Passive Mixing in Microchannels with GeometricVariations

by

Hengzi Wang

B.S. (Shanghai Jiao Tong University) 1990M.S. (Shanghai Jiao Tong University) 1993

A thesis submitted to Industrial Research Institute Swinburne (IRIS) infulfillment of the requirements for the degree of Doctor of Philosophy

SWINBURNE UNIVERSITY OF TECHNOLOGY,HAWTHORN, VICTORIA

AUSTRALIA

January, 2004

Passive Mixing in Microchannels with Geometric

Variations

Declaration

This thesis contains no material which has been accepted for the award of any other de-

gree or diploma at any university and to the best of my knowledge and belief contains no

material previously published or written by another person or persons except where due

reference is made.

Hengzi Wang September 19, 2004

i

Abstract

This research project was part of the microfluidic program in the CRC for Microtechnology,

Australia, during 2000 to 2003. The aim of this research was to investigate the feasibility of

applying geometric variations in a microchannel to create effects other than pure molecular

diffusion to enhance microfluidic mixing. Geometric variations included the shape of a

microchannel, as well as the various obstacle structures inside the microchannel.

Generally, before performing chemical or biological analysis, samples and reagents

need to be mixed together thoroughly. This is particularly important in miniaturized To-

tal Analysis Systems (µTAS), where mixing is critical for the detection stage. In scaling

down dimensions of micro-devices, diffusion becomes an efficient method for achieving

homogenous solutions when the characteristic length of the channels becomes sufficiently

small. In the case of pressure driven flow, it is necessary to use wider microchannels to

ensure fluids can be pumped through the channels and the volume of fluid can provide

sufficient signal intensity for detection. However, a relatively wide microchannel makes

mixing by virtue of pure molecular diffusion a very slow process in a confined volume of

a microfluidic device. Therefore, mixing is a challenge and improved methods need to be

found for microfluidic applications.

In this research, passive mixing using geometric variations in microchannels was stud-

ied due to its advantages over active mixing in terms of simplicity and ease of fabrication.

Because of the nature of laminar flow in a microchannel, the geometric variations were de-

signed to improve lateral convection to increase cross-stream diffusion. Previous research

using this approach was limited, and a detailed research program using computational fluid

dynamic (CFD) solvers, various shapes, sizes and layouts of geometric structures was un-

dertaken for the first time. Experimental measurements, published experimental data and

ii

analytical predictions were used to validate the simulations for selected samples. Mixing

efficiency was evaluated by using mass fraction distributions. It was found that the overall

performance of a micromixer should include the pressure drop in a microdevice, therefore,

a mixing index criterion was formulated in this research to combine the effect of mixing

efficiency and pressure drop. The mixing index was used to determine optimum parameters

for enhanced mixing, as well as establish design guidelines for such devices.

Three types of geometric variations were researched. First, partitioning in channels

was used to divide fluids into mixing zones with different concentrations. Various designs

were investigated, and while these provided many potential solutions to achieving good

mixing, they were difficult to fabricate. Secondly, structures were used to create lateral

convection, or secondary flows. Most of the work in this category used obstacles to disrupt

the flow. It was found that symmetric layouts of obstacles in a channel had little effect

on mixing, whereas, asymmetric arrangements created lateral convection to enhance cross-

stream diffusion and increase mixing. Finally, structures that could create complex 3D

advections were investigated. At high Reynolds numbers (Re = 50), 3D ramping or ob-

stacles generated strong lateral convection. Microchannels with 3D slanted grooves were

also investigated. Mixers with grooved surfaces generated helicity at low Reynolds num-

bers (Re≤ 5) and provided a promising way to reduce the diffusion path in microchannels

by stretching and folding of fluid streams. Deeper grooves resulted in better mixing effi-

ciency. The 3D helical advection created by the patterned grooves in a microchannel was

studied by using particle tracing algorithms developed in this research to generate streak-

lines and Poincare maps, which were used to evaluate the mixing performance. The results

illustrated that all the types of mixers could provide solutions to microfluidic mixing when

dimensional parameters were optimized.

iii

Acknowledgments

First of all, I want to thank my family. It is a long and difficult period for both my wife,

Hongtao and my son, Thomas. My wife takes care of all the family issues while she was in

Shanghai and many house-works. Thomas always reminds me his existence and I should

accompany with him more. He loves to play with my computer and my car. Except these,

he is a lovely boy and gives me lots of comforts.

I wish to thank my supervisors, Dr. Pio Iovenitti, Professor Erol Harvey and Associate

Professor Syed Masood for their advice and guidance to complete this research project.

All my colleagues have been friendly and helpful. They provided a comfortable envi-

ronment for researchers. We have about 30 colleagues now, and I may not be able to put

all the names here. But I would like to thank the colleagues who gave me advice about my

research project and on personal issues. Andrew Dowling and I have discussed all sorts of

problems and shared the joy of successes - when we have them. Yao Fu, Tony Liu and Irina

Simdikova are lunchtime chatting mates. Sometimes we do discuss research and science!

I consulted with Dr. Rowan Deam on fluid mechanics several times, and I thank him

for helpful advice. I wish to express my gratitude for the helpful discussion about chaotic

mixing with Dr. A. Stroock from Harvard University, and Dr. M. Rudman and Dr. G.

Metcalfe from CSIRO in Australia.

I visited microfluidic diagnostics laboratory in Stanford University, and I wish to thank

Dr. Santiago for spending much time on discussingµPIV systems, and also his students,

Michael Oddy, Klint Rose, Chuanhua Chen and Jung Byoungsok for talking about their

research projects and showing me around their laboratory.

Some papers in the references were not possible to obtain from the local libraries. I

should thank the authors for sending copies of their papers. Just name a few: Professor

David J. Beebe in University of Wisconsin and Dr. Jin-Woo Choi in University of Cincin-

iv

nati.

This research was mostly funded by CRC for Microtechnology, and I wish to thank

the CRC as well.

v

Contents

Abstract i

Acknowledgments iii

List of Figures xi

List of Tables xviii

Nomenclature xix

List of Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

List of abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii

1 Introduction 1

1.1 Background and Significance of the Research. . . . . . . . . . . . . . . . 1

1.2 Statement of the Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Objectives and Scope of the Research. . . . . . . . . . . . . . . . . . . . 3

1.4 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Literature Review 9

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

vi

2.2 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Fundamental Theories for Microfluidics. . . . . . . . . . . . . . . . . . . 11

2.3.1 Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.2 Conservation of Momentum. . . . . . . . . . . . . . . . . . . . . 13

2.3.3 Brownian Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.4 Taylor-Aris dispersion. . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.5 Chaotic Advection. . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Generation of Fluid Motion in a Microfluidic Device. . . . . . . . . . . . 19

2.4.1 Mechanically Actuated Micropumps. . . . . . . . . . . . . . . . . 20

2.4.2 Electrokinetic Micropumps. . . . . . . . . . . . . . . . . . . . . . 23

2.5 Active Micro-Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5.1 Planar Laminar Bubble Mixer. . . . . . . . . . . . . . . . . . . . 27

2.5.2 Periodic Perturbation Applied to a Main Fluid Stream. . . . . . . 27

2.5.3 Peristaltically Driven Micromixing . . . . . . . . . . . . . . . . . 29

2.5.4 Magnetohydrodynamic (MHD) Micro-mixers. . . . . . . . . . . . 30

2.5.5 Electrohydrodynamic (EHD) Micro-mixers. . . . . . . . . . . . . 32

2.5.6 Magnetic Micro-Stirrer. . . . . . . . . . . . . . . . . . . . . . . . 32

2.6 Passive Microfluidic Mixers . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.6.1 T-type mixers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.6.2 Splitting and Recombining. . . . . . . . . . . . . . . . . . . . . . 41

2.6.3 Focusing Fluid Streams. . . . . . . . . . . . . . . . . . . . . . . . 46

2.6.4 Inject Small Fluid Substreams into a Main Stream. . . . . . . . . 47

2.6.5 Passive Microfluidic Chaotic Mixers. . . . . . . . . . . . . . . . . 48

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

vii

3 Numerical Modelling and Experimental Methodology 57

3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2 Fundamentals of Computational Fluid Dynamics (CFD). . . . . . . . . . . 59

3.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2.2 CFD Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3 CFD Modelling in Microfluidics . . . . . . . . . . . . . . . . . . . . . . . 63

3.3.1 System of Units in Microfluidics. . . . . . . . . . . . . . . . . . . 64

3.3.2 Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . 64

3.3.3 Building Numerical Models for Micro-Flows. . . . . . . . . . . . 65

3.3.4 Simulation Results Analysis. . . . . . . . . . . . . . . . . . . . . 71

3.4 Experimental Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.4.1 Excimer Laser Micro-machining. . . . . . . . . . . . . . . . . . . 74

3.4.2 Microfluidic Packaging. . . . . . . . . . . . . . . . . . . . . . . . 75

3.4.3 Microfluidic Mixing Visualization. . . . . . . . . . . . . . . . . . 75

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4 Understanding Microfluidic Mixing 79

4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.2 Flow at Low Reynolds Number. . . . . . . . . . . . . . . . . . . . . . . . 80

4.3 Taylor-Aris Dispersion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.3.2 Dispersion of a Solute in Solvent Flowing Slowly Through a Rect-angular Cross-Section Microchannel. . . . . . . . . . . . . . . . . 86

4.3.3 Applying Taylor-Aris Dispersion to Enhance Mixing. . . . . . . . 89

4.4 Two Important Parameters for Designing a Microfluidic Mixer. . . . . . . 91

4.4.1 Peclet number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

viii

4.4.2 Aspect Ratio of a Rectangular Microchannel. . . . . . . . . . . . 95

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5 Microfluidic Mixing by Partitioning Flows 101

5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.2 Dividing Fluids into Discrete Concentration Zones. . . . . . . . . . . . . 102

5.3 Dollar Sign Shaped Micromixers. . . . . . . . . . . . . . . . . . . . . . . 104

5.4 3D Ramping Micro-mixers. . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.4.2 Simulations of 3D Ramping Micro-mixers. . . . . . . . . . . . . . 110

5.4.3 Discussion of Results. . . . . . . . . . . . . . . . . . . . . . . . . 110

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6 Optimizing Layout of Cylindrical Obstacles in Microchannels for EnhancedMixing 116

6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.2 Numerical Modelling of Mixing in a Microchannel with Cylindrical Ob-stacles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.3 Results and Discussions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.3.1 Simulation Results. . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.3.2 Experimental Results. . . . . . . . . . . . . . . . . . . . . . . . . 126

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7 Enhancing Mixing in Microchannels Using Square and Rectangular Obstacles129

7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.2 Numerical Modelling of Square and Rectangular Obstacles. . . . . . . . . 130

7.3 Results and Discussions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

ix

7.3.1 Number of Obstacles. . . . . . . . . . . . . . . . . . . . . . . . . 133

7.3.2 Size of Obstacles. . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.3.3 Angle of Obstacles. . . . . . . . . . . . . . . . . . . . . . . . . . 139

7.3.4 Obstacle Offsets from Channel Walls. . . . . . . . . . . . . . . . 143

7.3.5 Gap between Obstacles. . . . . . . . . . . . . . . . . . . . . . . . 145

7.3.6 Flow Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

7.3.7 Ratio of Height of Obstacles to Depth of Channel. . . . . . . . . . 150

7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

8 Numerical Investigation of Mixing in Microchannels with Patterned Grooves 154

8.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8.2 Numerical Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

8.3 A Particle Tracing Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . 158

8.3.1 Point location. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

8.3.2 Interpolate velocity. . . . . . . . . . . . . . . . . . . . . . . . . . 160

8.4 Results and Discussions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

8.4.1 Flow patterns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

8.4.2 Streaklines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

8.4.3 Poincare maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

8.4.4 Simulation of Mixing in Microchannels with Patterned Grooves. . 169

8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

9 Conclusions and Further Research 175

9.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

9.2 Contributions of the Research. . . . . . . . . . . . . . . . . . . . . . . . 175

9.2.1 Clarification of Related Theories to Microfluidic Mixing. . . . . . 175

x

9.2.2 Modifications to Serpentine-Shaped Microchannels. . . . . . . . . 178

9.2.3 Design Guidelines for Obstacle Structures. . . . . . . . . . . . . . 179

9.2.4 Design of Microfluidic Mixers with Patterned Grooves. . . . . . . 181

9.3 Limitations of the Research. . . . . . . . . . . . . . . . . . . . . . . . . . 182

9.4 Recommendations for Further Research. . . . . . . . . . . . . . . . . . . 183

9.4.1 Experimental Methods. . . . . . . . . . . . . . . . . . . . . . . . 183

9.4.2 Complex 3D Geometry. . . . . . . . . . . . . . . . . . . . . . . . 184

9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

References 188

Appendices 200

A List of Publications 200

A.1 Journal Publications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

A.2 International Conference Publications. . . . . . . . . . . . . . . . . . . . 201

A.3 Local Publications and Presentations. . . . . . . . . . . . . . . . . . . . . 202

B Selected Fortran Codes Developed to Study Microfluidic Mixing 203

B.1 Fortran Codes for Mesh Conversion. . . . . . . . . . . . . . . . . . . . . 203

B.2 Fortran Codes for Velocity Field in a Rectangular Duct. . . . . . . . . . . 211

B.3 Fortran Codes for Particle Tracing - Main Program. . . . . . . . . . . . . 215

xi

List of Figures

2.1 Statical fluid thermophysical point value. . . . . . . . . . . . . . . . . . . 12

2.2 Stirring of a ‘blob ’of marked fluid by 2D, time-dependent (time seriesfrom t0∼ t6), laminar flow . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Flow rates and pressure heads of micropumps could deliver - a review. . . 20

2.4 Disk piezoelectric reciprocating micropump. . . . . . . . . . . . . . . . . 21

2.5 A typical passive micro check valve. . . . . . . . . . . . . . . . . . . . . 22

2.6 Tesla micropump. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.7 2D chaotic advection microfluidic mixer. . . . . . . . . . . . . . . . . . . 28

2.8 Microfluidic mixer by periodical perturbation. . . . . . . . . . . . . . . . 29

2.9 Mechanical travelling waves induced by acoustic streaming. . . . . . . . . 30

2.10 A minute MHD micro-mixer. . . . . . . . . . . . . . . . . . . . . . . . . 31

2.11 (a). Schematic illustration of the active micro-mixer (b). Cross sectionalview of the active micro mixer. . . . . . . . . . . . . . . . . . . . . . . . 33

2.12 Mixing processes in the micro-mixer by applying electrokinetic instability. 34

2.13 Magnetic micro stirrer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.14 Mixing in microchannel by magnetic micro stirrer, (a). 0 rpm, (b). 150rpm, and (c). 300 rpm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.15 Mixing T structures, (a). Parallel configuration; (b). Serial configuration. . 37

2.16 Streamlines of engulfment flow. . . . . . . . . . . . . . . . . . . . . . . . 38

xii

2.17 Double-T micro-reactor. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.18 A schematic working principle of T-Sensor. . . . . . . . . . . . . . . . . 40

2.19 Parallel dividing into substreams.. . . . . . . . . . . . . . . . . . . . . . . 42

2.20 In-situ static micro-mixer by subdividing streams. . . . . . . . . . . . . . 43

2.21 Divided flow into substreams (a) vertical arrangement; (b) horizontal ar-rangement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.22 (a). Channel structures; (b). Schematic drawing of its split-recombine-principle; (c). Streamlines in a caterpillar mixer.. . . . . . . . . . . . . . . 44

2.23 Danfoss mixer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.24 Static Micro-mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.25 IMM super-focusing micro-mixer. . . . . . . . . . . . . . . . . . . . . . 47

2.26 Hydrodynamic focusing micro-mixer. . . . . . . . . . . . . . . . . . . . . 48

2.27 IMM-Mainz Mixing unit . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.28 Injection fluid into a mixing chamber. . . . . . . . . . . . . . . . . . . . . 49

2.29 (a). Mixing Chamber with Micro-plumes (b). Simulation of sub-jets ofreagent into main stream. . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.30 Serpentine-Shaped Micro-mixers. . . . . . . . . . . . . . . . . . . . . . . 51

2.31 Microfluidic chaotic mixer with grooved surfaces. . . . . . . . . . . . . . 52

3.1 Viscous flow in a rectangular micro-channel. . . . . . . . . . . . . . . . . 67

3.2 A typical meshed fluid volume in a rectangular micro-channel. . . . . . . 68

3.3 Normalized velocity profileu(y, zmid) in xOy plane . . . . . . . . . . . . . 68

3.4 Normalized velocity profileu(ymid, z) in xOz plane . . . . . . . . . . . . . 69

3.5 Mesh topology, (a). Gambitr 8-Hexahedral mesh; (b). MemCFDTM

8-Hexahedral mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.6 (a). Structured 8-node Hexahedral mesh by sub-mapping technique; (b).Irregular 8-node structured mesh by MemCFD

TM. . . . . . . . . . . . . . 70

xiii

3.7 Mesh uniformity study, (a). irregular mesh elements; (b). uniform meshelements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.8 Mixing of two fluids in a T-channel with grooved surfaces, top: simulationresult, two fluids are represented by blue and red, green colour is the mixedregion, bottom: cited experimental results by Stroock, et al. 2002a, twofluids are represented by gold and black colours.. . . . . . . . . . . . . . 73

3.9 A schematic illustration of Excimer laser micromachining. . . . . . . . . 74

3.10 Section of Y-channel machined in Polycarbonate by Excimer Laser (SEMimage) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.11 Laboratory setup for micro-mixing visualization. . . . . . . . . . . . . . . 76

4.1 Rectangular microchannel. . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2 Normalized velocity profile in a rectangular microchannel, w/H = 4;u∗ isnormalized velocity,y∗ = y/w andz∗ = z/H . . . . . . . . . . . . . . . . . 83

4.3 Normalized velocity profile by the maximum flow velocity plotted in XZplane fory = ymid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.4 Normalized velocity profile plotted in XY plane forw/H = 2 (O),w/H =5 (×) andw/H = 10 (+). . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.5 Ratio of Taylor dispersivity to pure molecular diffusion coefficient for dif-ferent Reynolds numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.6 Taylor dispersion along a straight channel: numerical simulation for vari-ous time intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.7 Experimental results: the solute dispersed at, (a) timet and (b) timet + δt . 91

4.8 Three microfluidic mixing zones defined by using the Peclet number. . . . 94

4.9 Pressure drop along a rectangular microchannel with the same cross-sectionarea, for different height to width aspect ratio, at various flow ratesq . . . . 97

4.10 Aspect Ratio and flow rate affect the total length to achieve complete mixing97

5.1 Different concentration gradient separated in the branches, (a). relativemass fraction distribution in percentage; (b). corresponding mass fractionin contour plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

xiv

5.2 Mass fraction in a planar$-mixer . . . . . . . . . . . . . . . . . . . . . . . 105

5.3 Streaklines mapped with mass fraction in the planar$-mixer . . . . . . . . 106

5.4 Mass fraction in 3D variation of the planar $-mixer. . . . . . . . . . . . . 107

5.5 Streaklines mapped with mass fraction in the 3D $-mixer. . . . . . . . . . 108

5.6 Two samples of 3D ramping structures made by Excimer Laser microma-chining. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.7 Computer model of a ramping structure meshed with 8-node Hexahedrons. 110

5.8 A 3D ramping mixer, at Re = 5. . . . . . . . . . . . . . . . . . . . . . . . 111

5.9 Mass fraction distribution in a 3D ramping mixer, at Re = 50. . . . . . . . 113

5.10 Streaklines in 3D ramping mixer mapped with mass fraction, at Re = 50. . 114

6.1 3D model of a T-channel with two arrays of cylindrical obstacles. . . . . . 118

6.2 Layout of square and triangular configuration of obstacle array. (a). con-figuration No.6. (b). configuration No.7 (c). configuration No.8. . . . . . . 118

6.3 Meshed fluid volume in a T-type channel with two arrays of cylindricalobstacles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.4 Relative mass fraction distribution in solution in a microchannel with twoarrays of cylindrical obstacles.. . . . . . . . . . . . . . . . . . . . . . . . 121

6.5 Relative mass fraction distribution in solution in a microchannel with twoarrays of cylindrical obstacles.. . . . . . . . . . . . . . . . . . . . . . . . 122

6.6 Evolution of design with obstacles in the Y-channel for configurations 1 to 8123

6.7 Finite element simulation results of (a) mixing efficiency, pressure dropbetween inlet and outlet versus number of obstacles, and (b) Mixing indexversus number of obstacles(configuration no.1∼ no.8). . . . . . . . . . . . 124

