buffer was then introduced through one reservoir and the time required for the current to reach a new plateau corresponding to the higher concentration of buffer was recorded. The electroosmotic mobility of the native-oxidized PDMS was determined to be -5.9 x 10 4 cm2/V-s compared with 2.3 x 10~4 cm2/V-s for the polybrene-coated surface, results that compared well with previous findings [70].

Figure 5.43 Schematic of selective patch patterning procedure (1) PDMS microchannel master featuring a channel configuration corresponding to a staggered pattern of heterogeneities and reversibly sealed to a glass slide. (2) PDMS master flushed with chargereversing solutions to selectively pattern charge heterogeneities. (3) Landmarking of surface heterogeneities achieved by aligning the corresponding photomask under 5x magnification with the PDMS master and firmly attaching it to the opposite side of the slide. (4) Removal of PDMS master and permanent sealing of oxidized T-channel to the glass slide such that the surface heterogeneities are appropriately positioned within the mixing channel. (5) Removal of attached photomask and completion of micro-mixer fabrication.


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The effect of the surface charge patterns on mixing efficiency was then examined in T-shaped microfluidic chips. To perform the mixing experiments, 25 mM sodium carbonate/bicarbonate buffer and 100 uM fluorescein, were introduced through either inlet channel. The system was illuminated by a mercury arc lamp equipped with a fluorescein filter set. The steady state transport of the dye was observed using a Leica DM LM fluorescence microscope with a 5x objective, and captured using a Retiga 12-bit cooled CCD camera. Digital images were obtained by QCapture 1394 and OpenLab 3.1.5 imaging software at an exposure time of Is. The acquired images were of resolution 1280 x 1024 pixels with each pixel representing a 2.5 micron square in the object plane. Images were exported in TIF format to MATLAB for digital processing. Dark field subtraction and bright field normalization was performed to eliminate anomalies introduced by the image acquisition system. Following image processing, concentration profiles prior to and subsequent to the heterogeneous region were developed directly from values of pixel intensities. Profiles were smoothed using a convolution filter and linearly scaled to range between 0 and 1. Mixing efficiency, 8, was calculated and compared with homogeneous systems (e.g., a uniform and negatively charged PDMS/glass system) using the following definition:

where Cx =0.5 corresponds with perfect mixing on a normalized scale, Co is the concentration distribution over the channel width, W, at the channel inlet and C is the concentration distribution at some distance downstream. In order to facilitate optimization of surface charge patterning, the BLOCS (Bio-Lab-On-a-Chip Simulation) finite element code (see discussions in the previous section) was used to simulate the experimental conditions and surface charge heterogeneity. Specifically, channel dimensions of 200 um in width and 8 um in depth were modelled along with the physiochemical properties of fluorescein, namely a diffusion coefficient D = 4.37 x 10~10 m2/s, and a electrophoretic mobility uep = 3.3 x 10~8 m2/Vs. The numerical model described in the previous section was used to optimize the pattern of surface charge heterogeneity and to characterize the effects of channel depth, zeta potentials, and diffusivity on mixing enhancement. Simulations indicated that a nonsymmetrical configuration of oppositely charge surface heterogeneities is optimal for mixing enhancement. Figure 5.44 presents the numerical results for

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the variety of configurations as detailed in Figure 5.42. For each configuration, the patch length and spacing parameters were selected to maintain a constant ratio of heterogeneous to homogeneous surface areas over a channel length of 1.8 mm. Evidently, the non-symmetrical patterns, namely the staggered and the diagonal, generated better mixing distributions in comparison to the symmetrical herringbone and in-line arrangements. With a theoretical mixing efficiency of 96%, the staggered configuration provided the greatest degree of mixing, outperforming the diagonal, the herringbone and the serpentine configurations by 8%, 31% and 36% respectively. In comparison with the homogenous case, the staggered configuration provided a 61 % increase in mixing efficiency. The staggered configuration was thus examined in more detail to determine the optimal patch length and spacing parameters. For a 1.8 mm mixing region length, the optimal patch length was determined to be 300 urn with a two-fold increase or decrease from this optimal value resulting in a 10% reduction in mixing efficiency. In comparison, additional design factors proved insignificant, with a two-fold and five-fold increase in channel depth resulting in a 1 % and 9% decrease in mixing efficiency respectively, while a difference of

Figure 5.44 Numerically simulated concentration profiles across the channel width for the staggered pattern subsequent to the mixing region for an applied potential of 280 V/cm for varying configurations of surface charge heterogeneity as defined in Figure 5.42.


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2.6% resulted from variations in spacing parameters. Values of electroosmotic motility in both the homogeneous surface and patch regions were critical for defining flow characteristics. As has been discussed in the previous section, for significant flow circulation, the patch electroosmotic mobility must be of opposite sign. Simulations revealed however that increasing the magnitude of the patch mobility above that generated by the polybrene layer does not significantly increase mixing efficiency, with a two fold increase resulting in a less than 2% change in mixing efficiency. Challenges of manufacturing may limit precise control of zeta-potentials, however as the presence of oppositely charged surface heterogeneities rather than their magnitude is the defining factor for mixing enhancement, charge altering treatments such as the polybrene coating used here should prove adequate. Based on the above analysis, the optimized micro-mixer consisting of 6 offset staggered patches (Figure 5.42B) spanning 1.8 mm downstream and offset

Figure 5.45 Images of steady state species concentration fields under an applied potential of 280 V/cm for (A) the homogeneous microchannel and (B) the heterogeneous microchannel with 6 offset staggered patches as obtained through numerical and experimental analysis. Arrows show the flow directions.

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10 um from the channel centerline with a width of 90 jam and a length of 300 um, was analyzed experimentally. Mixing experiments were conducted at applied voltage potentials ranging between 70 V/cm and 555 V/cm as corresponded to Reynolds numbers of 0.08 and 0.7 and Peclet numbers of 190 and 1500. As can be seen in Figure 5.45, experimental results compared well with numerical simulations with images of the steady state flow for the homogenous and heterogeneous cases exhibiting near identical flow characteristics and circulation at 280 V/cm. Qualitatively, experimental visualization of a staggered configuration of heterogeneities exhibited the formation of highly unsymmetrical concentration gradients indicative of flow constriction and localized circulation in the patterned region. Bulk flow was forced to follow a significantly narrower and more intricate route thereby increasing the rate of diffusion by means of local concentration gradients. Convective mechanisms were also introduced by local flow circulation that transported a portion of the mixed downstream flow upstream. Additionally, sharp, lengthwise gradients absent in the homogeneous case resulted in an additional diffusive direction and enhanced mixing as exhibited in Figure 5.46 which plots the centerline concentration against the

Figure 5.46 Experimentally measured centerline normalized concentration along the channel length with an applied potential of 280 V/cm for the optimized offset staggered micro-mixer design.


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Figure 5.47 Experimental concentration profiles (homogeneous channel: dotted lines; micromixer: dashed lines) and numerical (solid lines) across the channel width for both the homogeneous case prior to the mixing region (circular markers) and subsequent to the mixing region (star markers), and for the developed micro-mixer prior to the mixing region (triangular markers) and subsequent (square markers) for (A) 70 V/cm and (B) 280 V/cm.

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downstream channel length for an applied voltage potential of 280 V/cm. Clearly evident are these oscillating local concentrations induced by the surface charge patterning. Oscillations become substantially lower in magnitude at greater downstream distances as increased species homogeneity is achieved. Concentration profiles comparing the homogeneous and heterogeneous mixing channels at 70V/cm (Figure 5.47A) and 280V/cm (Figure 5.47B) indicate a substantial increase in species mixing with consistent correspondence between experimental and numerical results. Mixing was visibly enhanced for all voltages in comparison with a homogeneous T-channel, with a more marked improvement at increasingly higher voltages. Experimental and numerical profiles were consistent with values of mixing efficiencies within 2.5% at low voltage potentials (70 V/cm) and within 5% at higher potentials (555 V/cm). Using the developed micro-mixer, mixing efficiencies were improved from 75.3% to 97.2% at 70V/cm and from 22.7% to 90.2% at 555V/cm in comparison with the homogeneous system. Figure 5.48 presents a graph of the mixing efficiencies for varying voltage potentials for the developed micro-mixer compared with the homogeneous channel. Mixing is consistently enhanced with a more substantial improvement observable at higher

Figure 5.48 Mixing efficiencies for varying applied potentials for the homogeneous (square markers) channel and for the developed micro-mixer (triangular markers) as determined by experimental (dotted lines) and numerical (solid lines) analysis.


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voltage potentials where diffusive mechanisms are increasingly inefficient. As the mixing efficiency appears to become asymptotic at increasingly higher voltages in both experimental and numerical results, a maximum applied voltage potential of 555 V/cm was deemed sufficient to fully characterize micro-mixer performance. In terms of required channel lengths, numerical analysis indicated that at 2 80V/cm, a homogeneous microchannel would require a channel mixing length of 22mm for reaching a 95% mixture. By implementing the developed micromixer, an 88% reduction in required channel length to 2.6 mm was experimentally demonstrated. Practical applications of reductions in required channel lengths include improvements in portability and shorter retention times, both of which are valuable advancements applicable to many microfluidic devices. This enhanced micro-mixer technology can be applied to a wide range of lab-on-a-chip applications involving a variety of fluids ranging from fluorescent dyes with diffusion constants on the order of 10~10 m2/s to large molecules of DNA and proteins with diffusion constants on the order of 10~12 m2/s. Hence, the sensitivity to diffusion constants is of interest for the characterization of micro-mixer performance. This study was based on a fluorescein dye with diffusion constant D = 4.37 x 10~10 m2/s. With a staggered patch configuration, mixing efficiency was increased from 36 to 96% in comparison with the homogeneous case for an applied potential of 280 V/cm. Numerical simulations indicated that for a diffusion constant two orders of magnitude lower, mixing efficiencies are improved from 3.6% in the homogeneous case to 70% with staggered surface charge heterogeneities. Regardless of diffusion constant, the improvement in mixing efficiency over the purely diffusive case is substantial. As results are highly sensitive to diffusion coefficients however, required channels lengths for complete mixing will be largely application dependent. In summary, the passive electrokinetic micro-mixer with an optimized arrangement of surface charge heterogeneities can increase flow narrowing and circulation, thereby increasing the diffusive flux and introducing an advective component of mixing. Mixing efficiencies were improved by 22-68% for voltages ranging from 70 to 555 V/cm. For producing a 95% mixture, this technology can reduce the required mixing channel length by 88% for flows with Pec let numbers between 190 and 1500 and Reynolds numbers between 0.08 and 0.7.

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5-7


ANALYSIS OF ELECTROKINETIC FLOW IN MICROCHANNEL NETWORKS

Many microfluidic devices have a complex network of microchannels [72,73]. Different channel branches may have different size and surface properties. Controlling liquid flow in such microfluidic networks is critical to the functionality and performance of these microfluidic devices. Therefore, understanding of the electrokinetic flow characteristics in complicated microchannel networks is highly desirable. As a first step, a general analytical model [74] for one-to-multi-branch microchannels (a fundamental element in microfluidic networks) will be reviewed in this section. Two important parameters, the hydrodynamic conductance and the electrokinetic power, are defined to simplify the analysis and facilitate the understanding. The applications of this analytical model to a two-section heterogeneous microchannel and a oneto-two-branch microchannel system will be discussed. 5-7.1 General equations of electrokinetic flow in a single microchannel First, let's consider electrokinetic flow in a homogeneous microchannel of circular cross section of radius a (see Figure 5.49). For simplicity, we consider a microchannel filled with a symmetric electrolyte solution. The valence and the bulk ionic concentration are thus identical for the cations and anions. The ionic mobility is also assumed to be the same for the two types of ions. At steady state, the electrokinetic flow is described by the axial Navier-Stokes equation with an electrical body force

Figure 5.49. Electrokinetic flow in a homogeneous cylindrical microchannel of radius a and length /. The channel wall has a zeta potential C,. The volume flow rate is Q. The electrical current is /.


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where fi is the fluid viscosity, r the radial coordinate, u the axial fluid velocity, Vp the axial pressure gradient, and V^ the electric potential gradient parallel to the axis. The net charge density pe is given by the Poisson equation (42) where s is the dielectric constant of the fluid, s0 the permittivity of vacuum (8.854xlO~12 CV"1), and i// the EDL potential induced by the surface charge or the zeta potential C, at the channel wall. By substituting Eq. (42) into Eq. (41) and then integrating twice, the axial velocity u can be derived as Eq. (43) with the following boundary conditions: u(a) = 0, i//(a) = £ and «'(o) = y/'(o)= 0, where the prime indicates the first derivative with respect to r, (43) The current density j in an electrokinetic flow consists of two components: one is the convection current density due to the transport of ions with the bulk fluid flow, and the other is the conduction current density due to the motion of ions relative to the bulk fluid, (44) where m is the ionic mobility, z the valence of ions, e the electron charge (1.602xl0~19 C), n+ and n_ the volume densities of the cations and anions, respectively. The local ionic densities can be expressed in the form of the Boltzmann distribution, (45)

where n^ is the bulk ionic density, k^ the Boltzmann's constant (1.381x10 JK"1) and T the absolute temperature. Thus, Eq. (44) can be rewritten as

23

(46)

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where a = Imzen^ is the bulk conductivity of the liquid, and pe = -2ze« 00 sinh(zey// kfrT) is the very definition of the net charge density. Integrating Eqs. (43) and (46) over the channel cross-section gives the volume flow rate Q and the total current / (47) (48) where Ap = Vpl and A0 = V0/ are, respectively, the pressure difference and electric potential difference between the two ends of a microchannel with / being the channel length. L^ are the phenomenological coefficients from nonequilibrium thermodynamics of electrokinetic phenomena, and are expressed in terms of the properties of the microchannel and the liquid, (49a) (49b) (49c) where Gj (i= 1, 2, 3) are dependent only on the liquid properties and the channel size, the definitions of G, are given by [3]:


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where IQ and /j denote the zero order and the first order modified Bessel ( 1 1 /

V2

function of the first kind; K-\2Z e n^jes^k^T] is the Debye-Huckel parameter whose reciprocal {Ilk) indicates the characteristic thickness of EDL. As seen clearly from the above equations, Gy and G2 depends only on the nondimensional electrokinetic radius, K a. However, G3 is associated with the zeta potential of the channel wall, and is definitely no less than 1. As a result, the apparent conductivity for electrokinetic flow in a microchannel is higher than the bulk conductivity of the electrolyte (see the RHS terms of Eq. (49c)). This difference is attributed to the surface conductance due to the EDL effect. The equality of cross-coupling coefficients Z,12 a n d -^21 *s m accordance with Onsager's theorem. It should be noted that Eqs. (47) and (48) are applicable to electrokinetic flow in microchannels of arbitrary cross-section geometry. The cross-sectional shape of microchannels affects only the definitions of Li;. The model presented later in this paper, therefore, can be generalized to a variety of microchannels.

Figure 5.50. Electrokinetic flow in a one (superscript index 0) to n-branch microchannel (superscript indices from 1 to n) system. The subscript t represents the interface region from the main-channel to branch-channels. The hollow arrows show the moving directions (assumed positive) of the liauid in each branch of the channel.

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5-7.2 Electrokinetic flow in one-to-multi-branch microchannels Generally, a microfluidic network is composed of several one-to-multibranch channels and/or the opposite. Therefore, as a first step, we investigate only the electrokinetic flow in a one-to-multi-branch microchannel system. The same approach can be applied to more complicated microchannel networks. Figure 5.50 shows schematically such a one-to-n-branch microchannel. All branches (superscript indices from 0 to n) in this channel are assumed to be homogeneous and of circular cross-sections. In each branch, the phenomenological equations of the flow rate and the current, i.e., Eqs. (47) and (48), are transformed to (50a) (50b) (51a) (51b) where i varies from 1 to n, and all the symbols are labeled in Figure 5.50. According to Kirchoff s principle, the mass and electric current should be conserved when the liquid flows from the main-channel (superscript index 0) into n branches, i.e., (52)

(53) Pressure-driven flow For steady-state pressure-driven electrokinetic flow in a one-to-n-branch microchannel system, zero total electric current in each branch should be obeyed. This requires n constraints (i.e., one branch-channel gives one constraint) on the flow system, (54)


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Substituting Eqs. (50) and (51) into Eqs. (52)-(54), we obtain the following equations in the matrix form

(55)

where /^? and t are, respectively, the hydrodynamic pressure and electric potential at the junction from the main-channel to branch-channels (see Figure 5.50). <j>V' (i=l...n) denote the flow-induced electric potential at the end of each branch-channels. For this case, pt, (j)t and $S1' are the (n+2) independent variables to be determined. Using Cramer's rule, solutions of these quantities are derived as

(56)

(57)

(58)

The physical meanings of these parameters will be discussed later. It should be noted that <j>V' - gives the streaming potential generated in the passage from the main-

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channel to the fh branch-channel. The volume flow rates through the mainchannel and each branch-channel of this microchannel system are given by

(59a)

(59b)

From the above equations, we see that the flow rate is completely determined by the parameter gV' (z=O,l,...,w). If a conventional Poiseuille flow takes place in the same one-to-n-branch microchannel system, Eqs. (56)-(59) still hold except that g^1' is replaced by -L\\

in Eq. (49a) because the cross-coupling coefficients Z ^ vanishes.

Electroosmotic flow In electroosmotic flow, an externally applied electric field is used to transport the liquid while all ends of the microchannel system are usually exposed to atmosphere (i.e., p^' = 0 for i=0,\,..n). Consequently, Eqs. (52) and (53) are sufficient to determine the two independent variables pt and <j)t. As a result, the volume flow rate and the electric current through the /* branch of this microchannel system can be obtained as:

(60)


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5-7.3 Model analysis In the above derivations, we see two key parameters f*' and g^' involved in the general model. In order to understand the physical meanings of

and the Ohm's law (i.e., / = -^22^0 --na crA<j)/l) are readily recovered from Eqs. (47) and (48), respectively. It should be noted that, for a pressure-driven flow, A0 in Eqs. (47) and (48) is the flow-induced electrokinetic potential, i.e., the streaming potential. In this case, on the right hand side of Eq. (48), the first term represents the streaming current, and the second term represents the conduction current. At steady state, the net current along the microchannel should be zero, i.e., / = 0 in Eq. (48). This condition allows finding an expression for the streaming potential. That is,

channel system. For a pure electroosmotic flow with Ap = 0, combining Eqs. (47) and (48) gives the flow rate (63) As seen from the above two equations, / is an important parameter in electrokinetic flow. By analogy to the thermoelectric power (i.e., the so-called

306


Seebeck coefficient) defined in thermoelectrics, we may name this parameter as the electrokinetic power because electrokinetic effects can also be viewed as energy conversion (i.e., from pressure difference to electrical potential in Eq. (62)). From those definitions of phenomenological coefficients L;j in Eq. (49), we see that this electrokinetic power / can be negative or positive depending on the sign of zeta potential (negative C, corresponds to positive / ) , similar to that the sign of thermoelectric power is related to the polarity of ions (negative electrons correspond to negative thermoelectric power). Furthermore, Eq. (62) clearly indicates that the streaming potential is a linear function of the pressure drop. After substitution of Eq. (62) into Eq.(47), the flow rate in a pressure-driven flow can be expressed by (64) where g = (ij2^21 ~ A\^2l)l^22 > identical to the definition of g^> for the multi-branch channel system, can therefore be viewed as the hydrodynamic conductance at zero electric current in electrokinetic flow. The inequality LjjZ22 > ^12^21 must hold because of the positive entropy production requirement according to Prigogine's theorem. Therefore, g is a positive quantity. Also, g is independent of the sign of zeta potential. Combining Eqs. (62) and (64) yields a relationship between the streaming potential and the volume flow rate, (65) Under Debye-Huckel linear approximation and neglecting the variation of electric conductivity across the channel cross-section, the analytical forms of the electrokinetic power / and the hydrodynamic conductance g are given by [74]:

where


307

The first term on the RHS of the above denotes the hydrodynamic conductance of Poiseuille flow, and the second term reveals the negative effect due to electroviscosity resulting from EDL. The function F{KQ,C,) is dependent on both the electrokinetic radius and the zeta potential. Figure 5.51 shows the curves of / against the electrokinetic radius at different zeta potentials. The non-dimensional electrokinetic radius,

Figure 5.51. Electrokinetic power/with respect to the electrokinetic radius Ka at different zeta potentials. Pure water (lxl0~ 5 M concentration) is selected as the electrolyte solution. The working parameters include 7=298 K, Woo=6.022xl021 rrf3, £=80, ^=0.9xl0' 3 Kgm'V 1 ando=10~ 4 Sm~'.

308


Ka = a /(I/ k), is a measure of the channel size relative to the EDL thickness 1/K where K is the Debye-Huckel parameter. At C, = 0, / vanishes, equivalent to zero streaming potential in pressure-driven flow (Eq. (62)) or zero flow rate in electroosmotic flow (Eq.(63)). When C, ^ 0 , the magnitude of / increases with that of the zeta potential. Moreover, / becomes higher at a larger electrokinetic radius (e.g., a larger channel size or a higher ionic density or a thinner EDL), and approaches to constant for very large electrokinetic radius. In addition, / is independent of the channel length. Figure 5.52 displays the curves of hydrodynamic conductance g scaled by the one at £ = 0 (see the inset of Figure 5.52) against the electrokinetic radius. We see that g increases with the electrokinetic radius, but becomes lower in the presence of EDL (£ ^ 0 ) . The reason that g decreases with the zeta potential is due to the electro-viscous effect as mentioned above. The g curve attains a minimum when the electrokinetic radius is approximately 2. The g curves approach to one when the electrokinetic radius is larger than 100, indicating the electro-viscous effect becomes

Figure 5.52. Scaled hydrodynamic conductance with respect to the electrokinetic radius Ka at different zeta potentials. The microchannel is 1 cm long. The hydrodynamic conductance g is scaled by the one in the absence of EDL effect (C=0) which is plotted in the inset.


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negligible at high electrokinetic radii. Next, we will apply the general analytical model to two special channels. Pure water (lxl(T 5 M concentration) is selected as the electrolyte solution. 5-7.4 Two-section heterogeneous microchannel The two sections of a microchannel under investigation are assumed to have different zeta potentials. The microchannel of circular cross section has the same radius a in both sections, but the length of each section is not necessarily the same. Pressure-driven flow Setting n =1 in Eqs. (59) and (58), we can obtain the measurable volume flow rate and the streaming potential for pressure-driven flow in such a heterogeneous microchannel,

The denominator of the RHS term in Eq. (66) represents the total hydrodynamic resistance of the two-section microchannel, while the numerator is the total pressure drop applied onto the two ends of the channel. Moreover, the streaming potential in Eq. (67) is similar to that of a homogeneous channel (see Eq. (65)), and is the sum of the streaming potentials in the two sections. If we put Eq. (67) back into the form of Eq.(62), we find that the equivalent electrokinetic power /two-section of the two-section microchannel is given by

(68)

Figure 5.53 shows the distributions of flow-induced electric potential. Both the sign and the magnitude of zeta potential strongly influence this electric potential. This influence is due to the dependence of electrokinetic power / on the zeta potential. The lengths of homogeneous sections also affect the electric potential profile because they are involved in function of the hydrodynamic conductance g(l> (see Eq. (67)). The effect of zeta potential on the flow rate and

310


Figure 5.53. Distributions of flow-induced electric potential for pressure-driven flow in two-section heterogeneous microchannels: (a) /°WM).5 cm; (b) ^0)=0.25 cm and /(I>=0.75 cm. £0)= -50 mV, while: Case A, gl)= -50 mV; Case B, ^°= -100 mV; Case C, ^'WSO mV; Case D, i^'WlOO mV. a ( °W 1 ) = 5 urn. The applied pressure drop between the two ends of the channels is 1 atm.


311

the streaming potential are illustrated in Figure 5.54. The flow rate scaled by the flow rate without EDL effect is independent of the sign of zeta potential (symmetric about £X)= 0), but becomes lower at a higher zeta potential. For the cases shown here, the flow rate is only slightly affected by the magnitude of zeta potential (less than 2% in Figure 5.54. smaller channels should amplify this effect). As the zeta potential varies, the streaming potential changes significantly and even switches its polarity. It is not surprising that the streaming potential is almost a linear function of the zeta potential in one section of the heterogeneous channel. At a relatively large electrokinetic radius (e.g. 51.5 in Figure 5.54), the zeta potential has a minor effect on the hydrodynamic conductance g (see Figure 5.52). Hence, the streaming potential in Eq.(67) is mainly determined by the electrokinetic power/ which is approximately proportional to the zeta potential (see Figure 5.51). We also find that the pressure gradients in the two sections of the heterogeneous microchannel are not identical although very close to each other.

Figure 5.54. Scaled pressure-driven flow rate (thick lines) and the streaming potential (thin lines) against ^ of a two-section heterogeneous microchannel with 4=-50 mV. Case A, ^0)=0.5 cm and ^ 0 =0.5 cm; Case B, / through rough microchannels can be changed from Eqs. (6) and (7) into: (9)

OO)

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For electroosmotic flow through rough microchannels, three types of boundary conditions should be applied to Eqs. (9) and (10). They are called the fully-developed periodic conditions [12] applied at the inlet and the outlet of the computational domain, the slip boundary condition applied at the rough wall, and the symmetrical conditions applied at the two sides and the top plane of the computational domain (see Figure 6.3), respectively. It is worthwhile to note that the periodic conditions here are different from the conventional fully-developed periodic conditions. As no pressure difference is imposed on the inlet and the outlet of the rough microchannels, we have the following period conditions:

(11)

In the meantime, Veo and p at each point should satisfy the governing equations, Eqs. (9) and (10), especially at the inlet and the outlet planes of the computational domain. The slip boundary condition is applied at the channel wall (including the surface of roughness elements). According to the definition of the dimensionless governing equations, the dimensionless electroosmotic slip boundary velocity can be derived from Eq. (8): (12) At the rest computational domain boundaries, symmetrical conditions are applied. That is, for the velocity component parallel to the symmetric plane, the change of the velocity component normal to the plane is zero; for the velocity component normal to the plane, the velocity component itself is zero. Concentration field Study of the concentration field of sample species is of great importance in various microfluidics applications, for example, dye-based microflow velocity profile measurements, microscale species mixing, microfluidic dispensing, and surface reaction kinetic studies. Consider that a sample with concentration Co is released into the rough microchannel at time t0, the sample transport in time and space can be described by the law of mass conservation, which takes the form: (13)

Effects of Surface Roughness on Electrokinetic Flow

329

where Q is the concentration of /th species, Vep r- is the electrophoretic velocity of the /th species, and is given by Vepi = Hepj E, Dj and fiepj are the diffusion coefficient and the electrophoretic mobility of the z'th species, respectively. In Eq. (13), the first term on the left hand side is the transient term; the second term include two parts: the first part represents the convection effect resulting from the electroosmotic flow, the second part stands for the electrophoretic effect; the right hand side term is the diffusion term. By introducing the dimensionless group, (14) and define Peclet Number as Pe = VeosH I Dj, we can non-dimensionalize Eq. (13) as, (15)

Because releasing sample species into a microchannel is a transient process, initial condition and boundary conditions are required. The initial condition is: When t = 0, C,- (x ,y ,z ,0) = 0. The boundary conditions are: At the inlet plane, Cj (0, y , z , / ) = 1; at the outlet plane, we assume the concentration at each point is influenced only by its upwind control volume, and hence have VC7- = 0; at the symmetrical boundaries and the walls, no flux condition VC,- = 0 is applied. As seen from the above defined dimensionless parameters, the electrokinetic property of the microchannel surface (zeta potential C, ) and the properties of the solution (density and viscosity p and /S) are considered in the Reynolds number. For most electroosmotic flows in microfluidics, Re is in the range of 0.001-10. The property of the sample, the diffusion coefficient Dh is accounted in the Peclet number. Because the species electrophoresis and the liquid electroosmosis occur simultaneously under the same electrical field, the ratio nep,- / fieo instead of the individual values determines the relative electrophoretic velocity as Eq. (14) shows. Therefore, in addition to the parameters describing the surface roughness, Reynolds number is the basic

330


dimensionless parameter determining the characteristic of the electroosmotic flow; Peclet number and the ratio }xepi I neo are the basic dimensionless parameters determining the characteristics of the sample transport in the microchannels. The mobility ratio fiepj /fieo represents the ratio of the electrophoretic force to the electroosmotic force. If the two mobilities have a different sign (i.e., the electrical charge of the sample and the charge of the microchannel wall have different signs, one positive and the other negative), that is, juepj /jueo< 0, the sample molecules will move in the same direction as the electroosmotic flow and in a speed faster than the liquid. If juepi /jueo> 0 (i.e., the electrical charge of the sample and the charge of the microchannel wall have the same sign), the sample molecules will move in the direction opposite to the electroosmotic flow and in a speed slower than the liquid when ignoring the movement caused by diffusion. The absolute value of this ratio, | fiePii /jueo \ > 1 or | juepJ /jueo | < 1, signifies the speed of the sample's electrophoretic motion relative to that of the bulk liquid electroosmotic flow. Here we use the properties [13] of Rhodamine 6G as a representative sample to calculate the Peclet number and the ratio liepj Ineo, where Hepj= 1.4xlO~4 cm2/(Vs), and Dt = 3.0x1 (T6

ctn/(s). The above described model for the electrical potential field, the electroosmotic flow field and the concentration field during the species transport process was solved numerically by using a 3D-computation code based on the finite volume approach [14] and SIMPLEC algorithm [14]. As a part of the computational domain, the virtual space in the roughness is also solved using the extension of computational region approach [14]. That is, when solving the electrical potential field, a conductivity function is set up so that it will has a value of zero on the control volumes inside the non-conducting roughness prisms, and a value of one on the control volumes in the bulk liquid phase. When solving the electroosmotic flow field, we consider the control volumes in the roughness elements as a kind of fluid with extremely high viscosity. When solving the species transport process, we consider the control volumes in the solid roughness elements as species with a zero diffusion coefficient. 6-1.2 Electroosmotic flow and the associated sample transport in smooth channels As the same electrical potential difference is applied to both the smooth and the rough microchannels with the same channel length, the electrical potential drop over the length of each period, A<j> = (j)out - <j)jn, is the same for the rough and the smooth microchannels. It is known that in a smooth channel, the applied electrical potential varies linearly along the length direction, and hence the electrical field strength and electroosmotic flow velocity are uniform in the whole channel, as given below:


331

(16) (17) Referring to the non-dimensional groups used for the rough channels, the values Exs and Veo?s are the characteristic applied electrical field strength and the characteristic electroosmotic velocity. Therefore, the dimensionless values 0*, E and Ve0 in rough channels represent the relative values of electrical potential, the applied electrical field strength and the electroosmotic velocity at a certain point in the rough channel to the values of the smooth channel, respectively. Substituting the above Eqs. (16) and (17) into Eq. (14), the dimensionless governing equation for the associated sample transport through smooth channels can be derived from Eq. (15):

(18)

6-1.3 The applied electrical potential field and the electroosmotic flow field Due to the existence of the rough elements, the surfaces of the constant electrical potential are not parallel to each other. For example, the electrical potential field for the case of symmetrically distributed roughness elements is shown in Figure 6.4, where the colors represent the values of the dimensionless potential <j>. These isoelectric surfaces are distorted by the rough elements and therefore the electrical strength is not uniform in the microchannels. Thus the calculated local slip boundary velocities are different in the values and in the directions. As shown in Figures 6.5a and 6.5b, the top view of the slip boundary velocity on the bottom channel surface, the color scale shows the value of the dimensionless velocity component in x (the main flow) direction. The values bigger than one at certain regions indicate the slip velocity component in the main flow direction is larger than the electroosmotic velocity in the smooth channels. Figures 6.5a and 6.5b clearly shows that the slip velocity larger than one is mainly distributed in the lengthwise pathway formed between two neighboring rough elements (which will be called the "pathway" in the following), and the maximum value of ue0 is present around the corner in the pathway; while the slip velocity much less than one is mainly located at the gap between the front and the back surfaces of two neighboring rough elements (which will be called the "gap" in the following). It should also be noted that at

332


the rough element's surface, especially at the front and the backside of the prism's surface, see Figure 6.5c, the distorted electrical potential builds up a varying slip boundary velocity field. On the backside surface of the roughness elements, the velocity vectors are directed towards the center of the intersection line of the roughness element and the substrate surface. The magnitude of the velocity decreases gradually as it approaches this point. On the front side, the slip boundary velocities have the same distribution except with the opposite directions. This indicates the driven force distribution around the rough element's surface. The velocity field of the electroosmotic flow and the induced pressure field in a rough microchannel with symmetrically arranged roughness elements are shown in Figure 6.6. Even though no pressure differences are applied to the inlet and outlet of the whole channels, local pressure gradients are induced by the existence of the roughness elements in order to satisfy the flow continuity. Figure 6.6 shows that there are high-pressure zones at the upwind regions and the low-pressure zones at the tail regions of the rough elements, the pressure in the central flow above the roughness has little changes. The side view of the flow

Figure 6.4. The electrical potential field in the symmetrically arranged rough microchannels with a/H- 0.6, b/H= 1 and h/H= 0.2. The surfaces of the constant electrical potential are distorted due to the presence of the rough elements. The color represents the value of the dimensionless potential <j>.


333

Figure 6.5. The slip boundary velocity distribution on (a) the bottom plane in symmetrically arranged rough microchannel, (b) the bottom plane in asymmetrically arranged rough microchannel, and (c) the surface of one quarter of the rough element, with rough channel parameters: a/H= 0.6, b/H= 1 and h/H= 0.2. The arrows are local velocity vectors. The color represents the value of the dimensionless velocity.

334


and the pressure fields also shows that the maximum values of the velocity component ueo at each cross section all occur on the roughened wall. The similar phenomena are found for the electroosmotic flow in microchannels with asymmetrically arranged rough elements. Figure 6.7 shows the 3D streamlines of the electroosmotic flow through the symmetrically arranged rough microchannels. One can clearly see that the streamlines are curved with the roughness shapes. This is true for all the simulations conducted for cases of symmetrically and asymmetrically arranged roughness under various conditions. There are no flow recirculation zones in the gap, no matter what roughness element's sizes, separation distances, heights, the channel heights and the Reynolds numbers are. This implies that there is no sample molecule trap or accumulation of sample molecules in the gap regions between rough elements when sample transporting with the electroosmotic flows.

Figure 6.6. The top view (upper) and the side view (lower) of the electroosmotic velocity and the induced pressure distribution in symmetrically arranged rough microchannels with parameters: a/H = 0.6, b/H= 1 and h/H = 0.2. The arrows are local velocity vectors. The color represents the value of the local dimensionless pressure.

335


6-1.4 The influence on the electroosmotic flow rate and the induced pressure The roughness-induced pressure has a direct effect on the electroosmotic flow rate and the ability of removing the liquid from the gap to the flow pathway. In order to demonstrate the roughness effects on the flow rate, we define the corresponding electroosmotic flow rate in the virtual smooth channel as Qs = Veo,s^c,s • Since the pressure variations in x directions is key to the flow, we define the average dimensionless pressure in the cross section area as (19) where Ac is the cross section area in the rough microchannels. The effect of rough element's size on the induced pressure field Figure 6.8a shows the roughness size effect on the cross-section averaged dimensionless pressure (pave) in two-period-length symmetrically arranged rough channels. From Figure 6.8a, one can see that the induced pressure is periodically undulating along the rough channel, and the local maximum locates at the upwind region of the rough element, while the local minimum locates at the element's tail region. A higher maximum value of pave

indicates a larger

Figure 6.7. The 3D streamlines of the electroosmotic through symmetrically arranged rough microchannels with parameters: a/H= 0.6, b/H= 1 and h/H= 0.2.