6.8 Section of the channel illustrated the convective effect using velocity vectorfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.9 Channel with cylindrical obstacles (SEM image). . . . . . . . . . . . . . 126

6.10 Configuration no.1, channel without cylindrical obstacles (food colouring),Re = 0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

xv

6.11 Configuration no.5, channel with 3 cylindrical obstacles (Ferric Ammo-nium Sulfate and Ammonium Thiocyanate solutions), Re= 0.4. . . . . . . 127

6.12 Configuration no.8, channel with 18 cylindrical obstacles (food colouring),Re = 0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.1 T-channel with 4 rectangular obstacles, (a) 3D view; (b) layout.. . . . . . . 131

7.2 Mass fraction distributions, (a) symmetric layout of obstacles, (b) asym-metric layout of obstacles. . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.3 Mixing efficiency versus number of square obstacles. . . . . . . . . . . . 133

7.4 Pressure drop versus number of square obstacles. . . . . . . . . . . . . . . 134

7.5 Mixing index,midx, versus number of square obstacles. . . . . . . . . . . 135

7.6 Angle between flow axis and obstacle axis, two and three obstacles. . . . . 136

7.7 Mixing efficiency with two and three obstacles inside a microchannel ver-sus angle of layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.8 Pressure drop versus two and three square obstacles. . . . . . . . . . . . . 137

7.9 Mixing index,midx, versus two and three square obstacles. . . . . . . . . 138

7.10 Mass fraction distribution for two rectangular obstacle layouts with differ-ent size of obstacles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.11 Mixing efficiency versus size of obstacles. . . . . . . . . . . . . . . . . . 139

7.12 Pressure drop versus size of obstacles. . . . . . . . . . . . . . . . . . . . 140

7.13 Mixing index,midx, versus size of obstacles. . . . . . . . . . . . . . . . . 140

7.14 Mass fraction distribution, various angles of obstacle layouts. . . . . . . . 141

7.15 Mixing efficiency versus angle of rectangular obstacles. . . . . . . . . . . 141

7.16 Pressure drop versus angle of obstacles. . . . . . . . . . . . . . . . . . . . 142

7.17 Mixing index,midx, versus angle of obstacles. . . . . . . . . . . . . . . . 142

7.18 Mass fraction distribution, for different offsets of obstacles from channelwalls, (a). no offset from the center; (b). with offset.. . . . . . . . . . . . . 143

7.19 Mixing efficiency versus offset from the wall. . . . . . . . . . . . . . . . 144

xvi

7.20 Pressure drop versus offset of obstacles from wall. . . . . . . . . . . . . . 144

7.21 Mixing index,midx, versus offset of obstacles from wall. . . . . . . . . . 145

7.22 Mass fraction distribution for different obstacle gaps, (a). 160µm gap; (b).400µm gap; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7.23 Mixing efficiency versus gap between obstacles. . . . . . . . . . . . . . . 146

7.24 Pressure drop versus gap between two obstacles. . . . . . . . . . . . . . . 147

7.25 Mixing index,midx,versus gap,g, between two obstacles. . . . . . . . . . 147

7.26 Mixing efficiency versus flow rates. . . . . . . . . . . . . . . . . . . . . . 148

7.27 Pressure drop versus Reynolds number. . . . . . . . . . . . . . . . . . . . 149

7.28 Mixing index,midx, versus Reynolds number,Re . . . . . . . . . . . . . . 149

7.29 Mass fraction versus obstacle height to channel depth aspect ratio,h/H,(a). h/H = 0.2, (b). h/H = 0.9 . . . . . . . . . . . . . . . . . . . . . . . 150

7.30 Mixing efficiency versus height to depth aspect ratio of obstacles. . . . . . 151

7.31 Pressure drop versus obstacle height to channel depth ratio. . . . . . . . . 151

7.32 Mixing index,midx, versus obstacle height to channel depth ratio,h/H . . 152

8.1 Fluid volume in microchannel with grooved bottom surface.L0 is the entrychannel length,Lr is the grooved channel length andLm is the channellength to complete the mixing by diffusion. . . . . . . . . . . . . . . . . . 156

8.2 Meshed fluid volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

8.3 Hexahedral cell decomposed into 5-Tetrahedrons. . . . . . . . . . . . . . 160

8.4 Tetrahedron geometry.Ni, i = 1, 4. are the shape functions. . . . . . . . . 161

8.5 One periodic section of patterned grooves, with section planes labelledfrom 0 ∼ 2π, and particles advected from plane 0 to plane2π . . . . . . . . 163

8.6 Cross section velocity vector planes,0 ∼ 2π, Re = 5. . . . . . . . . . . . . 164

8.7 Streaklines of two neighboring fluid particles from two fluid streams re-spectively,α = 0.20, Re = 5 . . . . . . . . . . . . . . . . . . . . . . . . . 165

8.8 Gap between the two Streaklines in Figure 8.7. . . . . . . . . . . . . . . . 166

xvii

8.9 Poincare map,45o patterned grooves: (a)α = 0.05, Re = 5; (b)α = 0.30,Re = 5; (c)α = 0.30, Re = 0.01. . . . . . . . . . . . . . . . . . . . . . . . 167

8.10 Length of grooved channel to complete one circulation, forH/w = 0.5 . . 168

8.11 Two fluid streams flow into a T-type channel with patterned grooves,α =0.10 andRe = 5. (a). The interfacial line is indicated by the bright mixedregion. The helicity was measured by the angleΩ between this interfacialline and the channel axis, (b). The visualization of mixing. . . . . . . . . . 170

8.12 Interfacial regions whenα = 0.30 andRe = 5 . . . . . . . . . . . . . . . 171

8.13 Mean helicity measured by the angleΩ for H/w = 0.5 . . . . . . . . . . . 171

8.14 Helicity at different Reynolds numbers, numerical estimations forα = 0.10 172

B.1 Flow chart to convert gambit neutral mesh to universal mesh. . . . . . . . 205

B.2 Flow chart for viscous flow in a rectangular channel. . . . . . . . . . . . . 212

B.3 Flow chart to compute particle trajectories. . . . . . . . . . . . . . . . . . 216

xviii

List of Tables

3.1 Systems of units in microfluidics. . . . . . . . . . . . . . . . . . . . . . . 65

4.1 Values ofk in a rectangular-cross-section channel for various width todepth aspect ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.1 Configuration of microchannel with Cylindrical obstacles. . . . . . . . . . 117

6.2 Properties of water and ethanol at 20 ˚ C. . . . . . . . . . . . . . . . . . . 120

xix

Nomenclature

The symbols listed in the chapters follow the convention of international system of

units (SI), SI base units and SI derived units. In practice, some adjustment of the system of

units are described in Chapter3 for the convenience of microfluidic applications.

List of Variables

Symbol Description SI unit

A Surface of control volume m2

c relative mass fraction

dh Hydraulic diameter m

D Diffusion coefficient m2 s−1

D∗ Virtual (Taylor) dispersivity m2 s−1

E Electrical field V

e Internal energy Nm

F Force N

f Body force N

FD Friction force between liquid and wall N

g Gravitational acceleration m s−2

H Depth of channel m

xx

h Height of obstacles (or depth of groove) m

I Moment of Inertia kg m2

J Capillary force along the channel N

k constant in Equation4.6

Kn Knudsen number

l Characteristic length m

lm Channel length for a complete mixing m

L Length of the channel m

m Mass kg

meff mixing efficiency

midx mixing index

N number of molecules in a fluid point

n number of samples

Ni ith shape function

p Pressure Pa

p(t) particle position m

P Power W

Pe Peclet number

Pn Surface force N

q volumetric flow rate m3 s−1

qλ Conductive heat N m

qR Radiative heat N m

r radius of channel m

Re Reynolds number

t Time s

tc Convection time s

tD Diffusion time s

T Temperature K

xxi

TCE Thermal Coefficient of Expansion K−1

um Mean velocity in the channel m s−1

v Velocity field m s−1

v1 velocity along x-axis m s−1

v2 velocity along y-axis m s−1

v3 velocity along z-axis m s−1

w width of channel m

Wx Work from outside control volume N m

x x-coordinate m

y y-coordinate m

z z-coordinate m

α aspect ratio of half groove depth to depth of channel

β aspect ratio of channel depth to channel width

ε permittivity C V −1 m−1

γ Surface tension factor Nm−1

λ Mean free path of a molecule m

µ Dynamic viscosity Pa s

ν kinematic viscosity m2 s−1

ρ Density kg m−3

ρE Electric charge density C m−3

τ The lump of fluid in the control volume m3

Ω Angle degree

ψ The electrical potential V

φ Diameter m

ζEO electroosmotic zeta potential V

ζEP electrophoretic zeta potential V

xxii

List of abbreviations

The list of abbreviations used in this thesis is shown below.

Abbreviation Description

CAD Computer Aided Design

CFD Computational Fluid Dynamics

DRIE Deep Reactive Ion Etching

EDL Electrical Double Layer

FEA Finite Element Analysis

FEM Finite Element Method

FVM Finite Volume Method

IC Integrated Circuit

LIGA Lithographic Galvonoformung Abformung

(lithography, electroplating, molding)

MEMS MicroElectroMechanicSystem

MTTF Mean Time To Failure

µTAS Miniaturized total analysis systems

1

Chapter 1

Introduction

1.1 Background and Significance of the Research

This work was conducted in the Industrial Research Institute Swinburne (IRIS) dur-

ing the years from 2000 to 2003. It was part of the Microfluidics project supported by

Cooperative Research Centre (CRC) for Microtechnology in Australia.

Fluidic devices made by MicroElectroMechanicSystem (MEMS) technology can be

termed microfluidics, which have dimensions ranging from microns to a few millime-

ters (Gravesen, Branebjerg & Jensen 1993). Microfluidics research commenced several

decades ago. Early in the mid of 60s, pressure transducers started to utilize microfluidic

techniques. However, it is generally recognized that microfluidics originated at Stanford

University in the development a gas chromatograph system in 1975. Later in 1977, IBM

adopted the idea and incorporated it in the manufacture of inkjet printers (Petersen 1982).

In the years that followed, further research was done but at a quite moderate level . How-

ever, in 1990, Manz introduced the concept of miniaturized total chemical analysis systems

(µTAS) (Manz, Graber & Widmer 1990), which could shrink the whole chemical analysis

2

laboratory to an integrated microfluidic chip that could be held in a human hand. Since

then, the research in microfluidics has dramatically increased, showing its great potential

significance in a wide range of technologies, such as pressure sensors, medical diagnostic

systems, chemical synthesis.

TheµTAS concept provides an overarching vision and compelling motivation for de-

velopment of integrated microfluidic systems. However, before fully developed systems

can be put in the market place, sufficient development of all the necessary components to

build such systems needs to be achieved. Among these individual components, the unit

for mixing samples and reagents is critical for the subsequent chemical reactions and de-

tections. The mixing unit has to provide sufficient mixed solution in a confined length

of a micro-device, and before the point where detection is taking place. Mixing in mi-

crofluidic devices can be achieved by active mixing methods and passive mixing methods.

Passive mixing using geometric variations, which includes the shape of a microchannel

and other obstacle structures inside the microchannel, provides a versatile solution to sim-

plify the design and fabrication of microfluidic device, and is totally compatible with the

current microfabrication techniques. Hence, it is a cost-effective mixing method and has

great potential in a practicalµTAS. Therefore, as a contribution to building this critical unit

for µTAS, the author presents his research on passive microfluidic mixing with geometric

variations.

1.2 Statement of the Problem

To achieve a homogenous mixed solution of samples and reagents in a microfluidic

device, the dimensions such a device are too small to put reliable mechanical agitation to

stir the fluids. Naturally, the Reynolds numbers of microfluidic devices are low due to their

small dimensions. Hence, viscosity predominates the flows, especially for aqueous solu-

tions of most bio-materials. In such viscous flows, even though one can put mechanical

3

agitation into a microfluidic device, it is not likely to efficiently improve mixing perfor-

mance (Purcell 1977). Hence, mixing relies on the virtue of molecular diffusion.

On the one hand, microchannels are elegant and efficient for mixing fluids in narrow

channels with dimensions up to a few tens of microns. Mixing can be completed by dif-

fusion without any other stirring assistance. This makes microfluidics a very controllable

and predictable technique for rapid mixing. On the other hand, it is difficult to drive flow

through such small channels due to viscosity. Although fluid can be driven electrokineti-

cally, electrokinetically driven flows introduce some other problems, such as surface chem-

istry, gas bubbles created by electrolysis and Joule heating. Pressure driven mechanisms

have commonly been used in microfluidic devices, however, a relatively wider channel is

necessary to reduce the pressure drop. In addition, sometimes a sufficient volume of fluid

is needed for a detection stage, and this also requires a relatively large channel. For mi-

crochannels with dimensions over 100µm, diffusion is not generally effective within the

confined space of micro-devices, especially for bio-materials with very low diffusion coef-

ficients. These require a long channel for adequate residual time for diffusion to take place.

However, the pressure drop is proportional to the length, and it is difficult to drive the flow

through a long microchannel. For the above reasons, methods have to be found to reduce

the diffusion distance between fluid streams without restricting the flow.

In brief, microfluidic mixing is challenging and attracts the attention of many re-

searchers.

1.3 Objectives and Scope of the Research

The principal objective of this research was to improve the passive mixing efficiency

using geometric configurations in microchannels.

Although many researchers have been attempting to apply active mechanisms to im-

4

prove mixing performance in a microfluidic device, the complexity of active mechanisms

makes it difficult to fabricate and expensive to make. Passive mechanisms have been stud-

ied extensively, however, previous research on passive mixing has concentrated on typically

one of geometric factors, and an overall view of geometric configurations and their effect

on mixing is not clear. Therefore, it is the intention of this research to provide a compre-

hensive research on passive mixing with various geometric configurations.

To achieve this goal, this research systematically investigated the relation between var-

ious geometric configurations and their stirring mechanisms towards microfluidic mixing

by using computational fluid dynamic (CFD) simulations and experimental validations. As

microfluidics is so diverse that a particular design for mixing may be efficient for some

applications, it may not be efficient for other applications. Therefore, an overall systematic

investigation of geometric configurations is necessary to provide useful guidelines for the

design of microfluidic mixing components.

Many micro-fabrication processes create certain surface roughness or structures in-

side microchannel, these surface roughness or structures may have influences on mixing.

Some of these structures are actually obstacles to the flow. Obstacles can easily be micro-

fabricated, and some similar structures had been studied, therefore, this research chose

obstacles or similar structures in microchannels to study passive mixing. The various com-

binations of obstacle structures influence the flow pattern differently, therefore their effects

on mixing can be evaluated.

This research also includes other complementary objectives. These objectives are

necessary for the principal objective to be carried out.

• At the early stage of this research, the theories on microfluidics and microfluidic mix-

ing needed to be identified and clarified. From a certain point of view, microfluidics

is just another terminology for flow in a micro-domain with slow motion and not

necessarily a new technology. Therefore, existing theories for incompressible slow

5

motion flow may help the research on micro-flow and microfluidic mixing.

• Experimental methods needed to be designed to be able to visualize micro-flow and

microfluidic mixing, for the purpose of validation of the chosen theories.

• The major tool for the research is computational fluid dynamics (CFD) package. In

order to provide accurate simulation results, necessary customized coding needed

to be developed to improve the quality of CFD simulations. These include using

external mesh generator to provide uniform mesh and import to the CFD package for

simulation.

• In order to simulate the 3D convection, a set of particle tracing algorithms needed

to be developed to investigate mixing in spatially configured periodic structures in

microchannels. The algorithms can help to reveal the details of 3D convection in

such channels, and provide useful guidelines for the design of this type of micro-

mixer.

The scope of this research was restricted by several aspects. When width of a mi-

crochannel is of the order of a few tens of microns, diffusion mixing is sufficient without

other means to assist. WhenRe ¿ 1, the velocity in a microchannel is slow and the resid-

ual time is sufficient for diffusion mixing to take place. On the other hand, whenRe À 1,

this creates not only high a pressure drop, but also excessive volume of solution for the

subsequent detection. Nevertheless, one advantage of microfluidic devices is the low sam-

ples and reagents consumption, and using high Reynolds number reverses this advantage.

Therefore, high Reynolds numbers are outside of the scope of this research. The research

focused on mixing of two fluids in a microchannel with dimensions larger than100µm, and

with the Reynolds numbers were typically less than 10, but sufficient enough to produce a

detectable volume of solution.

6

1.4 Outline of this thesis

To achieve the objectives of this research, the very first step was to understand what

had been researched and why there were still problems. This was addressed in Chapter

2 by a review of literature and related theories to microfluidic mixing. Chapter2 also

demonstrates the complexity and diversity of microfluidic mixing. One outcome from the

literature review was that the periodic disruption to flow potentially favours mixing, be-

cause most of the micromixer designs applied periodic disruptive forms temporally and/or

spatially. However, the periodic disruption is only one of the necessary conditions that

should be satisfied, and might not be always correct, as later addressed in Chapters6 and

7.

In chapter3, the methodology to conduct CFD simulations and construct a simple

experimental rig are described. There are various geometric obstacles which may be incor-

porated in a microchannel, however, it is impractical to micro- machine each of the designs

and experimentally test them. Consequently, CFD was applied to extract useful information

for designing passive microfluidic mixers. To validate CFD simulations, selected designs

were micro-fabricated and tested, and compared to the simulation results. In addition, an-

alytical prediction of viscous flow in a rectangular channel was also used as benchmark of

the simulations.

Chapter4 identified the general issues surrounding microfluidic mixing. In a study

of all the parameter, flow rate was found to be the most important parameters. Flow was

measured by Reynolds and Peclet numbers, and the Peclet number is defined as the ratio of

bulk mass convection to molecular diffusion. Micro-flow can be recognized as flow at very

low Reynolds number, and as time is implicitly defined in flow rate, the residual diffusion

time is also defined. One of the general parameters for microchannels is the height to width

aspect ratio. For predefined flow rate, very high channel aspect ratios reduce the thickness

of fluid layers for fluid to diffuse through. However, high aspect ratio structures need spe-

7

cial microfabrication techniques, such as deep reactive ion etching (DRIE) or LIGA, which

are expensive and not generally available. Therefore, the research concentrated on low

aspect ratio planar micro-fabricated structures, which were much easier for microfabrica-

tion. Moreover, Taylor dispersion was found to have tremendous influence on microfluidic

mixing, and the understanding of Taylor dispersion also provided information about what

microfluidics could, and could not do in an economical sense.

Chapter5 introduced two novel designs in splitting and recombining fluid streaming

with low pressure drop. From Chapter6 to 8, obstacles or similar structures inside mi-

crochannels were explored by simulations and experiments. A very important outcome

from these chapters was that asymmetric structures provided more useful disruption of

flows to improve mixing, and that a symmetric layout of structures had little effect on

mixing performance. Chapter6 applied cylindrical obstacles to investigate influences on

mixing by optimizing the layout and number of obstacles. An extensive study of obsta-

cles was carried out in Chapter7. Rectangular obstacles were applied to provide necessary

parameters for the study. These parameters included size, gap, angle, obstacle height to

channel depth aspect ratio and position of obstacles. An interesting result was obtained

when the obstacle height to channel depth aspect ratio became negative, which meant the

obstacles actually became grooves in the channel wall. The grooved surfaces provided lo-

cal slip boundary condition, which induced uneven distributed resistance to the flow. The

uneven distributed resistance created lateral convection. A further study of grooved sur-

faces in microchannels was carried out by applying particle tracing algorithms in Chapter

8.

The thesis was concluded with conclusions and recommendations for future research

in Chapter9. This chapter highlighted a number of important outcomes emerging from the

present study. The investigation of geometric structures to achieve better disruption flows

to create useful convection for mixing provided useful design guidelines for microfluidic

mixing. The extensive utilization of CFD simulations validated selectively by experiments,

8

or published experimental data, was proven to be a rapid and reliable way to optimize the

performance of microfluidic devices.

9

Chapter 2

Literature Review

2.1 Overview

In this chapter, the fundamentals of microfluidics and theories related to mixing are

first reviewed in sections2.2and2.3to understand the speciality of mixing in micro-scaled

devices. Then, in section2.4 fluid drive mechanisms are briefly described. The drive

mechanisms determine the flow rate and pressure drop, and are used to confine the research

scope on microfluidic mixing. The existing microfluidic mixing techniques were catego-

rized into active and passive micro-mixers, which were covered in sections2.5 and 2.6

respectively. Section2.7summarizes the findings of the literature review and identifies the

areas of research which form the basis of the present research.