336


pushing force acting on the fluid element near the upwind region to push the fluid from the gap to the central flow. Similarly, at the tail region of the roughness element, the low-pressure attempts to suck the fluid from central flow into the gap. Such exchange of fluid between the gap and the central flows is referred as the even-out effect. From Figure 6.8a, one can clearly see that this even-out effect exhibits a maximum value at a/H = 0.4 when the roughness element's size changes. When a/H < 0.4, the electrical potential field distortion due to the presence of the roughness elements is rather small, and hence difference in the slip velocities and the induced pressure gradients at the front and the back faces of roughness elements is small, resulting in a relatively small even-out effect. When a/H > 0.4, the high-pressure zone and the low-pressure zone are very close to each other, the overlapping effect reduce the absolute value of the induced pressure in the gap, reducing the fluid exchange between the gap and the central flow. for the Figure 6.8b shows the roughness size effect on pave asymmetrically arranged rough microchannels. To show the pressure fluctuation clearly, only half flow period is shown in Figure 6.8b. Similar to the symmetrically arranged rough channels, the maximum of pave

occurs at the

upwind region and the minimum of pave occurs at the tail region of the roughness, as shown by the curve a/H = 0.2 in Figure 6.8b. It is obvious that there is a large difference between Figures 6.8a and 6.8b. This difference is originated from the roughness element's arrangement. For the asymmetrically arranged roughness as illustrated in Figure 6.5b, the locations of the high/low pressure zones on the opposite sides of the flow pathway are asymmetrical (i.e., at different x positions) along the flow direction. For the symmetrical arrangement, as seen in Figure 6.6, the high/low pressure zones on the opposite sides of the flow pathway are at the same x position. Thus the curve Pave ~ x for the asymmetrical arrangement is more complicated and the distance between the high-pressure zone and the next low-pressure zone is much smaller than that of the symmetrically arranged rough channels. The effects of the rough element's separation distance, height, and the channel height When varying each parameter of b, h and H in turn, the curves of pave ~ x have similar undulating characteristics, and thus we do not show the figures. However, it is worthwhile to note the following effects. When the separation distance between roughness elements, b, changes from lOum to 20um while keeping the other parameters fixed (a = 6um, h = 2um, H = 10am, symmetrical arrangement), the simulations show an increasing even-out effect (a larger


337

Figure 6.8. The roughness size effect on the induced cross-section averaged dimensionless pressure p*ave in (a) two periods of the symmetrically arranged rough microchannel, and in (b) half period of the asymmetrically arranged rough microchannel with parameters: b/H= 1 and h/H= 0.2.

338


maximum value of pave) caused by the pressure-pushing force in the pathway. When the roughness element's height, h, increases, for example from 2um to 5|^m, while the other parameters are fixed ( # = 1 0 um, b = 10 um, a = 6um), the even-out effect is smaller because the velocity component on the backside of the rough element's is smaller. When the rough channel's height, H, increases, for instance from 10um to 30u.m, the curves of pave~ x overlap with each other, this indicates the induced cross-section averaged pressure resistance, dpaveldx, is proportional to the \IH under the same Reynolds number. As the central flow above the roughness contributes little to the pressure undulation, the induced high (or low) pressure at the front (or back) side of the rough elements does not vary so much when changing only the rough channel's height. The roughness effects on the electroosmotic flow rate The above-mentioned roughness effects on the electrical field and on the induced pressure field ultimately influence the electroosmotic flow rate. Figure 8 shows the influences of the rough microchannel geometry parameters on the electroosmotic flow rate, where Q/Qs is the ratio of the flow rate through the rough channels to that of the virtual smooth channels. Figure 6.9 shows that the

Figure 6.9. The effect of roughness parameters a, b, h, H on the electroosmotic flow rate with default parameters: a = 6(^m, b = lO^m, H= lOîm, and h = 2um.


339

induced pressures dramatically reduce the electroosmotic flow rates through rough microchannels, and the effect increases as the roughness size or the roughness height increases, and decreases as the roughness separation distance increases. As the rough channel's height increases, the ratio of the flow rate to that of the smooth channel decreases at first and then increases. When H decreases from 20um to lOum, relatively more electroosmotic driven force is produced due to the relatively larger rough element's surface and the larger electrical field distortion. These two effects contribute to the slight increase of the flow rate compared to the electroosmotic flow rate in smooth channel. For all the asymmetrically arranged roughness, the flow rates are all larger than those of the symmetrically arranged roughness. This is because the asymmetrical arrangement produces more streamlined flows, the induced pressure resistance is divided into more steps while the fluctuation amplitude is smaller and the pressure resisting flow mostly happens in each step. 6-1.5 Sample mass transport in the rough microchannels The characteristic of sample mass transport in the rough microchannels Generally, sample mass transport is the result of three major mass transport mechanisms: convection, electrophoretic transport and diffusion. In view of the characteristics of the electroosmotic flow field and the electrical potential field in rough microchannels, sample mass transport in rough microchannels is very different from most reported studies where smooth channel walls are considered. Figure 6.10a shows the sample concentration field in the computational domain when a certain amount of sample is released into the computational domain at the inlet plane. In the simulations, the electrokinetic properties are chosen as /ieo = 4x1 CT8 m2/Vs and fj.ep = 1.4xl(T8 rn/Vs. The color scale from dark to light green represents the value of dimensionless concentration from zero to one. For a given quantity of sample released into the computational domain, Figure 6.10a shows the concentration field in the computational domain during a period of/* = 9.15. At the end of this period, 71.5% of the sample remains in the computational domain and the rest has flown out. As seen from Figure 6.10a, there's no 3D enclosed iso-concentration surface in the gap region between the roughness elements. This implies that there will be no micro sample molecules trapped or accumulated in the gap regions. Finite sized particle may be trapped in the gaps. However, such particle-liquid systems are well beyond the scope of this paper. The model presented here generally is not applicable to the (micron sized) particle-liquid systems. The sample-liquid systems in this study are limited to small molecule-solution systems. If there is a 3D enclosed recirculation flow zone in the gaps between roughness elements, such a recirculation flow zone is referred to as the liquid molecule trap. The even-out effect under the electroosmotic driven flow as discussed in this work avoids such

340


a recirculation flow zone in the gaps. It is also noticed from the outlet plane of the computational domain that the sample concentration at the near wall region is higher than the region above. This is due to the larger values of the electroosmotic velocity near the channel wall region, while the central velocity field above the roughness region was impaired by the induced pressure gradient. Figure 6.10b shows the sample concentration field on the rough microchannel surface under the same conditions as in Figure 6.10a. From Figure 6.10b one can see that the concentration in the pathway surface is the highest. The effect of electrokinetic mobility on sample transport To examine the effects of electrokinetic mobility, we will compare the

Figure 6.10. The sample concentration field in and h/H = 0.2 at the time when a given microchannels. The concentration field (a) in rough surface with fxeo = 4x10~8 nf/Vs and \iep =

a rough microchannel: a/H = 0.4, b/H = 1 amount of sample is released into the the computational domain and (b) on the 1.4xlO"8 m2/Vs.


341

sample mass transport through the microchannel for a given quantity of sample flowing into the computation domain. The quantity of sample flowing into the computation domain is determined by the original sample concentration Co, and the time of releasing. The amount of sample flowing into the computation domain is defined by: (20)

where Nin is the quantity of the sample molecule flowing into the computation domain during a period At; Nin is its dimensionless form, Nin = Njn / CQH ; Ain

Figure 6.10. The sample concentration field in a rough microchannel: a/H= 0.4, b/H= 1 and h/H = 0.2 at the time when a given amount of sample is released into the microchannels. The concentration field (c) in the computational domain and (d) on the rough surface with neo =

6xlQ-8 rn/Vs and nep = lAxl0's tn/Vs.

342


is the cross section area at the inlet of computational domain, At is the duration that the released sample Nin reaches a given value. It should be noted here that the amount of sample transferred into the inlet of the computational domain by diffusion is much smaller than that by electroosmotic convection and electrophoresis, and therefore is neglected. We focus on the effect of electroosmotic mobility neo and the electrophoretic mobility jj.ep on the sample transport through rough microchannels. Figure 6.10c shows the sample concentration field in the whole computational domain with an increased electroosmotic mobility value, jieo = 6x10~8 m/Vs, while all other parameters are the same as in Figures 6.10a and 6.10b. As the electroosmotic mobility increases the electroosmotic velocity increases proportionally, therefore the duration of releasing the same quantity of sample into the computational domain is shorter, t = 6.13 in Figure 6.10c (note

Figure 6.10. The sample concentration field in a rough microchannel: a/H = 0.4, b/H= 1 and h/H = 0.2 at the time when a given amount of sample is released into the microchannels. The concentration field (e) in the computational domain and (f) on the rough surface with jieo = 4xlO"8 rn/Vs and /iep = -1.4xl0" 8 m2/Vs.


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t* = 9.15 in Figure 6.10a). From Figure 6.10c one can see that the concentration in the gap region is higher than that in Figure 6.10a, and there are overall more sample molecules (76% comparing with 71.5% in Figure 6.10a) in the whole computational domain. Figure 6.10d shows the corresponding concentration field on the roughened microchannel surface. Clearly the surface concentration is more uniform. When \xep = -1.4xlO~8 rn/Vs, i.e., the same absolute value of the electrophoretic mobility as in Figures 6.10a and 6.10c but with a different sign, while keeping the rest parameters the same, one sees a further improved concentration filed in the whole computational domain as shown in Figure 6.10e, and on the channel surface as shown in Figure 6.1 Of. When the electric charge of sample molecules has a different sign from that of the channel wall, the electrophoretic motion is in the same direction as that of the electroosmotic flow, and the sample molecule velocity is larger than the electroosmotic velocity of the bulk liquid. This mechanism contributes significantly to enhance the concentration uniformity in the channel's cross section area, and to increase the amount of the sample molecules in the gap regions, and the surface concentration. The duration of releasing the same quantity of sample into the computational domain is reduced further to / = 2.36, and more sample molecules, 81.7%, are in the computational domain. 6-1.6 Summary The electroosmotic flow through microchannels is greatly influenced by the presence and characteristics of roughness on the channel walls. The electrical field is distorted by surface roughness elements, which makes the electroosmotic slip velocity non-uniform. Due to the presence of roughness elements in the flow passage, the electroosmotic flow in the rough microchannels induces a periodic pressure field that makes the central flow velocity smaller than that in the near wall region and hence the flow rate through the rough microchannel is significantly reduced. The induced high-pressure zone occurs at the upwind region of the rough elements and the low-pressure zone is at the tail region. The induced pressure field causes the exchange of liquid between the gaps and the central flow, which is referred to as the even-out effect. The roughness element's size, height and the separation distance have large influences on the even-out effect, the microchannel height has a small influence on the even-out effect. There is no sample concentration trap in the gap regions between roughness elements during the electroosmotic transport. The increase of the electroosmotic mobility or the decrease of the electrophoretic mobility can significantly improve the uniformity of the sample concentration field in a rough microchannel.

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6-2


EFFECTS OF 3D HETEROGENEOUS ROUGH ELEMENTS

As we have seen in the previous section, the 3D roughness elements will influence liquid flow and sample transfer in microchannels. If the surface of these 3D roughness elements is chemically modified, or the rough elements are made of different materials, the electrokinetic properties of these rough elements will be different from that of the substrate surface. This will bring more complication to the electrokinetic transport processes. This section will examine the influence of micron-sized heterogeneous rough elements on electroosmotic flow and the associated mass transfer in microchannels [15]. We consider electroosmotic transport in a slit microchannel (formed between two parallel plates) with numerous heterogeneous prismatic roughness elements on the microchannel walls. All 3D prismatic rough elements have the same surface charge or zeta potential, the substrate (the microchannel wall) surface has a different zeta potential. We consider two types of distributions: symmetrical and asymmetrical arrangements, as illustrated in Figure 6.2 in the last section. The electroosmotic flow field and the sample concentration field are investigated in terms of the influence of the rough element size, height, density, the element arrangement, the channel size and the ratio of the electroosmotic mobility of the rough element's surface to that of the substrate. 6-2.1 Mathematical model The model assumptions and the computational domain (see Figures 6.2 and 6.3) are the same as in the previous section. The mathematical models of the electrical field, flow field and the concentration field are also the same as in the case of homogeneous roughness channels in the last section, except the slip flow boundary velocities. Because the roughness elements in this case have a different zeta potential from the substrate surface, the slip flow boundary conditions are different between the substrate (channel wall) surface and the roughness elements' surface. This will make the characteristics of the electroosmotic flow in this case very different from that shown in the previous section. The slip boundary velocity on the substrate surface is given by:

The slip boundary velocity on the roughness elements' surface is given by:


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where s^ = jj,eOiR /fxeo is the ratio of the electroosmotic mobility of the rough elements' surface to that of the substrate surface. The influence of electrophoretic mobility on the electrokinetic flow through microchannels was studied in the previous section. Therefore, in this section, we focus on an approximately electric neutral species [16], Rhodamine B, as a representative sample in the electrokinetic species transport in heterogeneous rough microchannels. The electrophoretic mobility of Rhodamine B is negligible under certain pH values, therefore we simply take juepj/jueo = 0. The diffusion coefficient of Rhodamine B is taken as [16] £),• = 4.37x10" cm /(s).

Figure 6.11 The slip boundary velocity distribution on (top) the bottom plane in a symmetrically arranged heterogeneous rough microchannel, and (bottom) the surface of one quarter of the rough element, with roughness parameters: a/H= 0.6, b/H= 1, h/H= 0.2 and e^ = -0.5. The arrows are local velocity vectors. The color represents the value of the dimensionless velocity.

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6-2.2 The applied electrical potential field and the electroosmotic flow field Due to the existence of the rough elements, the surfaces of the constant electrical potential are not parallel to each other. The isoelectric surfaces are distorted by the rough elements and therefore the electrical strength is not uniform in the microchannels. Thus the local slip boundary velocities are different in values and in directions. As shown in Figure 6.11, the top view of the slip boundary velocity on the bottom channel surface and 3D view of the slip boundary velocity on the rough element surface, the color scale shows the value of the dimensionless velocity component in x (the main flow) direction. The values bigger than one at certain regions indicate the slip velocity component in the main flow direction is larger than the electroosmotic velocity in the smooth channels. Figure 6.1 l(top) clearly shows that the slip velocity larger than one is mainly distributed in the lengthwise pathway formed between two neighboring rough elements (the "pathway"), and the maximum value of ueo is present around the corner in the pathway; while the slip velocity much less than one is mainly located at the gap between the front and the back surfaces of two neighbor rough elements (the "gap"). It should also be noted that at the rough element's surface, especially at the front and the backside of the prism's surface, refer to Figure 6.11 (bottom), the distorted electrical potential builds up a varying slip boundary velocity field. On the front and the backside surface of the roughness elements, the velocity vectors have a centrifugal or centripetal distribution with the center at the central point of the intersection line of the roughness element and the substrate. The value of the local slip boundary velocity on the roughness element's surface is proportional to the electroosmotic mobility. The velocity field of the electroosmotic flow and the induced pressure field in a symmetrically arranged rough microchannel with ^ = -0.5 and s^= 1.5 are shown in Figure 6.12 and Figure 6.13, respectively. The electroosmotic mobility ratio sM = -0.5 implies that the surface charge of the rough elements has an opposite sign to that of the substrate surface, and the value of the zeta potential of the rough elements is 50% smaller than that of the substrate surface, which forms the negative (opposite to the x-axis or main flow direction) slip boundary velocities on the roughness surface (e^ = -0.5) and the positive (in the x-axis or main flow direction) slip boundary velocities on the substrate. Figure 6.12(a) shows that when e^ = -0.5 there are high pressure zones located near the front of the roughness and the entrance of the pathway, and low pressure zones located near the backside of the roughness and the exit of the pathway. As seen in this figure, the first high pressure zones are near the rough element's front surface. This is because the flow from upwind driven by the positive substrate electroosmotic force and the counter-flow caused by the negative electroosmotic driving force on the rough elements' surface congest in front of the roughness


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Figure 6.12 The electroosmotic velocity and the induced pressure distributions in symmetrically arranged heterogeneous rough microchannels with parameters: a/H= 0.6, b/H = 1, h/H= 0.2 and e^ = -0.5 on (a) the plane y = 0.007 (the bottom substrate surface); (b) the plane y* = 0.035 (a horizontal plane above the substrate surface); (c) the plane z = 0.18 (a cross-section cutting through two rough elements and the gap in between); and (d) the middle vertical plane z* = 0.5. The arrows are local velocity vectors. The color represents the value of the local dimensionless pressure.

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elements and form the high pressure zones. The next high pressure zones are located at the entrance corners of the pathway near the roughness element, which is caused by the countercurrents on the roughness side surfaces forming the pathway. In the gap between two rough elements (Figure 6.12(c), the vertical surfaces of the rough elements are perpendicular to the horizontally applied electrical field, and hence there is no active electroosmotic flow on these surfaces. However, the positive (moving in right or x-axis direction) flow on the substrate surface generates a high pressure zone in front of the second rough element and a negative pressure zone near the back surface of the first rough element. These high and low pressure zones produce a counter-clock-wise circulation flow in the gap, as shows in the side view of the flow field, Figure 6.12(c). Further investigation shows that the circulation not only locates in the gap, it also propagates into the intersection. Figure 6.12(d) shows the velocity and pressure distributions in the plane z = 0.5, i.e., the middle vertical plane of the computational domain. One can see a larger circulation with the center a little higher than that of the circulation in the gap. Therefore, there is a circulation flow crossing over the width of the heterogeneous rough microchannels. As the negative s^ value increases, lower circulation center and higher circulation strength were found. The electroosmotic mobility ratio s^ - 1.5, implies that, although the rough elements' surface has the same type of surface charge, the zeta potential of the rough element surface is 50% higher than that of the substrate surface. As seen in Figure 6.13, there are high pressure zones located near the front of the roughness and the exit of the pathway, low pressure zones located near the backside of the roughness and the entrance of the pathway at the plane close to the substrate surface. In comparison with Figure 6.12, the different

Figure 6.13 The electroosmotic velocity and the induced pressure distribution in symmetrically arranged heterogeneous rough microchannels with parameters: a/H = 0.6, b/H = 1, h/H= 0.2 and e^ = 1.5 on the plane y = 0.007 (the bottom substrate surface).


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position of the high pressure zone in this case is caused by the larger slip boundary velocity (e^ = 1.5) on the roughness element's surface, which forms the flow congestion in the pathway and induces the pressure resistance in the pathway. In the electroosmotic flow through heterogeneous rough channel with a positive roughness heterogeneity s^ >1, no flow circulation were found. 6-2.3 The influence of the heterogeneous roughness on the electroosmotic flow rate The synergic roughness and heterogeneity effect on the slip boundary velocity and on the induced pressure field ultimately influence the electroosmotic flow rate. The value Q/Qs is the ratio of the flow rate through the heterogeneous rough channels to that of the virtual smooth and homogeneous channels. Similar to the homogeneous rough microchannels, the heterogeneous rough elements' size, height, separation distance, arrangement and the channel height determine the excess driven force at the solid-liquid interface relative to that of the smooth channel, the position and the frequency of the local pressure zones and the total resistance to the flow. Additionally, the electroosmotic mobility ratio sM determines the relative value and the direction of the electroosmotic driven force on surfaces, which also influence the flow patterns, local pressure distribution and the total resistance to the electroosmotic flow and hence the flow rate in the heterogeneous rough channels. The coupled effect of the heterogeneity and the roughness on the flow rate is shown in Figure 6.14, with s^ = -0.5. From Figure

Figure 6.14 The effect of the heterogeneous roughness parameters a, b, h, H on the electroosmotic flow rate with parameters as: a = 6(im, b = lOum, H = lOum, h = 2|im and £„ = -0.5.

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6.14, one can see that the two roughness arrangements have almost the same flow rate for the given set of rough microchannel geometry parameters. The relative flow rate to that of the smooth channel increases linearly when the roughness separation distance or the channel height increases. This is because when the roughness separation distance or the channel height increases, the roughness resistance effect decreases and hence the flow and the flow rate becomes closer to that in the smooth microchannels. The relative flow rate decreases when the roughness size increases. These behaviors are similar to the flow in the homogeneous rough microchannels. It is worthy to note that when the roughness height increases, the value Q/Qs increases from negative to a value close to zero. This is because when increases the roughness height, the increase of the induced pressure resistance on the opposite flow exceeds the effects of the counter flow driven force on the roughness surface. The studies of the effects of the degree of the roughness heterogeneity or the electroosmotic mobility ratio e^ with identical rough microchannel geometry parameters show that the relative flow rate increases linearly when increases the electroosmotic mobility ratio eM in the range of (-1 ~ 1.5). For example, with the rough microchannel geometry parameters set as a/H'= 0.6, b/H = 1.0, h/H = 0.2 and H = lOum for symmetrical and asymmetrical arrangements, Q/Q value ranges from -0.2 to 0.5 when s^ increases in the range of (-1 ~ 1.5). This indicates the roughness always acts as a hindrance of the flow, no matter the flow is in its normal direction with positive s^, or the flow change its direction due to large negative driven force on the rough elements when the roughness heterogeneity £M is less than a certain negative value. 6-2.4 Sample mass transport in the heterogeneous rough microchannels Generally, sample molecule transport is the result of three major mass transport mechanisms: convection, electrophoretic transport and diffusion. When electrically neutral species are released into the heterogeneous rough microchannels, sample transport is controlled by two mechanisms: convection and diffusion. Figures 6.15(a), (b) and (c) show the concentration fields of an electrically neutral sample (Rhodamine B in this case) in the microchannels at a moment when a given (the same) amount of sample molecules enters the inlet surface of the computation domain. The non-dimensional time /* is defined in Eq. 14. It should be noted that sM = -0.5 in Figures 6.15(a) and (b), and £^ = 1.5 in Figure 6.15(c). The electroosmotic flow is much faster in the case of Figure 6.15(c), consequently the time required to transport the same amount of sample molecules is shorter in Figure 6.15(c). Figure 6.15 shows the concentration fields of the vertical middle plane z = 0.5 and the bottom plane y = 0 to present the sample transport in the pathway and in the intersection region. From Figures


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Figure 6.15 The sample concentration field in a heterogeneous rough microchannel on the bottom plane y = 0 and the middle vertical plane z = 0.5 with roughness parameters as a/H = 0.6, b/H = 1 and h/H = 0.2 for (a) symmetrical arrangement and e^ = -0.5 at time t = 39.9; (b) asymmetrical arrangement and en = -0.5 at time t = 39.9; (c) symmetrical arrangement and £„ = 1.5 at time t = 1.03.

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6.15(a) and 6.15(b), one can clearly see the flow circulation, as indicated by the concentration field contours. The sample transport in the heterogeneous rough microchannel forms a tidal wave like concentration distribution in the plane z = 0.5. For asymmetrical arrangement, it forms four circulation zones with less circulation strength in the computation domain. This is caused by the staggerly distributed roughness elements that provide one more T-shape intersection region in each flow period. For symmetrical arrangement, it forms two circulation zones with higher circulation strength. This is caused by the large cross-shape intersection and stronger counter flow driven force due to the larger roughness surface area than that of the asymmetrical arrangement. When the electroosmotic mobility ratio e^ is increased to 1.5, there is no circulation exist in the flow and the sample concentration profiles in the vertical plane is similar to that of the smooth channel. The concentration field in the gap region between two rough elements is shown in Figure 6.16 for the case of Figure 6.15(a), i.e., a/H'= 0.6, b/H'= 1 and h/H = 0.2 and e^ = -0.5 in symmetrical arrangement at t = 39.9. Figure 6.16 shows that the sample circulation in the gap region occurs close to the substrate. It also indicates that the sample molecules moving into the gap is mostly from the main flow pathway, few sample molecules move over the roughness elements and enter the gap. This behavior is very different from the sample transport in homogeneous rough channels, where sample transports from both top and side of the rough elements and fills in the gap region.

Figure 6.16 The sample concentration field in a heterogeneous rough microchannel on the vertical plane z* = 0.28 with roughness parameters as a/H = 0.6, b/H = 1 and h/H = 0.2 and e^ = -0.5 for symmetrical arrangement at time t = 39.9.


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REFERENCES [I] [2] [3] [4] [5] [6] [7] [8] [9] [10] [II] [12] [13] [14] [15] [16]

D. Long, H.A. Stone and A. Ajdari, J. Colloid Interface Sci., 212 (1999) 338. A.D. Stroock, S.K. Dertinger, G.M. Whitesides and A. Ajdari, Anal. Chem., 74 (2002) 5306. D. Erickson and D. Li, Langmuir, 18 (2002) 1883. B. Potocek, B. Gas, E. Kenndler and M. Stedry, J. Chromatogr. A, 709 (1995) 51. A. Ajdari, Phys. Rev. Lett., 75 (1995) 755. A. Ajdari, Phys. Rev. E, 53 (1996) 4996. A. Ajdari, Phys. Rev. E, 65 (2002) 016301. A.D. Stroock, M. Week, D.T. Chiu, W.T.S. Huck, P.J.A. Kenis, R.F. Ismagilov and G.M. Whitesides, Phys. Rev. Lett., 84 (2000) 3314. Y. Hu, C. Werner and D. Li, Anal. Chem, 75 (2003) 5747-5758. D. Erickson and D. Li, Langmuir, 18 (2002) 8949. S.V. Patankar, C.H. Liu and E.M. Sparrow, ASME J. Heat Transfer, 99 (1977) 180. R.S. Amano, ASME J. Heat Transfer, 107 (1985) 564. S.V. Ermakov, S.C. Jacobson and J.M. Ramsey, Anal. Chem., 70 (1998) 4494. S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, New York, 1980. Y. Hu, C. Werner and D. Li, Proceedings of 2nd International Conference on Microchannels and Minichannels, Rochester, New York, June 17-19, 2004. D. Sinton, D. Erickson and D. Li, J. Micromech. Microeng., 12 (2002) 898.

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Chapter 7

Experimental studies of electroosmotic flow Electro-osmotic pumping (i.e., transporting liquids in microchannels by electroosmosis) is a preferred method in a variety of Lab-on-a-Chip devices, because the electroosmotic pump has no moving mechanical parts, zero-volume (i.e., it requires only electrodes that can be embedded in the chip) and provides uniform cross-section flow (i.e., plug-type velocity profile). To design and to operate an electroosmotic pump, it is critical to know the volumetric flow rate or the velocity of the electroosmotically pumped flow. As the flow rates involved in the lab-chip devices usually are very small (e.g., 0.1 uL/min.), and the size of the microchannels is very small (e.g., 10-100 urn), it is extremely difficult to measure directly the flow rate or velocity of the electroosmotic flow in these microchannels. To study liquid flow in microchannels, various microflow visualization methods have evolved. Micro particle image velocimetry (microPIV) is a method that was adapted from well-developed PIV techniques for flows in macro-sized systems [1-5]. In the microPIV technique, the fluid motion is inferred from the motion of sub-micron tracer particles. To eliminate the effect of Brownian motion, temporal or spatial averaging must be employed. Particle affinities for other particles, channel walls, and free surfaces must also be considered. In electrokinetic flows, the electrophoretic motion of the tracer particles (relative to the bulk flow) is an additional consideration that must be taken. These are the disadvantages of the microPIV technique. Dye-based microflow visualization methods have also evolved from their macro-sized counterparts. However, traditional mechanical dye injection techniques are difficult to apply to the microchannel flow systems. Specialized caged fluorescent dyes have been employed to facilitate the dye injection by using selective light exposure (i.e., the photo-injection of the dye). The photo-injection is accomplished by exposing an initially non-fluorescent solution seeded with caged fluorescent dye to a beam or a sheet of ultraviolet light. As a result of the ultraviolet exposure, caging groups are broken and fluorescent dye is released. Since the caged fluorescent dye method was first employed for flow tagging velocimetry in macro-sized flows in 1995 [6], this technique has since been used to study a variety of liquid flow phenomena in microstructures [7-15]. The

Experimental Studies of Electroosmotic Flow

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disadvantages of this technique are that it requires expensive specialized caged dye and extensive infrastructure to facilitate the photo injection. As an alternative to photo-injection of fluorescent dye, a flow marker can be created in a uniform solution of fluorescent dye by local photobleaching [11,15,16]. In additional to these PIV and dye-based techniques, the electroosmotic flow velocity can also be estimated indirectly by monitoring the electrical current change while one solution is replaced by another similar solution during electroosmotic flow [12,17,18]. In this chapter, we will discuss the indirect techniques and direct techniques of measuring the electroosmotic flow velocity.

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7-1


MEASUREMENT OF AVERAGE ELECTROOSMOTIC VELOCITY BY A CURRENT METHOD

An experimental approach was developed [17] to evaluate the electroosmotic flow velocity by monitoring the current change in a process where one solution replaces another similar solution in a microchannel. In this method, a capillary tube is filled with an electrolyte solution, then brought into contact with another solution of the same electrolyte but with a slightly different ionic concentration. Once the two solutions are in contact, an electrical field is applied along the capillary in such a way that the second solution is pumped into the capillary and the 1st solution flows out of the capillary from the other end. As more and more of the second solution is pumped into the capillary and the first solution flows out of the capillary, the overall liquid conductivity in the capillary is changed, and hence the electrical current through the capillary is changed. When the second solution completely replaces the first solution, the current will reach a constant value. Knowing the time required for this current change and the length of the capillary tube, the average electroosmotic flow velocity can be calculated by (1) where L is the length of the capillary and At is the time required for the higher (or lower)-concentration electrolyte solution to completely displace the lower (or higher)-concentration electrolyte solution in the capillary tube.

Figure 7.1. Illustration of an experimental setup for measuring the current change during an electroosmotic displacing process.


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Figure 7.1 shows an experimental set-up used to measure the average velocity by monitoring the current change in the electroosmotic displacing process. In the experiments, polyamide coated silica capillary tubes of various diameters (Polymicro Technologies Inc., Phoenix, AZ) were cut to 10cm in lengths and used to connect the electrolyte solution in reservoir 1 to the electrolyte solution in reservoir 2. The reservoir 2 was filled with an electrolyte solution at a desired concentration. The capillary tube and reservoir 1 were filled with an electrolyte solution at a concentration lower than the concentration in reservoir 2. For example, the capillary tube and reservoir 1 were filled with an electrolyte solution at a concentration that is 95% of the concentration in reservoir 2. Immediately after connecting the two reservoirs by the capillary tube, a voltage difference between the two reservoirs was applied by setting reservoir 1 at ground potential and reservoir 2 to a high voltage power supply unit (CZE1000R, Spellman, NY) via Platinum electrodes. The applied electrical field results in an electroosmotic flow in the capillary tube. During electroosmosis, the higher-concentration electrolyte solution from reservoir 2 gradually displaces the lower-concentration electrolyte solution in the capillary tube. As a result, the overall electrical resistance of the liquid in the capillary tube changes. By monitoring the change in the electrical current under a constant applied voltage difference during electroosmosis, the change in electrical resistance can be monitored. An L-DAS8 data acquisition chip (Kieffley) was used to record the voltage (kV) and current (uA) as a function of time (seconds). Once the lower-concentration solution in the tube is completely replaced by the higher-concentration solution from reservoir 2, the

Figure 7.2. A typical result of current versus time for the specific case of capillary diameter D=100 urn and Ex=3500V/10 cm. KCL concentration in Reservoir 1 is C95%=0.95xl0"4 M. KCL concentration in Reservoir 2 is Cioo%= lxl(T 4 M.

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Figure 7.3 The measured average velocity versus the applied voltage and the capillary diameter, for a 10 cm long polyamide coated silica capillary tube.


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current will reach a maximum and constant value. The measured time for the current to reach such a plateau value is the time required for the solution from reservoir to travel through the entire capillary tube. The average velocity of the liquid flow can then be calculated by using Eq.(l). The average velocity of the liquid during electroosmosis is determined by Eq. (1). Figures 7.3a, 3b and 3c are plots of the average velocity versus the applied voltage and the capillary diameter for a polyamide coated silica glass capillary of 10 cm in length, for 3 different ionic concentrations. As seen in the figures, an almost linear relationship between the applied voltage and the average velocity is evident. In addition, for each plot it is clear that the average velocity is independent of the diameter of the capillary tubes, a relationship that has also been shown for rectangular microchannels in a numerical study [19]. In the experimental studies, each measurement was repeated at least three times for a given set of conditions. All experiments were conducted at room temperature (22°C). In Chapter 4, we have discussed the general electroosmotic solution displacing processes between two solutions of different electrolytes and very different concentrations. We may apply the model developed in Chapter 4 to examine the electroosmotic flow of one solution replacing another similar solution. The current-time relationship, the time evolution of diffusing zone between the two solutions, and the average velocity of the electroosmotic flow can be predicted in this way. The model predictions will be compared with the experimental data. When the two solutions are brought into contact in a microchannel under an applied electrical field, the mixing (involving convection and diffusion) occurs near the interface of two solutions. As discussed in Chapter 4, during the solution replacing process, the capillary can be divided into three sections according to the concentration distribution. The first section is filled with one solution, the second section is the diffusing zone and the third section is filled with another solution. The total electrical resistance of the capillary and the

Figure 7.4. Coordinator system for a cylindrical capillary tube. Z is the flow direction.

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electrical resistance of each section depend on the type of ions and the ionic concentration distribution in the capillary and vary with time. The electrical current, which is generated by the ions' motion under the applied electrical field, is axially uniform and varies with time. Consequently, the electrical field strength along the capillary (volts per meter) is different from location to location. If the two solutions are very different in terms of the ion type, the ionic valence, and the ionic concentration, the zeta potential, £ , and the ion distributions in the EDL field will be different in different sections. Therefore, the conditions of the electroosmotic flow are different in different sections. However, here we consider the electroosmotic replacing flow of two solutions that contain the same kind electrolytes but have a small difference in ionic concentration. For such a system, the zeta potential and the EDL field can be approximated as the same along the microchannel. As discussed before, if we consider a symmetrical electrolyte solution ( z : z = 1:1), the net charge density pe and the electrical potential \j/{r) in the electrical double layer field in a cylindrical capillary tube are described by the Boltzmann equation and the Poisson-Boltzmann equation, respectively. (2)

(3) where s is the dielectric constant of the solution and SQ is the permittivity of vacuum; ni<x> and zt are the bulk ionic concentration and the valence of type i ion, respectively; e is the charge of a proton, k^ is Boltzmann constant and T is the temperature. Introducing the dimensionless variables, (4)

where d is the diameter of the capillary tube and r is the radius of the capillary tube, the non-dimensional Poisson-Boltzmann equation can be written as


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(5)

Because of the symmetry of the EDL field in the cylindrical capillary, Eq. (5) is subjected to the following boundary conditions: (6)

where C, is the zeta potential. During the displacing flow, there are three sections of different ionic concentrations in this capillary. As shown in Figure 7.1, the solution on the left side (entering the capillary) has a relatively higher concentration, C2, the solution on the right side (leaving the capillary) has a relatively lower concentration, Cj, and between these two section, there is a diffusion zone due to the concentration difference. For the case interested here, we consider (C2 - Ci)IC2 < 10%. The liquid is considered as an incompressible fluid and there are no velocity components in r and 9 directions. That is,

Substituting the above conditions into the continuity equation for incompressible fluid: (7) yields, (8) It is obvious that the velocity component in z-direction uz is constant axially and can be calculated based on any section of the capillary. For simplicity, we calculate the velocity for a pure solution section. At a given time, the flow of a pure solution can be considered as a fully developed, onedimensional flow. The momentum equation for such a case is given by:

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Figure 7.5. Comparison of the experimentally determined current - time relationship with the model prediction under 35 KV/m.


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(9) where î is the viscosity, pei is the local net charge density and Ej is the electrical field strength applied to the z'-th section. Substituting the Eq. (2) for the net charge density into the Eq. (9) and introducing the following non-dimensional variables,

the non-dimensional equation of motion can be obtained as: (10)

Figure 7.6. Comparison of the experimentally determined average velocity - concentration relationship with the model prediction for KCl solution under an electrical field of 35 KV/m.

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which is subjected to the no-slip boundary conditions: (lla) (lib) Consider the electroosmotic flow for the system shown in Figure 7.1. During the electroosmosis, the high concentration solution gradually displaces the lower concentration solution in the capillary tube. Because there is difference in chemical potential between the high concentration solution and the low concentration solution, the diffusion appears across the interface, meanwhile, the convection also occurs due to the movement of the liquid. Consequently, the concentration in diffusion zone changes with time, and the diffusing zone length changes. The concentration distribution is governed by the conservation law, which for this case takes the form: (12) where C is the bulk ionic concentration in capillary tube, uz is the electroosmosis velocity of the liquid which can be determined by the equation of motion and D is the diffusion coefficient of the electrolyte in the solution. Introducing the following non-dimensional parameters:

where z represents the flow direction as shown in Figure 7.4. The equation of concentration can be non-dimensionalized as follows: (13)

The above equation is subjected to the following initial and boundary conditions: (14a)


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(14b)

(14c)

where Ltoto/ and rf are the total length and the internal diameter of the capillary tube, respectively. In this case, Ltoto/ is 10 cm and d is 0.01 cm. L\ is the length of the section filled with the low concentration (C/) electrolyte solution. Now we have the complete set of equations, Poisson-Boltzmann equation (5), motion equation (10) and concentration equation (13) and the matching initial and boundary conditions. In this model, the diffusing zone length is defined as follows: The position where C = 99.99%C2 will be the left boundary of the diffusing zone. The position where C - 99.99%Cj will be the right boundary of the diffusing zone. It is obvious that the length of the diffusing zone and the lengths of the other two sections change with time. Consequently, according to the equation of electrical resistance, (15) where Lj is the length of the z'-th section, C,- is the concentration of the i-th section, Xj is the bulk conductivity of the i-th section and A is the cross section area of the capillary, the overall electrical resistance of the liquid in the capillary tube will change with time during the electro-osmosis. At a given time, the total resistance is the sum of the resistance of the three sections, given by: (16) where R\ is the electrical resistance of the section with the low concentration (C/) electrolyte solution; R2 is the electrical resistance of the section with the higher concentration solution (C2), and Rmix is the electrical resistance of the diffusing zone where the ionic concentration is between C] and C2. When the total resistance and electrical voltage applied to the capillary are known, the electrical current through the capillary can be determined by:

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(17) where Vtotai is the total electrical voltage applied to the capillary. As the current keeps constant along the capillary and the electrical resistance varies from section to section (see Eq. (15)), the electrical field strength for each section will be different and can be calculated by (18) Using the above-defined equations, one can determine the concentration distribution, the electrical field strength for each section and the diffusing zone length. First, at a given time, a guessing value for the concentration is chosen, the equation of electrical potential Eq. (5) can be numerically solved to find the EDL potential y/(r). Once the EDL potential is known, the local net charge density pe(r) can be determined according to Eq. (3). With the guessing value for the concentration field again, the electrical resistance, the electrical current and the electrical field strength for each section can be determined by Eqs. (15) (18). The equation of motion, Eq. (10), can be numerically solved to find the velocity, uz(r). Using this velocity profile, the equation of concentration, Eq. (13), can be solved to obtain the concentration distribution in capillary, and hence the length of the diffusing zone can be determined based on the obtained concentration profile. Repeat this iteration procedure until the convergence of concentration is reached. Finally, with the results obtained in this time step, the above procedure is repeated for the next time step until the high concentration solution completely replaces low concentration solution in the capillary tube {i.e., the left boundary of the diffusing zone reaches the exit of the capillary tube). Figure 7.5 shows the experimentally measured current-time relationships. Because the electrical current depends on the applied electrical field and the electrical resistance of the solution, which in turn depends on the concentration and the conductivity, when the concentration in the capillary change, the current will change. Therefore, the electrical current in the capillary increases with time as the high concentration solution gradually replaces the low concentration solution. The increase in current continues until the capillary is completely filled with the high concentration solution. Thereafter the current reaches a steady and maximum value. Using the aforementioned numerical model, the complete set of non-linear differential equations, governing the diffusion and convection process of the two solutions in capillary, were solved to predict the diffusing process. As


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Figure 7.7. Concentration distribution in capillary tube at time t = 12 s, t = 24 s and t = 36s for 1.1 x 10"4 M KCl solution under an electrical field of 35 KV/m.

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seen from Figure 7.5, there is a good agreement between the experimental data and the model prediction. As explained before, the time for the current reaches a plateau value is the time required for the high concentration electrolyte solution to travel through the entire capillary tube. The average velocity of the liquid flow can then be calculated by using Eq. (1). From the data shown in Figure 7.5, the required time can be determined, therefore, the average velocity of the electro-osmotic flow can be obtained with the known capillary length. Figure 7.6 is the plot of the average velocity versus the ionic concentrations. As seen in Figure 7.6, an essentially linear relationship between the average velocity and the concentration is evident and a good agreement between the model predictions and experimental data was found. The model presented in this section also reveals the diffusing process. The concentration difference across the interface generates the diffusion. Initially, the length of the diffusing zone increases with time after the two solutions are brought into contact. However, the length of the diffusing zone will decrease when the low concentration solution completely leaves the capillary tube and the high concentration solution continues to move towards the end of capillary. This can be seen from Figure 7.7. The length of the diffusing zone at 24 seconds is bigger than that at 12 seconds, which shows the expansion of the diffusing zone. The length of the diffusing zone at 36 seconds is smaller than that at 24 seconds, showing the reduction of the diffusing zone at the end of the displacing process.


7-2

369

MEASUREMENT OF AVERAGE ELECTROOSMOTIC VELOCITY BY A SLOPE METHOD

In the last section, we discussed an experimental approach to evaluate the electroosmotic flow velocity by monitoring the current change in a process where one solution replaces another similar solution in a microchannel. However, practically it is difficult to determine the exact time required for one solution completely replacing another solution. This is because the gradual changing of the current with time and the small current fluctuations exist at both the beginning and the end of the replacing process. Figure 7.8 shows one of the typical results of the measured current-time relationship. These make it very difficult to determine the exact beginning and ending time of the current change from the experimental results. Consequently, significant errors could be introduced in the average velocity determined in this way. It has been observed that, despite the curved beginning and ending sections, the measured current-time relationship is essentially linear in most part of the process, as long as the concentration difference is small. Therefore, a new method was developed to determine the average electroosmotic velocity by using the slope of the current-time relationship [18].

Figure 7.8. Typical results of the measured current-time relationship for lxlO" 4 M KC1 solution in a microchannel of 10cm in length and 200 u.m in diameter, under an applied electrical field of 30 kV/m.

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When a high concentration electrolyte solution gradually replaces another solution of the same electrolyte with a slightly lower concentration in a microchannel under an applied electric field, the current increases until the high concentration solution completely replaces the low concentration solution, at which time the current reaches a constant value. If the concentration difference is small, a linear relationship between the current and the time is observed. This may be understood in the following way. In such a process, because the concentration difference between these two solutions is small, e.g., 5 %, the zeta potential, which determines the net charge density and hence the liquid flow, can be considered as constant along the capillary. Consequently, the average velocity can be considered as a constant during the replacing process. As one solution flows into the capillary and the other flows out of the capillary at essentially the same speed, and the two solutions have different electrical conductivity, the overall electrical resistance of the liquid in the capillary will change linearly, and hence the slope of current-time relationship is constant during this process. The slope of the linear current-time relationship can be described as: (19) where Ltotal is the total length of the capillary and uave is the average electroosmotic velocity, E is the applied electrical field strength, A is the cross section area of the capillary, ( A ^ - ^ J I ) *s t n e bulk conductivity difference between the high concentration solution and the low concentration solution. In Eq. (19), all the parameters are known and constant during the replacing process. If we group all the known parameters and denote 77 =EA(Xj,2 - H\)l^totah Eq(19) can be rewritten as: (20)

Eq.(20) indicates that the average electro-osmotic flow velocity can be determined by measuring the slope of the current-time relationship. To measure the slope of the current-time relationship, experiments similar to what described in the last section were performed. In the experiment, Polyamide-coated silica capillary tubes of different internal diameters (Polyamide Technologies Inc., Phoenix, AZ) were cut to 10 cm in length and used to connect the electrolyte solutions in reservoir 1 and reservoir 2. In the experiment, reservoir 2 was filled with a higher concentration solution 2. The capillary tube and reservoir 1 were filled with a lower concentration solution 1. All the solutions in our experiments were prepared by using Deionized


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ultrafiltered water (Fisher Scientific, Ontario), KC1 (Anachemia Science, Quebec), and LaCl3 (Fisher Scientific Company, New Jersey). Immediately after connecting the two reservoirs by the capillary tube, a voltage difference between the two reservoirs was applied by setting reservoir 1 at a ground potential and reservoir 2 to a higher voltage (HV power source: CZE 1000 R, Spellman, NY) via platinum electrodes.

Figure 7.9. Examples of the measured current—time relationship, for 1x10 3 M KC1 solution (top) and lxlO" 3 M LaCl3 solution (bottom) in a 10cm long capillary of 100 urn in diameter under an electrical field of 10 kV/m.

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Figure 7.10. The average velocity versus the applied voltage and the capillary diameter, for 1x10 3 M KC1 solution (top) and lxlO"4 M KC1 solution (bottom) in a 10cm long capillary tube.


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The applied electrical field results in an electroosmotic flow in the capillary tube. The electrolyte solution from reservoir 2 gradually displaces the electrolyte solution in the capillary tube. As a result, the overall electrical resistance of the liquid in the capillary tube changes, A PGA-DAS 08 data acquisition chip (OMEGA Engineering, Quebec) was used to record the voltage (KV) and current (uA) as a function of time (second). Once the solution in the capillary tube is completely replaced by the solution from the reservoir 2, the current reaches a constant value. In the experimental studies, each measurement was repeated at least five times for a given set of conditions. All experiments were conducted at a room temperature (25°C). Because the electrical current depends on the applied electrical field strength and the conductivity of the solution, when the concentration in the capillary changes, the conductivity and hence the current will change. Therefore, the electrical current through the capillary increases with time as the high concentration solution gradually replaces the low concentration solution in the capillary. The increase in current continues until the capillary is completely filled with the high concentration solution. Thereafter the current reaches a constant and maximum value. Some typical results of the current change with time are shown in Figure 7.9. All the experimental data shown in the figure is an average value of 5 independent measurements with an error approximately 5%. The average electroosmotic velocity can be determined by using the Eq. (2) and the measured slope. Figure 7.10 shows that the average velocity versus the applied voltage and the capillary diameter for lxlO"3 M and lxlO"4 M KC1 solutions in a glass capillary of 10 cm in length. As seen from the figure, a linear relationship between the applied voltage and the average velocity is evident. In addition, it is clear that the average velocity is essentially independent of the size of the capillary. Figure 7.11 shows that the comparison of the average velocity for lxlO"3 M KC1 solution and for lxl0" 3 M LaCl3 solution between the theoretical model prediction and the experimental results determined by the current—time method and by the slope method. Here the theoretical model has been described in the previous section. It can be seen from this figure that there is a better agreement between the model predictions and the experimental results determined by the slope method. The experimental results determined by using the estimated time for the complete displacing process deviate from the above two results. This is because of the inaccuracy in determining the required time (particularly the ending time) for one solution replacing another solution. In the current—time method presented in the last section, the total time required for one solution completely replacing another solution has to be found, which often involves some inaccuracy of choosing the beginning and the ending points of the replacing process. In this slope method, instead of identifying the

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Figure 7.11. Comparison of the model predicted average velocity with the experimentally determined average velocity by the current—time method and by the slope method for (top) lxlO"3 M KC1 solution, and (bottom) lxlO' 3 M LaCl3 solution.


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total period of time, only a middle section of the current-time data is required to determine the slope of the current-time relationship. This method is easier and more accurate than finding the exact beginning and the ending points of the replacing process. In addition, for some cases of high concentration solutions under low electrical field strength, the time required for the high concentration solution completely replacing low concentration solution is very long (e.g. 700 seconds). Therefore, the joule heating effects become significant in these cases. However, Using this new slope method, the joule-heating problem can be avoided since it is not necessary to complete the displacing process. Usually, a sufficient amount of data to determine the linear relationship can be obtained in less than 100 seconds, the average velocity can thus be determined and the experiments can be stopped before appreciable heating effect takes place.

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7-3


MICROFLUIDIC VISUALIZATION BY A LASER-INDUCED DYE METHOD

Although the current-monitoring methods discussed in the previous sections provide simple means to evaluate the average electroosmotic flow velocity in microchannels, they are indirect methods and cannot provide direct insight of microflows. This section will introduce a direct microfluidic visualization technique to measure the microflow velocity profiles. A major macroflow visualization technique is the particle image velocimetry (PIV) [1-5]. Application of PIV to microflows typically involves use of fluorescent microspheres. Taylor and Yeung [1] calculated fluid velocity from the streaks produced from fluorescent particles in a liquid flow in a capillary. PIV techniques involving double-frame cross-correlation algorithms have been successfully applied to microflows [2,3]. The hydrophobicity of latex particles, a common choice of marker in PIV, can cause particle aggregation and adhesion to the channel surface. The use of hydrophilic markers, such as the unilamellar liposomes employed by Singh et al. [4], can mitigate these effects. Extracting an unaltered local fluid velocity from an observed marker velocity, however, remains a fundamental challenge in PIV and flow visualization techniques in general. This is further complicated on the microscale by the addition of electrokinetic effects including electrophoresis, electrophoretic relaxation, and particle-wall interaction [20]. Dye-based microflow visualization techniques have also evolved from their macro-sized counterparts. A major difficulty with the dye techniques is how to directly inject the dye into the microscale flows without significantly altering the flow pattern. The conventional mechanical injection techniques are difficult to apply on the microscale without interfering with the flow field. One option is to replace a solution with a dye solution and track the concentration front [1]. Such a method is complicated by the relatively fast diffusion at the dye front, and the changes in the EDL due to differences between the dye solution and the original solution. In addition, accurate tracking usually requires high concentration gradients, which in turn generate high rates of diffusion relative to the mean flow velocity. Tracking points of zero concentration gradient (maxima) would minimize diffusion effects, however, these points are not clearly definable when one solution is gradually replaced with another. Specialized caged fluorescent dyes can serve to eliminate the problems associated with the mechanical dye injection. A photo-injection is accomplished by exposing an initially non-fluorescent solution seeded with caged fluorescent dye to a beam or a sheet of ultraviolet light. As a result of ultraviolet exposure, caging groups are broken and fluorescent dye molecules are released [21]. The free fluorescent dye may then be imaged using classical fluorescence techniques. Lempert et al. [6] applied caged dyes to flow visualization in a lcm internal


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diameter pipe as well as a large water channel facility. Paul et al. [7] applied the method to microflow visualization, they presented images of dye transport in pressure-driven flow and electroosmotic flow. Velocity profiles were calculated for the pressure-driven case using image pairs and a scalar image velocimetry (SIV) technique. That study outlined several key issues associated with cageddye techniques on the microscale, namely: decreasing signal to noise ratio near the wall; and predicting electrophoretic motion of the charged dye molecules (especially problematic in large molecular weight, multi-component dyes). Dyebased velocimetry requires knowledge of the electrophoretic mobility of the dye molecules when the method is applied to electrokinetic flows (analogous to PIV). Herr et al. [8] applied the method of Paul et al. [7] to study capillary flow with nonuniform zeta potential. Herr et al.[8] tracked the motion of the dye by fitting Gaussian curves to the axial concentration profiles. Dahm et al. [9] presented a SIV technique for four-dimensional, turbulent velocity field measurements. Recently, Sinton and Li [13] developed a microchannel flow visualization system and complimentary analysis technique by using caged fluorescent dyes. Both pressure-driven and electrokinetically driven velocity profiles determined by this technique compare well with analytical results and those of previous experimental studies. Particularly, this method achieved a high degree of near-wall resolution. This technique will be discussed below.

Figure 7.12. An imaging apparatus of the laser-caged dye microflow visualization method.

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Generally, in the experiment, a caged fluorescent dye is dissolved in an aqueous solution in a capillary or microchannel. The caged dye molecule cannot emit fluorescent light because it is encapsulated by a polymer group. Ultraviolet laser light is focused into a sheet crossing the capillary (perpendicular to the flow direction). The caged fluorescent dye molecules exposed to the UV light are uncaged and thus are able to shine. The resulting fluorescent dye is continuously excited by an argon laser and the emission is transmitted through a laser-powered epi-illumination microscope. Full frame images of the dye transport are recorded by a progressive scan CCD camera and saved automatically on the computer. In the numerical analysis, the images are processed and cross-stream velocity profiles are calculated based on tracking the dye concentration maxima through a sequence of several consecutive images. Several sequential images are used to improve the signal to noise ratio. Points of concentration maxima make convenient velocimetry markers as they are resistant to diffusion. In many ways, the presence of clearly definable, zeroconcentration-gradient markers is a luxury afforded by the photo-injection process. Imaging apparatus A labeled photograph of the system is provided in Figure 7.12. The two required laser beams were fixed and ran horizontal and parallel: the blue, excitation beam above; and the ultraviolet beam below. The ultraviolet light was provided by a 300uJ, A = 337nm, pulsed nitrogen laser (LSI, VSL -337NDS). This beam was reflected upwards by a stationary mirror and through a vertical optical rail assembly containing two consecutive, counter-oriented, cylindrical optics followed by a blank optic on which the capillary rested (all of which were fused silica). The first cylindrical optic (f= 100mm piano convex) focused the beam into a sheet. The second cylindrical optic (f= 50mm piano convex) condensed this sheet into the capillary in the cross-stream direction. All three optics were adjusted relative to each other on the vertical optical rail. The rail itself, including the flow module was mounted on a precision three-axis stage. A single-line, 200mW, A = 488nm argon laser (American Laser Corp., LS300B) provided a continuous flood of excitation light through the objective. The laser beam passed through a lOx beam expander before entering the epifluorescent microscope head (Leica DMLM). The original, lamp-based optics in the microscope was removed to facilitate collimated light, and no excitation filter was required as the light was monochromatic. The beam is reflected off a dichroic mirror (A = 510nm long pass) directly into the oil immersion, 25x objective with numerical aperture, NA = 0.75 (Leica PL Fluotar). High numerical aperture objectives are especially desirable in weak signal, epi-


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illumination applications. The received signal intensities vary with numerical aperture (A^) and magnification (mag) as follows:

(2D Other advantages of high NA include reduced depth of field and increased resolution, often at the expense of working distance [22]. The Abbe spatial resolution of this apparatus was approximately 1.2|am in the object plane, and the depth of field was calculated as 2.1 urn. A single drop of optical oil surrounded the windowed portion of the capillary and connected directly to both the objective and the blank base optic. This full immersion design provided index matching for imaging and both the excitation and ultraviolet light. Cargille (FF) optical oil was used due to its ultra-low fluorescence and an index of refraction, « re /= 1.485, similar to that of the fused silica, nre/ = 1.46. The received fluorescent signal passed through the dichroic mirror and through an additional suppression filter (A = 515nm long pass) to the 2/3" progressive scanning interline transfer CCD camera (Pulnix, TM-9701). Progressive scan architecture enabled all pixels in the CCD array to be exposed concurrently with all data being subsequently transferred (in contrast to even and odd field interlacing). This is desirable from an image quality perspective, however, when running in this mode the signal could only be viewed through the computer. The video monitor required signals in interlaced mode and could only be used for focusing and alignment. Images received by the camera were digitized and saved automatically through a PC-based image acquisition system. Full frame images could be collected at rates up to 30Hz. Image acquisition A four-channel delay generator (SRS, DG535) controlled the firing of the nitrogen laser, run frequency, and synchronization with the camera when necessary. The system could be run in two modes: 1.) The camera was asynchronously reset by the delay generator a set time after the laser had been fired; and 2.) The camera was run in video mode and the delay generator fired the laser at convenient intervals. To increase the amount of uncaged dye, the nitrogen laser could be pulsed multiple times (to a maximum rate of 60Hz) at each uncaging event. The number of pulses per uncaging event is restricted only by fluid velocity and the desired sheet thickness. The excitation laser ran unshuttered, continuously flood-illuminating the capillary. Image exposure durations were controlled using electronic shuttering modes in the camera. The acquired images had a resolution of 640 x 484 pixels. This corresponded to a viewing area of 346 x 261 um with each pixel representing a 0.54jj.m square. A

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lx c-mount was employed such that this magnification is only a consequence of the objective magnification and the size of the CCD array. Image Processing To remove any non-uniformities present in the imaging system, dark field image subtraction and bright field image normalization were performed [22]. A dark field image (or background image) for each run was taken immediately before the laser was fired. This dark field image contained a faint signal from stray background light as well as fluorescent emission from trace amounts of uncaged dye in the bulk fluid. Before a bright field image was taken, the flow was stopped and the nitrogen laser was fired repeatedly until uncaged dye had diffused well beyond the camera's field of view. Using a Matlab program, the dark field image was subtracted from all images including the bright field image. These images were then normalized with respect to this bright field image and smoothed using a distance-based 7x7 pixel kernel. This kernel size corresponds to a 3.2um square in the object plane, such that pixel intensity values could influence neighbors up to 1.62um away in each radial and axial direction. Finally, the resulting image series was linearly scaled by a single factor so as to fill the grayscale range. Flow systems A custom designed, ultra-low flow rate syringe pump provided pressuredriven flow. A DC micromotor was required to ensure a continuous flow rate, not achievable with commercial designs based on stepper motors. For electroosmotic flow, a separate, two-reservoir flow module was used. A 14cm long capillary liquid junction joined the small reservoirs with embedded platinum electrodes. The capillary and both reservoirs were filled with the same caged dye solution before any voltage was applied. All capillaries were fused silica with exterior polymide coating as supplied by Polymicro Tech. The nominal inner diameters of the capillaries were lOOum and 205um. However, using an electron microscope the inner diameters were found to be 102 urn and 205 um. Each capillary was prepared by flushing with pure water, followed by filtered buffer. The exterior polyimide coating was oxidized and removed to create a viewing window. Since only minute amounts were involved in uncaging events, most dye was used in filling the flow system. The cost of specialized caged dyes was such that careful design of these systems can reduce the operating costs significantly. Chemicals Two caged fluorescent dyes, supplied by Molecular Probes, were employed here: 5-carboxymethoxy-2-nitrobenzyl (CMNB)-caged fluorescein (826.81 MW); and 4,5-dimethoxy-2-nitrobenzyl (DMNB)-caged fluorescein


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dextran (10000 MW). Both dyes were dissolved in sodium carbonate buffer of pH = 9.0. The buffer was prepared by dissolving 39.8xlO~3 mol of NaHCC>3 and 3.41xlO~3 mol of Na2CO3 in 1 litre of pure water resulting in a solution of ionic strength, / = 0.05. Stock solutions of DMNB-caged fluorescein dextran and CMNB-caged fluorescein were prepared in 0.2 mM and lmM concentrations respectively. The solutions were aliquoted and stored in darkness at -20°C. Immediately before use, all solutions were filtered using 0.2um pore size syringe filters. Both caged dyes release fluorescein upon photolysis which has an absorption maximum at approximately X = 490nm and is well suited to excitation with the X = 488nm output line of the argon laser. DMNB caging groups absorb light most efficiently from X = 340nm to X = 360nm. CMNB caging groups, however, exhibit an absorption maximum at X = 334nm which is very well suited to the X = 337nm output of a nitrogen laser. To facilitate the multiple-image analysis technique, all results presented here were acquired using mode 2 in which the camera ran in progressive scan video mode and the delay generator controlled the ultraviolet laser. The nitrogen laser was triple-pulsed at 30Hz at each uncaging event. The overall run frequency was set to 0.15 Hz, providing more than sufficient time for uncaged dye to exit the field of view. The camera was run at 15 Hz resulting in 100 stored images per uncaging event. Individual exposure times were 1/125 sec (corresponding to shutter mode 8 on the TM-9701). The timescale over which the uncaging process occurs is also an important consideration. In general, this delay can vary from microseconds to seconds [21]. Here, dye release was effectively complete 50msec after the uncaging event, and only images taken after this period were used in velocimetry calculations. This delay corresponds to the first major timescale of dye release found by Lempert et al. [6]. Numerical Analysis Technique Although the mass-weighted average velocity, u, is of interest, only the transport of the uncaged dye ions is observed. The numerical analysis technique must infer the mass-weighted average velocity, u, from images of ion transport which can differ from the advective transport due to forced and ordinary —* diffusion. The molar flux of uncaged dye ions, _/,• , is governed by the NernstPlanck equation in the form (22)

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where ¥ is the electrostatic potential. If the flow is fully developed, and the axial gradient of the electrostatic potential is a known constant, Ex, the axial electrophoretic velocity, uPh, becomes a known constant and the ionic flux takes the form (23)

(with respect to the special case of pressure-driven flow in the presence of a thin EDL, the streaming potential becomes insignificant and the electrophoretic velocity, uPh, may be neglected.) In locations where the concentration gradient is zero, the observed ionic transport is only the addition of a known electrophoretic velocity and the desired, mass-weighted averaged velocity. In this method, each of these locations of zero axial dye concentration gradient were chosen as marker points, such that the cross-sectional velocity profile could be calculated directly from the observed velocity of these points as follows: (24)

Before each run, the camera was rotated such that the pixel grid was aligned with the radial and axial directions. After each run, the received images were processed and read into the analysis program in matrix format. Along each axial line of pixels, a zero gradient marker point was located using a weighted average of the highest intensity values on that line. The set of markers from each image formed a cross-stream, maximum concentration profile. Any pair of these profiles could provide a velocity profile by dividing the distance between them, Ax, by the corresponding time step, At. With 8 images, 28 velocity profiles could be constructed. In general, for a set of N images a set of n velocity profiles could be constructed where (25)

Here, all n displacements were calculated and an error-weighted average technique was applied to calculate the velocity profile. The basis of the weighting was that displacement measurements, Axh would all have similar error due to finite, random, signal noise. The size of the corresponding timestep, however, would scale the effect of that error on the velocity value. Weighting each measurement such that each contributed equally to the overall error resulted in the straightforward calculation below


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(26)

Pressure-Driven Flow In the absence of an applied electric field and any electro-viscous effects, pressure-driven flow in a circular cross-section capillary results in a parabolic profile of the form

(27)

where uP is the velocity at the centerline. Images of a pressure-driven flow in a 205 um i.d. capillary are shown in Figure 7.13. The DMNB-caged fluorescein dextran in the sodium carbonate buffer was used. Although it is difficult to calculate the exact concentration of dissociated sodium ions, the ionic strength of / = 0.05 would indicate a bulk counter-ion concentration on the order of 10~2 M at minimum. This is a relatively high ion concentration with respect to electrokinetic effects. An estimate of the Debye length is about 3nm. A conservative estimate of EDL thickness is four Debye lengths. This corresponds to a thickness on the order of lOnm, which is significantly smaller than the maximum achievable resolution of a light microscope. Such an EDL thickness is sufficiently thin to ensure negligible electro-viscous effects, so that the velocity profile can be expected to remain parabolic right up to the wall. Figure 7.14(a) shows plots of the relative signal intensity (along the centerline of the capillary) vs. axial distance for 8 consecutive images. The points of zero axial concentration gradient correspond to peaks in signal intensity. Profiles of

Figure 7.13. Images of the uncaged dye in a pressure-driven flow through 205u,m i.d. capillary at 133msec intervals.

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Figure 7.14. Data from 8 consecutive images (taken at 15Hz.) of uncaged dye in a pressuredriven flow through a 205um i.d. capillary: (a) Plots of relative signal intensity (along the centerline) vs. axial distance for all images; (b) Plots of concentration maxima for all images; and (c) Calculated velocity (points) plotted with the parabolic analytical solution.


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concentration maxima calculated along each row of pixels in each image are shown in Figure 7.14(b). Applying the numerical analysis technique to these profiles resulted in the velocity data (points) plotted with the parabolic analytical solution (line) in Figure 7.14(c). The parabola was calculated by using conventional laminar flow equation assuming a zero-slip velocity at the wall and a centerline velocity equal to that determined experimentally. The velocity profile is in good agreement with the parabolic analytical solution up to 8 microns from each wall (r/R = 0.92). Paul et al. [7] also showed a strong parabolic trend but only data for -0.5 < r/R < 1 is displayed due to an insufficient signal-to-noise ratio near the one wall. Near the other wall the reported profiles show a large scatter beyond r/R = 0.80. The symmetry of the full cross-section velocity profile given here in Figure 7.14(c) is also noteworthy and demonstrates the effectiveness of this method. Electro-osmotic Flow The two reservoirs in the flow module were carefully balanced in order to avoid pressure-driven flow. In addition, both reservoirs contained the dye solution so that the system could be left for several minutes and allowed to come into equilibrium. In early tests, it was found that the fluorescein dextran dye performed poorly in electroosmotic flows. Paul et al. [7] suggested that the multi-component nature of this dye make it unsuitable for these applications. Here, the CMNB-caged fluorescein with the sodium carbonate buffer and 102(im i.d. capillaries were used in all electroosmotic flows. Images of the uncaged dye transport in four different electroosmotic flows are displayed in vertical sequence in Figure 7.15. The dye diffused symmetrically as shown in Figure 7.15(a). Image sequences given in Figures. 7.15(b), (c), and (d) were taken with voltages of 1000V, 1500V, and 2000V respectively (over the 14cm

Figure 7.15. Images of the uncaged dye in electroosmotic flows through a 102u,m i.d. capillary at 133 msec intervals with applied electric field strength: (a) 0V/14cm; (b) 1000V/14cm; (c) 1500/14cm; and (d) 2000V/14cm.

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length of capillary). The field was applied with positive at left and negative at right. The resulting plug-like motion of the dye is characteristic of electroosmotic flow in the presence of a negatively charged surface at high ionic concentration. The cup-shape of the dye profile was observed in cases 7.15(b), (c), and (d) within the first 50msec following the ultraviolet light exposure. This period corresponded to the uncaging time scale in which the most significant rise in uncaged dye concentration occurs. Although the exact reason for the formation of this shape is unknown, it is likely that it was an artifact of the uncaging process in the presence of the electric field. Fortunately, however, the method is relatively insensitive to the shape of the dye concentration profile. Once formed, it is the transport of the maximum concentration profile that provides the velocity data. This also makes the method relatively insensitive to beam geometry and power intensity distribution. As in the pressure-driven flow case, the major counter-ions to the negatively charged wall were those offered by the buffer solution. The resulting EDL thickness on the order lOnm ensured that a flat electroosmotic velocity profile could be expected to extend right to the wall. Admittedly, there would be a steep velocity gradient in the EDL region, however, the thickness of the EDL prohibits resolution of this gradient with this or any other known velocimetry technique. This realization provided an opportunity to test the nearwall resolution of the present technique as follows: The point near the wall where the velocity profile falls from the electroosmotic velocity could be considered the closest significant datum. A diffusion coefficient of D = 4.37xlO~6 cm2/s was assigned, giving an electrophoretic mobility of vPf, = -3.3x10 4 c m V ' s " ' as suggested by Harrison et al. [23] and Paul et al. [7]. Due to the negative charge of the uncaged dye ions, their net velocity was necessarily less than the underlying electroosmotic velocity. Figure 7.16 shows velocity data for the four flows corresponding to the image sequences in Figure 7.15. Each velocity profile was calculated using an 8-image sequence and the numerical analysis technique. The velocity profile resulting from no applied field, appearing on the far left in Figure 7.15, corresponds closely to stagnation as expected. This run also serves to illustrate that, despite significant transport of dye due to diffusion, the analysis method is able to recover the underlying stagnant flow velocity. Although the other velocity profiles resemble that of classical electroosmotic flow [5], a slight parabolic velocity deficit of approximately 4% was detected in all three flows. This was caused by a small back-pressure induced by the electroosmotic fluid motion. In general, several factors can cause concave velocity profiles including: a narrowing or partial blockage anywhere in the capillary; the presence of charged particles in the fluid; capillary forces resulting from different curvatures of the


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free surfaces in the reservoirs; contact angle hysteresis of a bubble entrapped in the channel; a non-uniform wall surface charge distribution; and slight height differences between the free surfaces in the reservoirs. Sensitivity to these factors is highly correlated with the capillary radius. In this case, it was evaluated that an adverse pressure gradient of only 50 N/m3 would have been sufficient to generate the observed back-flow. Such a pressure gradient could be easily generated if the height of one reservoir exceeded the other by less than 1 mm. Velocity profiles plotted in Figure 7.16 indicate electroosmotic wall velocities of 3.8xlO"4 m/s, 6.0xl(T4 m/s, and 8.2xlO~4 m/s (from left to right) giving electroosmotic mobility values of vE = 5.32xlO~4 c m V s " 1 , vE = 5.60x10 4 cm2V"1s"',vE= 5.74xlO"4 c m ^ v V 1 respectively. These values are in good agreement with vE = 5.07xl0"4 cmV^s" 1 in pH 8.0 reported by Paul et al. [7], and vE = 5.87xlO"4 c m V s l in pH 8.5 reported by Harrison et al. [23] (both in fused silica). From the average mobility calculated here, an effective zeta potential for this system is calculated to be C= -80 mV. This is in keeping with silica oxide surfaces at KNO3 concentrations near 10"2 M at pH 9.0 as reported by Hunter [20]. It should be noted that the resolution of the determined velocity profiles in the electroosmotic flows is better than that in the pressuredriven flows. This is because the two flows differ with respect to cross-stream velocity gradients, which induce radial concentration gradients and radial diffusion. In pressure-driven flow, the dye sample is transported with the parabolic velocity field, which leads to an increased radial diffusion (in addition

Figure 7.16. Plots of velocity data from four electroosmotic flow experiments through a 102u.ni i.d. capillary with applied electric field strengths of: 0V/14cm; 1000V/14cm; 1500/14cm; and 2000V/14cm (from left to right).

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Figure 7.17. Velocity data of three electroosmotic flow experiments through a 102^m i.d. capillary with an applied electric field strength of 1000V/14cm: (a) Plotted over capillary cross-section; (b) Plotted in lO^m leading up to the wall (-0.8 > r/R > -1).


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to axial diffusion) and a loss of resolution. In electroosmotic flow however, the sample is mostly translated with the plug-like flow filed and only axial diffusion acts to disperse the sample. As shown in these figures, all electroosmotic flow profiles exhibit a strong degree of symmetry. The velocity profile remains relatively flat for 0.95< r/R 510 nm) is passed to the camera. In a similar manner, the blue excitation light from the microscope will be reflected at the bubble interface as illustrated in Figure 7.33. Figure 7.33(a) illustrates that uniform excitation light exposure in a liquid-only

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capillary would cause a spatially uniform rate of fluorescence, and in turn, a spatially uniform rate of photobleaching. The addition of an air bubble will alter the path of the excitation light depending on its angle of incidence with respect to the local interface normal. Thus, excitation light may be refracted or reflected at the interface as illustrated in Figure 7.33(b). Light incident at angles greater than the critical angle is reflected back into the liquid as shown on the right in Figure 7.33(b). Light incident at angles less than the critical angle is refracted at both interfaces and transmitted back into the liquid as shown on the left in Figure 7.33(b). In both ways the intensity of the excitation light is increased in the liquid near the bubble. This results in higher total excitation light exposure, initially higher fluorescence intensity, and a higher rate of photobleaching, which in time, results in lower fluorescence intensity. This local photo-destruction is the result of micro-bubble lensing induced photobleaching ((x-BLIP).