2.2 Background

In fluid dynamics, in order to study the flow pattern transition from laminar to fully

developed turbulent flow, the Reynolds number must exceed the first critical transitional

10

number (Recr1 ∼ 2300) and even the second critical number (Recr2 > 105). When the

Reynolds number is larger than the first transitional Reynolds number, the flow turns to tur-

bulent under the disturbance, however, as soon as the disturbance is removed, the flow turns

back to laminar flow. When the Reynolds number is beyond the second critical transitional

number, any disturbance to the flow causes turbulent flow, and remains turbulent even af-

ter the disturbance was removed (Pan 1988). For the majority of microfluidic devices, the

dimensions were well below 1mm, which made the Reynolds numbers well below the crit-

ical transitional thresholds. Therefore, it was generally accepted that flow in microfluidic

devices was laminar, especially when the working fluid was liquid.

Laminar flow was better understood and easier to manipulate than turbulent flow,

therefore, the laminar nature of microfluidics was an advantage. On the other hand, mi-

crofluidic mixing could not obtain any assistance from turbulence. This meant that any

mixing was to be performed by virtue of diffusion, and diffusion was driven by the concen-

tration gradient between two or more fluids, in accordance with Fick’s law (Bird, Stewart

& Lightfoot 1960). In fact, all the mixing processes for miscible fluids are completed by

diffusion eventually. The engineering challenge is to make the diffusion path small enough

that the process can take place rapidly.

In a macroscopic mixing process, mechanical agitations are normally applied to create

turbulent or chaotic advection. Turbulence stretch and break down large fluid streams into

smaller fluid filaments, so that the diffusion path can be reduced to a microscopic scale.

Eventually, the mixing process is completed by diffusion to create a homogenous solution.

However, it is impractical to bring mechanical agitation into a microchannel for two rea-

sons: first, it is difficult to fabricate; and secondly, it is not an effective approach due to the

low Reynolds number in a microchannel (Purcell 1977).

Diffusion in a microchannel up to a few tens of micron width was found to be effi-

cient for most applications. However, when the size of a microchannel was more than a

hundred microns, pure molecular diffusion could be very slow to achieve complete mixing.

11

The length of channel to achieve a complete mixing by virtue of diffusion would be very

long. As the resistance to the flow was proportional to the length of the channel, this also

meant increasing pressure drop and the micropump might fail to drive the fluids through

the channels. This makes it difficult to design microfluidic chips with such long channels,

therefore, it was necessary in the present research to investigate the techniques to improve

mixing within the confined space of microfluidic devices.

In the following sections, the fundamentals of microfluidics and mixing are discussed,

and the existing active and passive microfluidic mixing methods are reviewed.

2.3 Fundamental Theories for Microfluidics

When a fluidic channel is scaled down to micro-scale, the scaling effects make some

physical parameters, such as gravity and inertia, less important than parameters like surface

tension and viscosity. Conventional fluid mechanics still applies to study microfluidics,

however, some assumptions need to be clarified.

2.3.1 Continuum

In fluid mechanics, a fluid is a substance that deforms continuously under shear stress.

Fluids are composed of molecules in motion. A truly detailed picture of flow will pre-

sumably track each individual molecule. However, this was still impractical for current

computational or experimental resources. Instead of tracking individual molecules, the

fluid could typically be treated as a continuum, which could be infinitely divisible and had

a thermophysical property value at each point in space.

In practice, there was a smallest point that the fluid could be divided into. The volume

of this point should contain a sufficient number of molecules,N , that could give a statisti-

12

cally thermophysical value. According to the random process theory,N was approximately

104 to achieve a statistic point valueρ (Figure2.1). So the continuum assumption holds

true for average scale length L∼ 1µm for gases and L∼ 100nm for liquids. To simplify

Figure 2.1:Statical fluid thermophysical point value

(Santiago 2002)

the judgement of continuum of microfluidics, the Knudsen number could be used. Knud-

sen number (Kn) was defined as the ratio of the mean-free-path of the molecules (λ) to the

characteristic lengthL of the device. It could be stated as,

Kn =λ

L(2.1)

The flow regions could be divided into four zones in accordance with the Knudsen number.

They were continuum, slip, transition and free-molecular flow. For liquids with low molec-

ular weight, Knudsen numbers are approximately zero. Continuum was valid for such

liquid flow when the channel was over the micron scale (Gad-el Hak 1999). Therefore, the

continuity equation could be written in Equation2.2,

∇ · v = 0 (2.2)

wherev is velocity field.

13

2.3.2 Conservation of Momentum

The conservation of linear momentum is actually Newton’s Second Law. For a system

mass, the equation can be stated as,

∑F =

d(mv)

dt(2.3)

whereF is the force applied on the system,m is the mass,v is the velocity andt is the time.

Applying the conservation principle to a differential element volume of a fluid, a general

conservation equation for momentum can be derived:

ρDv

Dt= ρ(

∂v

∂t+ v · ∇v) (2.4)

whereρ is the density of the fluid. The applied forces to the system could include body

forces and surface force. Body forces include gravity, electromagnetic force (Lorentz force)

and others. Pressure is the most common surface force in fluid mechanics. For incompress-

ible fluid motion with uniform viscosity, the conservation of momentum can be written as,

ρDv

Dt= f −∇p + µ∇2v (2.5)

wheref is the body forces,p is the pressure andµ is the dynamic viscosity.

2.3.3 Brownian Motion

The discovery to Brownian motion was credited to Robert Brown, a botanist who ob-

served the random movement of pollen grains under a microscope. Brownian motion was

thus a continuous stochastic process which was normally distributed, and whose variance

14

increases the farther it gets from the origin. It was independent of its remote history, and a

change in its location at any increment in time was independent of a change anywhere else

over the same time increment. Albert Einstein gave the first mathematical demonstration

of the underlying normal distribution of Brownian motion (Einstein 1905). Though other

mathematicians described many of detailed mathematical properties of Brownian motion,

the present research limited its investigation to Einstein’s approach to understand the diffu-

sion phenomena in microfluidic devices.

Einstein applied molecular-kinetic theory and well-known diffusion theory - Fick’s

laws to prove that the probability distribution of the resulting displacements during an ar-

bitrary timet was thus the same as the distribution of random errors,

Fp(x, t) =n√4πD

e−x2

4Dt√t

(2.6)

whereFp is the probability displacement function inx direction,n is number of samples

andD is the diffusion coefficient. With this equation, Einstein related the diffusion coeffi-

cient to displacement by giving the root-mean-square displacement as,

tD =l2

2D, (2.7)

where l is the diffusion distance, andtD is the time required to diffuse across distance

l. This equation had been widely used in interpreting microfluidic mixing. Reducing the

length in the equation could reduce the diffusion time by a second order, and the relation

revealed one important method to improve mixing. This was that reducing the diffusion

distance between fluid streams improves mixing.

15

2.3.4 Taylor-Aris dispersion

Taylor (Taylor 1953, Taylor 1954) described the dispersion of solute in a circular pipe

along the longitudes (axial) direction rather than transverse direction. In a circular pipe,

the flow follows Poiseuille’s law - the flow pattern was a parabolic shape for a viscous

fluid. The fluid close to the wall had more time to diffuse than the fluid in the center of the

channel. When the flow was slow enough for a complete radial diffusion, but much faster

than axial diffusion, diffusion in axial direction was negligible. Therefore, the dispersion

of solute in such a flow acted like a slug dispersed axially without the obvious parabolic

flow pattern. The virtual dispersion could be written into mathematical form as,

D∗ =a2u2

48D,

4L

aÀ Pe À 7 (2.8)

whereD∗ is the virtual dispersivity or Taylor’s dispersivity,a is the radius of the circular

pipe,u is the mean velocity,D is the molecular diffusion coefficient,L is the longitudinal

length andPe is Peclet number, which is a measurement of convective mass transfer to

molecular diffusion, andPe = 2ua/D (Catchpole & Fulford 1966). Aris extended Taylor’s

research and gave a more general expression (Aris 1956),

D∗ = D +a2u2

48D, Pe <

4L

a(2.9)

Equation2.9 includes convective, axial and radial diffusion, and was particularly use-

ful in microfluidics, in which the Peclet number was very often larger than 100. And when

Pe À L/a, mixing was by pure convective dispersion.

There has also been extensive discussions about the Taylor-Aris dispersion in a rectan-

gular channel (Doshi, Daiya & William 1978, Chatwin & Sullivan 1982, Dutta & Leighton

2001). From these discussions, it was generally accepted that the apparent diffusion coef-

ficient D∗ (Taylor’s dispersivity) in a rectangular channel could be described in equation

16

(2.10),

D∗ = D +2

105

u2a2

Dfg(

b

a) (2.10)

whereD is the molecular diffusion coefficient,u is the mean velocity in the channel,a

is half the width of the channel,b is half of the height of the channel andfg is the geometric

function of a rectangular channel.fg is determined by the depth to width aspect ratiob/a.

The geometric functionfg is not unity, which is the case when ignoring the side walls

of the channel. However, the side walls of a microchannel cannot be omitted from the

above equation, therefore, instead of being unity,fg = 7.95 while b/a ¿ 0. For a Taylor

dispersivityD∗, the length between the dispersed leading and trailing edges of slug,lm,

was estimated by Equation2.11, which is rewritten from Equation2.7.

lm =√

2tD∗ (2.11)

Taylor dispersivityD∗ was much larger than the molecular diffusivityD, therefore,

for a fixed widthlm of a channel, the mixing timet was reduced.

Taylor-Aris dispersion mechanism was found to be an important factor in understand-

ing mixing, and is studied in more detail in the analysis of section4.3.

2.3.5 Chaotic Advection

In section2.3.3, Einstein’s interpretation of Brownian motion stated that the goal of

mixing was reduction of length scales to achieve the uniformity of concentration. It was

well known that the combined action of stretching and folding produces exponential area

growth and is a desirable goal in achieving efficient mixing (Figure2.2). The Taylor-Aris

dispersions can be interpreted as the combined effects of convection and diffusion. The

17

actual dispersion was several orders faster than pure molecular diffusion due to convection.

In this section, chaotic advection is introduced. Technically, how chaotic advection could

be created and the theoretical background are presented.

t0 t1 t2

t3 t4

t5 t6

Figure 2.2:Stirring of a ‘blob ’of marked fluid by 2D, time-dependent (time series from t0∼ t6), laminar flow

(Ottino 1989)

Chaotic was an apt description of stretching in truly turbulent flows. Chaotic advection

in laminar flow was a term introduced by Aref (Aref 1984). He stated in his paper: “Essen-

tially what was being proposed was the existence of a new advective region, intermediate

between turbulent and laminar advection, which one might call ‘chaotic advection’.”

18

In Aref’s study, the passive advection to emphasize that the particle was so light and

inert that it could do nothing but follow the fluid, and at any instant, its own velocity was

equal to that of the ambient flow,up = uf , whereup was the velocity of particle anduf was

the velocity of the fluid. Therefore, the particle advection could be calculated by solving the

Navier-Stokes equations for Newtonian flows. The condition that particle velocity (u, v, w)

equals fluid velocity then leads to the advection equations:

dx

dt= u(x, y, z, t)

dy

dt= v(x, y, z, t)

dz

dt= w(x, y, z, t)

(2.12)

From the vantage point of dynamic system theory, Equation2.12 was more than

enough for producing non-integrable or chaotic dynamics. From Equation2.12, to produce

chaotic particle motion, 3D steady flows or 2D with time-dependent flows (Figure2.2))

were the necessary conditions for chaotic advection. Steady 2D advection was integrable,

which means non-chaotic.

For an incompressible 2D flow, Hamiltons canonical equations for a one-degree-of-

freedom system can be stated as (Pan 1988):

dx

dt=

∂ϕ

∂ydy

dt= −∂ϕ

∂x

(2.13)

whereϕ is the stream function.

The coordinates of the particle,x andy, are the conjugate variables. Use of either

one as generalized coordinate, and the other Cartesian coordinate is then the conjugate

generalized momentum. Phase space in this problem was configuration space. Briefly,

2D kinematics of advection for an incompressible flow is equivalent to the Hamiltonian

dynamics of a one-degree-of-freedom system. This applies regardless of whether the dy-

19

namics of the fluid itself is viscous or inviscid. A time-dependent Hamiltonian system with

one degree of freedom can be chaotic (Aref 1984, Metcalfe & Ottino 1994, Rudman 1998).

A necessary condition for chaos was the “crossing ”of streamlines. In a 2D system,

this could be achieved by time modulation of the flow field, for example by motions of

boundaries or time periodic changes in geometry. In a 3D system, a periodic velocity in

space was necessary to create chaotic advection.

2.4 Generation of Fluid Motion in a Microfluidic Device

Pumping methods have a close relation with microfluidic mixing. For example, con-

tinuous microfluidic mixing could be induced by pulsatile micropumps. In this case, the

mixing was achieved by the combination of micropumps and micro check-valves (Deshmukh

2001). For many electrokinetically (EK) driven flows, the mixing channels could be nar-

rower than those of pressure driven flows so that mixing was sufficient by virtue of molec-

ular diffusion. This implied that it was necessary to study mixing in pressure driven flows

rather than electrokinetically driven flows. On the other hand, the flow rates and pres-

sure heads that could be achieved by current micropumps are two important factors to be

considered in the study of microfluidic mixing.

There were two major methods to drive flow in microfluidic devices: pressure driven

and electrokinetically driven pumping techniques. There were other mechanisms, such

as magnetohydrodynamic (MHD) (Hosokawa, Shimoyama & Miura 1993, Lemoff & Lee

2000, Jang & Lee 2000), which was driven by Lorentz force, electrohydrodynamic (EHD)

pumping (Bart, Tavrow, Mehregany & Lang 1990, Darabi, Rada, Ohadi & Lawler 2002),

which was driven by Coulomb force, and pumping by acoustic streaming, ie. ultrasonic

pumping (Nguyen, Meng, Black & White 2000). MHD, EHD and acoustic streaming

pumping had no moving parts and could deliver continuous flows. These pumping tech-

20

niques had close relation with microfluidic mixing and the mechanisms are reviewed in

section2.5. A brief review of pressure driven pumping and electrokinetic pumping would

be conducted in this section.

The most critical parameters for micropumps were found to be the flow rates and

pressure heads that micropumps could deliver. Some information about flow rates and

pressure drops of micropumps could be found in early review papers (Shoji & Esashi 1994).

In Figure2.3, the typical flow rates and pressure heads that could be delivered by current

micropumps are illustrated.

Figure 2.3:Flow rates and pressure heads of micropumps could deliver - a review

(Chen & Santiago 2002)

2.4.1 Mechanically Actuated Micropumps

There have been a few reviews on micropumps (Gravesen et al. 1993, Shoji & Esashi

1994, Nguyen, Huang & Chuan 2002). In this literature review, only a few are covered to

21

demonstrate the working principles of mechanical micropumps.

The fluids were driven by a pressure difference between the inlet and outlet in the

micropumps controlled by mechanical micro check-valves. According to the actuation,

they could be categorized into piezoelectric micropumps, pneumatic micropumps, thermo-

pneumatic micropumps, electrostatic micropumps and others. The maximum output pres-

sure of a micropump depended directly on the available force an actuator could deliver.

Figure2.4 illustrates a typical micropump. When the membrane was actuated by piezo-

electric force or other actuation forces, the membrane disk was deformed and enlarged the

volume of the cavity to create a vacuum. The inlet check valve was opened passively by

the vacuum and fluid flowed in. When the membrane was released from actuation, the

membrane recovered its shape and reduced the volume of the cavity to create a positive

pressure. The outlet check valve was opened passively and fluid flowed out. Figure2.5

shows a typical passive check valve used in micropumps.

Figure 2.4:Disk piezoelectric reciprocating micropump

(Shoji & Esashi 1994)

Mechanical micropumps without moving parts featured two complicated series of

loops for the intake and outlet valves designed to favour flow in one direction over the

other. Although this approach was not very efficient in terms of overall flow rate, it had the

advantage of being simple in construction and highly reliable because of no moving parts

in the pump (Figure2.6). The pump was machined by reactive ion etching (RIE) and pack-

22

Figure 2.5:A typical passive micro check valve

(Shoji & Esashi 1994)

aged with electrical field assisted bonding (anodic bonding) with Pyrex. The driving force

was from a piezoelectric driver glued to the Pyrex. Application of voltage potential across

the piezoelectric driver causes it to change its diameter resulting in bowing of the Pyrex

cover. The maximum flow rate that had been achieved was 1600µl/minute and pressure

heads of over 7 m water had been achieved.

Figure 2.6:Tesla micropump

(Forster, Bardell, Afromowitz, Sharma & Blanchard 1995)

23

2.4.2 Electrokinetic Micropumps

Electrokinetics was the general term describing phenomena that involved the interac-

tion between solid surfaces, ionic solutions, and applied electric fields. Those phenomena

were collectively defined as electrokinetic phenomena and included two important classes:

electroosmosis and electrophoresis. Electroosmosis described the movement of a liquid

relative to a stationary charged interface under the influence of an electric field. The fixed

surface would typically be a capillary tube or porous plug. Electrophoresis was the induced

movement of a charged interface (usually colloidal particles or macromolecules) relative to

stationary liquids under the influence of an applied electrical field.

It had became clear that electrokinetics was one of the important methods to drive

fluids through microchannels with dimensions in the order of 10 to 100µm and flow

rates in the nanoliter per second range for rapid mixing and detection (Bousse 1999).

Microfabricated capillary electrophoresis integrated systems had a clear benefit for rapid

separations (Harrison, Fluri, Seiler, Fan, Effenhauser & Manz 1993, Manz, Effenhauser,

Burggraf, Harrison, Seiler & Fluri 1994). Compared to conventional capillary electrophore-

sis, the separation times were reduced from an order of 10 minutes down to an order of

seconds and even less than one millisecond.

Electro-Osmotic (EO) flows in microchannels were critical to the design and process

control of various on-chip microfluidic devices such as modern instruments used in chem-

ical analysis and biomedical diagnostics.

To understand electrokinetically driven flows, it is necessary to describe the electrical

double layer model.

24

Electrical double layer

Most solid surfaces acquire a surface electrical charge when brought into contact with

an electrolyte liquid. When a polar liquid was placed in contact with another phase (like the

wall of the surrounding channel), a potential difference develops at the interface. Molecules

of the polar liquid (dipoles) would re-orient themselves near the phase boundary. Ions

or excess electrons present would cause a charge re-distribution at the interface, causing

formation of the so-called electrical double layer (EDL).

The thickness of electrical double layer was described by several models. An impor-

tant one was the Debye length, which could be stated as,

ψ = ψo exp(−redlz) (2.14)

whereψ was the electrical potential,z is the distance from the wall andredl was identified

as the reciprocal of the thickness of the electrical double layer, also commonly referred to

as the “Debye length”.

Electroosmosis

In the electrical double layer, the ionic concentration near the solid surface was higher

than that in the bulk liquid. In the compact layer, which was about 0.5 nm thick, the ions

were strongly attracted to the wall surface and were immobile. In the diffuse layer the ions

were affected less by the wall and were mobile. The thickness of the diffuse layer ranges

from a few nanometers up to several hundreds of nanometers, depending on the electric

potential of the solid surface, the bulk ionic concentration and other properties of the liquid.

When a liquid was forced through a microchannel under hydrostatic pressure, the ions in

the mobile part of the EDL were carried towards one end. This causes an electrical current,

called streaming current, to flow in the direction of the liquid flow. The accumulation of

ions downstream sets up an electrical field with an electrical potential. This field causes

25

a current, called conduction current, to flow back in the opposite direction. When the

conduction current was equal to the streaming current a steady state was reached. It was

easy to understand that, when the ions were moved in the diffuse double layer, the induced

viscous shear stress would pull the liquid along with them. This was the so-called Electro-

Osmotic (EO) flow (Rice & Whitehead 1965, Burgreen & Nakache 1964, Molho, Herr,

Kenny, Mungal, Deshpande, Gilbert, Garguilo, Paul, St. John, Woudenberg & Connell

1998).

When the Debye length is small, the equation of motion for steady flow with low

Reynolds number, and incompressible flow is:

µ∇2u = −ρEE +∇p (2.15)

whereµ is the dynamic viscosity,u is velocity,E is electric field,p is pressure andρE is

the electric charge density.

In the limit of a small Debye length, solving the above the equation yields the Helmholtz-

Smoluchowski equation for the electroosmotic velocity (Equation2.16).

u =−εζEOEx

µ(2.16)

whereε is the permittivity,ζEO is the electroosmotic zeta potential andEx is the electrical

field.

The Helmholtz-Smoluchowski equation predicts a ”plug-like” velocity profile when

the Debye length was much less than the capillary diameter. Electrophoresis describes the

motion of a charged surface submerged in a fluid under the action of an applied electric

field. The electrophoretic velocity of the dye was again described by the HelmholtzSmolu-

chowski equation (Equation2.17).

u =−εζEP Ex

µ(2.17)

26

whereζEP is the electrophoretic zeta potential.

The advantages of using the EO flow as a pumping method to transport liquid in mi-

crochannels in on-chip microfluidic devices include no moving part, no noise and easy to

control the electrical field. A key in the applications of EO pump was to be able to deliver

the precise amount of liquid. Two critical disadvantages for EO flow are the very high

electric voltage required and bubble formations inside microchannels due to electrolysis.

Electroviscous Effect

The EDL field at the solid surface exerts electrical forces on the ions in the liquid,

and hence, restricts the motion of these ions. Consequently, the presence of the EDL field

would reduce the liquid flow in comparison with the cases of no EDL effects. Further-

more, the apparent viscosity could be several times higher than the bulk viscosity of the

liquid (Kulinsky, Wang & Ferrari 1999, Li 2001).

2.5 Active Micro-Mixers

The above sections introduced the flow driven mechanisms and flow behavior which

have a close relation with the mixing mechanisms in a microchannel. In this section, the

utilization of active flow driven mechanisms to create secondary flow to reduce diffusion

path is introduced.

Due to the laminar nature of micro flows, applications of active micro-mixers also

try to create secondary flows, and in some cases, chaotic effects could be obtained. For

most active micro-mixers, this could be achieved by periodic perturbation of the flow fields

temporally, or spatially. This section reviews the magneto-hydrodynamic (MHD) (Bau,

Zhong & Yi 2001, Yi, Qian & Bau 2002) and electro-hydrodynamic (ED) (Choi & Ahn

27

2000) micro-mixers. Travelling waves and acoustic or peristaltic waves, were used to drive

flows and improve mixing in several applications (Yang, Goto, Matsumoto & Maeda 2000,

Selverov & Stone 2001), and could be categorized into active mixers. Mechanical agitation

was rarely used in micro-scaled devices, however, it was still possible to fabricate stirring

rods inside a micro-device (Lu, Ryu & Liu 2002).

In this section, micro-mixers with periodic perturbation of fluids spatially or tempo-

rally are reviewed first in sections2.5.1to 2.5.3. Then, mixing mechanisms by the contin-

uously generation of secondary flows are reviewed in sections2.5.4and2.5.5. Finally, a

design of mechanical stirrer is described in section2.5.6.

2.5.1 Planar Laminar Bubble Mixer

A microfluidic planar bubble mixer was fabricated with a five mask process (Fig-

ure 2.7). It was composed of bubble pumps and bubble check valves. Thermo-capillary

bubble valves operate by creating a vapor bubble within chamber, and then, using these

bubbles prevent flow though the chamber. Bubble pumps operate by turning on and off

micro-heaters to create or eliminate bubbles within the chamber, thus the volume thermo-

expansion or shrinkage created piston-type pumping. The pumping process created chaotic

advection to mix fluid streams in the planar, laminar chamber. A chaotic flow field was one

in which the path and final position of a particles placed within the field were extremely

sensitive to their initial position. In the chaotic flow field, particles initially close together

might become widely separated, and the flow as a whole becomes well mixed.

2.5.2 Periodic Perturbation Applied to a Main Fluid Stream

Periodic pressure or electrokinetic perturbation was sometimes declared as chaotic ad-

vection (Figure2.8). Without further investigation, it was not clear that it was in the chaotic

28

Figure 2.7:2D chaotic advection microfluidic mixer

(Evans, Liepmann & Pisano 1997)

region. Similar designs had been studied with a different interpretation (Deshmukh 2001).

The idea was to introduce slugs of solutes into the main stream. In accordance with the

concept of Taylor-Aris dispersion in a microchannel, the actual dispersion was much faster

than pure molecular diffusion (section2.3.4). Under the term of continuous microfluidic

mixing, pulsatile micropumps were successfully microfabricated. The periodic perturba-

tion by the pulsatile pumping produced alternated fluids slugs that created flow patterns

similar to Taylor-Aris dispersion. It was a versatile method to solve microfluidic mixing

problem. However, it should be noted that the claim about high flow rate mixing was not

clearly verified to fulfill the inequality of Taylor-Aris dispersion. For flow rates that were

too high, there was not sufficient residual time for the fluids to diffuse into each other.

High flow rates generate high pressure drops, which was a disadvantage for microfluidics

and should be avoided. Though, this kind of mixer contained moving parts, it was com-

plicated for fabrication and difficult for flow control. Nevertheless, mixing by applying

Taylor-Aris dispersion was found important for further study and is addressed in Chapter

4.

29

Figure 2.8:Microfluidic mixer by periodical perturbation

(Lee, Deval, Tabeling & Ho 2001)

2.5.3 Peristaltically Driven Micromixing

In the previous section (2.5.2), the periodic perturbations to the main flows were un-

der low frequencies. When the frequencies were high and the magnitudes of the waves

were low, this formed another mixing category - peristaltically travelling waves. Most of

the peristaltically type driven micro-mixing was induced by acoustic waves, and some by

mechanically driven peristaltic waves.

Although these mixing methods showed promising results, the mixing mechanisms

were described differently. Turbulence could possibly exist in such a system, as declared

in a previous paper (Yang et al. 2000). However, most researchers agreed that the fluid

motion induced by the acoustic waves caused certain recirculations that favour mixing (Zhu

& Kim 1998, Monnier, Wilhelm & Delmas 1999, Rife, Bell, Horwitz, Kabler, Auyeung &

Kim 2000, Vivek, Zeng & Kim 2000, Selverov & Stone 2001, Liu, Yang, Pindera, Athavale

& Grodzinski 2002, Liu, Lenigk & Grodzinski 2003).

In acoustic microfluidic mixers, piezoelectric acoustic transducers were attached to

the fluidic channels. The transducers converted radio-frequency (RF) electrical energy into

30

an ultrasonic acoustic waves. The acoustic waves were absorbed in the fluids, but generated

mechanical travelling waves ( flexible plate waves or surface acoustic waves (SAW)) at the

attached solid structures (Figure2.9). The mechanical travelling waves induce fluid mo-

tion. By controlling the direction of the streaming, pumping or mixing could be achieved.

For microfluidic mixing, mixing time across channels with characteristic lengthl would

beO(l2/D), whereD = kBT/6πµr was the Stokes-Einstein translational diffusion coeffi-

cient,kB was the Boltzmann constant,T was the absolute temperature andr was the radius

of the diffusing particle. Typical times for Peristaltic convective transport wereO(1/ε2ω),

whereε is the dimensionless amplitude of the travelling wave andω is the frequency of the

oscillation of the wall (see Figure2.9). This implies that frequencies should beO(D/ε2l2)

or larger. For typical microfluidic mixers driven by piezoelectric waves, convective trans-

port in aqueous solutions requires frequencies ofω À 103 Hz.

Figure 2.9:Mechanical travelling waves induced by acoustic streaming

(Nguyen et al. 2002)

2.5.4 Magnetohydrodynamic (MHD) Micro-mixers

The potential to apply electromagnetic force for fluids pumping had attracted atten-

tion (Hosokawa et al. 1993, Heng, Huang, Wang & Murphy 1999, Lemoff & Lee 2000, Jang

31

& Lee 2000), as mentioned in section2.4. MHD could be used not only for the purpose

of pumping fluids, but also to control fluid flow in microchannels and to induce secondary

flows that maybe beneficial for mixing. By depositing an array of transverse electrodes on

the bottom of a microfluidic conduit (Bau et al. 2001), the electrodes were connected alter-

nately to two terminals of a DC power supply. As a result, electric currents were induced

in opposite directions between adjacent pairs of electrodes. The interaction between these

currents and uniform magnetic field perpendicular to the bottom of the channel led to the

formation of Lorentz forces in opposing directions between adjacent pairs of electrodes.

This, in turn, led to the formation of convection. This convection could be used to fold and

stretch material lines and enhance mixing (Figure2.10). Yi et al. extended the study exper-

imentally and theoretically, and demonstrated chaotic stirring effects by MHD in a cylinder

cavity (Yi et al. 2002). However, steam bubbles induced in the MHD micro-mixers was a

major problem.

Figure 2.10:A minute MHD micro-mixer

(Yi et al. 2002)

32

2.5.5 Electrohydrodynamic (EHD) Micro-mixers

In electrically dominated flows, there is always some residual free charge present,

typically residing at the interface between two fluids with different electric conductivities

flowing side by side in a microchannel. The external electrical fields that are tangential to

the interface result in a tangential stress. Dragged by this shear stress, fluid motion can be

formed with the moving charges. This model which deals with fluid motion induced by

electric fields was firstly introduced by Taylor, and Melcher used it extensively to develop

electrohydrodynamics (Brenner & Stone 2000).

In the case of microfluidic mixing, the flow motion induced by an electrical field

transverse to the main stream, forms the secondary flow (Figure2.11). This secondary

flow stretches and folds material lines, and therefore, enhances microfluidic mixing. The

operation voltage was of a range from 7 to 27 volts.

In Figure2.11, the electrical field was perpendicular to the surface charges, so it could

form secondary flow. While the external electric field was parallel to the surface charges,

transverse secondary flow was not induced. However, when the working voltage was larger

than 1kV, it creates electrokinetic instability. Although the mechanism for the cause of the

electrokinetic instability was still not clear, its appearance was very much like turbulent

flow, even though the Reynolds number was low. By applying this instability, a micro-

mixer was fabricated in Stanford University (Figure2.12).

2.5.6 Magnetic Micro-Stirrer

Inspired by the macro-scaled magnetic bar mixers, a micro-scaled mechanical stirring

mixer was fabricated by surface micro-machining (Lu et al. 2002). In the process, a mag-

netic actuated rotary permalloy bar, or an array of these bars, on Pyrex 7740 glass wafer

was encapsulated in the microchannel formed on polydimethylsiloxane (PDMS) substrate

33

a

b

Figure 2.11:(a). Schematic illustration of the active micro-mixer (b). Cross sectional viewof the active micro mixer

(Choi & Ahn 2000)

34

0 sec 0.3 sec 1 sec 2.3 sec

Figure 2.12:Mixing processes in the micro-mixer by applying electrokinetic instability

(Oddy, Santiago & Mikkelsen 2001)

(Figure2.13). When the stir bar was rotated, it stirred the fluids and increased the contact

area between the two fluids, hence increased the rate of diffusive mixing between them.

The mixing efficiency would be further improved, if the rotating speed was increased (Fig-

ure2.14). However, viscosity in a microfluidic device plays a dominant role (Purcell 1977),

and the magnetic force on the stirrer had to overcome the viscosity of the fluids. Hence, it

is difficult to achieve very high rotating speeds. Nevertheless, it was predicted by Purcell

that the mechanical agitation for mixing was not effective at very low Reynolds numbers.

However, this micro-stirrer mixer might have an unique advantage on mixing fluids in a

chamber.

Figure 2.13:Magnetic micro stirrer

(Lu et al. 2002)

35

Figure 2.14:Mixing in microchannel by magnetic micro stirrer, (a). 0 rpm, (b). 150 rpm,and (c). 300 rpm

(Lu et al. 2002)

36

2.6 Passive Microfluidic Mixers

By definition, passive micro-mixers have no moving parts and other external mecha-

nisms to create secondary flows. Therefore, they are relative easy for fabrication. Because

no electrical and magnetic parts are involved, there is no special requirements for electric

and/or magnetic material properties, and passive micro-mixers have much broader materi-

als to choose from. Less material requirements also reduce the complexity of microfabri-

cation.

A passive micro-mixer can improve mixing by reducing the diffusion path sufficiently,

and by continuously moving fluids away from the interfacial areas, which is actually bulk

mass convection. In a microscopic scale, convection is always coupled with diffusion, due

to the low Reynolds number of microfluidics.

In this section, diffusion dominated passive mixing is first reviewed. This includes T-

type mixers, splitting and recombining mixing. Then, passive mixers coupled with convec-

tion and diffusion type mixers are reviewed. This category includes focusing fluid streams,

sub-injection, collision and passive chaotic mixers.

2.6.1 T-type mixers

In many microfluidic devices, to introduce reagents and samples into the devices, the

T-type shape of channels were often used. The “Y ”shape, cross, and other similar shapes

follow the same design principles and could be categorized into T-type channels. When

the size of microchannel was a few tens of microns, the contacting of two substreams in a

T-type channel was sufficient enough to achieve adequate mixing. Many of electroosmotic

driven microfluidic devices use channels that had the sizes less than 100µm, and could

contact reagent and sample directly in T-type mixer without any other mixing assistance.

37

Mixing Tee in Electrokinetically Driven Microchips

In Figure2.15, it was shown that in a compact chip architecture, and with a single

voltage source, parallel mixing with different mixing ratios of sample to buffer could be

accomplished by using a series of T-mixers (Figure2.15a), and the serial mixing device

was based on an array of cross-intersection and sample shunting (Figure2.15 b). The

channels had similar cross-sectional areas, which made channel resistances proportional

to length. Applying the same potential to both the sample and buffer reservoirs allowed

accurate and fast mixing.

a b

Figure 2.15:Mixing T structures, (a). Parallel configuration; (b). Serial configuration

(Jacobson, McKnight & Ramsey 1999)

38

Collision of Fluid Streams in Mixing-Tees

When collision of two fluid streams in a T-junction occurs, a flow pattern termed

engulfment flow could be created (Figure2.16). The swirling flow increased the contact

area between fluid streams and so did the mixing performance.

Figure 2.16:Streamlines of engulfment flow

(Kockmann, Foll & Woias 2003)

Similar in principle, a double mixing tee micro device was applied to improve mixing

(Figure2.17). The outlet of the first tee formed one inlet of the second mixing tee. This

geometry provided a so-called residence time channel of defined length separated by two

mixing tees. In the idealized case of short mixing length, this design would enable imme-

diate start of a reaction and control of the reaction time by variation of flow rate or length

of flow passage before stopping the reaction by a second mixing process. This structure

needed a high volumetric flow rate to ensure the good mixing by collision of two streams.

T-sensors

A T-type of micro-mixer, sometime termed T-sensor, is one of the micro Total Anal-

ysis System (µTAS) components. T-sensors had been investigated extensively by many

researchers (Weigl & Yager 1999, Kamholz, Weigl, Finlayson & Yager 1999, Kamholz &

Yager 2000, Hatch, Kamholz, Hawkins, Munson, Schilling, Weigl & Yager 2001, Kamholz,

Schilling & Yager 2001, Kamholz & Yager 2001, Kamholz & Yager 2002, Ismagilov, Ros-

marin, Kenis, Chiu, Zhang, Stone & Whitesides 2001, Ismagilov, Stroock, Kenis, White-

sides & Stone 2000). Two or more fluid layers enter the channel side by side, and mixing

39

Figure 2.17:Double-T micro-reactor

(Bokenkamp, Desai, Yang, Tai, Marzluff & Mayo 1998)

takes place between the fluids streams only by diffusion. The mixing zone could be quan-

titatively measured by optical methods (Figure2.18). For a pressure-driven viscous flow in

a microchannel, the experimental results showed that the flow patterns were parabolic-like

profiles in one direction or even both directions that were perpendicular to the flow axis

and the velocity was zero at the wall and maximum velocity in the center of the channel.

Due to the parabolic flow profiles, the fluids that were close to the wall moved slower,

hence, had more residual time for diffusion and were mixed first. To reduce the incon-

venience of the Taylor dispersion effect for sensing, the depth to width aspect ratio of

T-mixer was usually kept small for the solute to diffuse rapidly across the depth direction

to achieve uniformity before the detection zone. Scaling rules were also studied (Ismagilov

et al. 2000, Kamholz & Yager 2002). Near the wall of microchannel, the width of region

mixed by diffusion scales as the one-third power of the ratio of the axial distance, whereas

in the center of the channel, the diffusion scales by the one-half law. For a small depth to

width aspect ratio, reasonably accurate estimations using Einstein equation (Equation2.7)

could be made (Kamholz & Yager 2002).

40

Figure 2.18:A schematic working principle of T-Sensor

(Kamholz & Yager 2001)

41

The study of research on T-mixers provided a general understanding of how diffusion

acts in microfluidic devices and many novel applications were found, such as lab-on-a-chip

and medical diagnostics.

2.6.2 Splitting and Recombining

T-type micro-mixers provided a way to introduce two fluid streams into microchan-

nels, and the mixing was by virtue of diffusion. Mixing by diffusion requires a long mi-

crochannel to achieve complete mixing. To reduce the channel length and the space oc-

cupied by the mixing units, the diffusion path between fluid streams must be reduced to a

sufficiently small scale.

It was intuitive to split fluid streams into tens or hundreds of small substreams to

increase diffusion. In this way, large contact surfaces and small diffusion paths were gener-

ated (Koch, Chatelain, Evans & Brunnschweiler 1998, Koch, Witt, Evans & Brunnschweiler

1999, Koch, Schabmueller, Evans & Brunnschweiler 1999, Ehrfeld, Hessel & Lowe 2000)

(Figure 2.19)). These methods were simple, but also require a large amount of space,

which makes the device larger. Another concern was the uneven pressure distribution of

such mixers, which might cause partial mixing at some regions of the mixing unit.

In a microfluidic analytical system with flow rates of the order of nanoliters and/or

even picoliters, the splitting and recombining of fluid streams could be fabricated in-situ.

He et al. described the in-situ micro static mixer with a flow rate as low as 100pL, and

liquid A and liquid B enter a Y-channel along one side (He, Burke, Zhang, Zhang &

Regnier 2001, He, Tait & Regnier 1998). In this design, the flow was redirected later-

ally with shrinkage of channel width to introduce convective effect, and consequently, the

diffusion distance was reduced. The fluid streams were further divided partially by side

bypass channels and recombined at the other end (Figure2.20). For the electrokinetically

driven microfluidic device, the channel size was small, and the mixing efficiency was high.

42

Figure 2.19:Parallel dividing into substreams.

(Koch et al. 1999)

However, it may be difficult to apply this design in a pressure driven micro device because

of possible high pressure drop. Therefore, to determine the suitability of the micro-mixer

design for pressure driven flow, it was decided that an investigation be carried out and the

results reported in this thesis.

By 3D geometrically design of the micro-mixer (Schwesinger, Frank & Wurmus 1996,

Hessel, Hardt, Lowe & Schonfeld 2003, Nielsen, Branebjerg, Gravesen, Dyhr-Mikkelsen

& Gade 2001), the diffusion process could be arranged to form a multi-layer sandwich-like

structure to reduce the diffusion path (in Figure2.21). The fluids were split, stretched into

substreams and recombined repeatedly by the fork-shaped divided channels. The lamella

thickness for the diffusion could be reduced to a sufficiently small scale. Moreover, the

structures could also be used for immiscible liquids to produce emulsified mixtures.

Caterpillar micro-mixers were developed by IMM-Mainz. The splitting and recom-

bining was repeated periodically by using 3D structures (Figure2.22 a). The complex

geometric structures divided the streamlines repeatedly (Figures2.22b and c). However,

the complexity of the structures made it very difficult to fabricate and optimise.

The Danfoss design of a micro-mixer used very much a similar mixing method to the

Caterpillar micro-mixers (Nielsen et al. 2001), however, one stream passed underneath an-

43

Figure 2.20:In-situ static micro-mixer by subdividing streams

(He et al. 2001, He et al. 1998)

Figure 2.21: Divided flow into substreams (a) vertical arrangement; (b) horizontalarrangement.

(Schwesinger et al. 1996)

44

a b

c

Figure 2.22: (a). Channel structures; (b). Schematic drawing of its split-recombine-principle; (c). Streamlines in a caterpillar mixer.

(Hessel et al. 2003)

45

other stream and joined in one mixing chamber (Figure2.23). The recombining of streams

was in a vertical manner. The diffusion distance was reduced dramatically, due to the shal-

low mixing chamber and the broadening of channel width, which also reduced the thickness

of the fluid streams.

Figure 2.23:Danfoss mixer

(Nielsen et al. 2001)

Figure2.24illustrates the vertical splitting and recombining of fluids in a static micro-

mixer (Klaus Schubert, Wilhelm Bier, Erhard Herrmann, Thomas Menzel & Gerd Linder

2000). The guiding structures divided fluid into multiple smaller filaments that overlap

vertically. The patent claims to achieve uniformed mixture over the whole outlet area of

this mixer.