Figure 7.33. Illustrative schematics of optical phenomena: (a) uniform photobleaching in a liquid-filled channel, and (b) increased photobleaching in the near-bubble liquid due to the presence of a bubble.


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It is important to note that Figure 7.33 and the above explanation are simplifications of the real case. The two-dimensional model of the bubble cap is reasonable considering the hemi-spherical geometry of the interface. However, the excitation light beam will be, in reality, conical, due to the high numerical aperture objective used. This would add a layer of complexity to the optical analysis but the physics remains the same as that illustrated in Figure 7.33. In addition, lensing and increased photobleaching in the liquid film between the bubble and the wall (similar to wave-guide or fiber optic light transmission) is also expected. Due to the small thickness of the film however, the volume of liquid photobleached in this region does not contribute significantly compared to that photobleached at the bubble caps. For the same reason, the extent of photobleaching is essentially independent of bubble length. The image sequence in Figure 7.34 demonstrates the u-BLIP process. Figure 7.34(a) is an image of the uniform fluorescent emission of the dye-filled channel without a bubble. The image in Figure 7.34(b) was taken shortly after the bubble was moved into position. Although both caps contribute equally to the photobleaching, the camera's field of view is focused on the right cap, the point of which has a 'brightspot' reflection as discussed previously. In Figure 7.34(b) the fluorescent emission appears fairly uniform throughout the liquid at the level of the original dye concentration without a bubble (Figure 7.34(a)). The images in Figure 7.34(b-f) were taken in sequence at 20/15 s (1.33 s) intervals, and processed identically. Significant darkening in the near-bubble liquid is apparent. The radial orientation of the dark/bright ray pattern (believed to be caused by interference) is further evidence of the bubble lensing phenomena. After 5.3 s (Figure 7.34(f)), a significant photobleached region is formed around the bubble, where as the dye at the right-hand side of the camera view has retained the original intensity level. This is more apparent in the axial concentration profiles shown in Figure 7.35 where the five profiles correspond to the image sequence in Figure 7.34(b-f). As shown in Figure 7.35, the bulk of the photobleaching occurs during the first 2 seconds. The broadening of the dark region beyond that is mostly due to diffusion of the photobleached dye. Increasing the excitation light energy and correspondingly decreasing the exposure time could reduce this axial broadening. Allowing time for radial diffusion, however, creates a more uniform, axially symmetric photobleached region. For the most part, experimental parameters such as these (solutions, light intensities, exposure duration) were chosen out of convenience and it is likely that performance could be improved with optimization. One option would be to use high molecular weight dyes such as fluorescein-dextran conjugates (Molecular Probes) which are less susceptible to diffusion than standard dyes. The image in Figure 7.34(g) was taken after an axial electric field was applied to the channel. The resulting electroosmotic liquid flow has transformed the dye photobleached at both bubble caps into a dark, cross-stream, liquid flow

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Figure 7.34. (a) Image of fluorescent emission from a liquid-filled capillary, (b)-(f) Image sequence demonstrating the |a-BLIP process, (g) Image of photobleached dye advected with electroosmotic flow.


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marker. The electrokinetic transport of the bulk liquid in this system is calculated numerically. Scalar image velocimetry is then applied to a (i-BLIP generated flow marker for experimental determination of the cross-stream liquid velocity profile in the microchannel. In order to understand the underlying physics of the u-BLIP process, a series of numerical simulations were conducted using the BLOCS (Bio-Lab-Ona-Chip Simulation) finite element code. In these simulations we have considered a long, axially symmetric capillary, having a diameter equivalent to those used in the aforementioned experiments (lOOum), with a 200|0.m long stationary bubble located at the midpoint. The liquid domain geometry was discretized using 9-noded bi-quadratic elements, which were significantly refined in the region near the bubble and coarsened near the capillary entrance and exit. The first stage of the numerical analysis is the determination of the applied potential field in the liquid system which, for the case of a constant conductivity electrolyte, can be determined from the Laplace equation, (38) where (j> is the applied potential field. Requiring that the solution remain finite at the capillary axis and applying insulation conditions, d(j)/dn where n is the normal

Figure 7.35. Axial fluorescence intensity profiles corresponding to the image sequence in Figure 7.34(b)-(f). The bulk of the photobleaching occurs close to the interface and in the first few seconds of exposure.

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to the surface, were used along the bubble and capillary walls. The electric field lines generated for the case equivalent to 2600 V applied over the 15cm capillary (consistent with the experiments discussed here) are shown in Figure 7.36(a). Since the liquid cross sectional area is significantly reduced, and thus the local channel resistance is greatly increased, the electric field lines are concentrated in the thin film that surrounds the bubble. This has the effect of increasing the gradient of within this region and reducing it far away from the bubble. This is demonstrated in Figure 7.36(b) which shows the change in <j> along the length of the capillary for the no bubble case, a bubble with a .5jxm film thickness and a bubble with a . 1 îm film thickness. The presence of the bubble reduces dtydx. to 86% and 96% of that for the no bubble case for the 0.1 u.m film thickness and 0.5um film thickness respectively. Also of interest in Figure 7.36(a) is the shape of the iso-potential lines that assume a slightly curved shape beyond the thin film on either side of the bubble. At a distance less that one capillary diameter from the edge of the bubble however, they are nearly perpendicular to the channel wall. This limited influence of the bubble on the shape of the iso-potential field lines will facilitate the use of u-BLIP as a micro-flow visualisation technique. With the potential field solution developed the flow field can be determined using the low Reynolds number Stokes equations, Eq. (39a) and the compressibility condition Eq. (39b), (39a) (39b) where u is the fluid velocity, P is the pressure and n is the viscosity. In principal, Eq. (39a) should contain an electrical body force term resulting from the application of the electric field to the net charge density in the double layer. In this analysis however, we assume an electroosmotic slip velocity, ueo=veoE where veo is the electroosmotic mobility (taken as 6.2 xlO~8 m2/Vs [35]) and E is the gradient of 0 tangential to the surface, applied along the capillary wall. This simplification eliminates the double layer formulation and has proven accurate in transport cases, such as that examined here, where the double layer thickness is very thin compared with the capillary diameter. In this case the condition is slightly more stringent as the double layer thickness must be thinner than the film surrounding the bubble. In the case examined the ionic strength of the buffers is on the order of 10~2 M which gives us a double layer thickness on the order of 3 ran. This is several orders of magnitude smaller than the film thickness encounter here. Applying this slip boundary condition along the capillary wall, a boundedness condition along the capillary axis and a zero tangential shear


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Figure 7.36. (a) Numerically determined iso-potential lines in the presence of an insulating bubble, (b) Influence of bubble film thickness on the global applied electric field.

condition along the bubble [36] yields the near-bubble flow field shown in Figure 7.37(a). The velocity profiles at 0.1, 0.4 and 1.0 capillary diameters away from the edge of the bubble are shown in more detail in Figure 7.37(b). As can be seen, very near the bubble the velocity close to the capillary wall is slightly higher than that at the center, however it very rapidly evolves to the traditional plug type velocity profile expected for electroosmotic flow. This result will be used as a verification of the microflow visualization technique developed in the following section.

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Methods for inferring a bulk fluid velocity by analyzing a sequence of images of dye transport fall under the broad category of scalar image velocimetry. In general, the goal of these methods is to extract the bulk fluid velocity vector, u, from the advective terms of the mass-conservation equation satisfied by the imaged dye species: (40) where c is the concentration of dye species and D is diffusion coefficient. The analysis may be greatly simplified when applied to internal, fully-developed flows by using a discrete dye sample. This is because the motion of the locus of dye-concentration maxima is, for the most part, not affected by diffusion, and reflects the bulk motion of the fluid. This maximum concentration tracking method has been applied in caged-dye based microfluidic flow visualization studies as discussed in the previous sections. In those cases, a photo-injection of fluorescent dye provided the only non-zero dye concentration, c, in an otherwise non-fluorescent solution. In the case of u-BLIP, however, the solution contains a uniform dye concentration, c = c0, and a portion of the dye is photo-chemically destroyed. Since the fluorescent dye concentration, c, is a conserved quantity, the concentration of photo-destroyed dye, c , is also conserved. Thus Eq. (40) is satisfied for c = c', where, (41) and F is any constant. Thus 'maximum' concentration tracking methods can be directly applied to inverted and linearly scaled intensity images of a photobleached sample. Equivalently one may think of the conserved quantity as the photodestroyed fluorophores whose concentration is quantified by a lack of fluorescence. Once the near-bubble liquid was photobleached (as shown in Figure 7.34), an axial electric field was applied by setting the upstream (left) reservoir potential to 2600V, and connecting the downstream (right) reservoir to ground. The camera recorded the resulting transport of the photobleached sample. A five-image sequence taken at 1/15s intervals is given in Figure 7.38(a). The electroosmotic liquid flow has combined the dye photobleached at both bubble caps into a dark, cross-stream, band, which is advected downstream with the electroosmotic flow. The bubble itself is shown to have a finite velocity in the direction opposite to the electroosmotic liquid flow.


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Figure 7.37. (a) Numerically determined electroosmotic flow field in the near bubble region, (b) X-direction velocity profiles at various distances from the bubble edge. L denotes distance from the edge of the bubble.

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Figure 7.38. (a) An image sequence of the p.-BLIP generated flow marker in the electroosmotic flow, (b) Corresponding axial concentration profiles of the flow marker, (c) Corresponding cross-stream profiles of marker concentration maxima.


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The concentration field of photobleached dye (c ) was calculated for each image using in-house developed image processing software. Axial marker concentration profiles of the images shown in Figure 7.3 8(a) are given in Figure 7.38(b). The vertical lines on the left in Figure 7.38(b) indicate the motion of the interface, and the profiles on the right show the marker transport. A relatively clear concentration maximum is apparent in each profile. To determine the bubble velocity the distance between the vertical lines may be divided by the corresponding time step. Here a bubble velocity of iibubbie = —190 um/s was determined. To determine the velocity profile in the liquid region, the point of maximum concentration was located along each axial line of pixels, using a weighted average of the highest concentration values in the liquid phase. The set of maximum concentration points from each image formed the cross-stream, concentration maxima profiles shown in Figure 7.38(c). In a given sequence, any pair of concentration profiles could provide a velocity distribution by dividing the distance between them by the corresponding time-step. Here, all five profiles were used in an error-weighted average to determine the velocity data. Once calculated, this velocity data represents the 'observed' velocity of the marker, uob. Since the dye molecules are charged, the observed velocity of the concentration maxima will be the summation of the bulk fluid velocity, ueo, and the electrophoretic velocity of the dye, uph, as follows, (42) The electrophoretic velocity, uph, must be calculated directly from the electrophoretic mobility, vph, and the applied electrical field strength, E, as: (43) The electrophoretic mobility of fluorescein may be taken as vph = -3.3x10 8 (m2/Vs). The local axial applied electric field, however, requires special consideration here. Through the numerical simulations it was shown that the isopotential lines became almost totally radial less than one diameter away from each bubble cap (Figure 7.36(a)). Thus it is reasonable to assume that the electrophoretic velocity of the marker is purely axial (aligned with the potential gradient vector, E). In the case of a channel containing no bubble and filled with a uniform electrolyte solution, the magnitude of the axial electric field may be calculated simple by dividing the applied voltage differential, AV, by the length of the channel, L. As shown in the numerically determined axial potential profiles, Figure 7.36(b), the bubble adds resistance to the channel and hence causes a field reduction in the liquid far away from the bubble. To experimentally determine this reduced field strength in the liquid the electrical

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current was measured both with and without the bubble present and the following calculation performed: (44)

where / is the electrical current draw, subscript B indicates the presence of a bubble, and subscript NB indicates that no bubble is present. The bracketed current ratio in Eq. (44) is the factor by which the electric field is reduced from the overall value (AV/L) due to the presence of the bubble. A plot of the currents measured with and without the bubble present vs. overall applied electrical field strength is given in Figure 7.39. Both curves trend slightly upwards due to Joule heating induced increases in fluid temperature [14,32]. The current ratio (in Eq. (44)), however, was found to be relatively constant at 0.85 ±0.025 over this range of applied field. Thus according to these current measurements and Eq. (44), the applied voltage drop of AV = 2600 V over the L = 0.15 m length of capillary generated an electrical field strength of E = 14.7 kV/m in the liquid region. This is consistent with the with the decrease in E observed for the 0.1 um film thickness case shown in Figure 7.36(b), suggesting the film thickness is of

Figure 7.39. A plot of the current values measured vs. the overall electrical fields applied to the channel with and without a bubble present.


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this order. Using this E value to calculate the electrophoretic marker velocity in Eq. (43), and substituting the result into Eq. (42), gave the bulk liquid electroosmotic velocity profile shown in Figure 7.40. The flat plug-like profile observed is characteristic of electroosmotic flows and is in keeping with the corresponding numerically predicted velocity profile in Figure 7.37. From the wall velocity, the electroosmotic mobility was calculated to be veo = 6.8 xl(T 8 m2/Vs, in reasonable agreement with the value of veo = 6.2 xlO~8 m2/Vs reported by Duffy et al. [35]. The plug-like velocity profile achieved with the (a-BLIP generated flow marker extends to within 4 urn from each wall. This degree of near-wall resolution is comparable to, and in many cases improved over, that achieved with caged-dye based micro-flow visualization techniques that involve increased infrastructure and specialized chemicals. Another advantage of this technique is the ability to concurrently image the bubble geometry, bubble velocity and the local fluid velocity in multiphase systems. In this case, the effective film thickness was measured (through current measurements) and calculated (through numerical simulations shown in Figure 7.36) to be less than 1 um. Near the bubble caps, the interface is optically indistinguishable from the wall, indicating that the film in that area it is less than 1 um. However, the images in Figure 7.38(a) also show significant film thickening in the middle

Figure 7.40. Cross-stream electroosmotic liquid velocity profile obtained by applying scalar image velocimetry to the transport of the n-BLIP generated flow marker.

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region of the bubble. This thickening will result in lower liquid velocities in that portion of the film. By integrating the cross-stream velocity profile (Figure 7.40), and determining the bubble velocity from the interface movement in Figure 7.38(b), the liquid film velocity was determined to be 3.3 mm/s at the bubble midpoint and an estimated 30 mm/s near the bubble caps. This demonstrates the applicability of this technique to the study of coupled dynamic transport phenomena, characteristic of microscale multiphase systems. In summary, the micro-bubble lensing induced photobleaching ((i-BLIP) method takes advantage of the curvature and the step change in properties across a gas-liquid interface to create a lens/mirror optical arrangement in which light incident on the bubble is focused into the surrounding liquid resulting in a locally increased total light exposure. The effect is demonstrated experimentally by imaging the increased photobleaching rate of fluorophores in the near-bubble region. Based on these phenomena, a micro-bubble lensing induced photobleaching (u-BLIP) technique can then be applied as a method to inject a marker for flow visualization. A series of numerical simulations on the multiphase system demonstrated that both the iso-potential lines and the flow field are disturbed by the presence of the bubble however the effect is limited to the near bubble region (typically less than 1 capillary diameter from the edge of the bubble). Using the u-BLIP technique, the electrokinetic transport of the photobleached marker is analyzed to determine the cross-channel velocity profile of the liquid phase, and the liquid velocity in the film. The results are in good agreement with the numerical predictions and are consistent with velocity measurements from previous studies. This is a new technique for microfluidic flow visualization, particularly applicable to the study of multiphase microchannel flows.


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JOULE HEATING AND HEAT TRANSFER IN CHIPS WITH T-SHAPED MICROCHANNELS

A side effect of microchannel transport by electrokinetic means is the internal heat generation, i.e., Joule heating, caused by electrical current flow through the liquid (e.g., a buffer solution). Maintaining uniform and controlled solution temperatures is important for minimizing dispersion in electrokinetic separations [30,31] and for controlling temperature sensitive chemical reactions such as DNA hybridization [37]. Therefore, it is essential to understand Joule heating and the related heat transfer in microfluidic or lab-on-a-chip systems. Microfluidic or lab-chip systems must have the ability to rapidly reject this heat to the surroundings. Generally it is the ability to dissipate this heat that limits the strength of the applied electric field and thus the maximum flow rate, etc. Recently, more and more microfluidic systems and lab-chips are made from low-cost polymeric materials such as poly(dimethsiloxane) (PDMS) [3841], poly(methylmethacrylate) (PMMA) [42-44], and others (see Becker and Locascio [45] for a comprehensive review) as opposed to traditional materials such as glass or silicon. The primary attractiveness of using these materials is that they tend to involve simpler and significantly less expensive manufacturing techniques, they are also amenable to surface modification [46,47] and the wide variety of physiochemical properties allows the matching of specific polymers to particular applications. While the development of these systems has reduced the time from idea to chip from weeks to days, and the per unit cost by a similar ratio (particularly with the advent of rapid prototyping techniques [35]), the low thermal conductivity inherent in these materials (0.18 W/mK for PDMS which is an order of magnitude lower than that of glass) retards the rejection of internally generated heat during electroosmosis [48]. Studying the on-chip Joule heating and heat transfer requires techniques of temperature measurements in microchannels. A few techniques have been recently developed for making direct "in-channel" measurements of buffer temperatures in microscale systems, the advantages and disadvantages of many of which are discussed in detail by Ross et. al. [48]. While NMR [33], Raman spectroscopy [49], and the recently developed on-chip interferometric backscatter detector technique [32] have been used, the most popular techniques involve the addition of temperature sensitive probes (for example: thermochromic liquid crystals [50], nanocrystals [51], or special fluorescent dyes [52]) to the buffer solution and an observation of the spatial and temporal changes in the thermal field via some type of microscopy technique. Rhodamine B is a fluorescent dye whose quantum yield is strongly dependent on temperature in the range of 0°C to 100°C, making it ideal for liquid based systems. Recently Ross et. al. [48,53] developed a Rhodamine B based thermometry technique for monitoring temperature profiles in microfluidic systems, based on that developed

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by Sakakibara et. al. [54] for macroscale systems. A similar technique was used by Guijt et. al. [55] to experimentally examine chemical and physical processes for temperature control in microfiuidic systems. In general very little work has been done concerning microscale thermal analysis of microfiuidic based lab-chips on a "whole-chip" level (as opposed to examining just the fluidic domain for example), particularly with respect to the recently developed polymeric systems. This section will present a detailed experimental and numerical analysis of the dynamic changes in the in-channel temperature and flow profiles during electrokinetic pumping at a T-shaped microchannel intersection, using pure PDMS/PDMS and PDMS/Glass hybrid microfiuidic systems [56]. A T-intersection was selected as it represents a general structure of a microfluidic system and it provides an interesting theoretical case due to the inherent spatial gradients in current density and volumetric flow rate. Using a fluorescence based thermometry technique, direct measurements of the in-channel temperature profile are taken and the results are compared with a detailed "whole-chip" finite element model, which accounts for the effects of the temperature field on the local solution conductivity and fluid viscosity as well as thermal conduction through the substrate.

Figure 7.41. Computational geometry for PDMS/PDMS or PDMS/Glass hybrid microfiuidic system.


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Thermal model and simulation domain Unlike momentum and species transport analysis, which is confined to the fluidic domain, thermal modeling presents some unique challenges as the heat transfer necessarily requires the simulation domain to include not only the liquid domain but also a significant portion of the chip, if not the entire chip. Different from a macroscale system, where the fluidic domain is most often of comparable size to the solid regions, a microchannel system typically encompasses only a very small fraction of the substrate and thus heat transfer is typically governed by a large time scale thermal diffusion process through the solid region. As shown in Figure 7.41, the system of interest here comprises three coupled domains, the lower substrate (made from either glass or PDMS), fluidic domain (buffer solution) and the upper substrate that contains the channel (made from PDMS). Electroosmotic flow occurs when an applied driving voltage interacts with the net charge in the electrical double layer near the liquid/solid interface resulting in a local net body force that induces the bulk liquid motion. When this voltage is applied to a buffer solution with a finite conductivity, the resulting current induces an internal heat generation effect often referred to as Joule heating. Within the fluidic domain the non-dimensional energy equation takes the form, (45)

where PeF is the Peclet number for the fluidic domain (Pe = pCpLvJk where p is the density, Cp is the specific heat, L is a length scale taken to be the channel height in this case, v0 is the reference velocity and k is the thermal conductivity), 6 is the non-dimensional temperature (6 - (T - To)kL/X00max where Xo is the electrical conductivity at the reference temperature, To, <j)max is the maximum applied voltage and the subscript L signifies properties of the liquid domain), 8 is the non-dimensional time, V is the non-dimensional velocity (V = v/vo) which will be obtained via the fluidic analysis discussed below, A is the nondimensional buffer electrical conductivity (A = A/Ao) and is in general a function of temperature, (P is the non-dimensional applied electric field strength (CP = (t>/<j>max), and the ~ symbol over the V operator indicates the gradient with respect to the non-dimensional coordinates (X = x/L, Y = y/L and Z = z/L). While most thermal properties in the above formulation (i.e. Cp, k, etc.) remain relatively constant over the temperature range of interest, and thus have been assumed so for the purposes of this study, the buffer electrical conductivity has a strong temperature dependence which cannot be ignored. Here we assume that the

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buffer conductivity is linearly proportional to temperature (having slope a) as shown below, (46) Within the upper substrate (subscript US) and lower substrate (subscript LS) the energy equation takes on a simplified form in that convective effects and the internal heat generation term is absent leaving only the transient and diffusion terms as shown below, (47)

The dominant mechanism of heat rejection from the fluidic domain is diffusion into the solid substrate. For very short times the temperature field is confined to a small region around the channel. However, at longer times (approaching those which are required for the system to reach equilibrium) the temperature field can span an area several orders of magnitude larger than the

Figure 7.42. Measured temperature as a function of scaled fluorescence intensity of Rhodamine B dye. Solid line represents a second order polynomial fit to the data.


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channel size, due to lateral thermal diffusion. This poses significant computational problems as it introduces another length scale into the problem. While the height of the computational domain is fixed by the system geometry, the required width of the domain was found to be approximately 50 times the channel width through numerical experimentation. Choosing a smaller domain necessarily led to a significant underestimation of the lateral heat transfer. Boundary conditions on the thermal domain were selected to conform to how most chip-based microfluidic experiments are conducted. The lower surface of the substrate was assumed to have a fixed room temperature (as would be the case for a chip sitting on a relatively large flat surface) while the upper surface was assigned a free convection boundary condition, (48) where Bi is the Biot number (Bi = hL/kus where h is the heat transfer coefficient computed from the "heated upper plate" relation from Incropera and DeWitt [57] to be h = 10 W/m2K). Zero flux conditions were used along the side surfaces. Flow model Generally, the high voltage requirements limit most practical electroosmotically driven flows in microchannels to small Reynolds numbers therefore transient and momentum convection terms in the Navier-Stokes equations can be ignored and thus the fluid motion is governed by the Stokes and continuity equations as shown below, (49)

(50) where a is the non-dimensional shear stress (1000kV/m) [6]. Through

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numerical modelling, however, Ermakov et al. [7] reported that the tightest sample focusing may not be suitable for detection purposes due to dilution (dispersion) effects. This is because intensely focused samples contain less mass and exhibit high concentration gradients, and hence are dispersed quickly by diffusion. It was also found that electrokinetic focusing confines the stream spatially but does not increase the peak sample concentration. They simulated different degrees of focusing by altering the field strengths in reasonable agreement with a previous experimental study [5]. Ermakov et. al. [7] presented a 2-D mathematical model to investigate the electrokinetic focusing in a cross intersection of microchannels and the sample mixing in a T-shape microchannel. Based on this model, they [8] studied electrokinetic injection techniques in microfluidic devices. In this paper, the effect of the electric field distribution in the channels on the injected sample concentration was studied, however, they didn't investigate how to control the volume and the uniform distribution of the dispensed sample. Patankar et. al. [9] simulated a 3-D electroosmotic flow in two crossing microchannels. The focus of this model study is the electroosmotic flow of buffer solutions, and the sample transport was not considered. The loading and dispensing of samples that can be transported along the dispensing channel (rather than be directly separated electrophoretically) is very important. These processes are required to deliver discrete samples to points where they may be chemically modified, diluted or mixed, or coupled to other analytical techniques such as chromatography or mass spectroscopy. Recently, how to control the size, shape and concentration of the dispensed samples in straight-cross microchannel dispensers has been investigated numerically and experimentally [10-14]. In general, using a on-chip crossing microchannel dispenser, the sample injecting or dispensing may be realized as follows: Electrical fields are applied to the two microchannels perpendicular to each other as illustrated in Figure 8.1 via

Figure 8.1. Illustration of a sample dispensing process in a crossing microchannel.

Electrokinetic Sample Dispensing in Crossing Microchannels

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four electrodes. Initially the dominant electrical field along channel 1 is used to load sample solution 1 (white in Figure 8.1) into the intersection. The perpendicular channel 2 contains a solution 2 (black in Figure 8.1). Then the dominant electrical field is switched to be applied along channel 2, the solution 2 starts flowing. Because of the viscous shear force, the flowing solution 2 will "cut" a pocket of solution 1 from the intersection and carry it to the downstream in channel 2. A sample is dispensed in this way. The key in the dispensing process is how to control the size and the concentration of the dispensed samples. This chapter will outline the effects of buffer electroosmotic mobility, sample diffusion coefficient, sample electrophoretic mobility, the conductivity difference of the solutions, the applied field strength and the channel size on the loading and dispensing processes. By understanding the complicated electrokinetic processes involved in the sample dispensing processes, we can find optimal applied voltages and the improved methods to control the size, shape and concentration of the dispensed samples.

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8-1


ANALYSIS OF ELECTROKINETIC SAMPLE DISPENSING IN CROSSING MICROCHANNELS

To understand the physics of the electrokinetic sample dispensing process, this section will show how to model the loading and the dispensing processes in a crossing microchannel dispenser. The numerical simulation results of this model [10] will be discussed. The microfluidic dispenser examined here is formed by two crossing microchannels as shown in Figure 8.2. The depth and the width of all the channels are chosen to be 20 urn and 50 |uim, respectively, except indicated otherwise. There are four reservoirs connected to the four ends of the microchannels. Electrodes are inserted into these reservoirs to set up the electrical field across the channels. Initially, a sample solution (a buffer solution with sample species) is filled in Reservoir 1, the other reservoirs and the microchannels are filled with the pure buffer solution. When the chosen electrical potentials are applied to the four reservoirs, the sample solution in Reservoir 1 will be driven to flow toward Reservoir 3 passing through the intersection of the cross channels. This is the so-called loading process. After the loading process reaches a steady state, different electrical potentials are applied to the four reservoirs. The sample solution loaded in the intersection will be "cut" or dispensed into the dispensing channel by the dispensing solution flowing from Reservoir 2 to Reservoir 4. This is the so called the dispensing process. The volume and the concentration of the dispensed sample are the key parameters of this dispensing process, and they depend on the applied electrical

Figure 8.2. The schematic diagram of a crossing microchannel dispenser. Wx and Wy indicate the width of the microchannels.


467

field, the flow field and the concentration field during the loading and the dispensing processes. Electrical field In most on-chip applications, buffer solutions are used. Buffer solutions usually have high ionic concentration, e.g., 10~2M, correspondingly, the electrical doubly layer is very thin, on the order of several nanometers. Recall that only in the electrical double layer, there is a non-zero net charge. To simplify the analysis, we consider the electrical double layer filed with negligible thickness. In this way, the liquid in the microchannel will be considered electrically neutral, i.e., the number of positive ions is equal to the number of negative ions, or the net charge density is zero. According to the theory of electrostatics, the applied electrical potential, (f), in the liquid can be described by Poisson equation, (1) Because no electrical potential is applied in z-direction (the direction of the channel depth), the electrical potential profile in this direction is constant. Therefore, Eq. (1) can be rewritten as (2, Introducing nondimensional parameters

where O is a reference electrical potential and h is the channel width chosen as 50 (a.m, Eq. (2) can be non-dimensionalized as:

Boundary conditions are required to solve this equation. We impose the insulation condition to all the walls of microchannels, and the specific non-

468


dimensional potential values to all the reservoirs. Once the electrical field in the dispenser is known, the local electric field strength can be calculated by (4)

Flow field The basic equations describing the flow field are the continuity equation, (5) and the momentum equation, (6)

where Veo is the bulk electroosmotic velocity vector, p is the density of the liquid, P is the pressure in microchannels, n is the viscosity of the liquid and pe is the net charge density in the solution. As mentioned earlier, the driving force in the liquid flow is the electrical force, peE, which appears as the third term of the right hand side of Eq. (6). The electroosmotic velocity is constant everywhere in the channel except in the region very near the wall (i.e. within the double layer). However, when the concentration is higher (i.e. C > 10" mol/l), the electrical double layer is very thin (i.e., less than 10 run) as compared to the dimensions of the microchannel's cross-section (i.e., 50 urn). Therefore, we will neglect the electrical driving force term in Eq. (6) and consider the effect of electroosmotic flow as the slip wall boundary conditions to the equation of motion. The electroosmotic mobility is assumed to be fj.eo =5.5xl0~ 8 m2 l(V-s) in this section, except indicated Vh otherwise. In the electroosmotic flow process, Reynolds number, Re = — (v is v the kinematic viscosity, V is the electroosmotic velocity and h is the channel width), the ratio between the inertial forces and the viscous forces, is very small (e.g. Re < 0.1). Hence, the viscous forces prevail and define the characteristic time scale at which the flow field reaches a steady state. Therefore, the time scale for electroosmotic flow to reach a steady state can be evaluated by


469

(7) This time scale is very small as compared to the characteristic time scales of the sample loading and sample dispensing. Hence, the electroosmotic flow here is approximated as steady state. In addition, the velocity component in z-direction, w, is very small as compared to the velocity component in x-direction and y -direction, wand v. Therefore, the dispensing process in the dispenser can be considered as a twodimensional problem, which has been verified by Patankar [9]. Taking into account of the above considerations, Eq. (5) and Eq. (6) can be rewritten as: (8)

(9a)

(9b)

where ueo, veo are the electroosmotic velocity component in x and y direction, respectively. Introducing non-dimensional parameters:

where Pa is the atmospheric pressure, Eqs. (8), (9a) and (9b) can be nondimensionalized as: (10)

470


(lla)

(lib)

Boundary conditions are required in order to solve this set of equations numerically. The slip velocity conditions are applied to the walls of the microchannels, the fully developed velocity profile is applied to all the interfaces between the microchannels and the reservoirs, and the pressures in the four reservoirs are considered as the atmospheric pressure. Concentration field In order to obtain the information about the volume of the dispensed sample, the sample's concentration distribution has to be determined. The distribution of the sample concentration can be described by the conservation law of mass, which takes the form of

(12)

where C; is the concentration of the i-th species, ueo and veo are the components of the electroosmotic velocity of the i- th species, Dj is the diffusion coefficient of the i-th species chosen as Dj =1.0x10" 11 m /(V • s) in all the calculations unless specified otherwise, and uepi and vepi are the components of the electrophoretic velocity of the / - th species given by uepi - E[xepj, where /j.epi is the electrophoretic mobility. As will be shown in the next section, rhodamine dye will be used as a sample in the visualization experiments to verify the model predictions. Therefore, rhodamine dye is chosen as a sample in all the simulations reported here. The electrophoretic mobility for the rhodamine dye was

experimentally

determined

as

-2.46xlO~ 8 m /(V-s)[l5]

and

Vepi = -2-0 x 10~8 m /(V • s) was used in all computations in this section unless specified otherwise. We are interested in how to control the volume of the dispensed sample. This volume depends on the electroosmotic mobility of the solution, the


471

diffusion coefficient and the electrophoretic mobility of the sample, the applied electrical potential and the dimensions of the dispenser. Introducing nondimensional parameters:

where C is a reference concentration, we can non-dimensionalize Eq.(12) as:

(13)

Boundary conditions are required to solve the above equation. In the calculations, the concentration of sample in reservoir 1 is the applied sample concentration; the sample concentrations in reservoir 2 and 4 are zero, and the flux of the sample to reservoir 3 is zero. Since both loading and dispensing process are unsteady process, the initial conditions are required to solve the above equation. For the loading process, the initial concentration in reservoir 1 is the applied sample concentration, in the other reservoirs and the channels are zero. After the loading process reaches the steady state, the loaded sample in the cross intersection can be dispensed into the dispensing channel by adjusting the electrical potentials applied to all the reservoirs. Therefore, the concentration distribution for loading process at the steady state was used as the initial conditions for dispensing process. The complete set of non-dimensional equations, Eq. (3), Eq. (10), Eq. (lla), Eq. (lib) and Eq.(13), were solved using the semi-implicit method for pressure-linked equation (SIMPLE) algorithm developed by Patankar [16]. The algorithm is based on a finite control volume discretization of the governing equations on a staggered grid. In order to capture all the features near the four corners of the dispensers shown in Figure 8.2, the non-uniform grid is employed. The control volume size next to the wall is minimum. The size of successive control volumes away from the walls is increased by a factor of 1.2. In this implementation, the solution to this set of equations is obtained by an iterative procedure. During each iteration procedure, the discretized equations are solved by a line-by-line iteration method. Electrical field and flow field Once the electrical potentials are applied to the four reservoirs, the electrical fields are set up across the channels. These electrical fields exert the electrical force on the liquid to pump the solution containing the sample to flow

472


through the cross intersection of the channels (loading process) and then dispense the sample into the dispensing channel. We assume the chip is made of electrically insulation material, and applied the insulation conditions to all the channel walls. Therefore, when the applied electrical potentials change or the channel sizes change, the electrical field will change and hence the velocity field will in turn change. Figure 8.3 shows the typical electrical field and flow field for loading and dispensing process, respectively. In this figure, the non-dimensional applied electrical potentials are:

Figure 8.3. Examples of the applied electrical field (left) and the flow field (right) at the intersection of the microchannels in a loading process (top) and in a dispensing process (bottom).


473

Figure 8.4. Effect of the diffusion coefficient on the concentration field of the sample in the dispensing process. The black regions are the buffer solution; the white regions are the sample solution with a sample concentration higher than 80% of the original sample concentration. The diffusion coefficient: a) 1.0 x 10" 11 m2/s, b)1.0 x 10" 10 m2/s .

474


Figure 8.5. Effect of the electroosmotic mobility on the concentration field of the sample in the dispensing process. The black regions are the buffer solution; the white regions are the sample solution with a sample concentration higher than 80% of the original sample concentration. The electroosmotic mobility: a) 5.0xl0~ 8 /n 2 /(F-.s), b) 6.0xl0~ 8 m 2 /(V-s).