The above review showed that the diffusion distance could be reduced by fabricating

mixing unit in 3D structures, and the use of device space was decreased. The disadvantage

of this mixer was its complexity for fabrication and the high pressure drops that prohib-

ited their practical applications. For the structures that were made on two separate Silicon

wafers, the packaging process also required high accuracy to align two or more parts to-

gether. To reduce pressure drop in such devices, a modified structure was proposed (Mahe,

Tranchant, Tromeur & Schwesinger 2003). The modified design demonstrated comparable

mixing efficiency and much lower pressure drop.

46

Figure 2.24:Static Micro-mixer

(Klaus Schubert et al. 2000)

2.6.3 Focusing Fluid Streams

Reducing the diffusion distance between fluid streams could be also achieved by so

called “focusing ”. Focusing the fluid streams into a very thin layer could be done by

geometrically shrinking the width of the channel (Veenstra, Lammerink, Elwenspoek &

van den Berg 1999), or by hydrodynamic focusing (Knight, Vishwanath, Brody & Austin

1998). A super-focusing micro-mixer had been developed by combining the geometrically

splitting and recombining technique together with the focusing technique. First the main

fluid streams were divided into many substreams with reduced channel width, then the

substreams were converged into a channel with much smaller width than their total width

of the sub-channels (Figure2.25). Mixing in such a micro-mixer could be completed within

only a few milliseconds.

In a hydrodynamic focusing system for ultra-fast micromixing (Knight et al. 1998),

47

Figure 2.25:IMM super-focusing micro-mixer

(Hessel et al. 2003)

one stream was squeezed into a main channel by two side flows hydrodynamically (Figure

2.26). For incompressible flow, to keep the conservation of mass (continuity), the flow

rates must be constant. Therefore, by controlling the velocity of the middle fluid stream, in

accordance withv = (flow rate) / (Cross-section area), the cross-section area of the middle

stream was stretched into a thin layer to keep the flow rate constant. While the thickness

of this thin filament was reduced to a few microns, mixing could be achieved rapidly by

diffusing through this thin layer.

2.6.4 Inject Small Fluid Substreams into a Main Stream

In Figure2.27, the two fluids flowed towards each other and subdivided by the sinu-

soidal shaped slits. The substreams were recombined in a mixing chamber above these

slits, and reduced diffusion paths could be achieved. The main concern of this structure

was the control of the uniformity of the flow. If the mixing zone was significantly larger

than the microchannel width, a non-uniform flow would occur, resulting only in a partial

overlap of the sheets. The injection mixing schemes had another arrangement illustrated in

48

Figure 2.26:Hydrodynamic focusing micro-mixer

(Knight et al. 1998)

Figure2.28. However, instead of combining fluid streams above the channel, the sub-jets

from the bottom of the structure were guided so they could be combined with the main fluid

stream and flowed side-by-side into the mixing chamber.

Figure2.29 illustrates injection of sub-jets into a mixing chamber through a micro-

fabricated square-shaped array of nozzles (Elwenspoek, Lammerink, Miyake & Fluitman

1994). The injection of multiple sub-jets formed micro-plumes, which increased the con-

tact surfaces between two fluids.

2.6.5 Passive Microfluidic Chaotic Mixers

If one defines the axial flow as the primary flow, any modification of the speed or

direction of primary flow was termed secondary flow. In some cases, strong secondary

flows can create chaotic advection (covered in section2.3.5).

49

Figure 2.27:IMM-Mainz Mixing unit

(Ehrfeld, Golbig, Hessel, Lowe & Richter 1999)

Figure 2.28:Injection fluid into a mixing chamber

(Larsen 2001)

50

a b

Figure 2.29: (a). Mixing Chamber with Micro-plumes (b). Simulation of sub-jets ofreagent into main stream

(Elwenspoek et al. 1994)

Chaotic advection could stretch and fold the fluids dramatically. As a result, the con-

tacting interfacial area was increased and the thickness of the fluids layer was decreased

exponentially (Ottino 1989). In other words, the diffusion distance was reduced exponen-

tially. As the process continues, rapid mixing could be achieved.

One important strategy to solve microfluidic mixing problem was to create chaotic

advection. Chaotic advection, sometime called Lagrange chaos, had been studied inten-

sively in the macro-world (Aref 1984, Jones, Thomas & Aref 1989, Ottino 1989). The

knowledge could be borrowed for the design of microfluidic mixers. For instance, using a

twisted pipe, secondary flow occurs at the bends of the pipe (Jones et al. 1989). Chaotic

advection could be a result of these secondary flows. Inspired by the twisted pipe, 3D ser-

pentine channels were fabricated (Beebe, Adrian, Olsen, Stremler, Aref & Jo 2001, Liu,

Stremler, Sharp, Olsen, Santiago, Adrian, Aref & Beebe 2000) (Figure2.30). The flow in

3D serpentine channels demonstrated chaotic advection at high flow rates (Re ∼ 70) and

good mixing performance. One of the advantages of the serpentine design was to keep the

51

height-to-width ratio near one so that it minimizes the chances of clogging, fouling and

loss of sample in the biological analytical applications. However, the challenges of ser-

pentine mixers were the microfabrication of complex 3D structures and the need for a high

Reynolds number to stir the fluids to generate chaotic advection. Another drawback of this

design is the dead volume created in the perpendicular bends, as the fluid at the corner of

these bends keeps still, the only mass transport is diffusion.

Figure 2.30:Serpentine-Shaped Micro-mixers

(Liu et al. 2000)

Chaotic microfluidic mixers made by patterning grooves in microchannels were com-

patible with microfabrication processes and could be realized in general micromachining

52

centers (Johnson, Ross & Locascio 2002, Stroock, Dertinger, Ajdari, Mezic, Stone &

Whitesides 2002, Stroock, Dertinger, Whitesides & Ajdari 2002). The anisotropic pat-

terned grooves in the micro-channel could create a pressure component laterally to the

primary flow, which induced spiral circulation around the flow axis, and could stretch and

fold streams of liquids to complete mixing in a shorter channel length (Figure2.31). The

mixing effects were investigated over a range of Reynolds numbers. Both the serpentine

and patterned grooves chaotic mixing strategies showed that the design for chaos was one

of the effective solutions to enhance mixing in a micro-scaled device.

Figure 2.31:Microfluidic chaotic mixer with grooved surfaces

(Stroock et al. 2002)

Further study revealed that deeper grooves create stronger chaotic effects (Johnson

& Locascio 2002), and that the flow patterns for this type of micro-mixer were indepen-

dent of Reynolds number (Stroock et al. 2002). Similar to the slant grooved surface de-

sign, transverse recirculation could also be created by the so called transverse electrokinet-

ics (Ajdari 2002)

The obvious drawback of patterning grooves in microchannels for mixing was the

dead volume created by the grooves. While reducing dead volume is desirable, the deep

grooves were critical for generating helicity and research to date has not yet found any

53

direct solution to reducing the dead volume in such mixers. From an engineering design

point of view, the depth of the grooves is one of the design parameters of this type of mi-

cromixer and needs to be further investigated. This work is carried out in this research.

Nevertheless, the design was compatible with microfabrication processes, and works effi-

ciently, especially at low Reynolds number. Hence, this mixing strategy could be used in

many applications. For instance, disposable microfluidic devices for medical diagnosis and

other applications that did not need the grooves to be cleaned after using them. Under these

conditions, a lower flow rate normally means lower pressure drop, which makes it easier to

drive the flow through a microfluidic device.

2.7 Summary

In a typical microfluidic device, viscosity dominates flow, and as a result, the Reynolds

number is low and the flow is laminar. Therefore, the mixing of two or more fluid streams

in microfluidic devices is by virtue of diffusion, which is a slow process. On the other hand,

for microfluidic applications towards biomedical or chemical diagnostics, such as lab-on-a-

chip, it was very important to mix two or more reagents and test samples thoroughly before

detection can take place. This requires, a) the mixing needs to be done in a confined area;

b) The volume of fluids for detection should be adequate and sufficiently mixed.

Therefore, microfluidic mixing is challenging and device design issues need to be

solved, and many mixing methods have been developed. Based on the mixing mechanism,

micro-mixers have pure molecular diffusion at one end and chaotic mixing on the other.

Based on structures, micro-mixers can be categorized in either the active or the passive

mixing methods. Most of active micro-mixers enhance mixing by stirring flow. The stirring

can be done mechanically or by the means of magneto-hydrodynamic (MHD), electro-

hydrodynamic (EHD) or by acoustic streaming to create secondary flows. The secondary

flows can stretch and fold material lines to reduce the diffusion path between fluid streams,

54

and hence, enhance mixing. Most active micro-mixers are complex to fabricate and require

an external power source.

Some passive mixers reduce diffusion path between fluid streams by splitting and re-

combining. Recently, passive chaotic micro-mixers were investigated by using 3D serpentine-

type channels. The flow in 3D serpentine channels demonstrated chaotic advection at high

flow rates (Re ∼ 70) and good mixing performance. However, the challenges of serpentine

mixers are the microfabrication of complex 3D structures and the need of high Reynolds

number to stir the fluids to generate chaotic advection.

Passive micro-mixers with planar structures are relatively easy to fabricate and can be

made of most engineering materials. The mixers were found to be efficient over a range

of Reynolds numbers and compatible with microfabrication processes. However, most of

the research to date focused on one or a few passive structures, and a full understanding

of geometric structures, and their influence on mixing and flow patterns had not yet been

achieved.

In this research, in-situ passive microfluidic mixers with various planar or semi-planar

geometric structures were systematically studied to understand the influence of these struc-

tures on mixing performance and pressure drop.

It would be impractical to apply solely experimental method to undertake the system-

atic investigation of a series geometric variations. Furthermore, the nonlinear behaviour of

some optics used in visualisation of flow in the existing experimental methods also added

uncertainty to the accuracy of measurement. Therefore, to achieve the objective of this

research, advanced computational fluid dynamics (CFD) techniques were chosen as the

major tools for the investigation. In addition, a simplified experimental method was also

setup to validate selected numerical models used in the CFD simulations. By the experi-

mental validation, the numerical models could be verified, and confidence to use CFD for

further investigation could be achieved. Where applicable, published experimental and an-

55

alytical results were used for the validation of simulation results. The methodology for the

investigation is described in Chapter3.

For the mixing mechanisms reviewed in this chapter, there was not a clear interpreta-

tion of mixing in relation to geometric variations. To initiate the present research, a brief

description of mixing in a T-channel is introduced in Chapter4. The study of two-adjacent

fluid streams diffusing into each other was investigated previously. In Chapter4, further

discussion of Taylor-Aris dispersion is presented. Because pressure drop was also a criti-

cal parameter to the performance of a micro-mixer, the relation between pressure drop and

channel width to depth ratio was also investigated for the better understanding of microflu-

idic mixing.

In regards to periodic geometric structures, a serpentine shaped channel was investi-

gated previously. Serpentine shaped channel structure increases the total length of channel

on an area constrained microfluidic chip, and hence, increases residual time for diffusion.

However, for further improvement of mixing efficiency, it requires high Reynolds num-

ber and 3D serpentine shaped micro-mixers are complicated for fabrication. Chapter5

presents three methods to modify serpentine and similar structures to improve the mixing

performance of this type of mixers. The first method was to partition flow in a single-side

serpentine to create mixing zones with different concentrations. The second approach was

to short-circuit the serpentine by a straight branch to generate partial splitting and recom-

bining at the interface between two fluid streams. The last approach in this chapter was

inspired by the structures machined by Excimer laser, which had periodic ramping struc-

tures at the bottom of the channel. This structure was similar to 3D serpentine structures.

While the periodic ramping structures inspired further research, it was difficult to de-

fine the geometry of a 3D ramping in a numerical model. Therefore, the ramping structures

are simplified to cylindrical obstacles placed inside the microchannel. Obstacles could

be positioned in the microchannel arbitrarily so that mixing performance versus different

layout of obstacles could be studied. This work is reported in Chapter6.

56

Although cylindrical obstacles could provide the information about how the positions

act on mixing, other aspects of obstacles are still of the interest to this research. Therefore,

rectangular shaped obstacles were added and the results presented in Chapter7. Rect-

angular shaped obstacles provided other parameters for the study, such as size, degree of

blockage, offset, angle and height of the obstacles versus the mixer performance. When the

height of the obstacles became negative, the obstacles decayed into grooves at the bottom

of the channel, which resulted in the further study of grooved surfaces in Chapter8.

Past research on mixers with grooved channels covered experimental and analytical

prediction. Due to the relevance to the study of obstacle structures, numerical investigation

of micro-mixers with slantly patterned grooves was added in this research to determine

guidelines for the engineering design of this type of micro-mixer.

The research presented in this thesis links the existing knowledge on passive mixing

mechanisms with geometric configurations with simplified obstacle structures. The results

of the present research provide useful information for the design of passive micro-mixers

and identify further areas of research in this field. These are summarized in Chapter9.

57

Chapter 3

Numerical Modelling and Experimental

Methodology

3.1 Introduction

In chapter2, the literature review found that it was important and necessary to study

passive micromixers with in-situ geometric variations to enhance mixing. These geomet-

ric variations included channel shape, structures inside channel and geometric topology of

channel walls. For the large number of parameters involved, it was difficult to conduct

the research by experiments. Though some researchers delicately utilized micro particle

imaging velocimetry (µPIV) to measure the whole flow field, these techniques are limited

to particular applications, and typically, to slow shallow flow field due to the depth of field

of optical lens (Santiago, Wereley, Meinhart, Beebe & Adrian 1998). Moreover, Computa-

tional Fluid Dynamics (CFD) can neglect un-important factors in the real environment, and

focuses on the fundamental principles of the problem, therefore, CFD can be a versatile

tool for studying laminar incompressible flows.

58

The existing CFD softwares can be divided into three categories. One is the general

CFD packages for macroscopic scale analysis, such as Fluent, CFX, which are based on

the Navier-Stokes equations. One way to apply this kind of CFD softwares to microfluidics

is to understand the principles and the structures of the packages, and then modify some

parameters to model a microfluidic device by utilizing the user program interface embedded

in the software. This category of CFD package, in general, is well tested, and usually

implemented with a versatile CAD system for pre-processing and visualization tools for

post-processing. Therefore, the users can treat the packages as black boxes and concentrate

on the analysis. However, if the structure of the package needs to be changed for simulating

the special flow behavior in a microscopic scale, they may face problems. Because the users

normally cannot access the source code of the packages for modification, and even if they

can, it is a time consuming and a difficult task for the individual researcher.

The second group of CFD package is specially designed for microfluidics, such as

MemCFD from Coventor Inc. and CFDACE+ from CFDRC Inc. These softwares enable

users to do some quick simulations and designs, and hence, save time. However, com-

pared to the general CFD package, they have a much smaller user group, and a shorter

history. Furthermore, the theory of microfluidics was still under development. Therefore,

the users should be aware of the restrictions of these softwares. In addition, because the

surface/volume ratio is high in microfluidics devices, a CFD package should be able to

simulate the interfacial effects, such as capillary flow and electrokinetic flow.

The third group, eg. DiffPack and Matlab, contains some general tools, for exam-

ple, partial differential equation (PDE) solvers, which can complement CFD packages for

developing microfluidic models. This group has the greatest flexibility. Nevertheless, the

development and validation of numerical coding were expensive, time consuming, and

the performance of the simulation depends very much on the experience of individual re-

searchers.

For the above reasons, a microfluidic CFD package, MemCFDTM

, was chosen as

59

the primary tool to study the microfluidic mixing. MemCFD embeds Fluent and Fidap

solvers to solve Navier-Stokes equations and has functions to simulate multi-species mix-

ing, pressure-driven flows as well as electro-osmotic flows. Fluent and Fidap are the state-

of-art CFD solvers to date, and MemCFD also has integrated advanced modelling and

visualization tools, therefore, MemCFD can provide reliable and high quality simulations.

To complement and validate the CFD simulation results and the algorithms developed

as part of this study, experiments were conducted to validate typical models as well as the

use of published experimental data and analytical predictions. In the research, customized

codes, to improve simulation quality, and particle tracing algorithms were also developed

for an extensive investigation of 3D complex advection in micro-mixers with patterned

grooves (section8.3).

In this chapter, section3.2 introduces the CFD fundamentals. Section3.3 describes

the special requirements of microfluidic modelling. Complementary experiments were nec-

essary to understand the physical principles behind the numerical simulations, therefore, a

simple experimental method is introduced in Section3.4.

3.2 Fundamentals of Computational Fluid Dynamics (CFD)

3.2.1 Overview

The technological value of computational fluid dynamics has become undisputed. In

microfluidics, it might be very difficult to experimentally diagnose the flow behavior at

micron or sub-micron scales. However, it is still feasible for CFD to be used to simulate

micro-flow as long as the validation of continuity is maintained (in Section2.3.1). For the

understanding of the significance and limitation of CFD in microfluidics, a basic knowledge

is introduced in this section.

60

CFD simulation can be divided into three stages, pre-processing (modelling), analysis

(solve Navier-Stokes equations) and post-processing (results output). Among these CFD

modules, the solvers are the core of the problem.

3.2.2 CFD Solvers

The Navier-Stokes equations for flows of practical interest are usually so complicated

that an exact solution is unavailable, and it is necessary to seek a computational solution.

Computational techniques replace the governing partial differential equations with systems

of algebraic equations, so that a computer can be used to obtain the solution. Since the

governing equations for most cases of fluid dynamics are nonlinear, the computational so-

lution usually proceeds iteratively. Therefore, if the numerical algorithm that performs the

iteration is stable, and the discrete equations are faithful representations of governing equa-

tions, then the computational solution can be made arbitrarily close to the true solution of

governing equations by refining the grid. However, the refinement is restricted by com-

puter resources and accumulation of error during the iteration. Therefore, optimized grid

size should be applied to obtain sufficient accuracy.

In general, there are many methods for discretizing the physical domain, including fi-

nite difference method (FDM), finite volume method (FVM), finite element method (FEM)

and spectral method. FDM is very well developed and very straightforward to understand

the discretization of PDEs. However, FDM requires regular shape of mesh elements and

cannot be used to solve problems with complex geometric shape. In general, FDM is rarely

used in most of the contemporary CFD codes. FVM and FEM are found commonly used,

and are based on the theory of weighted residual methods (WRM).

A weighted residual method assumes that the solution can be represented analytically.

The starting point for a weight residual point is to assume an approximate solution, and

then the partial differential equations can be replaced by a system of algebraic equations or

61

a system of ordinary differential equations for time-dependent problems. By substituting

the exact solution, an equation residual is generated. The integral of the weighted residual

over the computational domain should be zero to determine the necessary coefficients of the

system of algebraic equations or the system of ordinary differential equations in general.

Different choices for the weight (trial) function give rise to different methods in the class

of methods of weight residuals. This forms the theoretical foundation of the finite element

and finite volume methods.

Finite Element Analysis

In the Finite Element Method, the solution domain can be discretized into a number

of uniform or non-uniform finite elements that are connected via nodes. The change of

the dependent variable with regard to location is approximated within each element by an

interpolation function. The interpolation function is defined relative to the values of the

variable at the nodes associated with each element. The original boundary value problem

is then replaced with an equivalent integral formulation. The interpolation functions are

then substituted into the integral equation, integrated, and combined with the results from

all other elements in the solution domain. The results of this procedure can be reformulated

into a matrix equation of the form, which is subsequently solved for the unknown variable.

The finite element method was developed initially as an ad hoc engineering proce-

dure for constructing matrix solutions for stress and displacement calculations in structural

analysis. The method was placed on a sound mathematical foundation by considering the

potential energy of the system and giving the finite element method a variational interpre-

tation. However, very few fluid dynamic (or heat transfer) problems can be expressed in a

variational form. For many situations, the Galerkin method is used to define the trial func-

tion or shape functions in the engineering literature for solving variational problems (Li,

Wang & Yi 1999). Consequently, most of the finite element applications in fluid dynamics

62

have used the Galerkin finite element formulation. The Galerkin finite element method has

two different features from traditional weighted residual methods. Firstly, the approximate

solution is written directly in terms of the nodal unknowns. The second important fea-

ture is that the approximating functions are chosen exclusively from low-order piecewise

polynomials restricted to contiguous elements.