475

where (i) represents the non-dimensional applied electrical potential to / - th reservoir. For this specific case, the electrical field and the flow field for loading process are symmetric to the middle line of the horizontal channel, and the electrical field and the flow field for the dispensing process are symmetric to the middle line of the vertical channel. Diffusion coefficient effect Under the same applied electrical field as specified in the above, two cases with

two

different

diffusion

coefficients,

L\ =1.0x10~ n m Is

and

Z)2 = 1.0x10~10/w Is, were tested. Figure 8.4 shows the concentration distribution during the dispensing process. In this figure, the black regions are the buffer solution; the white regions represent the region where the sample concentration is higher than 80% of the original sample concentration (the same for all other figures in this section). Generally, the dispensed sample size decreases with the increase of diffusion coefficient, because more samples will diffuse into the buffer solution when the diffusion coefficient is larger, consequently the high concentration region of the sample is smaller. However, the effect of the diffusion coefficient is not significant for this particular case. As seen from Figure 8.4, when the diffusion coefficient increases from 1.0xl0" n w 2 Is (Figure 8.4a) to 1.0xl0" 10 w 2 Is (Figure 8.4b), the volume of the dispensed sample decreases only by 2 % (i.e., from 390 pico-liters to 383 pico-liters in this particular case). Electroosmotic mobility effect Under the same applied electrical field as specified above, two cases with the

different

electroosmotic

mobility,

îeoj =5.0x10

m l(V-s) and

V-eoi ~ 6-0 x 10 m I(V • s), were examined. The simulation results show that when the electroosmotic mobility increases, the dispensed sample moves further downstream in the dispensing channel, as can be seen in Figure 8.5a and Figure 8.5b. This is because when the electroosmotic mobility increases, the average velocity of the bulk flow increases. Thus, during the same time period, the solution with high electroosmotic mobility is moved further than that with low electroosmotic mobility. The numerical results also reveal that when the electroosmotic mobility increases, the size of the dispensed sample is smaller. For example, at time / = 1.25s, the sample volume for the case of low electroosmotic mobility is 395 pico-liters and for the case of high electroosmotic mobility is 372 pico-liters. This is because the velocity in the upstream of the dispensing channel is higher than that in the downstream of the dispensing channel, as shown in Figure 8.3. When the sample is dispensed downstream in

476


Figure 8.6. Effect of the electrophoretic mobility on the concentration field of the sample in the dispensing process. The black regions are the buffer solution; the white regions are the sample solution with a sample concentration higher than 80% of the original sample concentration. The electrophoretic mobility: a) -2.0x10" 8 m 2 l(V-s), b) -3.5xlO" 8 /n 2 l(V-s).


477

the dispensing channel, the front of the sample (in the downstream of the dispensing channel) moves relatively slower than the back of the sample (in upstream of the dispensing channel), consequently, the sample is compressed. When the electroosmotic mobility increases, the difference between the velocity in the upstream and the velocity in the downstream of the dispensing channel is larger. Indeed, the numerical results show that the difference between the velocities in the downstream and the upstream increases when the electroosmotic mobility increases. Consequently, the dispensed sample is more compressed and the volume of the dispensed sample is smaller for the case of high electroosmotic mobility. Electrophoretic mobility effect Under the same applied electrical field as described previously, two cases with different electrophoretic mobility, /uep =-2.0x10 Hep =-3.5x10

m /(V-s),

m /(V-s)

and

were examined. Figure 8.6 shows that the

concentration distribution during the dispensing process. From this figure, we can see that when the absolute value of the electrophoretic mobility increases from 2.0xl0~ 8 m2 /(V-s) to 3.5xl0" 8 m2 I(V -s), the volume of dispensed sample at the same time (i.e. t = 2.0s) increases from 332 pico-liters to 406 picoliters. This is because, with the negative electrophoretic mobility, the electrophoretic motion of the charged sample species is against the electroosmotic flow. The net effect is a reduced bulk flow velocity. When the value of the electrophoretic mobility increases (from 2.0x10 8

m /(V-s)

in

2

Figure 8.6a to 3.5xlO~ w /(V-s) in Figure 8.6b), the net flow velocity of the sample decreases, and the velocity difference between the front of the sample (in the downstream of the dispensing channel) and the back of the sample (in upstream of the dispensing channel) is smaller. Consequently, the compressing effect (as discussed in the previous section) against the diffusion is reduced, and the volume of the dispensed sample increases. Channel size effect In the simulations of the channel size effect, the non-dimensional applied electrical potentials are:

478


Figure 8.7. Effect of the microchannel size on the concentration field of the sample in the dispensing process. The black regions are the buffer solution; the white regions are the sample solution with a sample concentration higher than 80% of the original sample concentration. The channel width ratio: a) Wy/Wx=1:1; b) Wy/Wx=2:1; c) Wy/Wx=l:2.


479

Three pairs of channel sizes were used to test the channel size effect on the dispensing process. The results are shown in Figure 8.7. In the first case (Figure 8.7a) the size of the horizontal channel is equal to that of the vertical channel. We describe this case by the ratio of these two channel sizes, Wy/Wx = 1:1, where Wy is the width of the horizontal channel and Wx is the width of the vertical channel. In the second case (Figure 8.7b), the size of the horizontal channel is twice that of the vertical channel, i.e., Wy/Wx = 2:1. In the third case (Figure 8.7c), the size of the horizontal channel is half that of the vertical channel, i.e., Wy/Wx = 1:2. All other parameters are same for these three cases. As seen from Figure 8.7, the channel size has significant effects on the dispensing process and on the dispensed sample volume. Under the same applied electrical field, the shape and the volume of the loaded sample at the intersection are very different for these three cases. The loaded sample volume in the case of Wy/Wx = 1:1 is the largest. Consequently, the shape and the volume of the dispensed sample are very different for these three cases. The dispensed sample volume in the case of Wy/Wx = 1:1 is the largest (the sample volume is approximately 113 pico-liters). This is because under the same applied electrical potentials and the same channel surface properties, if the channel size changes, the electrical field will change since we applied the insulation boundary conditions to all the walls. Since the flow field depends on the electrical field as shown in Figure 8.3, the concentration field, in turn, is strongly dependent on the electrical field. When the electrical field changes due to the change of channel size, the concentration distribution will change. Electrical field strength effect For a given set of properties of the sample and the buffer solutions and the specified microchannels' surface properties and dimensions, the applied electrical potentials are the key controlling parameters for the loading and dispensing processes. Two different combinations of non-dimensional applied electrical potentials were tested as shown in Figure 8.8 and Figure 8.9. The nondimensional applied electrical potentials for the two cases in Figure 8.8 are: In Figure 8.8a:

480


Figure 8.8. Effect of the applied electrical potentials in the loading process on the concentration field of the sample in the dispensing process. The electrical potentials for the dispensing process are the same for both case a) and b). The black regions are the buffer solution; the white regions are the sample solution with a sample concentration higher than 80% of the original sample concentration. The non-dimensional applied electrical potentials are shown at the bottom of this figure.


481

The non-dimensional applied electrical potentials for the two cases in Figure 8.9 are: In Figure 8.9a:

The simulation results show that the loading electrical field strength has very important effects on the volume of the dispensed sample, as shown in Figures 8.8a and 8.8b. Under the same dispensing conditions, if the applied electrical potential for the loading process is changed, the size of the loaded sample in the intersection of the microchannels changes and eventually the volume of the dispensed sample changes. Figures 8.8a and 8.8b clearly demonstrate the difference. The volume of the dispensed sample in the downstream of the dispensing channel changes significantly from 390 pico-liters in Figure 8.8a to 204 pico-liters in Figure 8.8b. Comparing Figures 8.9a and 8.9b, we see that under the same loading conditions, if the applied electrical potential for the dispensing process changes, the volume of the dispensed sample changes too. For example, during the dispensing process, while (j) (2) and (j) (4) keep constant, (l) and (j> (3) are increased from 0.2 in Figure 8.9a to 1.0 in Figure 8.9b, the volume of the dispensed sample increases (i.e. from 390 pico-liters in Figure 8.9a increases to 527 pico-liters in Figure 8.9b). This is because when <j) (l) and <j) (3) are bigger, the electrical field strength between the intersection and reservoir 1 and between the intersection and reservoir 3 is weaker. Consequently, less amount of the sample is drawn back toward these two reservoirs and a bigger portion of the sample is dispensed downstream in the dispensing channel (Figure 8.9b). Practical applications such as on-chip separation and detection require that the dispensed sample must distributed uniformly cross the channel cross-section, or have an evenly cut plug shape. However, the shape of the dispensed sample strongly depends on the dimensions of the cross microchannels and the combination of the applied electrical potentials. The even plug-shape of the dispensed samples presented in most figures in this section is the results of carefully choosing the optimal controlling electrical potentials. Figure 8.7b shows an example of non-uniformly dispensed sample.

482


Figure 8.9. Effect of the applied electrical potentials in the dispensing process on the concentration field of the sample in the dispensing process. The electrical potentials for the loading process are the same for both case a) and b). The black regions are the buffer solution; the white regions are the sample solution with a sample concentration higher than 80% of the original sample concentration. The non-dimensional applied electrical potentials are shown at the bottom of the figure.


483

In summary, the diffusion coefficient, the electroosmotic mobility, the sample's electrophoretic mobility, the applied potentials and the channel sizes all have different effects on the dispensing process. When the diffusion coefficient increases 10 times, the volume of the dispensed sample decreases by only about 2%. When the electroosmotic mobility increases, the average velocity of the bulk flow increases, and hence the sample is transported more quickly, and has less chance to diffuse into the buffer solution. When the absolute value of the sample's electrophoretic mobility increases, the region of high sample concentration increases and hence the dispensed sample volume increases. Because both the loading and the dispensing processes are electrokinetically driven processes, changes in the applied potentials and in the channel sizes will directly change the applied electrical field in the dispenser, and hence dramatically affect the effectiveness of the loading process and the size of the dispensed sample. Overall, the results discussed in this section show that the size and the shape of the dispensed sample can be controlled by the channels' dimensions and the applied electrical potentials. The model presented here can be used to find the optimal controlling parameter values for the loading and the dispensing processes.

484

8-2


EXPERIMENTAL STUDIES OF ON-CHIP MICROFLUIDIC DISPENSING

The theoretical model and the numerical simulation presented in the last section have revealed the characteristics of the electrokinetic dispensing processes associated with the crossing-microchannel dispenser. These theoretical predictions were verified by an experimental study of the dispensing processes by using a fluorescent visualization technique [11]. The results demonstrated the ability to load and dispense moderate and large sizes of samples on a crossing microchannel chip. During dispensing, smaller samples were found to disperse at a significantly higher rate than larger samples in similar conditions. In addition, the sensitivity of the loading step sample geometry to pressure disturbances was shown to increase dramatically with the extent of the sample. This section will review the experimental method and results, and compare the experimental results with the theoretical model predictions.

Figure 8.10. Schematic diagram of the imaging system and the crossing microchannel chip.


485

In the experiments, the sample employed was a fluorescent dye, and a fluorescent epi-illumination video microscope was used to visualize the process. A schematic of the imaging apparatus and the microfluidic chip is given in Figure 8.10. The dye was excited by a continuous flood of blue light provided by a single-line, 200mW, 488nm argon laser, through the microscope objective. A 32x, NA = 0.6 objective was used for magnified views of the intersection and a 16x, NA = 0.3 objective was employed for larger fields of view. The received signal was split by the dichroic mirror (51 Onm LP/ short-reflecting) and passed through an additional filter (515nm LP) and a 0.63x c-mount before reaching the camera. The chip was clamped to a precision 3-axis stage. The two horizontal positioning axes were used to position the field of view, and the vertical axis to focus the microscope. Such arrangements are flexible, but care must be taken to ensure that the objective remains sufficiently far/insulated from the electrodes. Images were captured and saved on the computer at a rate of 15Hz with individual exposure times of 1/125s. A progressive scan CCD camera was used to avoid image defects due to field-field interlacing. The acquired images had a resolution of 640x484 pixels and an 8-bit dynamic range. This corresponds to a viewed region of 1065x805 urn, and 550x416 um using the 16x and the 32x objectives respectively. The camera orientation was carefully adjusted before each run such that the pixel grid was aligned with the coordinate directions of the intersection. Digital image processing was. performed to remove any nonuniformity present in the imaging system and to relate the pixel intensity values to dye concentration. A background noise signal was subtracted, and brightfield image normalization was performed for each image. These images were then smoothed with a distance-based kernel, and scaled by a single factor such that the image series filled the grey scale range. The glass chips with crossing microchannels used in this study were manufactured by Micralyne Inc (Edmonton, Canada). A schematic diagram of the chip from above, and the shape of the channel in cross-section are given in Figure 8.10. The channels are 20u.m deep and 50u.m across, and the reservoirs are 2 mm in diameter. The D-shaped cross-section is a product of the manufacturing technique. The sample was loaded into reservoir Rl, the sample waste reservoir was R3, and the buffer focusing reservoirs were R2 and R4. The lengths of the channels from intersection to reservoir are 5 mm, 4 mm, 80 mm, and 4 mm for channels 1,2,3, and 4 respectively. Two fluorescent dyes, supplied by Molecular Probes, were employed here: fluorescein C2oH1205 (332.31MW); and rhodamine 110 (366.8MW). Both dyes have an absorption maximum near X = 490nm which is well suited to excitation with the X = 488nm output line of the argon laser. The dyes were dissolved in sodium carbonate buffer of pH = 9.0. The buffer was prepared by dissolving 39.8xlO"3 mol of NaHCO3 and 3.41xlO"3 mol of Na2CO3 in 1 litre of pure water

486


resulting in a solution of ionic strength, / = 0.05 M. Stock solutions of fluorescein and rhodamine were prepared in 0.1 mM and 0.2 mM concentrations respectively. The solutions were aliquoted and stored in darkness at -20°C. Immediately before use, all solutions were filtered using 0.2 urn pore size syringe filters. For fluorescein, a diffusion coefficient of D = 4.37x10~10 m2/s was assigned, giving an electrophoretic mobility of vphj = -3.3xl0"8 m2/(V-s). The diffusion coefficient for rhodamine was estimated to be the same as that of fluorescein, however, the electrophoretic mobility was found to be significantly less (less negative) than that of fluorescein. Both dyes were loaded and an onchip separation was performed to determine that rhodamine carried a charge of z R = - 1 at pH = 9.0 (as opposed to fluorescein, zF = -2) giving it an electrophoretic mobility of vpf, R = -1.65xlO"8 m2/(V-s).

Figure 8.11. The loading and dispensing of a focused rhodamine sample: a.) Processed images (experimental results); (b) Iso-concentration profiles at 0.1 Co> 0.3Co, 0.5Co, 0.7Co, and 0.9Co, calculated from the images; and (c) Corresponding Iso-concentration profiles calculated through numerical simulation.


487

Chips were prepared by rinsing with each of the following solutions for 20 minutes (in sequence): 1M filtered Nitric acid; 1M filtered Sodium hydroxide (NaOH); and filtered buffer. To pull these solutions through each channel, a vacuum was applied to R3 using a 30mL syringe. A tapered lOOOul pipette tip was cut to the appropriate diameter and friction fitted to join the syringe to the chip. Experimental and numerical results of the loading and dispensing of a focused rhodamine sample are presented together in Figure 8.11. The numerical simulation results were obtained by using the theoretical model presented in the last section under the specified experimental conditions. Processed images are displayed in Figure 8.11 (a) (the top row). The first image shows the steady state loading process. The following sequence images show the dispensing process at 2/15s intervals. Iso-concentration contours of the dye at 0.1Co, 0.3Co, 0.5Co, 0.7Co and 0.9Co (where Co is the concentration of the original sample solution) are given in Figure 8.1 l(b). Corresponding iso-concentration contours calculated using the theoretical model are given in Figure 8.1 l(c). With respect to all plots, the chip was oriented with channels from Rl, R2, R3, and R4, counter-clockwise from top. Orienting the loading process in this way took advantage of the inherent left/right symmetry such that the same voltage could be applied at R2 as R4. This simplified both the experiments and the numerical analyses. The voltage applied to Rl in the loading step in Figure 8.11 was V1L = 1295V. Applied voltages normalized with this value were 1.0, 0.91, 0, and 0.91 for s\i - sî respectively, (where e indicates normalized applied voltage and in the subscripts the numbers correspond to reservoir number and L denotes the loading step). The triangular shape of the sample was a result of the intersection of the buffer flow from R2 and R4 and the sample solution flow from Rl. Considering the applied voltages and the length of channels 1,2, and 4, the electroosmotic flow rates from each channel were very similar (in fact, the flowrates in channels 2 and 4 were only slightly higher than that in the sample channel 1). The result is a narrowing of the sample stream as it joins the two buffer streams that feed into the sample waste channel 3. The specific geometry of the sample is a combination of advective, diffusive, and electrophoretic transport. To dispense the sample into channel 4, potentials of strength 0.022, 0.063, 0, and 0 were applied for e\j) - s^p (in the subscripts the numbers correspond to reservoir number and the D means the dispensing step) respectively. Under this field, most of the fluid pumped from channel 2 enters the dispensing channel 4, however, some fluid is also pumped into the sample and sample waste channels (1 and 3). It is preferred to drive the sample streams back in this way to avoid leakage and 'cut' the sample. At t = t3, the experimental results in Figure 8.1 l(b) indicate that significant sample diffusion has occurred and the peak

488


concentration value is just over 0.7Co. The numerical results in Figure 8.1 l(c) show a similarly sized sample with a more triangular shape. At t3 there is a small amount of sample at 0.7Co, although most the peak region is between 0.5Co and 0.7Co. The discrepancies between experimental results and numerical results may be due to the assumption of two-dimensionality and/or errors in measuring the applied voltages, diffusion coefficients and mobility values (required as input to the model). Nonetheless, both numerical and experimental results indicate significant loss of sample at the peak concentration only a few channel diameters downstream of the intersection. The results in Figure 8.12 show the effect of decreasing the applied field strength in channels 2 and 4 during the loading stage. These voltages were

Figure 8.12. The loading and dispensing of a less-focused rhodamine sample: a.) Processed images (experimental results); (b) Iso-concentration profiles at 0.1 Co, 0.3Co, 0.5Co, 0.7Co, and 0.9Co, calculated from the images; and (c) Corresponding Iso-concentration profiles calculated through numerical simulation.


489

reduced to £21 =£\i= 0.88 (a 3% decrease) and all other run parameters were identical to those pertaining to Figure 8.11. The orientation, formatting and time-steps used in Figure 8.12 are the same as Figure 8.11. As a result, the loading sample geometry became significantly broader. When dispensed, the experimental sample profile is considerably more uniform in the cross-stream direction, and at t3 it contains over 20um of sample at the peak concentration. The numerical solution in Figure 8.12(c) also predicts a significant increase in dispensed sample, although again, the sample geometry is not as uniform in the cross-stream direction and thus not as concentration-dense as the corresponding experimental result. Both experimental and numerical results here demonstrate the ability to alter the sample size and concentration distribution through only minor adjustments to the applied potentials. Figure 8.13 shows the loading and dispensing of a sample of fluorescein dye, in contrast to the rhodamine dye (corresponding to the results in Figures 8.11 and 8.12). Although fluorescein and rhodamine are similar molecules, at pH = 9.0 fluorescein carries a higher charge and hence has a more negative mobility. Since the sum of the electroosmotic mobility and the electrophoretic mobility is greater than zero, the sample moves in the direction of the bulk flow (like rhodamine) but at a reduced rate. The voltage applied to Rl in the loading step was V1L = 1300V, and the reservoir potentials normalized with this value are 1.0, 0.88, 0, and 0.88 for s\i -£41 respectively. This field was effectively the same as that applied in the rhodamine-loading step in Figure 8.12. The fluorescein sample, however, is not as wide as the corresponding rhodamine sample during the loading step. Although it is likely that the total mass flowrates of solution are similar to that of the rhodamine case, the mass flux of the fluorescein species entering the intersection is significantly lower than that of rhodamine due to the difference in their electrophoretic mobilities. With less mass flux, diffusion into the buffer channels is reduced and the sample stream is narrowed as it enters the sample waste channel. This narrowing results in the dispensed sample geometry shown in Figure 8.13, which is smaller than that in Figure 8.12 and less uniform in the cross-stream direction. The dispensing potentials applied in this case were 0.055, 0.127, 0, and 0 for S\D -£40 respectively. The observed velocity of the fluorescein sample in the dispensing stage is similar to that of the rhodamine sample because the electric potential, e 2Z>> w a s doubled. Both numerical and experimental results presented in Figures 8.11, 8.12 and 8.13 demonstrate that: sample shape and concentration profile can be controlled through minor adjustments to the applied field; and similar sample geometry for samples with dissimilar mobility may be achieved through appropriate adjustments to the applied field. However, in all the three cases the numerically determined sample shapes were more focused than those determined

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experimentally. Thus, the numerical samples were less uniform in the crossstream direction and contained less sample. These differences were amplified in the dispensing process due to diffusion. It is expected that these systematic differences were due to an underestimation of the diffusion coefficients and the corresponding electrophoretic mobility values (related through the NernstEinstein relationship). Methods for more accurately determining these coefficients are needed. Processed images of the dispensing of a focused, large, and very large axial extent rhodamine sample are shown in Figure 8.14 (a), (b) and (c) respectively. To demonstrate the transport process, a wider field of view was used as well as higher dispensing potentials of 0.106, 0.193, 0.01, and 0 for

Figure 8.13. The loading and dispensing of a focused fluorescein sample: a.) Processed images (experimental results); (b) Iso-concentration profiles at 0.1Co, 0.3Co, 0.5Co, 0.7Co, and 0.9Co, calculated from the images; and (c) Corresponding Iso-concentration profiles calculated through numerical simulation.


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6

ID ~ SAD respectively. In each case, the dispensed sample is shown (at right) at approximately t = 4/15s after the dispensing field was applied. As expected, the peak concentration of the smaller sample was reduced during transport while the large samples maintained a significant portion of the original sample concentration. The largest sample, shown in the loading stage in Figure 8.14(c), extended well beyond the intersection. The width of the sample, measured as the full-width-at-half-maximum concentration, was over two channel widths (> 100(am). It was found that this was the largest axial extent sample that could be produced repeatably under these conditions. To further increase sample size, the electroosmotic flow from the buffer reservoirs (R2 and R4) would have to be

Figure 8.14. Processed images (experimental results) of the loading step (left) and dispensing step (at t £ 4/15s, right) of: (a) A focused rhodamine sample; (b) A large axial extent rhodamine sample; and (c) A very large axial extent rhodamine sample.

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further reduced, reducing the mass-based Peclet number (the ratio of advection to diffusion). Eventually diffusive forces dominate and the sample fills both focusing channels (2 and 4). It is important to note that a modified Peclet number is require here, because it is the sample's velocity which is pertinent. This sample velocity, USAMPLE, is a summation of bulk electroosmotic velocity, uEo, and electrophoretic velocity of the sample, uPH, as follows: (14) and the modified Peclet number, PeM, may be defined as: (15) where L is the characteristic length of the sample and D is the diffusion coefficient. Experimentally it was found that the sample would first leak into one sample stream or the other rather than both concurrently. This was due to a slight velocity bias caused by pressure forces originating in the reservoirs. Like diffusive effects, these pressure disturbances take on a greater role as the electroosmotic flow rate is decreased. In Figure 8.14(c), the center of the loaded sample is shifted slightly to the right of the intersection. This breaking of symmetry is a subtle indicator of the presence of a small pressure driven flow component. These pressure effects are investigated next. To investigate the sensitivity of large axial extent samples to pressure disturbances, a pressure difference, AP, was applied between R2 and R4. Reducing the fluid contained in R2 and maintaining R4, as shown in Figure 8.15, accomplished this. With this pressure gradient in place, the potential of R2 and R4 were varied to achieve the three steady state loading sample geometries in Figure 8.16. A top voltage of V iL = 1320V was applied, with &2L =£4L = 0.924, 0.923, and 0.904 corresponding to Figure 8.16 (a), (b), and (c) respectively. The focused sample shown in Figure 8.16 (a), is not significantly effected by the pressure gradient, although the narrowed sample stream in the sample waste channel 3, is visibly shifted to the left, indicating a larger flow rate from R2 than from R4. By reducing the electroosmotic buffer flow by 1%, the sample becomes larger and significantly asymmetrical, as shown in Figure 8.16 (b). Finally, reducing the electroosmotic flow in channels 2 and 4 a further 1%, the sample is washed into the focusing channel as shown in Figure 8.16 (c). Two conclusions may be drawn here: the axial extent of the sample is very sensitive to


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Figure 8.15. Schematic diagram illustrating reservoir liquid levels as a source of pressure driven flow.

changes in the focusing potentials; and larger axial extent samples are significantly more sensitive to pressure disturbances that focused samples. The source and magnitude of such pressure disturbances in these microfluidic chip applications may be analyzed as follows. Consider the pressure-driven flow between two parallel plates separated by 2h = 20iam (which is the depth of the microchannels employed here). The average velocity generated by a pressure gradient may be expressed as, (16) where ji is the dynamic viscosity, and AP is the pressure differential applied over length L, in the downstream direction. In order for such a pressure driven flow component to significantly influence the sample geometry, the velocity generated must be comparable to the combined electroosmotic and electrophoretic velocity of the sample.

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(17) Substituting Eq. (17) into Eq. (16), and substituting representative values for the velocities, viscosity, and channel dimensions gives AP = \00Pa. The hydrodynamic head, H, required to generate this pressure may be calculated as (18) where p is the mass density and g is the acceleration due to gravity. The hydrodynamic head generated by depleting R2 is shown schematically in Figure 8.15. Eq. (18) gives a required head of H = 10 mm, which is not reasonable, especially considering the reservoirs are only 1.1 mm in depth. Considering Laplace pressure effects, the pressure across a curved liquidgas interface may be calculated by:

Figure 8.16. Iso-concentration lines at 0.1Co, 0.3Co, 0.5Co, 0.7Co, and 0.9Co, calculated from images of loading sample geometries under the influence of an applied pressure gradient: (a) A focused sample; (b) A less-focused sample; and (c) A large axial extent sample causing washing of sample into the buffer stream.


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(19) where yLV is the surface tension, taken here to be that of water. If R4 is maintained with negligible curvature, and the meniscus of R2 is on the order of the reservoir radius, o = 1 mm (as shown in Figure 8.15), Eq. (18) gives a pressure differential AP = 140Pa. This is more than sufficient (>100Pa) to overcome the combined electroosmotic and electrophoretic velocity in the focusing channels. Thus it is found here that differential Laplace pressures generated in the reservoirs were the most significant cause of pressure-based disturbances in these flows. One solution may be to increase reservoir size (decreasing curvature) and/or increase channel lengths (increase L). Unfortunately these size increases may mitigate some of the key benefits of onchip processing, derived from small sample volumes, and short processing times. In summary, the loading and dispensing of sub-nanolitre samples is achievable by using a microfluidic chip with a crossing microchannel. The ability to inject and transport large axial extent, high-concentration samples was demonstrated experimentally. Both experimental and numerical results indicate the shape, cross-stream uniformity, and axial extent of the samples were very sensitive to changes in the electric fields applied in the focusing channels. In the dispensing process, larger samples were shown to disperse less than focused samples, maintaining more solution with the original sample concentration. In addition, a cross-chip pressure gradient may influence the sample dispensing. Larger samples were found to be more sensitive to pressure disturbances than the more focused samples.

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8-3


DISPENSING USING DYNAMIC LOADING

In many on-chip applications, longer dispensed samples (samples of greater axial extent) are required. This section will discuss a dispensing method to produce longer dispensed samples. Most on-chip sample dispensing/injection techniques have involved a two-step process. In a cross microchannel dispenser, as described in the previous section, the sample is focused through the intersection of the crossing microchannels by steady flows of buffer from the adjacent dispensing and buffer supply channels. To dispense the sample, a field is applied along the dispensing channel. Common on-chip dispenser/injector configurations include the cross, the tee, and the offset twin-T (or equivalently the 'double-T'). In the offset twin-T configuration, the sample and dispensing channel run together for the short distance between T-intersections resulting in an effectively larger injector length.

Figure 8.17 The three-step injection process illustrated with an image sequence at left and a schematic of the flow directions at right.


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This is used to obtain dispensed samples of greater length than previously achieved in crossing microchannel chips. However, the flowing of buffer and sample adjacently in the injector channel causes increased diffusion and results in a diluted dispensed sample. Furthermore the sample size is greatly determined by the specific injector channel length of the chip and cannot be varied during a given process. Liu et al. [19] conducted sequencing separations of single stranded DNA in a cross and offset twin-T chip injectors with injector lengths of 500 urn, 250 yun, and 100 yun. The best one-color and four-color separations were observed with the 250 pm, and 100 um injectors respectively. Using larger injector lengths was found to increase signal at the expense of resolution. Wallenborg [17] separated and detected a mixture of explosives by micellar electrokinetic chromatography using a straight-cross injector and offset twin-T

Figure 8.18. Image sequences of sample injections with dynamic loading periods of: (a) t D L = 0s (no dynamic loading); (b) t DL = 0.02s; and (c) t D L = 0.05s.

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injectors with injector lengths of 250 \im and 100 îm. It was found that the 250 l^m injector gave a 35% increase in peak signal over the straight cross. Although this was an improvement, it was less than the expected based on loaded sample size. Further investigation showed a major loss of sample during the onset of the separation step due to pullback flow induced in the offset channels. Other advances in injector geometry have been reported, such as a significant improvement in resolution, column efficiency and sensitivity using narrow sample channels [20]. In the previous sections, the formation of longer samples in crossing microchannel dispenser by reducing focusing buffer flow was discussed. In that technique, reduced focusing buffer velocities permit broadening of the sample during the steady-state loading step. This was found to produce significantly longer samples than more focused sample injections. However, since the broadening was effectively diffusive, the sample concentration was non-uniform in the axial direction and the sample contained relatively little solution at the original concentration. Experimentally it was found that the reduced focusing buffer velocities also increased the sensitivity of the injection to reservoir-based pressure disturbances. Slight pressure gradients were found to cause sample distortions, and in some cases excessive sample leakage. To generate longer dispensed samples with good control over the sample's size and concentration, a three-step injection procedure for use in crossing microchannel chip was developed [12]. The technique is a variation of the injection process described in the previous section with the addition of an intermediate dynamic loading step. In Figure 8.17, the three steps are shown in an image sequence at left with the flow directions illustrated schematically at right (a relatively long sample was chosen to illustrate the process). Results for both short and long dynamic loading periods will be presented, and compared with the focused and the less-focused crossing microchannel injections. In the experiments, the experimental setup, imaging processing and analysis, the buffer solutions, the dyes, and the crossing-microchannel chips are the same as described in the previous sections. In general, a three-step process requires independent control of three potential levels for each of the four electrodes (12 settings). For this purpose, a Glassman high voltage power supply was used in conjunction with a custom-made voltage controller. The timing of each switch was controlled via a Stanford Research Systems digital delay generator. The voltage controller was programmed for default outputs corresponding to the steady-state loading step. Rising edge triggers from the delay generator commenced dynamic loading and dispensing steps. The period over which the dynamic loading voltages are applied is termed the dynamic loading period, tî. Sample injections with short dynamic loading periods were conducted in an effort to increase peak signal strength with minimal


499

Figure 8.19. The measured Centerline concentration profiles corresponding to injections imaged in Figure 8.18. The profile of the steady-state loading step (common starting point of each injection) is delineated with "o" symbols. The following two profiles were obtained during the dispensing step (at 3/15 s intervals).

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sample length increase. All electrical potentials were normalized with respect the largest potential, which was 1355V. Normalized potentials applied during the steady-state loading step, sL, the dynamic loading step, eî, and the dispensing step, S£>, are given below:

(20)

Figure 8.20. The measured iso-concentration contours of the dispensed samples corresponding to injections imaged in Figure 8.18 (contours plotted at 0.1 Co, 0.3Co, 0.5Co, 0.7Co, and 0.9Co).


501

Figure 8.21. The numerically determined iso-potential contours plotted in the intersection for: (a) the steady-state loading step, (b) the dynamic loading step, and (c) the dispensing step.

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where subscripts 1-4 correspond to reservoirs 1-4, respectively. The results are shown in the image sequences in Figure 8.16. A sample injection with no dynamic loading, tpi = 0, is shown in Figure 8.16(a). The first image in the sequence is of the steady- state loading step, and the second and third images are during the dispensing step (taken 3/15 s apart). Although the injected sample is diluted from the original sample concentration, it is reasonably well defined, which is due to the application of a pull-back (or equivalently 'cut-off) voltage at reservoir 1 during the dispensing step. The effect of introducing a very short dynamic loading step of tDi = 0.02s is shown in Figure 8.16(b). Images in Figure 8.16(b) were chosen to approximately match the axial position of the samples with those of Figure 8.16(a). The matching is not exact, however, because the camera was not synchronized with the dispensing signal. The images show that the dynamically loaded sample is significantly brighter than that without dynamic loading. The effect of increasing the dynamic loading period to tpi = 0.05s is shown in Figure 8.16(c). The dispensed sample is threechannel-widths (150 um) long, measuring the full width at half maximum (FWHM). The peak height in the last image is 95% of the original sample concentration. Centerline concentration profiles obtained from the images in Figures 8.16(a), (b) and (c) are plotted in Figures 8.19(a), (b) and (c), respectively. The profile of the steady-state loading step (the starting point for all the injections) is delineated in ' o ' symbols. The original sample intensity level was taken as the signal level at the top of the intersection in the sample channel, and the images were processed such that the signal intensity shown is directly proportional to dye concentration (based on a previously determined, camera response characteristic). The small jump above the background signal, visible at x = 40 um, is an artifact of the imaging process and does not represent sample concentration. This figure illustrates the increase in peak height possible with dynamic loading. As can be seen in the profiles, the steady-state loading profile contains only a small portion of sample at the original sample concentration. The peak height of the dispensed focused sample is 64% (Figure 8.19(a)). This was increased to over 95% using a dynamic loading period of only t£)L = 0.05s (Figure 8.19(c)). The corresponding length increase was from 100 um to 160 um (FWHM). It was also found that the dynamically loaded samples exhibited a greater degree of cross-stream uniformity than the focused samples. The focused pinched-valve injection resulted in a dispensed sample with a higher sample concentration on the sample channel side of the dispensing channel. This is apparent in the iso-concentration contours plotted in Figure 8.20(a) (corresponding to the dispensed sample shown in Figure 8.18(a)). Concentration contours are plotted at concentrations of 0.1Co, 0.3Co, 0.5Co, 0.7Co, and0.9Co, where Co is the original sample concentration. In focused injections, the degree


503

Figure 8.22. Processed Images and corresponding centerline concentration profiles for: (a) a long sample created by reducing focusing velocities; (b) a long sample created by further reducing focusing velocities (note the sample drift); and (c) a long sample created by a tDL = 0.2s dynamic loading step.

504


of non-uniformity correlates with the degree of focusing potential applied. This is somewhat unfortunate because it creates larger cross-stream concentration gradients in small samples that, by nature of their size, are more susceptible to dilution. Iso-concentration contours for the dynamically loaded samples, corresponding to Figure 8.18, are given in Figure 8.20. Unlike the focused sample, both dynamically loaded samples exhibit a significant region in the top concentration range. In the case of the longer dynamic loading time (Figure 8.20(c)), this range extends across the channel, which represents a significant improvement in concentration density and cross-stream uniformity over the sample injected without dynamic loading (Figure 8.20(a)). To gain insight into the driving force behind the dynamic loading step, the potential fields for the loading, dynamic loading, and the dispensing steps were calculated using the numerical model described in Section 8-1. Iso-potential contours near the intersection are plotted in Figure 8.21. It is interesting to note that the steady-state loading steps have similar electric field patterns. In both cases, the highest potential values occur in the sample channel (at top). In the loading case (Figure 8.21 (a)), the curvature of the potential lines in the intersection shows how the sample and both buffer streams are electroosmotically pumped into the sample waste reservoir. In the dynamic loading case (Figure 8.2l(b)), the opposite curvature shows how the sample is electroosmotically pumped into both focusing channels and the sample waste channel. As expected, the electric field in the dispensing step, shown in Figure 8.2l(c), is mostly uniform throughout the dispensing channel. In addition to short concentration-dense samples, longer samples (samples of greater axial extent) are also of interest. In Figure 8.22(a), a reduced-focused sample is shown in the loading stage, and the corresponding centerline concentration profile is shown at right. This sample was produced by setting S£_2 =£L-4 =0.935. This 6.5% reduction in focusing potential resulted in a two-fold increase in sample length. The results of a further reduction to Si_2 =££-4 =0.918 are shown in Figure 8.21(b). The drift to the right, apparent in the figure, is likely due to undesired pressure disturbances originating from Laplace pressure acting on the free surfaces in the reservoirs. These effects only become significant when the electroosmotic buffer velocities are reduced. This presents a practical constraint on the length of samples that can be produced by this reduced-focusing technique. Fundamental to that technique is the use of diffusive forces to broaden the sample. Although this can create larger samples, it also results in an increased pressure sensitivity and sample dilution. Using dynamic loading, however, sample broadening is achieved by direct advection of sample into the dispensing channel. Thus, well-defined samples of high concentration density can be produced at any length. This is demonstrated in Figure 8.22(c) with a sample at the end of a dynamic loading period of tjji =


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Figure 8.23. The measured centerline concentration profile sequences of the dispensing of the samples shown in Figure 8.22. In each case, the left-most profile is that of the loaded sample, delineated in ' o ' symbols. The following two profiles were obtained during the dispensing step (at 3/15 s intervals).

506


0.2s (prior to dispensing). The flat-topped concentration profile indicates the concentration of the bulk of the sample is at the level of the original sample concentration. The high concentration gradients that sharply define the sample are a product of the unsteady nature of the dynamic loading process, and would not be possible with previous steady-state loading methods. Centerline concentration profiles obtained during the dispensing of these samples are given in Figure 8.23. In each case, the left-most profile is that of the loaded sample, delineated in ' o ' symbols. Regarding the dispensed sample profiles (right-most profiles), the peak height of the samples injected by the reduced-focusing technique (Figures 8.23(a) and (b)) are shown to have decreased significantly from their original values. In contrast, the dynamically loaded sample (Figure 8.23(c)) has maintained much of its volume at the original sample concentration.

Figure 8.24. The measured iso-concentration contours of the dispensed samples corresponding to injections of Figure 8.23 (contours plotted at 0.1Co, 0.3Co, 0.5Co, 0.7Co, and 0.9Co).


507

high percentage of the volume contains the original sample concentration has two important implications. Firstly, the net concentration density of the sample is increased. Secondly, as concentration gradients exist only at the edges of the sample, the plateau region serves to forestall diffusive effects. This is in contrast to Gaussian shaped samples in which diffusion occurs immediately throughout the sample volume. The iso-concentration contours for the dispensed samples in Figure 8.23 are shown in Figure 8.24. Although the dynamically loaded sample (Figure 8.24(c)) may be too long for some applications, the degree of cross-stream and axial concentration uniformity achieved is noteworthy. The ability to dispense samples in which a high percentage of the volume contains the original sample concentration has two important implications. Firstly, the net concentration density of the sample is increased. Secondly, as concentration gradients exist only at the edges of the sample, the plateau region serves to forestall diffusive effects. This is in contrast to Gaussian shaped samples in which diffusion occurs immediately throughout the sample volume. To demonstrate the ability to inject very long concentration-dense samples, the dynamic loading of a millimeter-sized sample was performed. Because the dynamic loading period here, tDL = 0.8 s, was relatively long, the imaging system was able to capture the symmetric growth process. An image sequence of the process, at 4/15 s intervals is given in Figure 8.25(a). Centerline concentration profiles corresponding to the dynamic loading step, at 2/15s intervals, are shown in Figure 8.25(b). Imaging artifacts at the surface of the chip caused the noise visible on the far right-hand side of the later images and the final two concentration profiles. At its final size, the sample is over 0.9 mm long, of which 88% contains the original sample concentration. This large sample was chosen to demonstrate the methods potential usefulness in applications that previously employed offset twin-T configurations with gap lengths of 250 urn and 500 um. The dynamic loading method has two key advantages over twin-T injection methods. Firstly, the size of the sample is not coupled to the geometry of the chip, and can be varied simply by varying the potentials applied or the dynamic loading period, tpi. Secondly, the use of the dynamic loading method does not incur the sample dilution inherent in the steady-state focusing step in twin-T injections (where focusing buffer and sample run adjacently in the injection length). As a final note, when large samples are injected using the dynamic loading method, the pull-back voltage(s) applied during the dispensing step (to 'cut' the sample) can lead to sample length reduction (sample loss). This loss occurs while the sample loaded into the buffer channel 2, is drawn through the intersection and into the dispensing channel. Thus, to achieve the desired sample length in the dispensing channel, the dynamic loading period (or potentials) must be set with this taken into account.

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Figure 8.25. The injection of a millimeter-sized sample: (Top) an image sequence at 4/15 s intervals showing the dynamic loading and dispensing of the sample, and (Bottom) a centerline concentration profile sequence, at 2/15s intervals, of the dynamic loading process.


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In summary, the dynamic loading method for discrete sample dispensing/injection in cross microchannel chips introduces a dynamic loading step between the steady-state loading and dispensing steps. Injected samples were shown to be more concentrated and more uniform in the cross-stream direction than traditional pinched-valve injections. Short dynamic loading times were shown to increase the peak height of short discrete samples. Long dynamic loading times resulted in samples exhibiting a constant concentration plateau at the original sample concentration, extending over the bulk of the sample. With a single cross microchannel, injections of well-defined samples with lengths varying from two channel widths (100 um), to twenty channel widths (millimeter sized) can be realized. In applications such as high-speed capillary zone electrophoresis, this injection technique may provide a desirable increase in signal with tolerable increase in sample length. This technique may also be preferred in many other applications where large samples are required.

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8-4


EFFECTS OF SPATIAL GRADIENTS OF ELECTRICAL CONDUCTIVITY

The dispensing processes discussed in the previous sections all use the same solution (the same chemical composition and concentration) as the driving buffer and the sample-carrying buffer, did not consider any spatial gradient of electric conductivity in the liquid, and assumed negligible effects of samples on the bulk conductivity. In many microfluidic systems, the gradient of electric conductivity exists in the liquid. For example, when a sample solution containing different sample species is required to be transported through microchannels without sample separation [21], a lower flow velocity in the sample zone is desirable during the loading process so that the electromigration and the separation can be minimized by the slow motion. This can be achieved by using a high conductivity buffer solution as the sample-carrying buffer (to have a lower electroosmotic velocity) and a low conductivity buffer solution as the driving buffer (to have a higher electroosmotic velocity). In such a case, the spatial gradient of electric conductivity exists. In other applications where a sample is required to be stacked in the system [22-25], the use of lower conductivity buffer solution in the sample region and higher conductivity buffer in the driving liquid region is a good choice to meet this need. This is because a higher velocity will be generated in the sample region and thus the sample can be stacked by the lower velocity buffer at the front of sample region. The objective of this section is to show the effects of spatial gradients of the electrical conductivity on the onchip microfluidic dispensing processes [13].

Figure 8.26. The schematic diagram of the crossing-microchannel.


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A crossing microchannel dispenser is shown in Figure 8.26. The depth and the width of all the channels are 20 um and 50 um, respectively. There are four reservoirs connected to the four ends of microchannels, in which the electrodes are inserted to set up the electrical field across the channels. The sample reservoir is Rl, where a sample solution (the sample-carrying buffer) is initially loaded. The sample waste reservoir is R3, and the buffer focusing reservoirs are R2 and R4. Reservoirs 2 and 4 and the channels are filled with the driving buffer solution initially. The lengths of the channels from intersection to reservoir are 2.5 mm, 10 mm, 2.5 mm, and 10 mm for channels 1, 2, 3, and 4 respectively. The sample-carrying buffer and the driving buffer have the same chemical composition, but different ionic concentrations. When a set of chosen electrical potentials is applied to the reservoirs of the dispenser shown in Figure 8.26, the sample solution (the buffer solution carrying the sample) in reservoir 1 will be flow toward reservoir 3 passing through the intersection of microchannels and the driving buffer solutions in channel 2 and 4 are also driven to flow into the waste channel. Consequently, the sample solution is loaded into the intersection. Once the loading process reaches the steady state, the dispensing step is initiated by adjusting the applied voltages and the loaded sample in the intersection will be dispensed downstream towards reservoir 4. In order to consider the conductivity difference, the mathematical description of the loading and dispensing processes is different from that discussed in Section 8-1. A set of 2D governing equations describing the potential field, the flow field and the concentration field during the injection process will be introduced below. Electric potential field When an electric field is applied along the microchannel, the current is setup along the channel and the local electric current vector is given by: (21)

where / is local electric current density vector, u is the local velocity vector and 0 is the local electrical potential. The jth ionic species has a valence zj, a diffusion coefficient Dj and an ionic number concentration nj. Here, e, kj, and T represent the fundamental elementary charge, Boltzmann constant and the system absolute temperature, respectively. For large KO, channel, where the electric double layer is very thin, electro-neutrality is assumed to dominate in the channel and the first term of Eq. (21) defining the current transport due to

512


convection can be neglected. Here a is the hydraulic radius and K is the Debye double layer thickness. Usually, one can neglect the second term of Eq.(21) which defines the current flow due to diffusion and is small as compared with the third term. Consequently, for large K a flows, one can write the electric current in terms of molar concentration as (22)

where Na is Avogadro number and the molar concentration is given by Cj = rij jNa . For a given electrolyte solution, Eq. (22) can be rewritten as (23) where X is the electric conductivity of electrolyte solution and takes the form of (24) Eq.(24) is an expression of Ohm's law for electrically neutral dilute solutions or solutions in a microchannel having large Ka, where C,- can be determined by a set of concentration equations. The charge conservation in the liquid has to be satisfied, which is described by: (25) Substituting Eq.(23) into Eq.(25), the equation of the electrical potential field is obtained as: (26) With the given concentration field and proper boundary conditions, Eqs. (24) and (26) can provide the distribution of the applied electric field, <j), in the crossing microchannel. It should be pointed out that if there is no spatial conductivity gradient, X is a constant, and hence Eq.(26) is reduced to: (26a)


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Flow filed The flow field is described by the modified Navier-Stokes equation and the continuity equation as follows: (27)

(28) where V stands for the mass average velocity vector, p is the density of liquid, P is the pressure, JJ. is the viscosity of liquid, and pe is the net charge density in the solution. For high ionic concentration solutions commonly used in on-chip microfluidic dispensing processes, the electric double layer usually is very thin, i.e., less than 0.2% of channel width. The thin electric double layer results in a plug-like electroosmotic velocity profile cross the channel except in the double layer region where velocity dramatically decreases to zero at the wall. For simplicity, we may treat the electroosmotic velocity as the boundary slip velocity and use it as the boundary condition for the momentum equation, Eq.(27). Mathematically, this slip boundary condition removes the last term in Eq. (27) and includes its effects in the boundary condition: (29) where neo is the electroosmotic mobility of the buffer solution. Concentration field We consider that the spatial gradients of buffer concentration exist and the gradients are time dependent, which have significant effects on the electrical potential field and the flow field during the dispensing process. Therefore, the buffer concentration field must be modeled simultaneously at each time step. The concentration field is governed by the mass conservation and can be described by: (30)

(31)

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where Vepj is the electrophoretic velocity of the ith species, given by Vepi = Vepi^'t'' where fxepi is the electrophoretic mobility of this species. Please note that Eq.(30) and Eq. (31) describe the concentration fields of the sample and the buffer, respectively. Numerical scheme The complete set of equations, Eqs. (24), (26), (27), (28), (30) and (31), were normalized and numerically solved using the semi-implicit method for pressure-linked equation (SIMPLE) algorithm developed by Patankar [16]. The algorithm is based on a finite control volume discretization of the governing equations on a staggered grid. In order to capture all the detailed features near the four corners of the intersection shown in Figure 8.26, the non-uniform grid system is employed. The control volume size next to the wall is minimum. The size of successive control volumes away from the walls is increased by a factor of 1.2. In this implementation, the solution to this set of equations is obtained by an iterative procedure. During each iteration procedure, the discretized equations are solved by a line-by-line iteration method. Due to the unsteady loading and dispensing processes, the numerical calculation of the potential field, flow field and concentration field must be converged at each iteration step and each time step. The conductivity gradient effects analyzed by the model simulation In the calculations, Rhodamine 110 standard is used as a sample with an electrophoretic mobility - 2 . 0 x 1 0 " m /(V-s)

and a diffusion

coefficient

^buffer = 1 -0 x 10~ m / s , respectively, which are close to the measured value . The driving buffer and the sample-carrying buffer have the same chemical composites, sodium carbonate, the same pH value (pH = 9.0), the same diffusion 22

coefficient, Dûffer =1.0 x 10

m /s, but the different ionic concentrations and

hence different electrical conductivities. Two scenarios are studied here to investigate the effects of the conductivity gradient on the transport phenomena of the sample injection processes. In the first scenario, the driving buffer has a concentration of 10 mM and an electroosmotic mobility of 5.5 x 10~ m f(V • s). The concentration of the sample-carrying buffer is chosen to be 50 mM (5 times that of the driving buffer) and its electroosmotic mobility is assumed to be 5.0x10" m /(V-s). In this case, the driving buffer has a lower conductivity than that of the sample-carrying


515

buffer and the relationship of the conductivity between these two solutions is approximated as {Xsampie buffer)1(^driving buffer) = 5 - I n t h e second scenario, the driving buffer has a concentration of 50 mM and the electroosmotic mobility is assumed to be 5.0x10" m /(V-s). The sample-carrying buffer has a concentration of 5 mM (1/10 times of the driving buffer) and the electroosmotic mobility is assumed to be 5.56x10" m /{V • s). The relationship of the conductivity between these two solutions is approximated as (^sample buffer ) ^driving

buffer ) = 1/10 .

Figure 8.27. The effects of the conductivity difference on the applied electrical field in the dispenser. The iso-potential contours in the intersection in the steady-state loading step (left) and the corresponding dispensing step (right) for: (a) (^sampie buffer^1^driving buffer^5' <W ^sample

buffer^1 ^driving

buffer^1-

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If the spatial gradient of the electrical conductivity in the liquid is not considered, the electric potential distribution is not affected by the conductivity (mathematically, the conductivity is not involved in the applied electrical potential equation, such as Eq.(26a)) and only need to be solved once prior to solving momentum and concentration equations. This simplicity, however, is not valid in this case where the spatial gradient of the conductivity exists. The equation of the electric potential, Eq.(26), must be solved at each time step when the concentration distribution changes and in turn the conductivity distribution changes. Figure 8.27(a) shows the normalized electric potential distribution of the first scenario {{lsampie buffer) ^driving buffer) = 5 ) i n t h e loading and the dispensing step and Figure 8.27(b), however, shows the normalized electric potential distribution without the consideration of spatial gradient of conductivity (i.e., (^sampie buffer) Wdriving buffer) = 1) i n t h e loading and dispensing steps. In all the plots, the chip was oriented with channels from Rl, R2, R3, and R4, counterclockwise from left and the same configuration is used for all the other plots in this paper, except stated otherwise. The normalized potentials applied to the reservoirs 1 - 4 are the same for all the plots in Figure 2, sy -1.0, £j = 1.0, £3 = 0.0 and £4 =1.0 in the loading step and £j = 0.2, £2 = 2.0, £3 = 0.2 and £4 = 0.0 at the dispensing step, respectively, where e indicates the normalized applied voltages and subscripts correspond to reservoir numbers. The lines shown in these plots are equal potential lines. The potential drop between two lines is the same through all the plots and the electrical field strength can be evaluated by the ratio of the potential drop to the distance between the lines. The variation of the electric field strength is clearly demonstrated between the situations with and without the conductivity gradient. During the loading process, the sample solution is pumped into channel 1 from the reservoir 1. Figures 8.28(a) and (b) show the concentration distribution of the sample near the intersection for the two scenarios, ^sample buffer)^driving

buffer) = 5and {^sample buffer)/(^driving buffer) = 1/10,

respectively. In all the plots, the concentration increases from black to white and the iso-concentration lines are 10%, 30%, 50%, 70% and 90% of the original sample concentration. The sequence of plots shows the unsteady loading of the sample at different time. One can see that the conductivity gradient has significant effects on the unsteadying loading process. The time to reach the steady state is different for the two scenarios, tstea(jy=\2s for the case of (^sample buffer)^driving buffer) =5 a n d * steady =6s f o r t h e c a s e o f (^sample buffer)^driving buffer) =ino- T h a t indicates that the average bulk velocity is affected significantly by the spatial gradient of conductivity. In addition, the concentration contour of the sample is distorted backward to the

517


Figure 8.28. The effects of the conductivity difference on the loading process: the sequences of the sample concentration profile of the unsteady loading process for: (a) (^-sample buffer)'(^driving

buffer)^5

and

0>)

(^-sample buffer)^driving

buffer)=Vl(> •

The

concentration decreases from white to black and the iso-contours are plotted at 0.1, 0.3, 0.5, 0.7 and 0.9 of the original sample concentration

518


sample reservoir (Rl) in the case of (lsamph buffer) ^driving buffer) = 1/10, as shown in the plot of t = 5s in Figure 8.28(b). This may be understood as follows. Because the conductivity of the sample region (the sample-carrying buffer and the sample) is much lower than that in the driving buffer region (-1/10), the voltage drop or the electric field strength in this region is bigger than that in the downstream of waste channel (from the intersection to right side reservoir 3). Additionally, the sample region has a lower ionic concentration and hence a higher electroosmotic mobility. The combination of the high electrical field

Figure 8.29. The sequences of the sample concentration profile of an unsteady loading process at Is intervals at the horizontal centerline (left) and the vertical centerline (right) for: (a) ^sample buffer)1 ^driving buffer) = 5 »a n d ( b ) ^sample buffer) W driving buffer)=l/i0 • The shown in the plots indicates the direction of time increase.

arrow

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Figure 8.30. The effects of the conductivity difference on the loading and the dispensing processes under the same applied voltages. The loaded sample at steady-state (la-3a) and the corresponding dispensed sample at t = 1.25s (lb-3b) for: (1) ^sample buffer) ^driving 3

l

( ) ^sample buffer)'( driving

buffer)

= 5

> @) (^sample buffer) ^driving

buffer) = • •

buffer)=l/l°

.

an

d

520


strength and the high electroosmotic mobility yields a much higher local electroosmotic velocity than that in the downstream. This intends to generate a higher flow rate in the upstream (sample-carrying buffer region) than the flow rate in the downstream (driving buffer region). For an incompressible liquid, the continuity condition requires the same flow rate throughout the microchannel. In order to achieve the same flow rate in both the upstream and the downstream, a negative pressure gradient (the pressure decreases in the flow direction) is induced in the downstream to increase the local flow rate, and a positive pressure gradient (the pressure increase in the flow direction) is induced in the sample region to decrease the local flow rate. This induced positive pressure gradient (the pressure increase in the flow direction) in the sample-carrying buffer region will add a backward pressure driven flow to the electroosmotic flow (plug-like velocity profile), resulting in the distorted backward velocity profile. Consequently, the sample concentration profile is distorted backward to reservoir 1 as well, because the sample concentration profile is dependent on the convection transport (bulk velocity). One can also find that for the second scenario ((^sample buffer) ^driving buffer) =l/l())> t n e sample is loaded slowly before t = 4s (i.e. there are no sample present at the intersection before this moment); however, the sample is suddenly pumped into the intersection when t = 5s and reaches the steady state at t = 6s. This is because the sample-carrying buffer has a lower conductivity than that of the driving buffer in this case. Initially, the entire channel is filled with the driving buffer solution (high conductivity and low resistance), which results in a low average velocity due to its low electroosmotic mobility. At the earlier stage of the loading process, when a small section of the sample channel (channel 1) is filled with the sample-carrying buffer (low conductivity and high resistance), the major potential drop still occurs in the region filled with the driving buffer and the average velocity is still low due to the fact that the liquid motion in most part of the microchannel is slow. However, when more and more of the sample-carrying buffer are pumped into the sample channel, the major potential drop occurs in the region filled with the sample-carrying buffer (low conductivity and high resistance), resulting in a high local electric field strength in this region. The combination of a high electric field strength and a high electroosmotic mobility (low ionic concentration) gives rise to a much higher cross-sectional average velocity than that at the earlier time. That is the reason that the sample is pumped slowly at the first 4 seconds and is quickly pumped into the intersection after that moment. It should be noted that the moment after which the sample is pumped much faster than before depends on the conductivity gradient. When the conductivity gradient changes, this time period changes.


521

In order to gain the insight into the transport phenomena behind the unsteady loading process, the sample concentration distribution at the centerline of the horizontal channel (from Rl to R3) and the vertical channel (from R2 to R4) are plotted in Figure 8.29 for both scenarios. The time between each line is 1 s and the direction of the time increase is indicated in the figure. Figure 8.29(a) shows the sample concentration at the centerline of horizontal channel (left) and vertical channel (right), for the scenario of (lsampie buffer) ^driving buffer) = 5> and Figure 8.29(b) shows that for the scenario of (^sample buffer) ^driving buffer) =1/10• F r o m Figures 8.29(a) and (b), one can see that the sample is gradually pumped into channel 1 ( 0 < x < 5 0 ) until a steady state is reached. A concentration drop is found at the inlet of the waste channel ( 5 1 < J C < 1 0 1 ) , which is due to the flow of the driving buffer solutions in channel 2 and 4 into the waster channel and the focus of the sample at the intersection. Also, the sample concentration at the vertical centerline is symmetric to the center point, which is because the same voltages are applied to R2 and R4 and channel 2 and 4 have the same length. It is also clearly shown that the sample is focused by the driving buffers at the intersection of microchannels (200 < j < 2 0 1 ) . Note that the extent to which the sample is focused depends on the combination of the four applied voltages and can be adjusted. A significant increase of the sample concentration with time is also found in the case of {Xsampie buffer)'(^driving buffer) = 1/10, as shown in the right plot of Figure 8.29(b). This is due to the speed of pumping the sample is much faster than earlier after a specific moment (i.e. t = 4s). This can be understood in the same way as explained previously for the phenomena that the pumping before t = 4s is very slow and the sample is pumped much faster after that moment. As discussed above, the loading processes in the presence of the spatial conductivity gradient is different from that without the spatial conductivity gradient. It can be expected that the corresponding dispensing processes under the same applied voltages will be different. Figure 8.30 shows the comparison of the loaded sample at the steady state (top row) and the dispensed sample at t = 1.255 (bottom row) between the two cases with the conductivity difference and the case without the conductivity difference. For all the three cases, the normalized applied potentials s\ ~s^ are 1.0, 1.0, 0.0, 1.0 in the loading step and 0.2, 2.0, 0.2, 0.0 in the dispensing step. In all the plots, the white region shows the sample solution with a concentration higher than 80% of the original sample concentration. Figures 8.30(la) and (lb) show the loaded sample and the dispensed sample for the case of (hsample buffer) ^driving buffer)l = 5> Figures 8.30(2a) and (2b) show that for the case of

522


{^sample buffer) ^driving buffer) = 1 / 1 ° a n d Figures 8.30(3a) and (3b) show that for the case of (Xsampie buffer) ^driving buffer) = 1 • The sensitivity of the loaded sample size (Figures 8.30(1 a)—(3a)) and the dispensed sample size (Figures 8.30(1 b)—(3b)) to the spatial conductivity gradient is clearly demonstrated. There is essentially no sample present in the dispensing channel shortly after the dispensing for the situations with the conductivity gradient. This is because the loaded sample size in the intersection is small (Figures 8.30(1 a) and (2a)) and

Figure 8.31. The variation of the loaded sample's size and shape at the intersection for different conductivity difference and different applied voltages. The loaded sample at steadystate with (Xsampie buffer)^driving buffer)=i under the applied voltages, s1 ~ e 4 , of: (a) 1.0, 1.0, 0.0, 1.0, (b) 1.0, 0.2, 0.0, 0.2. The loaded sample at steady-state with ^sample buffer^'^driving buffed = 1 / 1 0 u n d e r t h e applied voltages, el ~ £ 4 , of: (c) 1.0, 1.0, 0.0, 1.0, (d) 1.0, 0.2,0.0,0.2.


523

the major part of the sample in the intersection is driven to the reservoir 1 and 3 when the dispensing is initiated (see Figures 8.30 (lb) and (2b)) due to the small potentials applied to reservoir 1 and 3 in the dispensing step (i.e. (ej = £3 = 0.2 compared to £2 = 2.0). As seen from Figures 8.30 (1) and (2), with a strongly focused sample in the loading step, one cannot dispense or inject any sufficiently large sample in the dispensing channel. In order to obtain a sufficiently large dispensed sample, the size of the loaded sample at the intersection must be large at the end of the loading process. The applied voltages were found to be the most important parameter controlling the size of the loaded sample and the dispensed sample. Figures 8.31 (a) - (b) show the effects of the applied voltages on the steady-state loaded sample for the case of {Xsampie buffer)^driving buffer) =5- Figures 8.31(c)-(d) is for the case of dsample buffed ^driving buffer>XI™• T h e normalized applied voltages, s\ - £ 4 , for Figures 8.31 (a) and (c) are 1.0, 1.0, 0.0, 1.0, and for Figures 8.3 l(b) and (d) are 1.0, 0.2, 0.0, 0.2, respectively. When the same normalized voltages applied to the reservoirs 2 and 4, £2 = £4, are decreased from 1.0 (Figures 8.31 (a) and (c)) to 0.2 (Figures 8.3 l(b) and (d)), the loaded sample increases significantly for both scenarios because of the less focusing effects from channel 2 and 4 on the loaded sample in the intersection. Figure 8.32 shows the dispensed sample at different time for the case of (^sample buffer) ^driving buffer) = 5 b a s e d o n t h e different sizes of the loaded sample. The normalized applied voltages in the dispensing process are 1.1, 2.0, 1.1, 0.0 for s\ ~ £4 in all the plots. Figure 8.32(a) shows a loaded sample, and the corresponding dispensed sample at t = 0.9s and t = 1.8s, respectively. Figure 8.32(b) shows a bigger loaded sample and the corresponding dispensed sample at t = 0.9s and 7 = 1.8s, respectively. One can see that, the bigger the loaded sample, the bigger the dispensed sample at the same dispensing time and under the same applied voltages. Indeed, the loaded sample size and hence the applied voltages in the loading step are critical in determining the dispensed sample size. After the loading process reaches a steady state, the dispensing process can be initiated by changing the applied voltages to the four reservoirs. During the dispensing process, the loaded sample in the intersection is expected to be driven or dispensed into channel 4 and be surrounded by the buffer solutions from both the upstream and the downstream, providing a cut of sample for the subsequent chemical and biomedical analysis. This requires that the main flow direction in the dispensing process should be from R2 to R4 (perpendicular to the loading flow direction from Rl to R3), and the flow in channel 1 and 3 could be either from Rl (or R3) to R4 helping the major flow from R2 to R4 to provide a big sample, or from R2 to Rl (or R3) splitting the major flow from R2 to R4 to

524


Figure 8.32. The dependence of the dispensed sample size on the size of the loaded sample (or the loading voltages). The loaded sample at steady-state and the corresponding dispensed sample with the normalized applied voltages, E\ - £ 4 , of: (a) 1.0, 0.5, 0.0, 0.5 in the loading step and 1.1, 2.0, 1.1, 0.0 in the dispensing step, and (b) 1.0, 0.2, 0.0, 0.2 in the loading step and 1.1, 2.0, 1.1, 0.0 in the dispensing step.


525

provide a clear cut of sample in the downstream of channel 4. In order to perform this process, normally the voltage applied to reservoir 2 must be bigger than that applied to reservoir 1 and 3. Otherwise, there will exist a flow from Rl to R2 (R3 to R2 as well) besides the flow from Rl to R4 (R3 to R4 as well since R4 is grounded). This will split the loaded sample in the intersection to R2 and R4 and can not provide a cut of sample (for the purpose of the dispensing) because the sample is continuously driven to R2 and R4 from the sample reservoir (Rl). However, as we discussed earlier, when the voltages applied to reservoir 1 and 3 are much smaller than that to reservoir 2 during the dispensing, there will be too much loaded sample flowing to reservoirs 1 and 3. This will results in a small dispensed sample size. In order to verify this, the same loaded sample as shown in Figure 8.32(b) is dispensed using a new set of applied voltages s\ = 0.2,£ 2 = 2.0,£3 = 0.2,£4 = 0.0, where the applied voltages to reservoir 2 and 4 are the same as that used in Figure 8.32(b), however, the applied voltages to reservoir 1 and 3 are much smaller (i.e. s\ = £3 = 0.2) than that used in Figure 8.32(b) (i.e. ^ = £ 3 =1.1). We found that there is essentially no sample dispensed in the dispensing channel at t = 0.95 when using this new set of applied voltages {s\ =£3 =0.2) in the dispensing step. Since no sample is present in the downstream of channel 4, the simulation results are not shown here. This indicates that when the applied voltages to reservoirl and 3 are too low in the dispensing process, a significant amount of sample will be driven to reservoir 1 and 3 when the dispensing process is initiated. The left small amount of the dispensed sample easily diffuses into the buffer solution in channel 4, consequently, no sample with a sufficiently high concentration appears. Therefore, in addition to the loaded sample size, the combination of the applied voltage in the dispensing process is another very important controlling parameter for the dispensed sample size. From the above discussions, it is clear that the presence of the conductivity gradient has a strong effect on the sample loading and dispensing. The applied voltages can control the shape and the size of the loaded sample and hence the shape and the size of the dispensed sample. For a given sample-buffer system, there exists an optimal set of the controlling voltages in order to obtain a desired dispensed sample size. The model presented here is able to find such a set of the optimal voltages through the numerical simulation.

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8-5


CONTROLLED ON-CHIP SAMPLE INJECTION, PUMPING, STACKING WITH LIQUID CONDUCTIVITY DIFFERENCES

The net velocity of an individual ionic species in an electroosmotic flow is a combination of the bulk fluid velocity and the specific electrophoretic velocity of the species. An initially discrete multi-component sample can therefore be separated into bands based on the charge-to-mass ratios of the individual species. The rate of separation of the sample can be changed using the electrical conductivity difference between the sample and the running buffer. Although the sample and the choice of buffer are typically determined by the application, the relative buffer concentrations can often be varied (to some extent) to enhance or inhibit separation. An understanding of these systems is also required in less ideal cases where extremely high or extremely low sample concentrations are necessitated by the application. Conductivity differences alter the shape of the electric field, which is the driving force behind both the electroosmotic bulk flow and the electrophoretic velocity of individual species. In most Lab-on-a-chip applications the channel length greatly exceeds the sample length, and thus the sample velocity is dictated primarily by the electroosmotic flowrate in the running buffer (regardless of the electroosmotic velocity developed at the sample/wall interface). Thus the bulk fluid velocity developed by electroosmosis is constrained by the conservation of mass requirement. The electrophoretic velocity of a charged species in the sample, however, is not constrained, and responds directly to the local electrical field gradient. The two cases of interest are illustrated in Figure 8.33. The pumping case, in which sample separation is reduced, is illustrated in Figure 8.33(a). In this case, the sample conductivity, ASj is higher than that of the running buffer, Ao and the electric field strength in the sample, Es, is lower than in the running buffer, Eo. The result is reduced electrophoretic velocities in the sample region, and hence a reduced rate of separation. Although the electroosmotic velocity of the sample, veo.s, is likewise reduced, pressure forces induced by the comparatively high running buffer electroosmotic velocity, veo.o, pull the sample along. The opposite is observed in the stacking case (field amplified stacking), illustrated in Figure 8.33(b). In the stacking case, separation in the sample is enhanced using a sample with a lower electrical conductivity than that of the running buffer. The increased electric field leads to high electrophoretic velocities in the sample region. The mean velocity of the sample, however, is constrained by the comparatively slow electroosmotic velocity in the running buffer. This combination results in rapid separation of species in a relatively slow moving sample. A key aspect of the stacking case is that it is possible to obtain peaks with concentrations of charged components higher than in the original sample solution. Increasing the peak


527

Figure 8.33. Schematic diagram of pumping (high conductivity sample transport) and stacking (low conductivity sample transport) cases. In the pumping case, the electric field strength, and electrophoretic velocities are reduced in the sample region. In the stacking case, the electric field strength and electrophoretic velocities are increased in the sample region.

528


height allows for more accurate detection, and in some cases extends the analytical capabilities of microfluidic chips. To achieve significant sample concentration for either pumping or stacking, many studies have employed larger samples [26-30], in contrast to focused samples employed for direct separation by capillary zone electrophoresis [5,6]. Stacking on chips was first demonstrated by Jacobson and Ramsey [26] using a gated injection scheme. Due to the circulation generated (see Figure 8.33a), they suggested that a compromise between stacking enhancement (due to either increased conductivity difference or sample size) and separation efficiency must be reached. Haab and Mathies [28] and Vazquez et al. [29] employed offset twin-T chip configurations where the sample and buffer run adjacently in the dispensing channel for a short distance. Unfortunately, the running of buffer and sample together in this way tends to dilute the sample. To avoid this, Lichtenberg et al. [30] developed a new injector (essentially an offset twincross). Although more complicated, the new configuration was shown to successfully resolve the adverse effects of sample pinching in the offset twin-T. Maintaining sample purity is of particular importance because the degree of stacking or pumping achievable is critically dependent on the difference in conductivity of the injected sample relative to that of the running buffer. Any dilution of the sample during injection results in a reduction in the conductivity difference and a decrease in performance [30]. A new three-step technique [12] for discrete sample injection in straightcross microfluidic chips was discussed in Section 8-3. The technique has an intermediate dynamic loading step in which sample is pumped directly into the intersection and three connecting channels. A key feature of this technique, especially in the context of pumping and stacking, is the ability to inject welldefined samples of high concentration density. Another key feature of this loading technique is that samples of any axial length can be injected, and in contrast to offset twin methods [28-30] the sample size is not coupled to the chip geometry. The dynamic loading method has been demonstrated in Section 8-3 using only conductivity-matched liquids [12]. In this section, we will present applications of this dynamical loading method in on-chip injection and electrokinetic transport of samples differing in conductivity from the running buffer by fluorescence-based visualization [14]. A sample containing both a neutral dye (rhodamine-B) and negatively charged dye (fluorescein) is employed. While the charged dye responds to the field gradients induced by the conductivity difference between the sample and running buffer, the neutral dye simply tracks the location of the original sample buffer. Both electroosmotic pumping and stacking cases are studied. The coupled flow phenomena inherent in these situations are investigated and discussed in the context of microfluidic chip applications.