The finite element method discretizes flow domain in two stages. First a piecewise

interpolation is introduced over discrete or finite elements to connect the local solution

to the nodal values. The second stage uses a weighted residual method to obtain alge-

braic equations connecting the solution nodal values. Both stages introduce errors. Small

element size can achieve more accurate interpolation, however, this induces more interpo-

lation steps and the error of each interpolation is accumulated, therefore, a restriction on

the smallest element size should be placed in the computation.

Finite Volume Analysis

Finite volume analysis divides the computational domain into many sub-domains, and

the weighted function is assigned unity if inside a sub-domain or zero if outside. The fi-

nite volume method calculates the values of the conserved variables averaged across the

sub-domain and provides an appropriate framework for enforcing conservation at the dis-

cretized equation level. In this way the conservation properties inherent in the governing

equations are preserved. This is a particular advantage in obtaining accurate solutions for

internal flows.

Finite volume method, which has been used for both incompressible and compress-

ible flow, has two major advantages. First it has good conservation properties, as mentioned

above. Second it allows complicated computational domains to be discretized in a struc-

tured and/or unstructured manner.

63

In summary, the finite volume method is preferable to other methods as a result of

the fact that boundary conditions can be applied non-invasively. This is true because the

values of the conserved variables are located within the volume element, and not at nodes

or surfaces. Finite volume methods are especially powerful on coarse non-uniform grids

and in calculations where the mesh moves to track interfaces. For the above reasons, the

finite volume method was used extensively in this research.

3.3 CFD Modelling in Microfluidics

For the dimensions of the microfluidic devices and the fluids investigated in this re-

search, the Knudsen number (Section2.3.1) is approximately zero and continuum holds

true. Therefore, the computational fluid dynamic tools can be applied in the investigation

of passive microfluidic mixing. However, viscosity is critical and cannot be ignored for a

successful simulation.

In this study, a CFD package, known as MemCFDTM from CoventorWare, was ap-

plied. MemCFD provides both FVM and FEM solvers to simulate microfluidic problems.

For solving Navier-Stokes equations coupled with convection-diffusion equations, finite

volume method was used extensively for its advantage of mass conservation.

Although there is little significant change of governing equations in CFD solvers for

micro-flows, there are practical issues needed to be considered for a simulation in a micro-

channel. First, it is convenient to use a standard system of units and a modified system of

units for microfluidics was introduced in MemCFD. In this section, an analytical bench-

mark for CFD modelling and mesh sensitivity study are also introduced to calibrate an

accurate CFD simulation. To improve the performance of MemCFD, a mesh converting

program was developed to use an advanced meshing package to create a uniform struc-

tured mesh. The analysis of the simulation results are done by the post-processing tools

64

within MemCFD. Fortran programs were developed for further analysis of the results ob-

tained from the CFD simulations.

All simulations were performed on Win NT4.0 with PentiumIII 800MHz CPU and

512MB memory. The maximum number of mesh elements (cells) in the models was up to

300,000 for the limitation of this class of workstation.

3.3.1 System of Units in Microfluidics

As the characteristic lengths of microfluidic devices are of the order a few microns to

a few millimeters, it is practical to use micron as the unit of length. The units of length (L),

time interval (T ), temperature (Θ) and electrical current (A) are regarded as the reference

(M-L-T-Θ-A) system, and other units are derived from them. Therefore, the dimensional

system for microfluidic are mass (kg), length (µm), time interval (second), temperature

(Kelvin) and current (pico Ampere). When applying the reference magnitudes to derive

other ones, the dimensional homogeneity principles must be followed. Then, we can derive

the following units used in microfluidics (Table3.1). Other units, which are not listed in

this table, can be derived by the same rules.

3.3.2 Boundary Conditions

Boundary conditions are necessary for the CFD solvers to initialize the simulation

and achieve rational results. In microfluidic applications with liquids inside the channel,

the Reynolds number is low and the flow is generally laminar as stated in section2.3. As

viscosity dominates the flow domain, the velocity close to the wall is zero. For most of

the simulations run in this research, the boundary conditions were set as steady, laminar,

Newtonian, with one fluid (or two fluids to solve diffusion-convection problems). The

inlets were assigned volumetric flow rate/velocity, or pressure boundary conditions.

65

Parameter Symbol Microfluidic UnitsMass m kgCurrent C pALength l µmTime interval t sTemperature T KPressure p MPaDensity ρ kg µm−3

Diffusivity D µm2 s−1

Force F µNMoment of Inertia I kg µm2

Power P pWStress σij MPaThermal Coefficient of Expansion TCE K−1

Velocity v µm s−1

Viscosity µ MPa s

Table 3.1:Systems of units in microfluidics

3.3.3 Building Numerical Models for Micro-Flows

MemCFD provides an integrated environment to include all of the three basic CFD

modules (Section3.2.1). However, it does not give users sufficient control of the software,

especially the pre and post-processes. For some complex structures, MemCFD cannot cre-

ate a uniform structured mesh. However, a high quality CFD simulation depended very

much on the pre-processing stage, which includes meshing and boundary condition set-

tings. Therefore, mesh quality is very critical to the accurate simulations. More uniform

mesh and aspect ratio of mesh elements can improve the accuracy of numerical simulations.

To calibrate the CFD models, mesh sensitivity and uniformity studies were performed.

66

Mesh Sensitivity Study

For a general CFD solution of the governing equations over a range of significantly

different grid resolutions should be presented to demonstrate grid-independent or grid-

convergent result. Though it is not possible to use this method to address the truncation

error of the MemCFD solver, it is critical for an accurate CFD simulation. The empiri-

cal mesh density was estimated at least 20 element per edge for flows in a microchannel.

Denser mesh elements can give better accuracy, however, the solving and post-processes

would be slowed down and sometimes halt the system’s operation. To further determine

the size of mesh elements, an analytical benchmark of viscous flow in a rectangular micro-

channel was used to calibrate the mesh quality. Viscous flow in a rectangular micro-channel

also has practical value in studying microfluidics, and would be studied in Chapter4.

To give a sample of benchmarking process, a viscous flow in a rectangular micro-

channel with width to depth aspect ratio (w/H) of 4 was used to calibrate the CFD model

(Figure3.1). Figure3.2shows the structured mesh of the numerical model. The mesh was

constructed uniformly and had denser elements at walls.

For a given flow rate, the velocity field for a fully developed flow in a rectangular

channel could be predicted analytically by Equation4.4described in Section4.2or it can be

simulated by CFD solvers. The comparison of analytically prediction and CFD simulation

result gives good indication of the quality of numerical model. The comparisons could

be found in the line plots in Figures3.3 and3.4. The analytical prediction agreed with

the simulation results very well. The maximum error existed close to the centre of the

channel, and was less than 2.5%. As both the analytical predictions and CFD simulation

could contribute to errors, it was acceptable to determine the mesh density to be applied

when the difference was less than 5% in this research.

67

2b

x

z

y

Fluid in

Fluid out

2a

Figure 3.1:Viscous flow in a rectangular micro-channel

Mesh Uniformity Study

Because of the limitation for user to control the pre-processes in MemCFD, a state-of-

art mesh generation package, Gambitr from Fluent Inc., was used to improve the meshing

quality. Gambit could build a structured mesh by mapping or sub-mapping techniques.

However, the mesh topology in Gambit is different from that used in MemCFD (Figure

3.5). To use numerical models generated by Gambit, the mesh topologies needed to be

converted to the format that MemCFD could import. A Fortran program was coded to

convert the Gambit neutral file to universal neutral file compatible with MemCFD. The

algorithm in words is described below,

68

Figure 3.2:A typical meshed fluid volume in a rectangular micro-channel

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5

y/2a

u(y

, zm

id)

CFD simulationAnalytical prediction

Figure 3.3:Normalized velocity profileu(y, zmid) in xOy plane

69

0.0

0.2

0.4

0.6

0.8

1.0

-0.5 0 0.5

z/2b

u(y

mid

, z)

CFD simulation (z plot)analytical prediction

Figure 3.4:Normalized velocity profileu(ymid, z) in xOz plane

70

1

2

68

7

3

4

5

4 1

2

67

8 5

3

a b

Figure 3.5: Mesh topology, (a). Gambitr 8-Hexahedral mesh; (b). MemCFDTM

8-Hexahedral mesh

a b

Figure 3.6:(a). Structured 8-node Hexahedral mesh by sub-mapping technique; (b). Irreg-ular 8-node structured mesh by MemCFD

TM

startread Gambit neutral file

while ( < total number of elementsN )exchange node 3 and node 4exchange node 7 and node 8

endwhilewrite to universal neutral file

stop

The mesh illustrated in Figure3.6a is more uniform than the mesh in Figure3.6b

for a complex structure. The meshing uniformity of mesh is critical for an accurate CFD

simulation. Uniform mesh elements provide more accurate simulations and use less CPU

time.

71

Although the CFD prediction of velocity field with an irregular or unstructured mesh

was accurate, solving of convection-diffusion problem could be a problem. For instance,

to simulate a convection-diffusion problem in a microchannel of sizeH × w = 200µm ×100µm, two fluids were assigned to two inlets respectively, and each with a volumetric flow

rate of 2×108µm3 s−1 (Re = 5). The diffusion coefficient was4 × 10−6cm2 s−1, which

was calculated according to the chemical groups of a food colouring dye (Reid, Prausnitz

& Poling 1987). The simulation results with uniform and irregular mesh are shown in Fig-

ure3.7. Even with the same boundary conditions, Figure3.7a shows inaccurate simulation

results using irregular mesh elements. Two cross sections ((1) and (2) in Figure3.7b) of

mass distribution are shown in Figure3.8, and are compared with the published experimen-

tal measurement by Stroock et al. (Stroock et al. 2002). The flow patterns of experimental

measurements and simulation results have the same trend, which indicate the simulation is

accurate and reliable.

3.3.4 Simulation Results Analysis

CFD simulations create a large amount of data to be analyzed. MemCFD provided a

visualization tool to post-process the results directly and a feature to export the results for

further analysis. The relevant functions included scalar plots of flow velocity, mass fraction

distribution, vector plots of velocity field, and flow trajectory (streaklines). Generally, the

visualizer displays the flow pattern inside a micro-channel, and the performance of micro-

mixer is evaluated by the comparison of mixing efficiency integrated from the mass fraction

distributions.

For some applications, the existing post-processing functions in MemCFD were not

adequate. Nevertheless, MemCFD provided a feature to export the entire simulation results

into a text file. This text file can be analyzed in other visualization tools or analyzed by

customized program. In Chapter8, the velocity fields in a periodic grooved micro-channel

72

(a)

2

1

(b)

Figure 3.7: Mesh uniformity study, (a). irregular mesh elements; (b). uniform meshelements

73

Section 1 Figure 3.7b

Section 2 Figure 3.7b

Figure 3.8: Mixing of two fluids in a T-channel with grooved surfaces, top: simulationresult, two fluids are represented by blue and red, green colour is the mixed region, bottom:cited experimental results by Stroock, et al. 2002a, two fluids are represented by gold andblack colours.

were exported into text files and analyzed by a particle tracing program that was developed

to calculate the flow trajectories to investigate the folding and stretching of fluid streams.

3.4 Experimental Method

No matter how powerful the CFD solver is, if the numerical model or the bound-

ary condition is not correctly defined, the simulation result will not reflect the reality. In

this aspect, experiments are necessary to verify the numerical model. Because the various

numerical models used for simulating convection-diffusion problems in this thesis were

consistent with each other, only selected numerical models were tested by experiments.

Moreover, the simulation results of flow patterns in microchannels with patterned grooves

were compared with published experimental data, which were used to verify the CFD sim-

ulations of this type of micro-mixers.

In this section, an overview of micro-fabrication of micro-mixers is provided, followed

by description of an experimental setup used for the visualization of micro-mixing.

74

3.4.1 Excimer Laser Micro-machining

Excimer laser micro-machining was used to fabricate the micro-mixers in this re-

search, however, other micro-machining techniques would also be able to create the same

structures in a compatible manner but with silicon as the material. The micro-mixers were

fabricated in the microtechnology laboratory in the Industrial Research Institute Swinburne.

An Excimer Laser (ExiTech) was used to ablate channels onto polymer substrates. The

standard operation procedures could be found in early papers published by the labora-

tory (Harvey, Rumsby, Gower & Remnant 1995). In brief, the pulsed UV laser beam, with

a wavelength of 248nm, is projected through the designed features on a chrome-on-quartz

mask, and then the laser beam is focused to pattern the feature on the substrate (Figure3.9

). The Excimer laser machined structures had a depth resolution of the order of0.1µm and

spatial resolution of the order of1µm.

Plane of uniformity

& Mask

Lens Array Homogenizer

Input Beam248 nm (KrF) or 193nm (ArF)

Projection Lens

Workpiece step motion

Figure 3.9:A schematic illustration of Excimer laser micromachining

(Harvey et al. 1995)

75

Figure 3.10: Section of Y-channel machined in Polycarbonate by Excimer Laser (SEMimage)

3.4.2 Microfluidic Packaging

Figure3.10shows the Excimer laser ablated Y-channel on a polycarbonate substrate.

Following the ablation process, the channel was packaged in a lamination process. A thin

PET plastic foil coated with a melting adhesive film layer was pressed by a heated lami-

nation roller (at 150oC) onto the structure as the lid. MelinexR© Polyester film type301

(Dupont Teijin films) having a thickness of30µm was used in our fabrication, and the

lamination equipment was a MEGA dry film laminator model305 (Mega Electronics).

3.4.3 Microfluidic Mixing Visualization

Most of the conventional fluid field measurement methods were not suitable for mi-

crofluidic environments. Even though theµPIV technique has been researched since 1998

(Santiago et al. 1998), its complexity forbids it being widely used in general microfluidic

research centers. For studying microfluidic mixing, fluorescent dyes, food colouring dyes,

and ultraviolet absorption techniques have been used for measurement and evaluation of

76

Observation window

CCD camera

Image capture computer Optical microscope

Figure 3.11:Laboratory setup for micro-mixing visualization

77

mixing performance. In this research, food colouring dyes are used to visualize the flow

and mixing process and is reported in Section6.3.2.

Two colouring aqueous solutions have been applied in the study. One was mixture of

water and2.4% food dye E102 and E122 (Yellow). Another one was mixture of water and

2.1% food dye E133 (blue) (Queen Fine Foods Pty Ltd, Australia). The two aqueous solu-

tions were respectively introduced to the channel from two inlets by capillary effect. The

viscosity of the food dye/water solution was estimated as being approximately the same as

water, and the diffusion coefficient was calculated according to the chemical groups (Reid

et al. 1987). The wastes from the outlet were collected by porus paper, and by measuring

the net weight before and after wetting in a known interval of time, hence, flow rates could

be calculated.

Figure3.11shows the laboratory testing devices for visualizing mixing performance.

The micro-mixer was filled with two different food colouring dye aqueous solutions, and

the broadening of mixed region was observed under an optical microscope. The image

was captured by a charge-coupled device (CCD) camera, and then transferred to the image

capture card in the computer.

3.5 Summary

To study passive mixing with various geometric structures, there were many param-

eters. Because the design of experiments became too complicated for all the experiments

to be carried out, it was impractical to investigate all these parameters by an experimental

approach.

On the other hand, most of micro flows are laminar with non-slip boundary conditions.

For laminar incompressible viscous flow, advanced microfluidic CFD techniques can be

applied to provide accurate solutions to microfluidic problems. To achieve a reliable CFD

78

prediction of fluid behaviour, the model needed to be built carefully with the assistance

of a sensitivity study. To calibrate the CFD models, Hele-Shaw flow in a finite aspect

ratio rectangular micro-channel was used. As the Hele-Shaw model was well-developed,

it could be used to validate the velocity field predicted by CFD simulation. To achieve a

reliable CFD simulation, the mesh density should no less than 20 elements per edge. Even

the velocity field was accurately predicted, and the simulation of mass transport was very

sensitive to the mesh quality. A uniform, evenly distributed aspect ratio mesh needs to be

applied with sufficient mesh density. To achieve this, an advanced mesh generator was used

and a mesh could be translated and imported into the MemCFD solvers using a program

developed in Fortran. Typical simulations were selected to be validated by experiments and

existing published experimental data. Simple experimental testing equipment was setup

for carrying out tests. The equipment included an optical microscope, a CCD camera and

an image capture computer. The simple laboratory configuration could give qualitative

comparisons between experiment visualizations and numerical simulations. A fabrication

technique to produce simple micromixers in microchannels was also described.

With the numerical and experimental methods developed, the subsequent research on

microfluidic mixing is presented in the following chapters. In Chapter4, typical microflu-

idic flows in a rectangular microchannel were investigated to provide a necessary under-

standing of flow behaviours and mixing performance in a microscopic scale.

79

Chapter 4

Understanding Microfluidic Mixing

4.1 Introduction

To commence the research on microfluidic mixing, it is necessary to understand the

fundamentals of the scaling effects on the behaviour of microfabricated fluidic devices. The

scaling down of the characteristic length in microfluidic device resulted in low Reynolds

numbers, which initiated research to study fluid dynamics under low Reynolds numbers in

microfluidic devices.

The scaling down of a micro-device gave rise to the diffusion-controlled systems (Manz

et al. 1990). In such a device, the down-scale to1/10 of size reduced the diffusion time

to 1/100. If the Reynolds number remained constant, the pressure drop increased100

times. Therefore, the characteristic length of a micro-device was critical to the diffusion-

controlled system.

Taylor-Aris dispersion is an effect for dispersion of slugs of solute in a small tube with

a diameter of500µm. It is an unwanted effect for most analytical processes, however, it

had unique ability to enhance mixing.

80

The aim of this chapter is to identify the mechanisms behind microfluidic flow be-

haviour, and therefore determine the necessary parameters, such as characteristic length

scale and flow rate, for the design of a micro-mixer. Section4.2 introduces flow at low

Reynolds number, which is a characteristic feature of microfluidics. The understanding of

Taylor-Aris dispersion is essential to the design of miniaturized mixers, and is covered in

Section4.3. Critical design parameters of microfluidic mixers were discussed in Section

4.4. Finally, Section4.5summarizes the outcome of this chapter.

4.2 Flow at Low Reynolds Number

The Reynolds number is used to relate the inertial forces to the viscous forces, and is

given by equation4.1,

Re =ρul

µ=

ul

ν, (4.1)

whereρ is the density,u is the average velocity,µ is the dynamic viscosity,l is the charac-

teristic length of the microchannel andν is the kinematical viscosity.

In microfluidic devices, due to the low velocity of the flow and the micron size chan-

nels, the Reynolds number is low. This means that the viscosity plays a dominant role

in microchannels rather than inertia, and turbulence cannot exist in such a viscous flow.

One classic paper described fluid dynamic differences at the low Reynolds number (Purcell

1977). Purcell stated that whenRe ¿ 1, the inertia dies off from the Navier-Stokes equa-

tion, and gives the governing equation at such low Reynolds number as,

µ∇2u = ∇p (4.2)

wherep is the pressure andu is the velocity.

Equation (4.2) indicates that the velocity is determined only by the pressure distribu-

tion and all motion is symmetric in time at low Reynolds number, as described in a paper by

81

Brody (Brody, Yager, Goldstein & Austin 1996). This means that the flow can be reversed

back to its original status.

H

Fluid in

Fluid out

w

x

z

y

Figure 4.1:Rectangular microchannel

Microchannels with a rectangular section were widely used in the microfluidic appli-

cations. The slow viscous flow in a rectangular channel is very similar to a Hele-Shaw flow.

The literature review found vast literature on the Hele-Shaw flow, however, this was out of

the scope of this research. Nevertheless, the results from studying Hele-Shaw flow could

be used to study the flow behaviour in rectangular microchannels, which were the most

commonly used shape in microfluidic applications. Considering the velocity distribution

for the flow in a rectangular channel shown in Figure4.1, based on conservation of mass,

82

or continuity in other words, the velocity remains the same at thex direction for a fully

developed incompressible flow. The 2D velocity distribution,u(y, z), satisfies Poisson’s

equation4.3, which is rewritten from equation4.2,

∇2u(y, z) = − 1

µ

dp

dx(4.3)

whereµ is the dynamic viscosity anddp/dx is pressure gradient and assumed an absolute

constant (Lamb 1993).