529

The glass chips used here were manufactured by Micralyne (Edmonton, Canada). The crossing microchannels are 20 |am deep (at the center) and 50 urn wide (across the top), and the reservoirs are 2 mm in diameter. The D-shaped channel cross-section is a product of the manufacturing technique. The sample was loaded into reservoir Rl, the sample waste reservoir was R3, and the buffer focusing reservoirs were R2 and R4. The lengths of the channels from the intersection to the reservoir are 5 mm, 4 mm, 80 mm, and 4 mm for channels 1, 2, 3, and 4 respectively. The chip was clamped to a precision 3-axis stage. The two horizontal positioning axes were used to position the field of view, and the vertical axis to focus the microscope. The chips were prepared by rinsing with each of the following solutions for 20 minutes (in sequence): 1M filtered Nitric acid; 1M filtered Sodium hydroxide (NaOH); and filtered buffer. To pull these solutions through each channel, a vacuum was applied to R3 using a 30 mL syringe. A tapered 1000 ul pipette tip was cut to the appropriate diameter and friction fitted to join the syringe to the chip. The fluorescent dyes employed here were rhodamine-B (478.68 MW) as supplied by Fisher, and fluorescein (332.31 MW) as supplied by Molecular Probes. The dyes were dissolved in sodium carbonate buffer of pH = 9, also used as the running buffer. Immediately before use, all solutions were filtered using 0.2 ^m pore size syringe filters. Rhodamine-B at pH = 9 was found to be neutral. Fluorescein, however, carries a valence charge of z = -2 at pH = 9.0, with an electrophoretic mobility of vph F = -3.3x10"8 mV^s" 1 . The excitation source (488 nm) is well suited to excite fluorescein with an adsorption maximum at 490 nm, and less suited to excite rhodamine-B with an adsorption maxima at 570 nm. The result is a consistent, albeit less efficient, excitation of the rhodamine. The dye concentration loadings were adjusted to compensate for that effect somewhat, although the peak height of the fluorescein was intentionally made higher than that of the rhodamine. The dye concentrations, sample and buffer ionic strengths, employed in each run are summarized in Table 1. The electrical conductivities of the 10, 25, and 50 [mM] ionic strength buffer solutions were estimated to be A = 0.04, 0.10 and 0.20 [S/m]. Liquid temperatures in the chip were maintained at the ambient temperature (25 °C). A fluorescent epi-illumination video microscope was employed to visualize the process. The dye was excited by a continuous flood of blue light provided by a single-line, 200 mW, 488 nm argon laser (American Laser Corp.), through a 32x, NA = 0.3 microscope objective (Leitz). The received signal was split by the dichroic mirror (510 nm LP/ short-reflecting) and passed through an additional filter (515 nm LP) and a camera mount with 0.63x magnification before reaching the camera. The three-step process requires independent control of three potential levels for each of the four electrodes (12 settings). Here, a high voltage power

530


supply (Glassman PS/EH20R02.0) was used in conjunction with a custom-made voltage controller. The timing of each switch was controlled via a digital delay generator (Stanford Research Systems). The voltage controller was programmed for default outputs corresponding to the steady-state loading step. Rising edge triggers from the delay generator commenced dynamic loading and dispensing steps. Switch frequencies up to 200 Hz were verified, indicating a transition delay of under 2.5 ms. Since the transition delay after each trigger is expected to be consistent, the critical dynamic loading period, tDL, between triggers can be specified with considerably less uncertainty (estimated here < 1 ms).

Figure 8.34. Iso-intensity coutour plots during the dispensing of a two-analyte sample with: (a) a relatively high conductivity sample (pumping case, y = 0.2), (b) a sample with conductivity matching the running buffer (y = 1), and (c) a relatively low conductivity sample (stacking case, y = 5). No dynamic loading was employed. Contours are plotted at 5 even intervals.


531

Images were captured and saved on the computer at a rate of 15Hz. A progressive scan CCD camera (Pulnix, TM-9701) was used to avoid image defects due to field-field interlacing. The acquired images had a resolution of 640x484 pixels and an 8-bit dynamic range. This pixel matrix corresponded to a viewed region of 550x416 ^m. The camera orientation was carefully adjusted before each run such that the pixel grid was aligned with the coordinate directions of the intersection. Digital image processing was performed to remove any non-uniformity present in the imaging system and to relate the pixel intensity values to dye concentration. A background noise signal was subtracted, and brightfield image normalization was performed for each image.

Figure 8.35. Iso-intensity contour plots of the injection of a two-analyte sample with conductivity matched to that of the running buffer (y = 1): (a) the steady-state loading step (t = 0 s), (b) the dispensing step (t = 4/15 s), and (c) later in the dispensing step (t = 8/15 s). No dynamic loading was employed. Contours are plotted at 10 even intervals.

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These images were then smoothed with a distance-based kernel, and scaled by a single factor such that the image series filled the grayscale range. The period over which the dynamic loading voltages are applied is termed the dynamic loading period, tDL. For a given solution/chip combination and electrode potential settings, the dynamic loading period determines the axial length of the injected sample. The degree of stacking or pumping is dependent on both the initial size of the sample, and the conductivity of the running buffer relative to that of the sample buffer, y. The dye concentrations, sample and running buffer ionic strengths, and normalized electrode potentials employed in each case are summarized in Table 1. The effect of varying the conductivity difference (keeping initial sample size approximately constant) is shown in Figure 8.34. Iso-intensity profiles obtained during the dispensing of a two-dye mixture are shown for each of the three cases: pumping, uniform conductivity, and stacking (7 = 0.2, 7 = 1 , and 7 = 5 respectively). No dynamic loading was used, and the injected samples were of similar initial size. The two dyes in the relatively high conductivity sample (7 = 0.2, Figure 8.34a) do not separate, and in that sense, the sample was effectively 'pumped'. However, the sample is severely distorted by mismatch in electroosmotic velocities (see Figure 8.34a), and non-uniformity in the crossstream direction due to the nature of the injection. The effects of both these factors can be mitigated using larger initial sample lengths, as will be shown later. In contrast to the pumping case, the two dyes separate readily in the uniform conductivity case ( 7 = 1 , Figure 8.34b). The two dyes are even more rapidly separated in the relatively low conductivity sample (7= 5, Figure 8.34c). The fluorescein band in this case is thinner than that in the uniform conductivity case due to field amplified sample stacking. Figure 8.34 highlights the very significant role conductivity differences play in electrokinetic sample transport. Although the effectiveness of sample transport (pumping case) and sample separation (stacking case) are shown to improve somewhat over that of the uniform conductivity case, the injections shown in Figure 8.34 are far from ideal. The underlying phenomena and the effect of initial sample length for each case (uniform conductivity, y = 1, sample stacking, y = 5, and sample pumping, y = 0.2) will now be discussed in turn. When the conductivity of the sample matches that of the running buffer (y = 1) neither sample stacking or pumping occurs. In such cases, individual analytes in the sample move independent of the original sample-buffer solution at a velocity equal to the summation of the bulk electroosmotic velocity and their specific electrophoretic velocity. Upon dispensing, neutral rhodamine and negatively charged fluorescein are separated readily as shown in the sequence of iso-intensity contour plots in Figure 8.35. Regarding the size of the analyte bands, the fluorescein band remains in the intersection much longer than the


Table 1 Run parameters for each case studied. maximum applied voltage.

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All potentials listed were normalized with the

Uniform Conductivity

Pumping Case

Stacking Case

Figures

4b, 5,6

4a, 7,8

4c, 9, 10

Relative Conductivity, y

1

0.2

5

Rhodamine-B Concentration [uM]

100

100

200

Fluorescein Concentration [uM]

25

25

50

Running Buffer Ionic Strength [mM]

25

10

50

Sample Buffer Ionic Strength [mM]

25

50

10

Maximum Voltage Applied, Vmax [V]

1415

1415

1415

Normalized Reservoir Potentials: Loading Step 1

1.000

1.000

1.000

2

1.000 0.911

1.000

3

0.000 0.000 0.000

4

1.000 0.911

1.000

Dynamic Loading Step: 1

0.964 0.963 0.964

2

0.707 0.708 0.710

3

0.000 0.000 0.000

4

0.707 0.708 0.710

1

0.014 0.049 0.008

2

0.124 0.127 0.067

3

0.000 0.000 0.000

4

0.000 0.000 0.000

Dispensing Step:

534


rhodamine band. This positioning results in a loss of fluorescein into the sample and sample waste channels due to the application of pull-back (or equivalently 'cut-off) voltages. When a dynamic loading step was applied, the sample stream was electroosmotically pumped directly into the dispensing channel. The effect of dynamically loading the same sample mixture as employed previously (Figure 8.35), under the same conditions (y = 1) is shown in Figure 8.36. Images with samples at similar locations were chosen from the dispensing step of each run and processed identically. Dynamic loading is shown to increase the peak height of both analytes. As dynamic loading time was increased, however, there was little increase in the peak height of the rhodamine band as it approached a plateau concentration value equal to that of the original sample stream. The dynamic loading step, where neutral ions travel much faster in the dispensing channel than anions, amplifies the injection bias. This bias was evidenced by the larger

Figure 8.36. Centreline axial intensity profiles obtained during the dispensing of a twoanalyte sample with conductivity matched to that of the running buffer (y = 1). Profiles for dynamic loading times of t DL = 0 s, 0.1 s, and 0.2 s are superimposed with corresponding isointensity contour plots inset.


535

increase in rhodamine bandwidth (compared to that of the fluorescein) accompanying a dynamic loading period of just 0.1 seconds. In this uniform conductivity system, the benefit of increased peak height obtained with dynamic loading is at least partially offset by the cost of delayed separation due to increased band length. To facilitate the electroosmotic pumping case for reduced separation, a fluorescein/rhodamine sample mixture was prepared with buffer five times more conductive than the running buffer, y = 0.2 (run details in Table 1). The dynamic loading of the high conductivity sample is shown in Figure 8.37. In order to pull the high conductivity sample to the intersection, the focusing potentials in the steady state loading step were decreased from that of the previous run. In the

Figure 8.37. An image sequence of the three-step dynamic loading injection process of a two-analyte sample of higher conductivity than the running buffer (pumping case, y = 0.2). The dynamic loading time was t D L = 0.3 s.

536


dynamic loading step, electroosmotic flows of the running buffer pulled the comparatively slow sample stream into the dispensing channel. Upon application of the dispensing voltages, the sample takes on a curved shape in the dispensing step, characteristic of the pumping case (see Figure 8.33a). No separation was observed in the limited field of view shown in Figure 8.37c. This result is in contrast to the rapid separation observed in the y = 1 case (Figure 8.35). To investigate the affect of initial sample length in the electroosmotic pumping case, the microscope was repositioned 550 |j,m (11 channel widths) downstream of the intersection in the dispensing channel. The central pixel in the CCD array was chosen as a point fluorescence detector, and the signal

Figure 8.38. The signal recorded (at a location 550 (j.m downstream of the intersection) over time through a series of seven injections. The sample contains two-analytes and has a higher conductivity than the running buffer (pumping case, y = 0.2). The first injection employed no dynamic loading (t DL = 0 ) , the following injections employed increased dynamic loading times and the final injection was a repeat of the no dynamic loading case. Images of the dynamically loaded samples at the point of detection are shown as the inset.


537

recorded over time through a series of seven injections is shown in Figure 8.38. The first injection employed no dynamic loading (tDL-0), the following injections employed increased dynamic loading times and the final injection was a repeat of the no dynamic loading case. Note that the although the scale of the time axis is accurate, detection was paused after each dispensing step to allow the analytes to return to the intersection and fill into the sample waste reservoir. In the first and last case, which employed the shortest samples, the sample had begun to separate into two bands, the rhodamine peak arriving at the detector before the fluorescein peak. With a dynamic loading step of tDL= 0.1 s, the peak height more than doubled, although the sample is shown to be fronting in time (or 'trailing' in space). This fronting is primarily due to cross-stream diffusion in the wake of the convex sample shape, not to be confused with fronting due to mobility differences between the analyte and that of the running buffer co-ion. Due to diffusion, the characteristic shape is only barely noticeable in the inset images of the samples as they crossed the detection point in Figure 8.38. As the initial sample length was increased (by increasing tDL), the peak height increased greatly. The band length, however, was not similarly increased. Although the tDL= 0.8 s sample was the largest sample injected, it was the sample with the smallest band length at the detector, and thus the most effectively 'pumped'. This is because in multi-component, high conductivity samples, the separation rate is coupled to the concentration density of the sample buffer. Dilution occurs at the sample ends due to an internal circulation generated by the mismatch in electroosmotic mobilities between the sample and the running buffer (Figure 8.33a). The rate at which that dilution affects the average concentration of the sample is a function of sample length. Thus, larger samples experience a reduced rate of overall change than shorter samples under similar circumstances. By maintaining the conductivity difference, larger samples maintain lower internal electrical potential gradients and hence experience less broadening due to the separation of analytes. The similarity of the first and last (tDL =0) runs verifies that the increase in pumping effectiveness observed, evidenced in an 8-fold increase in peak height and a decrease sample length, was attributable to the dynamic loading of the high conductivity sample. To facilitate the electroosmotic stacking case for enhanced sample separation, a fluorescein/rhodamine sample mixture was prepared with buffer one-fifth as conductive as the running buffer, y = 5 (run details in Table 1). In Figure 8.39, iso-intensity contour plots and centerline intensity profiles are plotted at 4/15 s intervals during the dynamic loading and separation process. In the steady state loading step (Figure 8.39a), the sample is transported to the intersection by the relatively high electroosmotic velocity in the sample channel,

538


Figure 8.39. Plot sequence of the dynamic loading and dispensing of a two-analyte sample of lower conductivity than the running buffer (stacking case, y = 5). Iso-intensity contour plots at 4/15 s intervals are shown at left, and corresponding centerline axial intensity profiles are shown at right. The dynamic loading time was t DL = 0.2 s. Contours are plotted at 5 even intervals.


539

and experiences a relatively high electrical field strength. With application of the dynamic loading potentials, the sample pumps into the dispensing channel for tDL= 0.2 s (Figure 8.39b). The sample entering the dispensing channel is noticeably depleted in fluorescein. This depletion is due to the high electrical field strength and reduced bulk flow in the sample stream. Essentially, the sample-stacking mechanism is being applied to the whole sample channel, and fluorescein is being 'stacked' toward the sample reservoir. Also apparent is an M-shaped centerline intensity profile in Figure 8.39b at right. It is suspected that during the steady-state loading step, the fluorescein concentration was increased near the intersection as was found by Jacobson [5]. This local high concentrated portion was split during the dynamic loading step forming the M-shaped profile. With application of the dispensing potentials (Figure 8.39c), the fluorescein in the sample rapidly stacked to the rear (left) of the sample, creating a large peak in intensity. Because of the symmetry of dynamic loading, the stacked peak is formed to the left of the intersection (x < 0), partially insulated from the sample and sample waste channels. In contrast, the neutral rhodamine did not stack, marking the location of the low conductivity sample buffer. In addition, the front of the rhodamine band exhibited the concave shape characteristic of the

Figure 8.40. Centreline axial intensity profiles obtained during the dispensing of a twoanalyte sample of lower conductivity than the running buffer (stacking case, y = 5). Profiles for dynamic loading times of t D L = 0 s, 0.1 s, 0.2 s, and 0.3 s are superimposed.

540


stacking case (Figure 8.33b). As the fluorescein and the rhodamine further separate, some fluorescein was consumed by the sample and sample waste channels due to the pull-back voltages applied. Although the analytes were totally separated in Figure 8.39e, the two separate bands were distinguishable almost immediately after the application of the dispensing potentials (Figure 8.39c). This rapid separation is due to the high field strength developed in the sample. To investigate the role of dynamic loading in sample stacking, injections with varying dynamic loading times were performed. Images from each run (taken at a similar time period following the application of the dispensing potentials) were processed identically. The superimposed centerline intensity profiles are shown in Figure 8.40. Being neutral, rhodamine behaved here as in the uniform conductivity case, reaching a plateau concentration value equal to that of the original sample stream. The peak height and band length of the fluorescein band, however, are shown to increase significantly with dynamic loading. A dynamic loading period of tDL= 0.3 s resulted in a three-fold increase in fluorescein peak height over that produced through sample stacking without dynamic loading. Also, as the dynamic loading time is increased the sample length increases, moving the resulting location of charged analyte stacks further from the intersection. Imposing this distance has the benefit of insulating the analyte from the sample and sample waste channels, further increasing peak height. In summary, the electrical conductivity differences between sample and running buffer streams can greatly influence the transport of individual analytes in electrokinetically driven microfluidic systems. The two situations discussed in this section are sample pumping (where bulk transport is increased and separation of charged analytes is delayed using a relatively high conductivity sample), and sample stacking (where bulk transport is decreased and separation of charged analytes is expedited using a relatively low conductivity sample). It was shown that by employing the conductivity differences alone, the effectiveness of either sample transport or sample separation was improved over the uniform conductivity case. Then it was demonstrated that increasing the sample length, through dynamic loading, further increased the effectiveness of sample pumping, evidenced in an eight-fold increase in peak height as well as a decrease in total sample length at a downstream detector. Dynamic loading the in sample stacking case was shown to increase peak height (three-fold) in rapid separations. The success of these applications of dynamic loading was attributed to the ability to inject concentration dense samples of any length. Although these processes would benefit from a formal optimization, it was demonstrated that the dynamic loading technique used in conjunction with strategic conductivity differences significantly extends the capabilities of microfluidic chips.


541

REFERENCES [I] [2] [3] [4] [5] [6] [7] [8] [9] [10] [II] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

D. Harrison, A. Manz, Z. Fan, H. Ludi and H. Widmer, Anal. Chem., 64 (1992) 19261932. K. Seiler, D. Harrison and A. Manz, Anal. Chem., 65 (1993) 1481-1488. C. Effenhauser, A. Paulus, A. Manz and H. Widmer, Anal. Chem., 66 (1994) 2949-2953. A.J.P. Martin and F. Everaerts, Proc. Roy. Soc. London, 316 (1970) 493. S.C. Jacobson and J.M. Ramsey, Anal. Chem., 69 (1997) 3212-3217. S.C. Jacobson, C.T. Culbertson, J.E. Daler and J.M. Ramsey, Anal. Chem., 70 (1998) 3476-3480. S.V. Ermakov, S.C. Jacobson and J.M. Ramsey, Anal. Chem., 70 (1998) 4494-4504. S. Ermakov, S. Jacobson and J.M. Ramsey, Anal. Chem., 72 (2000) 3512-3517. N.A. Patankar and H. Hu, Anal. Chem, 70 (1998) 1870-1881. L. Ren and D. Li, J. Colloid Interface Sci., 254 (2002) 384-395. D. Sinton, L. Ren and D. Li, J. Colloid Interface Sci., 260 (2003) 431-439. D. Sinton, L. Ren and D. Li, J. Colloid Interface Sci., 266 (2003) 448-456. L. Ren, Ph.D. thesis, University of Toronto, 2004. D. Sinton, L. Ren, X. Xuan and D. Li, Lab on Chip, 3 (2003) 173-179. P.H. Paul, M.G. Garguilo and D.J. Rakestraw, Anal. Chem, 70 (1998) 2459-2467. S. V. Pantakar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corp, New York, 1980. S.R. Wallenborg and C.G. Bailey, Anal. Chem, 72 (2000) 1872-1878. S.C. Jacobson, L.B. Koutny, R. Hergenroder, A. M. Moore Jr., J.M. Ramsey, Anal. Chem, 66 (1994) 3472-3476. S. Liu, S. Yining, W.J. William and R.A. Mathies, Anal. Chem, 71 (1991) 566-573. C. Zhang and A. Manz, Anal. Chem, 73 (2001) 2656-2662. L. Bousse, C. Cohen, T. Nikiforove, A. Chow, A.R. Kopf-Sill, R. Dubrow and J.W. Parce, Annu. Rev. Biophys. Biomol. Struct, 29 (2000) 155-181. D.S. Burgi and R. Chien, Anal. Chem, 63 (1991) 2042-2047. W. Thormann, C.X. Zhang, J. Caslavaska, P. Gebauer and R.A. Mosher, Anal. Chem, 70 (1998) 549-562. J. Olgemoller, G. Hempel, J. Boos and G. Blaschke, J. Chromatography B, 726 (1999) 261-268. J.H. Lee, O.K. Choi, H.S. Jung, K.R. Kim and D.S. Chung, Electrophoresis, 21 (2000) 930-934. S.C. Jacobson and J.M. Ramsey, Electrophoresis, 16 (1995) 481-486. J.P. Kutter, R.S. Ramsey, S.C. Jacobson and J.M. Ramsey, J. Microcolumn Separations, 10(4) (1997) 313-319. B.B. Haab and R.A. Mathies, Anal. Chem, 71 (1999) 5137-5145. M. Vazquez, G. McKinley, L. Mitnik, S. Desmarais, P. Matsudaira and D. Ehrlich, Anal. Chem, 74 (2002) 1952-1961. J. Lichtenberg, E. Verpoorte and N.F. de Rooij, Electrophoresis, 22 (2001) 258-271.

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Chapter 9

Electrophoretic motion of particles in microchannels Consider a charged particle suspended in a bulk liquid. When an electrical field is applied, the particle will move in the liquid towards either the cathode or the anode depending on the sign of the surface charge of the particle. Such a particle motion in a stationary liquid phase is called the electrophoresis. Electrophoresis is a major subject in colloid and interface science, because it is one of the most widely used separation techniques in various engineering and science applications [1,2]. Electrophoretic motion of rigid particles in unbounded aqueous electrolyte solutions has been investigated extensively, and mathematical models have been developed to describe this phenomenon in detail. Excellent summaries can be found elsewhere [3-5]. In the electroosmosis, the solid — the channel wall — is stationary, the liquid is moving under an applied electrical field. The electrophoresis is the reversed process: the liquid is stationary, the solid - the particle - is moving under the applied electrical field. The physical nature of these two processes is the same. Therefore, some analyses of electroosmosis can be applied to the electrophoresis. For example, in Chapter 4-1, we derived the electroosmotic flow velocity for the case of thin electrical double layer (i.e., KCI = CI/(1/K) is large) as, (1) where the £waU is the zeta potential of the channel wall. Applying Eq.(l) to the electrophoresis, we have the particle's electrophoretic velocity as: (2) where C,p is the zeta potential of the particle. Eq.(2) is usually referred to as the Smoluchowski equation. In the literature, the electrophoretic mobility is defined as:

Electrophoretic Motion of Particles in Microchannels

543

(2a) By the definition, it is the electrophoretic velocity per unit applied electrical field strength, characterizing how fast a particle will move in an electrical field. The mobility is proportional to the zeta potential of the particle. For a more general treatment, we can derive the electrophoretic velocity by balancing the electrical force and the flow frictional force acting on the particle. The flow frictional force is given by the Stokes equation:

where a is the radius of a spherical particle. The electrical force is given by:

where Q is the total charge of the spherical particle. By using Eq.(34) in Chapter 2,

the force balance will give us: (3) When Ka « 1, Eq. (3) is reduced to (4) Apparently, Eq.(2) and Eq.(4) are different, Eq.(2) has a constant 1 and Eq.(4) has a constant 2 / 3 . The difference reflects the consideration of the socalled electrophoretic retardation effect. Realizing the existence of the EDL around the particle, the action of the applied electrical field on the excess counterions in the EDL region will generate electroosmotic flow. For example, consider a negatively charged particle. Its electrophoretic motion will be towards the positive electrode. The excess counterions in the particle's EDL region are positive ions. Under the same applied electrical field, the electroosmotic flow of

544


these positive ions will be the direction towards the negative electrode. Thus, the electroosmosis in the EDL region of the particle will cause a reduction in the velocity of the particle's electrophoretic motion. This is called the electrophoretic retardation effect. The underline assumption in the Smoluchowski equation, Eq.(2), is that the electroosmosis is the dominant force and the particle's motion is equal and opposite to the liquid motion. That is why the constant in Eq.(2) is 1. Eq.(4), however, is valid only for very small Ka values (or very thick EDL) and considers that the main retardation force to the electrophoresis is the flow frictional force. That is why the constant in Eq.(4) is - and not 1. A more general formulation, the Henry's equation,

(5)

unites the Eqs.(2) and (4). Here /(red) is the Henry's function, and it approaches 1 for small Ka and 3/2 for large Ka. The specific expressions of the Henry's function can be found elsewhere [3,4]. In addition to the above-mentioned retardation effect, the particle's electrical conductance and the particle shape will determine the local distortion of the applied electrical field due to the presence of the particle. Interested readers can find the related information from the literature [3,4]. Electrophoretic motion of rigid particles in unbounded aqueous electrolyte solutions has been studied extensively. In contrast, much less work has been done on the boundary effects on the electrophoretic motion of a particle. The boundary effects on the electrophoretic motion of particles need to be considered in cases where a particle moves in a microchannel whose size is close to the particle's size (e.g., electrophoretic motion of a protein through a porous membrane, electrokinetic transport of biological cells and bacteria in microchannels). With the emergence of bio-lab-on-chip technology, electrophoretic separation of particles in microchannels is often required. Therefore, studies of the boundary effects on the electrophoretic motion of particles are important. Keh et al. [6], Ennis et al. [7] and Shugai et al. [8] examined boundary effects on electrophoretic motion of a sphere for the following cases: a sphere near a non-conducting planar wall with an electric field parallel to the wall; a sphere near a perfectly conducting planar wall with an electric field perpendicular to the wall; and a sphere on the axis of a cylindrical pore with an electric field parallel to the axis. Ennis, et al. [9] experimentally investigated the electrophoretic mobility of proteins in membrane. They found that the protein mobility is identical to the free protein mobility when the size of the protein is


545

small relative to that of the pore, and the mobility is significantly reduced as the pore radius approaches the protein radius. In addition, Zydney [10] and Lee, et al. [11] investigated boundary effects on the electrophoretic motion of a charged spherical particle in a spherical cavity. Their results indicate that the boundary effect is weak for thin double layers but significant for thick double layers. They also found that the charge on the boundary alters the particle motion through the development of an induced charge on the particle and through the generation of an electroosmotic recirculation flow in the spherical cavity. In this chapter, we will discuss how to model and simulate the electrophoretic motion of particles in microchannels filled with an aqueous electrolyte solution [12-14]. Two idealized particle shapes will be considered: rigid spherical particles and rigid circular cylindrical particles with hemispherical ends. By studying the electrophoretic motion of these particles, possibilities of separating the particles in aqueous solutions and manipulating the particle motion in microchannels according to their sizes will be discussed.

546

9-1


SINGLE SPHERICAL PARTICLE WITH GRAVITY EFFECTS

Let's consider a spherical particle moving in a rectangular microchannel filled with an aqueous electrolyte solution. The microchannel is connected to two solution reservoirs that open to the atmosphere, i.e., there is no pressure difference along the microchannel. Figure 9.1 (a) shows the geometry of the microchannel, which has a height of 2H, a width of 2JFand a length of L. Cartesian coordinates (x,y,z) are used with the origin located at the center of the entrance cross-section. The height of the microchannel is much less than its width so that the side effect is negligible and the motion of the sphere is on the x-z plane. Figure 9.1 (b) shows the cross-section in the x-z plane, with the sphere entering at the center of the entrance cross-section (PI).

Figure 9.1. Illustration of electrophoresis of a sphere in a microchannel: (a) the geometry of the microchannel; and (b) the x-z cross-section plane of the system.


547

Generally, the particle and the carrying electrolyte solution have different densities. For example, the density for protein is about 1350kg/m3, which is 1.35 times higher than that of the water. The gravity force tends to pull the heavier particles near the lower wall of the channel where the particles will experience different flow resistance from that in the center of the channel. This may influence the electrophoretic motion of particles in microchannels. Therefore, the gravitational effect should be considered. Here we assume that the density of the sphere is higher than that of the solution. Both the sphere surface and the microchannel surface carry uniform negative charges that are characterized by their respective zeta potentials: C, p and C,w. When an external electric field, with strength Ex, is applied uniformly along the channel, an electroosmotic flow will be developed toward the cathode. Generally, the motion of the sphere in the microchannel can be divided into Phase I ( P I - P 2 ) and Phase II (P2-P3). In Phase I, the sphere moves at a constant velocity £/(/), which has both a x-component and a z-component of velocity due to the electrostatic force in the x-direction and the gravity force in the z-direction. LI is a x-component of the distance between PI and P2. The sphere will reach an equilibrium height (a separation distance d) above the lower channel wall at the position P2. In Phase II, the net force acting on the sphere in the z-direction vanishes, so there is no particle motion in the zdirection. The sphere moves at a constant velocity £/(//) parallel to the lower channel wall. The dashed spheres in Figure 9.1(b) illustrate the possible trajectories of the sphere motion in the microchannel. To determine the velocities of the sphere in Phase I and in Phase II and the separation distance d, and to capture the main characteristics of the electrophoretic motion under the gravitational field, the following assumptions are made: • Both the sphere and the microchannel are rigid and non-electrically conducting; • The electrolyte solution is Newtonian and incompressible. And the electrolyte is symmetric, that is, n^ - n_00 = « +00 and z = z+ = -z_; • The fluid flow field is steady, and the sphere moves at constant velocities respectively in Phase I and in Phase II. The flow is slow enough so that it can be considered as a Stokes flow; • The Brownian motion and the effect of the electrokinetic lift force are negligible; • The wall effect in Phase I are negligible; the upper wall effect in Phase II are negligible; and • The sphere moves in a uniform electroosmotic flow field.

548


The separation distance d When the sphere moves very close to the lower wall, the electric double layer interaction force and the van der Waals force [15] become effective, and they will balance the gravity force and the buoyancy force. In this system, we define the separation distance h (shown in Figure 9.2(c)) as the distance from the center of the sphere to the lower wall when the forces in the z-direction are balanced, and the separation distance d as d-h-a. Thus, a force balance is used to determine d. When the sphere moves at a constant velocity UX(II) parallel to and very close to the lower wall (shown in Figure 9.2(c)), there are four forces (shown in Figure 9.2(d)) acting on the sphere in the z-direction: the gravity force (FQ), the buoyancy force (Fg), the electric double layer interaction force (Fe) and the van der Waals force {FV(^W). These forces are subject to the following force balance equation: (6) The gravity and buoyancy forces can be calculated by the following equations:

Figure 9.2. The sphere's motion in Phase I and Phase II: (a) the motion of the sphere in the zdirection in Phase I; (b) the motion of the sphere in the x-direction in Phase I; (c) the motion of the sphere in Phase II; (d) the forces acting on the sphere in the z-direction in Phase II.


549

(7)

(8) where pp is the density of the sphere, p/ is the density of the electrolyte and g is the gravitational accelerating rate. Eqs. (7) and (8) reveal that FQ and Fg are dependent on the sphere's size, the sphere's density and the solution's density. Hogg, et al. [16] derived the electrical double layer interaction potential between two spheres: (9) where VR (d) is the electrical double layer interaction potential, a\ and a^ are the radius of the two spheres respectively, q\ and q^ a r e m e zeta potentials of the two spheres respectively, d is the separation distance between the two spheres, and K: is the reciprocal Debye length. For the symmetric electrolyte, K is given by (10) where s is the dielectric constant of the electrolyte solution, e 0 is the permittivity of vacuum (8.85xlO~ 12 C7F-m), z is the valence of the ion species, e is the charge on an electron (1.602 xlO~19C), nx is the ionic number concentration (m~3), k/, is the Boltzmann constant (1.381 x 10~23 JIK), and T is the absolute temperature (K). From Eq. (9), we can derive the electrical double layer interaction potential between a sphere and a flat plate by setting a\ =a, 02=°°, q\=qp (the particle) and £2 = gw (the wall or the plate). We obtain

(11)

550


The electric double layer interaction force Fe can be derived by using the following relation: (12)

As seen from the above equation, Fe is dependent on the sphere's size, the separation distance, the Debye length, the zeta potentials of the sphere and the channel wall. Generally, the van der Waals force can be attractive or repulsive depending on the properties of the sphere and the channel wall and those of the solution. Fvc{w can be approximated by the following equation [15]: (13) 1-20-

where A is the Hamaker constant. In this work, we choose A = 0.83x10 J, that is, FV(jw is an attractive force. This force is dependent on the sphere's size and the separation distance. Substituting the Eqs. (7)—(13) into Eq. (6), we have

(14)

Combining the gravity force and the buoyancy force into a net gravity force FgQ, we have (15) Thus Eq. (14) can be rewritten as:


551

(16)

The above equation reveals that the separation distance depends on the sphere's size, the density difference between the sphere and the solution and the electrochemical properties of the solution. The motion of the particle in Phase I As mentioned before, we assume that the flow is the Stokes flow, that the sphere moves in a uniform electroosmotic flow field and that the wall effects in Phase I are negligible. The motion of the sphere can be decomposed into the motion in the z-direction and the motion in the x-direction (shown in Figures 9.2(a) and (b)), where Uz (/) and Ux (I) are the velocity components in the zdirection and in the x-direction respectively.

Figure 9.3. The separation distance between the sphere and the lower channel wall as a function of sphere's radius with Ex = 30V/cm, p« = 1200 kg/m , C, „ = -40mV and C w = -20mV.

552


Figure 9.2(a) shows the forces acting on the sphere when it moves at a constant velocity Uz (I) through a bulk solution. In this direction, two forces act on the sphere: FgQ and F^Z{I). F^Q is the net gravity force given in Eq. (15); F/jZ(7) is the hydrodynamic force that can be determined by Stokes' law of resistance. The force balance on the sphere in the z direction yields (17) That is, the sedimentary velocity Uz (I) is proportional to the square of the sphere's radius and the density difference between the sphere and the solution. Figure 9.2(b) shows the forces acting on the sphere when it moves at a constant velocity Ux (I) through a uniform electroosmotic flow field (U eo ) in the x-direction. Fg is the electrostatic force, and Ffa (I) is the hydrodynamic force. For this system, we have (18)

Figure 9.4. The z-component of the sphere velocity in Phase I with respect to sphere's radius

(Ex = 30V/cm, pp=l200kg/m3,

C,p = -40mF and C w =-20mV).


553

where Uep is the electrophoretic velocity of the sphere. By Henry's solution, Uep can be expressed as

(19) where \x is the viscosity of the electrolyte solution, and / ( K H ) is Henry's function, which was derived by Henry [17] and was simplified recently by Ohshima [18]. For a sphere, an expression for f(Ka) is given by [18]

(20)

Note that the electroosmotic flow velocity of the solution is given by: (21) Therefore, we have (22) Eq. (22) indicates that UX(I) is proportional to Ex and it is dependent on the sphere's size, the electrochemical properties of the solution and the zeta potentials of the sphere and the channel wall. When the sphere velocity components in the x-direction and the zdirection are determined, we can calculated the time tl needed for the sphere moving from PI to P2 and the distance between them in the x-direction: (23)

(24)

554


Figure 9.5. The x-component of the velocity in Phase I and the velocity in Phase II as a

The motion of the particle in Phase II As mentioned earlier, in Phase II, the forces acting on the sphere in the zdirection are balanced, so there is no motion in the z-direction and the sphere moves at a constant velocity U(II) parallel to the lower wall through the uniform flow field (Ueo) (shown in Figure 9.2(c)). Ennis, et al. [7] studied the electrophoretic motion of a charged sphere parallel to a flat wall and considered the electrophoretic retardation effect. They used the method of reflections and derived the particle velocity. Their solution is valid for low zeta potentials and any value of ka. Using Ennis' solution, we have

(25)


555

where X is the ratio of the particle radius to the distance from the boundary, and the functions of fx(x), L(x), M(x), Wj,{x), Wp are the translational velocity and the angular velocity of the particle respectively, xp is the position vector on the particle surface, and X „ is the position vector of the particle center. Eq.(57) is the slipping flow boundary condition, i.e., the electroosmotic flow velocity, at the microchannel wall. Eq. (58) is the velocity boundary condition at the particle surface. In Eq. (58), the 1st term is the particle's translational velocity, the second term represents the particle's angular velocity and the last term is the electroosmotic velocity of the liquid around the particle. Particle motion Generally, the Newton's second law governs the particle's motion. The translational motion is described by:

where mp is the mass of the particle, Vp is the translational velocity, and Fnet is the net force acting on the particle. Since the particle carries uniform surface charge, there is electrostatic force acting on the particle by the applied electric field. At the same time, the flow field around the particle exerts a hydrodynamic force on the particle surface. The net force acting on the particle is:

where Fg is the electrostatic force acting on the particle and Ff, is the total hydrodynamic force acting on the particle. The hydrodynamic force can be divided into two components:

584


where Ff,jn is the hydrodynamic force acting on the particle surface due to the flow field in the inner region (i.e., electroosmotic flow in the EDL region around the particle), and FfjO is the hydrodynamic force acting on the particle surface due to the flow field originated in the outer region. In this model, we assume the EDL around the particle is so thin that its thickness can be neglected in comparison with the size of the particle (1-30 |j.m) and the size of the channel (~100 um). For example, in a solution with a high electrolyte concentration such as 10~2M, the EDL thickness is approximately 3 nm. Thus, we don't consider the detailed flow field in the EDL—the inner region, and simply replace the flow field in the thin EDL by the electroosmotic velocity as a slipping flow boundary condition for the outer region flow field. In this way the flow field around the particle is the flow field originated in the outer region and subject to the slipping flow boundary condition at the particle surface. Consequently, Fô is a hydrodynamic force acting on the particle surface. It can be shown that Fg and Fîn have the same value but operate in the opposite directions, therefore, the net force acting on the particle becomes:

Consequently, the equation governing the particle's translational motion becomes:

(59)

(60) The rotational motion of the particle is governed by: (61) where J is the moment of inertia of the particle, and T), is the torque (about the particle centre) on the particle by the flow field in the outer region. The hydrodynamic force and the torque are given by

585


(62) (63) where a is the stress tensor that is given by (64)

The initial conditions are given by: (65) In order to non-dimensionalize the equations, we chose / = a {a is the particle as a radius) as a characteristic length, the electric potential 0 = y/ r •MB

(66) (67) (68)

(69) (70)

586


(71) (72) where y is the ratio of the zeta potential of the channel wall to that of the particle. The quantities with a star in the above equations are the dimensionless variables.

is the density of the particle, we can write the dimensionless governing equations for the particle motion as follows:

(73)

(74)

(75)

where

which is a coefficient.

Finally, the initial conditions

become (76) Numerical method Due to the complicated geometry in the T-shaped junction region, temporal effects have to be considered. Therefore, the particle motion, the flow field and the electric field are coupled in the following way: the particle motion provides the moving boundary for the flow field and the electric field; the flow field exerts the hydrodynamic force and toque on the particle; and the electric field provides the slipping flow boundary conditions for the flow field. In order to solve the complicated system of coupled equations, numerical simulation is required.


587

Generally, two challenges exist for simulating the liquid-particle flow: (1) accurately calculating the hydrodynamic interactions (i.e., hydrodynamic force and torque); and (2) tracking the position of the particle. For a complicated geometry, it is extremely difficult to calculate accurately the hydrodynamic interactions. A direct numerical simulation method is preferable since it evaluates the hydrodynamic interactions with no averaging or approximation. For example, Hu [20] and Glowinski et al. [21] employed a generalized Galerkin finite element method. This method incorporates both equations of the fluid flow and equations of the particle motion into a single variational equation where the hydrodynamic interactions are eliminated and the explicit calculation of the hydrodynamic interactions is not required. Ritz et al. [22] assumed solid particles to have a very high viscosity and developed a continuous hydrodynamic model that leads to a unique system of equations for both phases. By this method, no explicit calculation of the hydrodynamic interactions is required. To track the position of the moving particle, two strategies have been developed. One is the moving mesh method that considers the relative position of the particle as a boundary of the fluid domain. Re-eshing is then needed at each time step. The other method is the fixed mesh method that tracks the position of the moving particle over time. As an example of the moving mesh method, Hu [20] adopted an arbitrary Lagrangian-Eulerian (ALE) technique to deal with the motion of the particles. The advantage of this method is that the interface of the liquid-particle is clearly defined at each time step. As an example of the fixed mesh method, Glowinski et al. [21] used a finite element based fictitious domain approach where the solid-liquid interface is defined by control points. At each of these points, a kinematic condition is imposed using Lagrange multipliers. The motion of these control points is governed by the action of the fluid on the surface of the particles. The advantage of this method is that no remeshing is needed. However, the interface of the liquid-particle is not clearly defined at each time step. In the numerical simulations presented in this section, the method of a generalized Galerkin finite element formulation [20,21] was used to incorporate both equations of the fluid flow and equations of the particle motion into a single variational equation where the hydrodynamic interactions are eliminated. In addition, for the system considered here, both the electric field and the slipping flow boundary conditions for the liquid phase require that the interface of the liquid and the particle be defined clearly. In this work, an ALE method [23] is used to track the position of the particle at each time step. We used the same procedures as used by Hu [20]: The nodes on the particle surface and in the interior of the particle are considered to move with the particle. The moving velocity of the nodes in the interior of the liquid is computed using Laplace's equation to guarantee a smoothly varying distribution of the nodes. At each time step, the grid is updated according to the motion of the particles and is checked for element degeneration. If unacceptable element distortion is detected, a new

588


finite element grid is generated and the flow fields are projected from the old grid to the new grid. The first order Euler backward method is used to discretize the temporal term. We use GAMBIT (a mesh generator developed by Fluent Inc.) to generate unstructured tetrahedral meshes. A program based on Taylor-Hood tetrahedral element [24,25] has been developed to solve the above set of dimensionless equations. To verify the numerical simulation programs, we considered a test case: a non-conducting spherical particle carrying uniform surface charge is freely suspended in the center of a large circular cylindrical channel. An external electric field is applied along the axis of the channel. The channel wall carries no surface charge and the radius of the channel, b, is 30 times larger than the radius of the particle, a. Therefore, the channel wall has a negligible boundary effect on the particle motion. Furthermore, we assume that the electrical double layer surrounding the particle is very thin, i.e., Ka -> oo, and that the double layer is not distorted by the applied electric field. The conditions of this test case match that of Henry's solution for the electrophoresis of a spherical particle in unbounded solution. For KM—»oo, according to Henry's formula, the electrophoretic velocity of the particle is given by: (77) Using this Uf, as a characteristic velocity, the calculated dimensionless velocity for the spherical particle in the large channel is Vp = —^- = 0.9988664. The discrepancy between the calculated result and Henry's solution is approximately 0.113%. Furthermore, the numerical simulation program was tested against Keh's solution. Keh et al [6] considered the electrophoretic motion of a rigid sphere with a radius a along the axis of a circular cylindrical pore of a radius b. Considering a thin electrical double layer, they derived the following approximate solution by a method of reflections:

(78)


589

Figure 9.25. Effect of the applied electric potentials on the electric field: (a) y/* = C* = 0.0 and (b) y/* =C* =1.0, with y/" = A' =1.0, y/* = B* =0.0 and a =1.0.

590


where U^ is the velocity of the spherical particle. From the above equation, the following dimensionless particle velocity can be derived:

(79)

It can shown that the discrepancy between the numerical solution results and Keh's solution is less than 1.0% when — is within the range from — to — . b 4 30 In the system studied here, the electric field (i.e., the gradient of the electric potential) is the driving force for both the particle motion and the flow field in the channel. In addition, since the size of the channel is close to that of the particle, the boundary effects are significant and the particle motion is dependent on its size relative to that of the channel. In order to study the particle size effects, we considered two particles of different sizes: one has a radius a and the other has a radius 2a. For both cases, we used the same characteristic length scale l~a, thus the dimensionless particles' radii become a -\ and a -2, respectively, for these two particles. The dimensionless sizes of a T-shaped rectangular channel used in the numerical calculations are shown in Figure 9.23. As seen in the figure, the channel is composed of three sub-channels with a cross-section of 10x10. The initial position of the particle center is set at (19,0,0). This particle-channel system is symmetrical to both the z = 0 plane and to the y = 0 plane. When three electrical potentials are applied at the inlet and two outlets respectively, the electric field (i.e., the gradient of the electric potential) will be in x and y directions, and so will the particle motion. Therefore, in the following discussion, we present the electric field and the flow field on the z = 0 plane. Effects of the applied Electric field In the numerical calculations, the applied electric potentials at the inlet and the outlet 1 are set at i// -A

= 1.0 and y/ = B = 0.0 respectively, while the

applied electric potential at the outlet 2 (i// = C ) changes from 0.0 to 1.0. A particle with dimensionless radius a' = 1 is suspended in the channel. The electric field on the z = 0 plane is shown in Figure 9.25, where the lines with arrows denote the direction of the electric field, and the gray levels denote the


591

Figure 9.26. Effect of the suspended particle on the electric field: (a) without a particle and (b) with a particle (whose dimensionless radius a' = 2.0) suspension, with y/* = A' =1.0, V/* = B' = 0.0 and y/* = C* = 0.5.

592


magnitude of the electric field with the darker areas representing the weak electric field and the lighter areas representing the strong electric field. It is shown that the applied electric potentials have a great influence on the electric field in the T-shaped junction region. The electric field is symmetrical in Figure 9.25(a) while it is unsymmetrical in Figure 9.25(b). As mentioned before, the geometry of the particle-channel system is symmetrical to the y - 0 plane. When the electric potentials applied at two outlets are of the same value, the electric potential distribution is symmetrical to the y - 0 plane. On the other hand, when the electric potentials applied at two outlets are of the different values, the electric potential distribution is not symmetrical to the y - 0 plane. Effect of the suspended particle The suspended particle provides a non-conducting boundary for the electric field. Figure 9.26 shows the electric fields on the z = 0 plane for two different cases: (a) without a particle, and (b) with a particle of a dimensionless radius a = 2. In these two cases, the same electric potentials are applied at the inlet and the two outlets: y/ -A =1.0, y/ =B = 0.0 and y/ =C =0.5, respectively. Comparing Figure 9.26(a) with Figure 9.26(b), it is seen that the presence of a particle distorts the electric field in the region adjacent to the particle, which will have an effect on the particle motion. Particle motion As mentioned before, the particle motion is a combined result of the particle electrophoresis and the electroosmotic flow field in the channel. Both the electrophoresis of the particle and the flow field depend on the applied electric field. In addition, since the size of the channel is close to that of the particle, the effects of the non-conducting boundaries, i.e., the channel wall and the particle surface, on the electric field and the flow field will affect the particle motion in the T-shaped microchannel. The initial position of the particle center is at (19,0,0) and the particle initial velocity is zero. When the electric potentials are applied at the inlet and two outlets of the channel, the particle will begin to move. As mentioned before, in this particle-channel system, the geometry is symmetrical to the z - 0 plane. When three electrical potentials are applied at the inlet and two outlets, the electric field will be in x and y directions, and so will the particle motion. Therefore, the particle translates on the z = 0 plane. In the following sections, we will discuss the track of the particle motion in the channel on the z = 0 plane and the decomposed translational velocity components of the particle in x and y directions. In the calculations, the following conditions are used: s = 80.1,


593

Figure 9.27. Flow velocity vector plot on the z = 0 plane at t* = 2 4 with y/' = A' =1.0, yz* =B' = 0 . 0 , i//' =C* = 0 . 5 , y = 5.

594


Thus the coefficient in Eq. (75) is a = 1.0. It should be noted that, in the calculations, the total nodal number ranges from 40,000 to 50,000 and the number of the tetrahedral elements ranges from 28,000 to 33,000. It is very time-consuming to calculate the elemental matrices, solve the electric field and the flow field, calculate the mesh moving velocity and update the mesh, etc., at each time step. Since our goal here is to examine the influence of the different parameters on the trend of the particle motion, we will not calculate the particle motion from its initial position to the exit of the channel. We only calculate the particle motion with limited time steps as far as the trend of the particle motion under different influences can be shown. Figure 9.27 shows the flow velocity vector plot on the z = 0 plane at t =24 with

Figure 9.28. Effect of the applied electric potential on the particle motion. The lines carrying different symbols denote the path of the particle moving in the junction region during a period ofdimensionlesstime t* = 44, 1//* = A' = 1.0, y* = B' = 0.0, a" = 1.0, y = 5.


595

Effect of the applied electric potentials The applied electric field is the driving force for the electroosmosis and the electrophoresis. Without the electric field, there is no liquid flow and particle motion in the channel. To calculate the influence of the applied electric potentials on the particle motion, we chose a particle with a dimensionless radius a = 1, and we set the ratio of the zeta potential of the channel to that of the particle as y =gw/gp

=5. The time step (Vt ) is Vt =2.0. The applied

electric potentials at the inlet and the outlet 1 are chosen as y/ -A

=1.0 and

y/ = B =0.0 respectively, while the applied electric potential at the outlet 2 (y/ =C ) changes from 0.0 to 0.25, 0.5 and 1.0. The trace of the particle motion under the different applied electric potentials are shown in Figure 9.28,

Figure 9.29. Effect of the zeta potential ratio on the particle motion. The lines carrying different symbols denote the path of the particle moving in the junction region during a period of dimensionless time t* = 66, y/* =A* =\.0, y/* = B' = 0 . 0 , y>* =C* = 0.5 , a =1.0.

596


where the lines carrying different symbols denote the path of the particle moving in the junction region during a period of dimensionless time t - 44. As shown in the figure, the particle motion is greatly influenced by the applied electric potentials. It can be concluded that the direction of the particle motion in the Tjunction can be controlled by the direction of the local electric field and that the particle moves faster when the local electric field is stronger. Effect of the zeta potentials of the channel and the particle The zeta potentials of the channel and the particle are critical for the electroosmosis and the electrophoresis. In the extreme case, if both the channel and the particle carry no surface charges (i.e., £ w = 0 and C,p =0), there is no liquid flow and the particle motion in the channel even if an electric field is applied along the channel. The effect of the zeta potentials of the channel and the particle on the particle motion can be seen from the effect of the zeta potential ratio, 7 = Cw ^p • Three cases are considered here: (a) y = 5.0; (b) y = 3.0; and (c) y = 0.5. The applied electric potentials at the inlet and the two outlets are the same for all three cases, i.e., y/ = A =1.0, y/ = B - 0.0 and y/ = C =0.5 respectively. A particle with a dimensionless radius a = 1 is chosen. The time step (Vt ) is set as Vt =2.0. The trace of the particle motion under the different ratios are shown in Figure 9.29, where the lines carrying different symbols denote the trace of the particle moving in the junction region during a period of dimensionless time t = 66. As shown in the figure, the ratio, y, has a great influences on the particle motion. For the first two cases (i.e., y - 5.0 and y - 3.0) where the value of the zeta potential of the channel is bigger than that of the particle, the particle moves in the same direction as the electroosmotic flow, and the bigger the ratio, the faster the particle moves. However, for the case of y = 0.5 where the value of the zeta potential of the channel is smaller than that of the particle, the particle moves in the opposite direction to the electroosmotic flow. This can be understood as follows: the particle motion is the result of the electrophoresis coupled hydrodynamically with the electroosmosis. When both the channel and the particle carry negative surface charges, electrophoresis will be in the direction against the electric field while the electroosmosis will be along the electric field. For thin EDL, the electrophoresis mobility of a particle in an unbounded solution is given by \xeph = sC,p //J., and the electroosmotic mobility of the liquid in the channel is given by pieof = sC,w /JX. Therefore the ratio y =C,JC,P is also the ratio of mobility neof/fJ-eph- By definition, the mobility is the velocity of electrophoretic or electroosmotic motion under a unit applied electrical field strength. When y >1.0, i.e., the magnitude of the electrophoretic mobility is less


597

than that of the electroosmotic mobility. The electroosmotic flow velocity is larger than the particle's electrophoretic velocity. Therefore the particle will by carried by the liquid electroosmotic flow to move in the electroosmotic flow direction but with lower speed than the electroosmotic flow. When y pi, a>p2, Xpi, xp2, Xpi, Xp2, Cpi and C,p2 are the translational velocity, the angular velocity, the position vector on the particle surface, position vector of the particle center and zeta potential for particle 1 and particle 2, respectively. Particle motion Similar to the analysis presented in the last section, we can show that the equations governing the particles' translational motion are given by:

(83)

(84)

(85)

(86) The rotational motion of the particle is governed by: (87)

(88)


603

Figure 9.32. Different particle positions on the j*=4 plane: (a) Case 1; (b) Case 2; (c) Case 3; (d) Case 4; (e) Case 5; and (f) Case 6.

604


where J 1 ; F/JOJ and T/,j are respectively the moments of inertia, the hydrodynamic force and the torque (about the particle centre) on particle 1 by the flow field in the outer region, and J2, Fhol anc^ ^hl a r e respectively the moments of inertia, the hydrodynamic force and the torque (about the particle centre) on particle 2 by the flow field in the outer region. The hydrodynamic force and the torque are given by (89) (90) (91) (92) where a is the stress tensor that is given by (93)

In order to non-dimensionalize the equations, we chose l-a\ {a\ is the radius of particle 1) as a characteristic length, the electric potential <j> = y/ r as a '•in BE A

(h

characteristic electric potential, and U = — - £ w — as a characteristic velocity. l 1 Letting x = lx , v = Uv , p = ^—p , I/A = 0y/ and t = ——t , we can derive /

77

the following dimensionless equations for the electrical field and the flow field: (94) (95) (96)


605

The dimensionless boundary conditions becomes

where / j and y 2 are respectively the ratio of the zeta potential of the particles to that of the channel, and A is the ratio of radius of particle 2 to that of particle 1, i.e., A = alla\.

mpl = ppl3mp*,

Defining J2 - A ppl

J

(pp

is

the

mp2 = A3 ppl3mp*, density

of

the

particles),

Jx=ppl5J*, a = -—a ,

ho\ =rlUlFhoX, Fho2 =r)UlFlo2, fhx =t]Ul2f^ and fh2 =rjUl2fh2,we can write the dimensionless governing equations for the particle motion as follows:

(97)

(98)

(99)

(100)

606


(101)

(102)

The quantities with a star in the above equations are the dimensionless variables. Numerical method For such a two-particle system, the particle motion, the flow field and the electric field are coupled in the following way: the particle motion provides the moving boundaries for the flow field and the electric field; the flow field exerts the hydrodynamic force and toque on the particles; and the electric field provides the slipping flow boundary conditions for the flow field. In order to solve the complicated system of coupled equations, numerical simulation is required. To solve the above couple equations, the method of a generalized Galerkin finite element formulation [20,21] was used to incorporate both equations of the fluid flow and equations of the particle motion into a single variational equation where the hydrodynamic interactions are eliminated. For the system considered here, both the electric field and the slipping flow boundary conditions for the liquid phase require that the interface of the liquid and the particles be defined clearly. An ALE method [23] is used to track the position of the particles at each time step. The following procedures were used: The nodes on each particle surface are considered to move with the particles respectively. The moving velocity of the nodes in the interior of the liquid is computed using Laplace's equation to guarantee a smoothly varying distribution of the nodes. At each time step, the grid is updated according to the motion of the particles and is checked for element degeneration. If unacceptable element distortion is detected, a new finite element grid is generated and the flow fields are projected from the old grid to the new one. The first order Euler backward method is used to discretize the temporal term. GAMBIT (a mesh generator developed by Fluent Inc.) was used to generate unstructured tetrahedral meshes. A program based on Taylor-


607

Hood tetrahedral element [24,25] has been developed to solve the above set of dimensionless equations. Here we will consider two kinds of initial particle velocity conditions: one is zero velocity and the other is the steady-state velocity. The steady-state velocity is determined by solving Eqs. (80)~(82) without the temporal term in Eq. (82). In the system studied here, the electric field (i.e., the gradient of the electric potential) is the driving force for both the particle motion and the flow field in the channel. In addition, since the size of the channel is close to that of the particle, the boundary effects may be significant and the particle motion may depend on its size relative to that of the channel. The dimensionless parameters of a straight rectangular channel (16x8x11) used in the numerical calculations are shown in Figure 9.31. In the calculations, the following conditions are used:

the dimensionless time via / = 2.5x10" t s. The centers of the particles are initially located on the j*=4 plane. Since the electric field is applied along the channel (in x direction), and the particle-channel system is symmetrical to the y*=4 plane, both centers of the particles will move along the y*=4 plane. Therefore, in the following discussion, we present the electric field and the flow field on the y*=4 plane. Figure 9.32 shows the initial positions of the particles for different cases. Electric field The suspended particles provide non-conducting boundaries for the electric field. Figure 9.33 shows the electric fields on the y*=4 plane for three different cases: (a) one particle, (b) two particles with the separation d*=1.0 (d* = d/al, where d is the separation distance between two particles, and al is the radius of particle 1) and (c) two particles with the separation d*=2.0. In this figure, the lines with arrows denote the direction of the electric field and the grey levels denote the magnitude of the electric field. Comparing Figure 9.33(a) with Figure 9.33(b-c), it can be seen that the electric field in the region adjacent to the bigger particle is distorted by the presence of the smaller particle. In addition, by comparing Figure 9.33(b) and Figure 9.33(c), one can see that the influence of the smaller particle on the electric field adjacent to the bigger particle is weaker when the separation distance between two particles is larger.

608


Figure 9.33. The electric field of (a) one particle, (b) two particles with the separation d*=l.O, and (c) two particles with the separation d*=2.0. The lines with arrows denote the direction of the electric field and the grey levels denote the magnitude of the electric field. The other parameters used in this figure are A = 2, yx = y2 = 0.9 and Ex = 62.SWIm.


609

Figure 9.34. The flow field of (a) one particle, (b) two particles with the separation d*=\.O, and (c) two particles with the separation d*=2.0 where the vectors denote the flow velocity and the lines with arrows denote the streamlines. The other parameters used in this figure are A = 2, y, =y 2 =0.9 and Em =62.5kV/m.

610


Flow field It is well known that the electroosmotic flow is directly dependent on the electric field. In this model, the electroosmotic flow is described by the slipping flow boundary velocity. The slipping flow boundary velocity for the flow field will change with the change of the electric field. In addition, the presence of moving particles provides moving boundaries for the flow field. Therefore the flow field depends on the electric field, the particles' positions and the particles' motion. Figure 9.34 shows the flow field on the y*=4 plane for three different cases: (a) one particle, (b) two particles with the separation d*=1.0 and (c) two particles with the separation d*=2.0, where the zeta potential ratios (the zeta potential of the particle to that of the channel wall) were chosen as 7j = Y2 ~ 0-9 • In the figure, the vectors denote the flow velocity and the continuous lines with arrows denote the streamlines. As is shown, the streamlines and velocity vectors surrounding a single particle are symmetric in Figure 9.34(a) but unsymmetrical in Figure 9.34(b-c). The flow field on the right side of the bigger particle is distorted by the presence of the smaller particle. In addition, by comparing Figure 9.34(b) and Figure 9.34(c), it can be seen that the influence of the presence of the smaller particle on the flow field adjacent to the bigger particle is weaker as the separation distance between two particles increases. Particle motion As discussed above, the presence of one particle exerts influences on the electric field and the flow field adjacent to the other particle, which will influence the particle motion. The effects of particle size, particle zeta potential, particle position and the separation distance on the particle motion are discussed below. Effect of particle size To discuss the effects of particle size on particle motion, Case 2 (Figure 9.32(b)) is considered: the centers of Particle 1 and Particle 2 are initially set at (7, 4, 3) and (7, 4, 7) respectively, and The radius of Particle 2 is twice of Particle 1. We assume the particles' initial velocity is zero, and the zeta potential ratios yl=y2= 0.2. After the electrical field (62.5KV/m) is applied, the particles start moving. The dimensionless translation velocities for Particle 1 and Particle 2 in the x direction reach 0.04831 and 0.04803 respectively at f=80. In dimensional terms, the velocities are 2.400mm/s for the small particle (al = 5\im) and 2.386mm/s for the larger particle {a.2 - lOum) at 0.002s. The smaller particle (Particle 1) moves a little faster than the bigger one (Particle 2). Though the difference between these two velocities is small, after a certain time, the two


611

particles can be separated in the x direction. For example, at /*=8000, i.e., t=0.2s, these two particles can be separated by d*=2.6 (ord= 13 |^m) in the x direction. Effect of particle zeta potential The zeta potentials of the channel wall and the particles are critical for the electroosmosis and the electrophoresis. Under the same electric field, the electroosmosis and the electrophoresis will be in the opposite direction if both the channel wall and the particles are negatively or positively charged. The particle motion is the result of the electrophoretic motion coupled with the electroosmotic flow field in the channel. Case 3 (Figure 9.32(c)) is considered: the centers of Particle 1 and Particle 2 are initially set at (7, 4, 3) and (7, 4, 8) respectively, Particle 2 has the same size as Particle 1. The zero velocity is used as the initial condition, and we set the zeta potential ratios as /j = 0.2 and Y2 = 0.4. At t*=64, the dimensionless translation velocities for Particle 1 and Particle 2 in the x direction reach 0.04960 and 0.03706 respectively. In dimensional terms, the velocities are 2.464mm/s for Particle 1 and 1.841mm/s for Particle 2 after 0.0016s. Particle 1, with the lower zeta potential, moves faster than Particle 2 in the electroosmotic flow direction. When C,p < £w (i.e., y < 1),

Figure 9.35. Paths of Particle 1 and Particle 2 for Case 3 from /*=0 to /*=288, where the dash circles and the solid circles respectively show the initial and the final positions of the particles. The parameters used in this figure are X = 1, yx = 0.2, y 2 = 0.4, and Em = 62.5kV/m.

612


the electroosmotic mobility is larger than the electrophoretic mobility, thus the particle motion will be in the same direction as the electroosmotic flow. It can be concluded that, within the range of 0 < y < 1, the smaller the value of y, the bigger the particle moving velocity. Figure 9.35 shows the tracks of Particle 1 and Particle 2 from /*=0 to f=288. It can be seen that at t*=288, the position of Particle 1 is ahead of Particle 2, and the separation distance between two particle's centers is 4.5um. Effect of the particle position To discuss the effects of the particle position on the particle motion, Case 4 and Case 5 (Figure 9.33(d-e)) are considered. The larger Particle 2 is ahead of the smaller Particle 1 in Case 4 and the larger Particle 2 is behind of the smaller Particle 1 in Case 5. The centers of two particles are initially set at (5, 4, 5.5) and (9, 4, 5.5) respectively. The radius of Particle 2 is twice of Particle 1. The separation distance for both cases is d*=l.O. We used the following zeta potential ratios: y\ = 0.2 (fixed) and Y2 changes from 0.1, 0.2 to 0.8, respectively. The velocities for Particle 1 and Particle 2 are shown in Table 2. It can be seen that the difference between particle velocities is very small (within 0.1%). It can be concluded the effect of the particle position, i.e., the bigger particle ahead or behind of smaller one, has a negligible effects on the particle moving velocity. It can also be seen from Table 2 that Particle l's velocity is influenced by the change of the velocity of Particle 2: Particle l's velocity increases with Particle 2's velocity. Table 2 A list of particle velocities for different particle positions with y, = 0.2. Case 4 (Figure 3d)

Case 5 (Figure 3e)

0.1

0.05092

0.05446

0.05092

0.05442

0.2

0.04995

0.04852

0.04995

0.04847

0.8

0.04410

0.01291

0.04412

0.01290


613

Effect of the separation distance To examine the effects of the separation distance, Case 1 and Case 4 (Figure 9.33(a, d)) are considered, where The radius of Particle 2 is twice of the Particle 1 and the separation distances are respectively d*=0.3 and d*=\.O. We let 7] = 0.2 and y2 change from 0.1, 0.2 and 0.8 respectively. Table 3 shows the calculated particle velocities for different separation distance. In order to compare the effects, we calculated the velocity for Particle 1 in the channel without the presence of Particle 2, and the velocity is 0.04993. Comparing this velocity with the velocities listed in Table 3, one can see that the separation distance does affect the particle motion. With the increase of the separation distance, the effect of the separation distance weakens. In addition, this effect on Particle 1 's velocity is dependent on Particle 2's velocity. For example, when the difference between Vp2 and Vp\ is small as in the case of y 2 = 0.2, the presence of Particle 2 has a negligible effect on Particle l's velocity. However, when the difference between Vp2 and fp\ is big as in the case of y2 = 0.8, the velocity of Particle 1 is significantly influenced by the presence of Particle 2. A faster particle climbing and passing a slower particle In this part, we want to investigate and demonstrate how one particle passing the other particle. Case 6 (Figure 9.33(f)) is considered: the initial positions of the two particles' centers are set at (7, 4, 4.8) and (3, 4, 7.2) respectively and the radius of Particle 2 is twice larger than that of Particle 1. Table 3 A list of particle velocities for different separation distances with yx = 0.2. Case 1 (d*=0.3)

Case 4 (d*= 1.0)

72

VpX

Vp2

Vpl

Vp2

0.1

0.05159

0.05429

0.05092

0.05446

0.2

0.04924

0.04854

0.04995

0.04852

0.8

0.03512

0.01404

0.04410

0.01291

614


The steady state velocity is used as the initial condition and we set j \ = 0.2 and Y2 = 0-8. The results show that Particle 1 moves faster than Particle 2. The paths of two particles during a climbing and passing process from t*=0 to t*=880 are shown in Figure 9.36. It can be seen that Particle 1 climbs up and passes Particle 2, and then climbs down, and Particle 2 climbs down a little during the first half process and then climbs up a little during the second half process. Here the phenomena of climbing up and climbing down would be explained by the unsymmetrical flow field surrounding the particle. Figure 9.37(a) shows the flow field at the state of t*=272 during the first half process. It can be seen that the flow field surrounding the particles is not symmetrical in the z-direction, and the unsymmetrical flow field pushes Particle 1 up while pushes the Particle 2 a little down in the z-direction. Figure 9.3 7(b) shows the flow field at the state of t*=600 during the second half process. As is shown, the flow field surrounding the particles is not symmetrical in the z-direction, and it draws Particle 1 down while draws the Particle 2 a little up in the z-direction.

Figure 9.36. The tracks of particles during a climbing and passing process from t*=0 to t*=880. The dash circles and the solid circles respectively show the initial and the final positions of a small particle and a large particle. The curves denote the tracks of the particles' center positions during the process. The other parameters used in this figure are X = 2 , =62.5kV/m. ft =0.2, n =0.8, and £ „


615

Figure 9.37. The flow field: (a) t*=272, and (b) t*=600, where the vectors denote the flow velocity and the lines with arrows denote the streamlines. The other parameters used in this figure are A = 2, yl = 0.2, y2 = 0.8, and Em = 62.5W I m.

616


REFERENCES [I] [2] [3] [4] [5] [6] [7] [8] [9] [10] II1] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

H. Shintani and J. Polonsky, "Handbook of Capillary Electrophoresis Applications", Blackie Academic & Professional, London, 1997. P.D. Grossman and J.C. Colburn, "Capillary Electrophoresis: Theory and Practice", Academic Press, San Diego, 1992. R.J. Hunter, "Zeta Potential in Colloid Science Principal and Applications", Academic Press, New York, 1981. J. Lyklema, "Fundamentals of Interface and Colloid Science", Academic Press, San Diego, 1991. T.G.M. van de Ven, "Colloidal hydrodynamics", Academic Press, San Diego, 1988. H.J. Keh and J.L. Anderson, J. Fluid Mech., 153 (1985) 417. J. Ennis and J.L. Anderson, J. Colloid Interface Sci., 185 (1997) 497. A.A. Shugai and S.L. Carnie, J. Colloid Interface Sci., 213 (1999) 298. J. Ennis, H. Zhang, G. Stevens, J. Perera, P. Scales and S. Carnie, J. Membrane Sci. 119, (1996)47. A.L. Zydney, J. Colloid Interface Sci., 169 (1995) 476. E. Lee, J.-W. Chu and J.-P. Hsu, J. Colloid Interface Sci., 196 (1997) 316. C. Ye and D. Li, J. Colloid Interface Sci, 251 (2002) 331-338. C. Ye, D. Sinton, D. Erickson and D. Li, Langmuir, 18 (2002) 9095-9101. C. Ye and D. Li, J. Colloid Interface Sci, (in press). J.N. Israelachvili, "Intermolecular and Surface Forces", Academic Press, San Diego, 1991. R. Hogg, T.W. Healy and D.W. Fuerstenau, Trans. Faraday Soc, 62 (1966) 1638. D.C. Henry, Proc. Roy. Soc, London, 133 (1931) 106. H. Ohshima, J. Colloid Interface Sci, 168 (1994) 269. J. Happel, Chapter 7 in Low Reynolds Number Hydrodynamics (2nd ed.), Noordhoff International Publishing, Leyden, 1973. H. Hu, Int. J. Multiphase Flow, 22 (1996) 335-352. R. Glowinski, T.-W. Pan, T.I. Hesla and D.D. Joseph, Comput. Methods Appl. Mech. Engrg.,25(1999)755. J.B. Ritz and J.P. Caltagirone, Int. J. Numer. Mech. Fluids, 33 (1999) 1067. T. Nomura and T.J.P. Hughes, Comput. Methods Appl. Mech. Engrg, 95 (1992)115. C. Johnson, "Numerical Solution of Partial Differential Equations by the Finite Element Method", Cambridge University Press, New York, 1987. O.C. Zienkiewicz and R.L. Taylor, "The Finite Element Method", ButterworthHeinemann, Oxford, Boston, 2000. C. Ye and D. Li, Proceeding of 2nd International Conference on Microchannels and Minichannels, Rochester, New York, June 17-19, 2004.

Microfluidic Methods for Measuring Zeta Potential

617

Chapter 10

Microfluidic methods for measuring zeta potential Zeta potential is an electrokinetic potential at the shear plane (the boundary between the compact layer and the diffuse layer) near a solid-liquid interface where the liquid velocity is zero. Zeta potential is a very important interfacial electrokinetic property to a huge number of natural phenomena, such as electrode kinetics, electrocatalysis, corrosion, adsorption, crystal growth, colloid stability and flow characteristics of colloidal suspensions and electrolyte solutions through porous media and microchannels. For example, zeta potential is a key parameter in determining the interaction energy between particles and hence the stability of colloid suspension systems. In microfluidics applications, as we have demonstrated in the previous chapters, particularly for electroosmotic flows through microchannels, the zeta potential will critically influence the velocity. From the modeling and simulation point of view, we must know the zeta potential values used in the boundary conditions. The surface conductance, another important parameter, usually is referred to the electrical conductivity through the electrical double layer region, i.e., a thin liquid layer near the solid-liquid interface where there is a net charge accumulation due to the charged solid-liquid interface. It is not difficult to understand that surface conductance may have a significant effect on streaming potential (e.g., lOOmV/cm) in pressure-driven flow of dilute aqueous electrolyte solutions in small microchannels. However, for electroosmotic flow, the effect of surface conductance on the electrical field in the microchannel is generally negligible due to the fact that the applied electrical field is usually high (e.g., lOOV/cm). In many cases, knowing the surface conductance is a must in order to evaluate the zeta potential and other electrokinetic properties correctly. Measurement of the surface conductance, therefore, is important in the studies of electrokinetic phenomena. Most techniques for measuring zeta potentials are based on electrophoresis and streaming potential measurements [1-3]. Within the scope of this book, two methods based on electrokinetic flows in microchannels to measure the zeta potential will be reviewed in this chapter. One of these methods involves measurement of the streaming potential in pressure-driven flow, and the other involves electroosmotic flow.

618


10-1 STREAMING POTENTIAL METHOD As discussed in Chapter 2, the rearrangement of the charges on the solid surface and the balancing charges in the liquid is called the electrical double layer (EDL). Immediately next to the solid surface, there is a layer of ions that are strongly attracted to the solid surface and are immobile. This layer is called the compact layer, normally about several Angstroms thick. Because of the electrostatic attraction, the counterions concentration near the solid surface is higher than that in the bulk liquid far away from the solid surface. The coions' concentration near the surface, however, is lower than that in the bulk liquid far away from the solid surface, due to the electrical repulsion. Therefore, there is a net charge in the region close to the surface. From the compact layer to the uniform bulk liquid, the net charge density gradually reduces to zero. Ions in this region are affected less by the electrostatic interaction and are mobile. This region is called the diffuse layer of the EDL. The thickness of the diffuse layer is dependent on the bulk ionic concentration and electrical properties of the liquid, usually ranging from several nanometers for high ionic concentration solutions up to several microns for pure water and pure organic liquids. The boundary between the compact layer and the diffuse layer is usually referred to as the shear plane. The electrical potential at the solid-liquid surface is difficult to measure directly. The electrical potential at the shear plane is called the zeta potential,

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