With necessary wall boundary conditions,uwall = 0, a Fourier series solution of ve-

locity field u(y, z) in a rectangular channel size of−w/2 ≤ w/2 and−H/2 ≤ H/2 can be

written as:

u(y, z) =4w2

µπ3

(−dp

dx

) ∞∑n=1

(−1)(n−1)

[1− cosh((2n− 1)πz/w)

cosh((2n− 1)πH/2w)

]cos((2n− 1)πy/w)

(2n− 1)3,

(4.4)

where the velocity varies in the channel width to depth aspect ratiow/H. As hyperbolic

functioncosh approaches infinity withn, it is non-trivial to achieve exact solution of above

equation4.4. However, as the sum converge very rapidly, a finite but sufficient largen

value can be used to estimate the solution by numerical computation (see AppendixB.2).

By integrating equation4.4 over the area of the channel section, the volumetric flow

rate can be obtained and stated as,

q =8Hw3

µπ4

(−dp

dx

) ∞∑n=1

[1

(2n− 1)4− 2w

(2n− 1)5πHtanh((2n− 1)πH/2w)

](4.5)

whereq is the volumetric flow rate,dp/dx is the pressure gradient alongx. For conve-

nience, the pressure gradient can be related to the mean velocityu and stated as,

dp

dx= −4kµu

H2(4.6)

wherek is a constant related to the aspect ratio of a rectangular channel, and the value of

k, which can be calculated by equations4.5and4.6with a finiten, is given in Table4.1.

83

Using equations4.5 and4.6, for a known volumetric flow rate, the velocity field for

a fully developed flow in a rectangular channel was computed. Figure4.2 shows the flow

field in a rectangular channel with width to depth aspect ratio (w/H = 4).

−0.5

0

0.5

−0.5

0

0.50

0.2

0.4

0.6

0.8

1

y*z*

u*

Figure 4.2: Normalized velocity profile in a rectangular microchannel, w/H = 4;u∗ isnormalized velocity,y∗ = y/w andz∗ = z/H

By cutting the normalized velocity profile from Figure4.2 along the mid ofy and

project it on the XZ plane, Figure4.3 shows that this velocity profile is very close to a

parabolic shape in shallow channel. Using the same method but for variousw/H ratios,

Figure 4.4 shows the velocity profiles projected on the XY plane, and that the velocity

profile moves to a plug-like flow when thew/H ratio becomes larger. From Figure4.4, it

can be concluded that the side-wall effects on velocity profile are negligible when the ratio

of w/H →∞, which is the Hele-Shaw flow.

It is important to notice that the velocity profiles are not restrained by Reynolds num-

ber when flow is in the laminar region, therefore, the results can be applied to low Reynolds

number microfluidic flows.

84

w/H 1 2 3 4 5 10 20 100 ∞k 28.454 17.492 15.191 14.244 13.731 12.807 12.3905 12.076 12.002

Table 4.1:Values ofk in a rectangular-cross-section channel for various width to depthaspect ratios

0.0

0.2

0.4

0.6

0.8

1.0

-0.5 0.0 0.5

z/w

u(y

mid

, z)

Figure 4.3:Normalized velocity profile by the maximum flow velocity plotted in XZ planefor y = ymid

85

y/H

u*(ymid,z)

Figure 4.4:Normalized velocity profile plotted in XY plane forw/H = 2 (O), w/H = 5(×) andw/H = 10 (+).

4.3 Taylor-Aris Dispersion

4.3.1 Overview

The nature of microfluidic mixing is similar to a process outlined by Taylor (Taylor

1953). Therefore, it is important to understand the background knowledge of Taylor Dis-

persion. Section4.3.2reviews dispersion of slowly moving solute in a rectangular cross-

section microchannel, which can be termed Taylor-Aris dispersion. Section4.3.3investi-

gates the application of Taylor-Aris dispersion to enhance mixing in a microchannel with

rectangular cross-section.

86

4.3.2 Dispersion of a Solute in Solvent Flowing Slowly Through a Rect-

angular Cross-Section Microchannel

In Taylor’s work, a section of one fluid was introduced into the flow of another fluid

in a small circular cross-section tube (φ500µm). The flow was laminar, with a parabolic

velocity profile, where the velocity in the center of channel was maximum and the velocity

at walls was zero. Therefore, the introduced fluid streams in the tube are stretched. Be-

cause the velocity closest to wall is slower, the fluid streams have more time to diffuse into

each other. When the mean velocity is slow and tube is small enough, the diffusion will

sample all the streamlines in the tube. That is the reason that the fluid spread around a

cross-sectional plane moving at the mean fluid velocity is similar to fluid diffusing from

such a plane in still fluid. This spreading of solute could be characterized by a dispersion

coefficient instead of a pure molecular diffusion coefficient. While Taylor’s work dealt

with a round tube, a similar analysis based on his ideas was applied to a 2-D rectangular

channel (Doshi et al. 1978, Chatwin & Sullivan 1982), which can be useful for mixing in a

microchannel.

The concentration distribution in a rectangular channel can be described by the con-

vective diffusion equations,

∂C

∂t+ u(y, z)

∂C

∂x= D

[∂2C

∂x2+

∂2C

∂y2+

∂2C

∂z2

](4.7)

87

along with the following boundary conditions:

C(0, y, z) =

0 0 ≤ y ≤ w/2,

1 −w/2 ≤ y < 0,

∂C

∂t= 0

∂C

∂x

∣∣∣∣x→∞

= 0

∂C

∂y

∣∣∣∣y=0,±w/2

= 0

∂C

∂z

∣∣∣∣z=0,±H/2

= 0

C(∞, y, z) = constant

(4.8)

whereC is concentration distribution.

The analytical solution to the above problem is non-trivial. Although there was attempt

to use Taylor’s approach to give the analytical prediction, the result created controversy

because the Taylor-Aris dispersion in rectangular microchannel was different from that

used in the original study of dispersion in a small circular pipe (Beard 2001, Dorfman &

Brenner 2001). Most of the microfluidic mixers researched introduced confluent streams

into the microchannel, and flows were steady (∂C/∂t = 0). The concentration gradient at

any position of the channel does not change with time, and this is a fundamental difference

to the Taylor-Aris dispersion.

The Taylor-Aris dispersion could and did inspire the study of microfluidic mixing,

due to the virtual diffusion coefficientD∗ (Taylor’s dispersivity) being several orders larger

than the pure molecular diffusion coefficient. Micro-mixers had been introduced to apply

Taylor-Aris dispersion using an active pulsed source (Niu & Lee 2003), and was claimed

as chaotic mixer. Though it is true the Taylor dispersion is a chaotic system with time

variable in a 2D Hamilton system, it is much easier to comprehend with well-developed

Taylor-Aris dispersion theories. Nevertheless, the Taylor-Aris dispersion is a special case

88

of the convection-diffusion problem and points out that introducing lateral convection is a

way to enhance microfluidic mixing.

Taylor’s dispersivity in a rectangular channel can be described in equation4.9,

D∗ = D +1

210

u2w2

Df(

H

w) (4.9)

whereD is the molecular diffusion coefficient,w is the width of the channel,u is the mean

velocity in the channel,H is the depth of the channel andf is the geometric function of a

rectangular channel, decided by the height to width aspect ratioH/w. While H/w ¿ 0,

assuming no side-walls, it gives that the geometric functionf a unity value, however, with

the consideration of side-walls,f = 7.95. Therefore, the side-walls of a microchannel

have to be considered while investigating mixing problems. For a Taylor dispersivityD∗,

the length between the dispersed leading and trailing edges of slug,lm, is estimated by the

Einstein-Smoluchowski relation (lm =√

2D∗t).

1

10

100

1000

10000

0 200 400 600 800

u (µm/s)

D*/

D

Figure 4.5:Ratio of Taylor dispersivity to pure molecular diffusion coefficient for differentReynolds numbers

Taylor dispersivityD∗ is much larger than the molecular diffusivityD (Figure4.5),

89

therefore, for a fixed width of channelw, the mixing timet is reduced. Analytical results

can evaluate Taylor’s dispersivity for rectangular channels with different depth to width

aspect ratio (Doshi et al. 1978). The transient behavior of most practical applications are

difficult to interpret by analytical approaches, however, it is much easier to interpret through

numerical simulations.

4.3.3 Applying Taylor-Aris Dispersion to Enhance Mixing

The simulation results in Figure4.6 demonstrate a time series of mass fraction of

a slug of fluid 1 dispersed in fluid 2 for a time period of 1 to 20 seconds. Boundaries

conditions were kept the same as for Taylor-Aris’s analytical interpretation, the flow was

very slow (Re = 0.005) and all wall boundaries were assigned zero velocity. Therefore,

the simulation could be verified by comparing with the analytical prediction.

Figure4.6 clearly shows that the length of the dispersed fluid is longer than a pure

diffusion length, because of the convection. As Taylor’s dispersivity coefficient is a func-

tion of mean velocity,D∗ = f(u), the inference is to have a large flow rate for fast mixing.

However, as velocity increases, there will be less than sufficient time for the diffusion to

sample all the streamlines.

In practice, the velocity in a microchannel is mostly too large to satisfy Taylor-Aris

requirement for an analytical solution. However, the effects of Taylor-Aris dispersion still

exist in a faster moving stream with a more complex flow pattern.

Simple experiments were carried out to demonstrate mixing of two fluids in a Y-type

microchannel, illustrated in Figure3.10. Initially, the two fluids flowed parallel to each

other without significant mixing. When one fluid supply was stopped, the remaining of

this fluid was embedded into the main stream and its dispersion into the second fluids

90

1sec

2sec

5sec

8sec

10sec

12sec

15sec

18sec

20sec

Figure 4.6:Taylor dispersion along a straight channel: numerical simulation for varioustime intervals

was enhanced dramatically. Later in the research, it was found that the phenomena was

termed Taylor dispersion in the literature. Nevertheless, the experimental results are shown

in Figure4.7, where the green region was the mixed fluids between fluid 1 (yellow) and

2 (blue). The dispersed solute was initially a narrow parabolic shape at timet1, and then

developed into a wider parabolic shape at timet2 in a short distance. Although there is lack

of quantitative measurement for evaluating the mixed volume of the two fluids, the enlarged

area of mixed green region and lengthened parabolic shape indicated the enhancement of

mixing. In addition, this phenomena was repeated several times during the experiments

and demonstrated similar results.

91

500µm 500µm

(a) (b)

Figure 4.7:Experimental results: the solute dispersed at, (a) timet and (b) timet + δt

The simulation results shows the same as experimental results that the broadening of

the slug of fluid1 in the axial direction, and the result demonstrated the actual dispersivity

was much faster than pure molecular diffusion. The matching of simulation results and

experimental results also shows the accuracy of the simulation and prove CFD can be a

reliable tool in studying microfluidic mixing.

4.4 Two Important Parameters for Designing a Microflu-

idic Mixer

Unlike chamber mixing, passive microfluidic mixing is an in-situ process in a mi-

crochannel. The mixing performance is measured by the length to achieve complete mix-

ing, which can be measured by the Peclet number. From the literature review, it was found

that the Peclet number is a critical parameter for the design of a micro-mixer, and hence,

discussed first. In addition, the aspect ratio of a rectangular microchannel is another impor-

tant parameter regarding the diffusion path and pressure drop, and is also discussed in this

section.

92

4.4.1 Peclet number

For mixing at low Reynolds number, diffusion is dominant and mechanical agitation

is ineffective. Molecular diffusion is a random process, and the diffusion timetD to cross

a distancel was given by Einstein in his paper on Brownian motion (Einstein 1905). For

typical small molecular liquids with a diffusion coefficientD of 103µm2/s, ie. H2O, the

diffusion time to cross a 100µm distance is about 5 seconds. However, for 1mm distance,

the mixing time is more than 8 minutes. For most bio-molecules with diffusion coefficients

one order or even two orders lower than that of water, diffusion time is of the order of hours,

which is not considered realistic in a microscaled device. For mixing to be completed in

milliseconds, the diffusion path should be of the order of 1µm.

Considering microfluidic devices with microchannels of the order of a few hundred

microns, diffusion is not very efficient due to the large diffusion path. However, for low

Reynolds number flow in a microchannel (Re< 10), there is no turbulence to assist mixing,

therefore convection is the only method for mass transport in laminar flows other than pure

molecular diffusion (Purcell 1977). The convection time through the channel can be noted

as,

tc =Lm

u(4.10)

whereu is the mean velocity, andLm is the length of the channel for mixing.

The molecular diffusion timetD through a distance ofa can be stated as,

tD =a2

D(4.11)

wherea is diffusion path (half of the width of channel in this research) andD is the diffu-

sion coefficient. Therefore, iftc ≥ tD, the fluids can be diffused into each other completely

by the end of the channel. Hence, the least channel length for complete mixing can be

stated as,Lm,

Lm ≥ a2

Du (4.12)

93

Lm can be refined as,

Lm ≥ u · tD = Pe · a. (4.13)

where Peclet number,Pe, is the ratio of bulk mass transfer (Convection or Stirring) to

molecular diffusivity,

Pe =au

D(4.14)

According to Equation4.13, it would be desirable to have a small Peclet number in

order to have a better mixing. When the flow velocity is very slow to have a small Peclet

number, therefore, fluids have sufficient residual time for diffusion to take place. A com-

plete mixing can be achieved without further assistant. However, it can be found that flow

velocity is a constrained design parameter to supply sufficient mixture for certain applica-

tion. In such cases, to reducea in Equation4.14is one way to reduce Peclet number. From

previous review, convection induced virtual diffusion coefficientD∗ (Taylor’s dispersiv-

ity) is several folds higher than pure molecular diffusion coefficient, to introduce Taylor’s

dispersivityD∗ is another way to reduce Peclet number in Equation4.14.

Using the Peclet number, microfluidic mixing can be divided into three zones (Figure

4.8). To clarify Figure4.8, the diffusion zone means diffusion dominates and convection

is neglected, and the convection zone means the diffusion can be neglected. Figure4.8

was used to define the scope of this research. If the maximum Peclet number for effective

mixing was defined asPet. WhenPe > Pet, complete mixing at end of the channel is not

achievable. In this case, a method has to be introduced to reduce the Peclet number. Ifa

(in equation4.14) is redefined as the distance for fluids to diffuse through andD is defined

as the virtual diffusion coefficientD∗, the upper limit of Peclet number is restricted by

the ability of a micromixer to reducea and enlargeD∗ to bring the Peclet number back to

the convection-diffusion region in Figure4.8. Therefore, for a specific microchannel with

knownL/w ratio and velocity, the design of a micromixer should be able to reducea and

enlargeD∗ to makePe ≤ Pet, wherePet is defined as the threshold Peclet number.

94

101

102

103

104

10-6 10-4 10-2 100 102 104 106

Péclet Number

L/w

Diffusion

Convection- Diffusion

Convection

Pe Pet

Figure 4.8:Three microfluidic mixing zones defined by using the Peclet number

When the flow is sufficiently slow, the Peclet number is less than unity, the flow is

in the diffusion zone, mixing can be achieved without further assistance. And when the a

Peclet number that is larger than 106, because the flow velocity is so high that it is difficult to

transport in micro-scale and the dispersion of solute is by pure convection forPe À a/D

according to the Taylor-Aris description (section2.3.4). For such a high flow velocity,

microfluidics loses its advantage for mixing. The relation between Reynolds number and

Peclet number can be calculated by,

Re =Pe ×D

ν. (4.15)

From Equation4.15, range of Reynolds numbers can be calculated, and Reynolds numbers

0.01 to 100 were chosen as the scope of this research for the above reasons.

95

4.4.2 Aspect Ratio of a Rectangular Microchannel

The width to height aspect ratio is one of the critical technical specifications for fabri-

cation of microchannels. For a given flow rate, the mixing performance acts differently for

different aspect ratio. For two fluids flowing parallel in a channel, the diffusion occurs at

the interface of the two fluids (Figure4.1). The fabrication of low aspect ratio microstruc-

tures is generally easier, however, for mixing, low aspect ratio microfluidic devices give

small contact area between fluids.

Gobby, et al. investigated the aspect ratio with constant width (Gobby, Angeli &

Gavriilidis 2001), and we extend this investigation to a constrained area. For simplicity, we

assume the bi-fluids travel at same velocityu or flow rate,q, through a channel with cross

section areaA. Thus,

q = Au, (4.16)

where

A = wH, (4.17)

and the channel depth to width ratioβ is

β =H

w, (4.18)

then,

u =q

βw2(4.19)

Einstein’s expression on dispersion of particles in a fluid stream can be stated as equa-

96

tion 4.20,

t =(w/2)2

2D=

w2

8D(4.20)

Using equation4.20and4.19, we can obtain the travel distanceLm for mixing in timet,

Lm = ut =q

8βD. (4.21)

Therefore, if the flow rate is constant, the minimum length of the channel for a com-

plete mixing,Lm, is identified as the reciprocal of the aspect ratio of the channel.

Equation4.21is an estimation of mixing length, and satisfies equation4.13. Because

chemical analysis requires low noise to signal ratio, therefore, a small height to width as-

pect ratio for sensing applications is widely used to achieve low optical signal distortion,

as a result, a long mixing length (Lm) is required (Kamholz & Yager 2001). Conversely,

large channel height to depth aspect ratio increases the contact area of confluent fluids to

mix fluids rapidly to have high productivity without concerning about any signal distor-

tion. Therefore, large channel height to depth ratio is preferable for chemical synthesis

productivity.

Using equation4.6 and the values in Table4.1 computed by the program described

in AppendixB.2, the relation between pressure drop and channel aspect ratio can be cal-

culated. Figure4.9 shows how the aspect ratio of a rectangular microchannel affected the

pressure drop. When designing a microchannel for mixing applications, the aspect ratio

should be optimized to favour the mixing, however, the pressure drop needs also to be con-

sidered. Under any condition, the least pressure drop happens at unity aspect ratio. Pressure

drops for channel aspect ratiosβ < 1 have a symmetric relation withβ > 1 againstβ = 1,

and were omitted in Figure4.9.

When the mixing is due to pure molecular diffusion, equation4.21holds true when the

flow is fully developed. Figure4.10shows shorter channel required for large channel width

to depth (β) ratio, because the diffusion works efficiently for flow in narrower channels.

97

1.0E-11

1.0E-10

1.0E-09

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1 10 100

Channel depth to width ratio, β

Pre

ssu

re g

rad

ien

t (M

Pa

/ µm

)

Re = 0.2

Re = 2

Re = 20

Figure 4.9:Pressure drop along a rectangular microchannel with the same cross-sectionarea, for different height to width aspect ratio, at various flow ratesq

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1.0E+05

1.0E+06

1 10 100

Channel depth to width ratio, β

Min

imu

m c

han

nel

len

gth

fo

r co

mp

lete

mix

ing

( µ

m)

Re = 0.2

Re = 2

Re = 20

Figure 4.10:Aspect Ratio and flow rate affect the total length to achieve complete mixing

98

For other arbitrary cross-section shaped microchannels, the pressure gradient,dp/dx,

for fully developed laminar flow is approximately (Pan 1988),

dp

dx=

128µq

πd4h

, (4.22)

where the hydraulic diameter dh is given by the equation4.23

dh =4× Area

wetted perimeter(4.23)

4.5 Summary

In this chapter, mass transport at low Reynolds number was introduced. At low

Reynolds number, the inertial component in the Navier-Stokes equation is relaxed and vis-

cosity is predominant in the flow. Because rectangular Microchannels were used regularly

in Microfluidic devices, the behaviour of a viscous incompressible flow in a rectangular

was studied.

The Taylor-Aris dispersion in a small channel is well-known in the analytical chem-

istry for its unwanted features, however, its effect may be a way to disperse solutes con-

vectively in the channel. The Taylor dispersivity is normally a few orders greater than

pure molecular diffusion coefficient, therefore, it is effective to create Taylor dispersion

in a micro-mixer. Section4.3.3described the dispersion of a solute slug in a straight mi-

crochannel by simulation and experiment. The result demonstrated that it was a promising

method to broaden the dispersion of solute to a fluid in a microchannel. To apply Taylor

dispersion in a micromixer, solute slugs need to be introduced into the microchannel con-

tinuously to form a sandwiched structure, which may be achievable by an active mechanical

shutter inside a microchannel. However, any active mechanism inside a micro-device will

unavoidably increase the complexity of the system, and the complicated control system

99

to cope with diffusion coefficient with the pulse frequencies and external energy source

requirement make it a non-preferred mixing method. In addition, for most of the microflu-

idic mixers which introduce confluent fluid streams into the channels, it is difficult to apply

Taylor dispersion using an active mechanism.

Taylor-Aris dispersion is a special case of convection enhanced mixing, and the intro-

duction of convection effects forms an important microfluidic mixing method. To evaluate

the convection effect, the Peclet number, which is the bulk mass transport to diffusion mass

transport ratio, was introduced. For a microfluidic device, it was found that only a range

of Peclet numbers need to be investigated. At very low Peclet numbers, mixing is in the

pure molecular diffusion zone, and mixing methods other than diffusion are not required.

Also, at very high Peclet numbers, mixing is in the pure convection zone. Because all the

mixing processes require diffusion to complete the mixing, therefore, a very high Peclet

number is also not preferable for mixing in microfluidic devices. Moreover, a high Peclet

number induces high pressure drop and can make it difficult to drive the fluids through the

microchannel. Peclet number is an important parameter for the design of a micromixer,

and was also used to form the scope of this research.

Another important parameter is the geometry of a micromixer. Because many mi-

cromixers are structured with rectangular shaped cross-sections, the microchannel depth

to width aspect ratio, (β), was investigated. It was shown in Section4.4 that mixing with

a constrained flow rate and channel transversal area was affected byβ, and the increas-

ing value ofβ benefited mixing. Therefore, one can expect good mixing performance in

high depth to width aspect ratio microchannels. On the other hand, it was found that mi-

crofluidic analytical devices, ie.µTAS, required a smallβ value to reduce the distortion of

flow pattern. A smallβ value increases the width for the fluids to travel, and hence, it has

poor mixing performance. Therefore, the attempts in the following chapters were made to

find methods to enhance mixing performance in a microfluidic diagnostic device with low

aspect ratio microchannels.

100

In Chapter5, three variations in design of serpentine structures or serpentine-like

structures are investigated to try to overcome the shortcoming of using high Reynolds

numbers in the 3D serpentine micro-mixers. In these designs, transverse convection was

brought into the main stream in the longitudinal direction. Hence, the convection could

disturb the main flow to induce bulk mass dispersion.

On the other hand, Taylor-Aris dispersion is a special case of convection enhanced

mixing. Inspired by the Taylor-Aris dispersion, methods need to be found to introduce

convection into enhance mixing.

101

Chapter 5

Microfluidic Mixing by Partitioning

Flows

5.1 Introduction

It may be intuitive to use a narrow channel to mix two confluent fluid streams rapidly,

because of the reduction of diffusion distance, however, for pressure driven flow, the pres-

sure drop in a narrow channel is high. Therefore, wide channels are usually used and

sometimes divided into multiple narrow sub-channels. Then, fluid streams could be intro-

duced into these multiple narrow sub-channels alternately like sandwich structures (Koch

et al. 1998). This method of mixing was termed the splitting and recombining technique,

and it occupied a large area of a microchip and such designs were too complicated to fab-

ricate. In some of these structures, the pressure drop was so high that it was very difficult

to drive the fluids through the channels (Schwesinger et al. 1996).

In this chapter, three types of serpentine-like designs were studied to improve the split-

ting and recombining techniques to reduce diffusion path. In addition, a guideline for these

102

designs was to introduce the transverse convection. The studies were carried out by sim-

ulating mixing in all three types of mixers using MemCFD. In section5.2, a single-side

serpentine shaped channel was divided into a number of mixing zones, where concentra-

tions were different. In section5.3, a dollar-shaped serpentine channel is introduced. This

serpentine channel was short-circuited by a straight channel. Inspired by the complex 3D

structures that were machined by an Excimer laser, a 3D periodic ramping structure is

investigated in section5.4. Section5.5summarizes the findings of Chapter5.

5.2 Dividing Fluids into Discrete Concentration Zones

In many chemical or biological diagnostic or analysis applications, it was not always

necessary to achieve a homogenous solution. In such applications, a mixed zone that satis-

fies the concentration requirement might be sufficient for conducting an analysis. Inspired

by the splitting and recombining techniques in microfluidic mixing (section2.6), a single-

sided serpentine channel with splitting zones was designed. However, instead of dividing

streams parallel to the flow direction, the main streams were divided into multiple sub-

streams laterally to the main flow. For the continuity of incompressible flow, the velocity

in the divided branches had to slow down to keep a constant volumetric flow rate. There-

fore, the design provided more residual time for diffusion to take place in certain divided

zones. The divided zones also reduced the diffusion path, and homogenous mixing can be

achieved in some of the zones.

The results in Figure5.1 show that two fluid streams coloured red and blue were

pumped into a T-type mixer via two inlets. The mass fraction of fluid 1 was determined

as a relative percentage to fluid 2. A mass fraction of 0.0 indicates 100% concentration of

fluid 1, and a mass fraction of 0.5 denotes a complete mixture with equal quantities of fluid

1 and 2. A mass fraction of 1.0 denotes a 100% concentration of fluid 2. The main streams

were divided into eight zones. As the concentration of two streams equaled each other, the

103

0.0%10.0%20.0%30.0%40.0%50.0%60.0%70.0%80.0%90.0%

100.0%

0 500 1000 1500 2000 2500 3000 3500

position ( µm)

Mas

s F

ract

ion

4 3 2 1 8 7 6 5

4 3

2

1 8 7

6

5

(a)

(b)

300µm

Figure 5.1:Different concentration gradient separated in the branches, (a). relative massfraction distribution in percentage; (b). corresponding mass fraction in contour plot

104

mass fraction of homogenous solution would be 50%. Zones 2, 6 & 7 were approximately

homogenous, and other zones had different mass fraction of fluid 1.

5.3 Dollar Sign Shaped Micromixers

In the fork shaped micromixers reviewed in section2.6.2, there were two major con-

cerns: pressure drop and pressure distribution. The high pressure drop limited its practical

application and it also had difficulty in distributing pressure evenly to the fork modules. In

addition, the serpentine shape microchannels were commonly used to increase the residual

time of fluids for sufficient diffusion. The serpentine and the fork-shaped mixing designs

were combined and a by-pass channel was introduced to short circuit the fluid streams. This

by-pass channel disrupted the main flow to create lateral convection, and also, partially split

the main fluid streams into the by-passing shortcut channel. This design is referred to as

the dollar sign shaped micromixer ($-mixer) and is shown in Figure5.2, and was developed

as part of the present research.

The width of the bypassing channel was smaller than the width of serpentine channel,

therefore, only some of the fluids flowed into the by-passing channel. The remaining fluid

stream formed a thinner layer close to the wall. Figure5.3shows the splitting and recom-

bining of fluids streaklines inside the channel, mapped with mass fraction distribution. As

explained in Section4.2, the flow pattern in a rectangular channel was nearly parabolic.

The velocity closest to the wall was slower than the flow in the center of the channel. This

gave fluid streams that were closest to the walls more residual time for diffusion to take

place. At the next turn of serpentine, the other fluid stream was partially split and the pro-

cedure repeated itself. In the simulation, because of the limitation of computer memory

and CPU speed, the size of the numerical model was restricted and only one section of

serpentine was illustrated in this research. Practically, many turns of serpentine could be

applied to achieve further mixing.

105

200µm

Figure 5.2:Mass fraction in a planar$-mixer

106

Figure 5.3:Streaklines mapped with mass fraction in the planar$-mixer

107

To further improve the performance of $-mixer, a multi-layer structure could be ap-

plied. Figure5.4shows the mass fraction distribution with a 3D variation of planar dollar-

mixer. The by-passing fluid stream could pass underneath the main stream. By doing this,

the by-passing fluid could join the other fluid at other side, which is impossible in the pla-

nar structures. Illustration of the flow using streakines, where the splitting and recombining

process was very much like plaiting fluid filaments into a pigtail is shown in Figure5.5.

The plaiting blended two fluid streams into each other, and reduced the diffusion path as

well as increasing residual time, which was similar to planar structures.

200µm

Figure 5.4:Mass fraction in 3D variation of the planar $-mixer

108

200µm

Figure 5.5:Streaklines mapped with mass fraction in the 3D $-mixer

109

5.4 3D Ramping Micro-mixers

5.4.1 Overview

Using the Excimer Laser micromachining process, very complex channels structures

could be fabricated. Figure5.6, shows two ramping structures made by Excimer laser mi-

cromachining in the microlithography laboratory in the Industrial Research Institute Swin-

burne (Hayes, Harvey, Ghantasala & Dempster 2001). These ramping structures were

created by periodically moving the machine platform and mask rotation. Therefore, the

ramping structures also had the periodic configurations. To some extent, the ramping struc-

tures could be recognized as serpentine-like structures. While the design intention was

to improve mixing, it had not been demonstrated whether these novel ramping structures

could actually benefit microfluidic mixing or not. Therefore, a study was carried out in this

section.

a b

Figure 5.6: Two samples of 3D ramping structures made by Excimer Lasermicromachining.

(courtesy of Dr. J.P. Hayes)

110

5.4.2 Simulations of 3D Ramping Micro-mixers

The ramping structures described above were very complex for computer modelling.

Therefore, the geometric model was simplified but the critical features were kept. Then,

the geometric model was discretized for CFD simulations to examine its effect on mixing

(Figure5.7). To achieve rational interpretation of simulation results, mesh elements must

be structured and have high uniformity. The mesh elements were created by mapping and

sub-mapping techniques (described in Section3.3).

100µm

Figure 5.7:Computer model of a ramping structure meshed with 8-node Hexahedrons

5.4.3 Discussion of Results

The ramping structures created 3D convective effects. The interfacial region between

two fluid streams was moved by convection created by the ramping structures (Figure5.8).

When the interfacial area was in the slow velocity region closest to the wall, the fluid had

more residual time to diffuse across the streamlines. Because diffusion is a irreversible

111

Figure 5.8:A 3D ramping mixer, at Re = 5

112

process against time, hence, mixed fluids remain mixed even when the region is moved

back to the high velocity region closer to the center of the channel due to the convection. A

periodic structure was necessary to repeat this process to achieve a required concentration.

When the Reynolds number was high (Re ∼ 50), this created strong secondary convection

(secondary flows). If the periodic structure of 3D ramping could generate periodic velocity

field, the strong secondary flows may lead to recirculations around the flow axis (Figure

5.9). Figure5.10shows the irregularity of flow pattern at a high Reynolds number. The

flow pattern demonstrated that the mixing not only relied on the diffusion across stream-

lines, but also and more importantly relied on the cross-streamline advection. In some

literature reviewed in Chapter2, it was mentioned that the high flow velocity may give rise

to chaotic advection. However, as stated in the previous Section4.4, this research concen-

trated on investigating mixing at low Reynolds number. Moreover, chaotic advection is not

a necessary condition for ideal passive microfluidic mixing, because while it reduces the

diffusion residual time, it requires longer channels for complete mixing.

The results suggest that chaotic mixing in a channel with periodic 3D ramping struc-

ture provide a basis for further research.

5.5 Summary

The work in this chapter showed that it was straight forward to split and recombine

fluid streams directly by using geometric topology. Three design concepts were presented

that could generate different concentration gradients.

A modification of the serpentine microchannel design concept showed that by-passing

channels could be added to create partial flows. The partial flow reduced the amount of

fluid entering the main channel, the fluid stream became thinner and closer to the wall.

Under this condition, the diffusion path was reduced and residual time was increased.

113

Figure 5.9:Mass fraction distribution in a 3D ramping mixer, at Re = 50

114

Figure 5.10:Streaklines in 3D ramping mixer mapped with mass fraction, at Re = 50

115

The last approach was to use 3D ramping structure to enhance mixing. It was difficult

to fabricate 3D ramping structures using techniques other than Excimer Laser microma-

chining. The ramping structures could create convective effects to enhance mixing, and at

high Reynolds numbers (Re = 50), irregular flow pattern was created. While mixing using

a high Reynolds number was not the focus of this research, for the sake of completeness,

the results were included and considered useful for future research.

The 3D ramping structure was novel and could be used to enhance mixing in a mi-

crochannel. However, it was difficult to define parameters for the study. Therefore, the 3D

ramping structures needed to be simplified.

In Chapter6, these ramping surface structures are further simplified to cylindrical

obstacles. It was envisaged that the cylindrical obstacles could provide disturbances to

the flow that was similar to the ramping structures, and resistance and disruption could be

represented by number and position of obstacles. By studying distribution of mass fraction

and velocity field affected by the obstacles in the channels, the effect of disturbance could

be revealed. Ideally, the disruption to flow velocity field alters the flow direction to create

lateral convection, and increases the contact area between the confluent fluid streams, and

hence, enhance mixing.

116

Chapter 6

Optimizing Layout of Cylindrical

Obstacles in Microchannels for

Enhanced Mixing

6.1 Introduction

In the previous chapter, it was found that the geometric topology of a microchannel

could create useful disturbance for microfluidic mixing. This chapter focuses on several

aspects of passive micro-mixer design and the optimization of layout of cylindrical obsta-

cles. MemCFDTM

was used as the principal tool to study the mixing of two liquids in

a Y-channel (or T-channel) with complementary experimental verification of mixing per-

formance. Obstacle structures used in this study were totally compatible with the current

micromachining techniques and were realized by Excimer laser.

It is emphasised here that obstacles in microchannels do not generate turbulence when

the Reynolds numbers are much lower than the threshold number for a turbulent flow,

117

however, the obstacles still disrupt the fluid to create lateral bulk mass transport to improve

the mixing.

In this chapter, the simulations of cylindrical obstacles inside a microchannel are first

described, followed by the experimental validation. In section6.2, the detailed numerical

models and the method to calculate mixing efficiency from mass fraction distribution are

introduced. In section6.3, the simulation results and experimental validation are discussed.

Section6.4summarises the outcomes of Chapter6.

6.2 Numerical Modelling of Mixing in a Microchannel with

Cylindrical Obstacles

The obstacle structure in the T-type channel is illustrated in Figure6.1. The T-type

channel has two inlet and one outlet ports. The width of the inlets is 200µm, the width of

the outlet channel is 300µm, and the height of the channel is 100µm. In order to reduce

computing time, we used a channel length of only 1.2mm, except for the two obstacle

arrays (configuration no.8) with a length of 2 mm. The diameter of the obstacles is 60µm.

Table6.1gives the number of the obstacles in eight different designs. Figure6.2illustrates

the layout of design no.6, no.7 and no.8, and the spacing for other designs are the same.

Configuration number No.1 No. 2 No. 3 No. 4 No.5 No.6 No.7 No.8Number of obstacles 0 1 1 2 3 9 9 18

Table 6.1:Configuration of microchannel with Cylindrical obstacles

Using the numerical modelling techniques described in section3.3.3, the geometric

models could be discretized into finite element meshes (Figure6.3). Boundary conditions

for flow at the walls of the microchannel and walls of obstacles were set to zero velocity,

due to the nature of viscous flow in a microchannel. Volumetric flow rate boundary con-

118

Fluid 2

Fluid 1

Mixture

Cylindrical obstacles

Figure 6.1:3D model of a T-channel with two arrays of cylindrical obstacles

(a).

φ

(b).

φ

(c).

φ

Figure 6.2:Layout of square and triangular configuration of obstacle array. (a). configura-tion No.6. (b). configuration No.7 (c). configuration No.8

119

ditions, q = 2 × 107µm3/sec, were assigned to both two inlets, which is equivalent to

Reynolds number,Re = 0.4.

100 µm

Figure 6.3: Meshed fluid volume in a T-type channel with two arrays of cylindricalobstacles

Simulations were performed using MemCFDTM

on Win NT4.0 with Pentium III 800

MHz CPU and 128MB memory. It was non-trivial to apply uniform structured mesh with

multiple cylindrical obstacles, and to improve the simulation quality, parabolic elements

were used instead of linear elements. The finite element solver required considerable com-

puter memory, and even for modest size 3D CFD models, it consumed the memory rapidly

120

and used the hard disk as virtual memory. This made the simulation very slow. For the

series of simulations to be run, 3D finite element analysis was very time consuming. How-

ever, the structures of the microchannels were planar with low aspect ratio, and diffusion is

rapid in the depth direction when it is a slow flow. In addition, the velocity and the deriva-

tion of velocity in the depth direction was zero. Hence, the consideration of dispersion in

the depth direction could be relaxed for these reasons. Therefore, it was possible to ap-

ply 2D finite analysis to achieve accurate simulation without losing the focus on the major

problem. The accuracy of the simulation will depend on the height to width aspect ratio.

For a slow flow in a low aspect ratio channel, the diffusion is rapid across the streamlines

in the depth direction (z-axis in Figure4.1) of the fluids due to the parabolic shape (see

Taylor dispersion in section4.3). The mixing problem would be on the interfacial region

between two fluid streams (Figure4.1). The width to depth aspect ratio of the channel was

1/3 for the purpose of demonstrating the concept. The sample fluids used in the simulation

were water and ethanol (Table6.2) . Steady flow with a Peclet number of 200 was assumed

in the simulation.

Fluids Viscosity (kgµm−1 s−1) Diffusivity (µm2s−1) Density (kgµm−3)Water 9.0e-10 1.2e3 9.998e-16

Ethanol 1.2e-9 1.2e3 7.89e-16

Table 6.2:Properties of water and ethanol at 20 ˚ C

6.3 Results and Discussions

Using the numerical models described above, CFD simulations were carried out. Se-

lected experiments provided qualitative verification and also presented in this section.

121

6.3.1 Simulation Results

The simulation results demonstrated the evolution of the design layouts (Figures6.2)

using cylindrical obstacles in the microchannels. In Figure6.4, the red colour represented

one fluid (Ethanol) and blue colour represented another fluid (water) and the green colour

was for the mixed solution. The wider the green area at the outlet, the better the mixing.

Figure6.4shows the simulation result of relative mass fraction distribution with cylin-

drical obstacles inside microchannels. Figure6.5 shows the line plots of mass fraction

distribution at different position along the microchannel width at location Y-Y.

Y

Y

Figure 6.4:Relative mass fraction distribution in solution in a microchannel with two arraysof cylindrical obstacles.

The mass fraction distribution illustrated in Figure6.5 indicates the degree of mixing.

However, to quantitatively evaluate the performance of mixing, the mixing efficiency can

122

0.0% 10.0% 20.0% 30.0% 40.0% 50.0% 60.0% 70.0% 80.0% 90.0%

100.0%

0 50 100 150 200 250 300

channel y - axis (µm)

mas

s fr

acti

on

(%

)

Figure 6.5:Relative mass fraction distribution in solution in a microchannel with two arraysof cylindrical obstacles.

be integrated from Figure6.5and calculated by equation6.1(Jeon, Dertinger, Chiu, Choi,

Stroock & Whitesides 2000),

meff =

1−

w∫0

|c− c∞|dx

w∫0

|c0 − c∞|dx

× 100%, (6.1)

wherec is the mass concentration distribution across the transverse direction at the out-

let, c∞ is the concentration of a complete mixing, andc0 is the initial distribution of the

concentration before any mixing.

Because of the disruption of obstacles in a microchannel, the pressure drop is affected

by the blockage of obstacles. The pressure drop is a crucial factor to the design of a mi-

crofluidic device, therefore, the overall performance of microfluidic mixer should include

the evaluation of pressure drop. This overall mixing performance is termed mixing index,

midx, and used in this research to evaluate the overall performance of a passive microfluidic

mixer. The mixing index is defined in Equation6.2.

midx = meff × ∆pmax

∆p(6.2)

123

wheremeff is mixing efficiency,∆p is pressure drop between outlet and inlet of a mi-

cromixer and∆pmax is the maximum pressure drop in the same set of simulation.

Figure6.6shows the evolution of the design with cylindrical obstacles for eight con-

figurations.

1 2 3 4

5 6 7 8

Figure 6.6:Evolution of design with obstacles in the Y-channel for configurations 1 to 8

The simulation results of mixing efficiency and pressure drop in the Y-channel were

calculated and plotted in Figure6.7a, and the mixing index is shown in Figure6.7b.

The simulation results (Figures6.6 and6.7) show that mixing efficiency increases as

the number of obstacles increases. The increased mixing can be interpreted as the obstacles

causing the parabolic velocity distribution to be evened out and give more time for diffusion

at the interface of the two liquids. However, the improvement of mixing performance is not

proportional to the increased pressure drop or the number of obstacles. Configurations no.2

and 3 have approximately the same pressure drop, but no.3 has much more mixing. Con-

figurations no.6 and 7 have the same number of obstacles, but no.7 has lower pressure drop

124

a

12

3 4 5

6

78

1 2 3 4 5

6

7

8

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

Number of obstacles

Mix

ing

Eff

icie

ncy

(%

)

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

1.4E-03

1.6E-03

Pre

ssu

re D

rop

(M

Pa)

mixing efficiency Pressure Drop

b

0.01.02.03.04.05.0

0 2 4 6 8 10 12 14 16 18

Number of obstacles

mid

x

1

2

4

3 5

76 8

Figure 6.7: Finite element simulation results of (a) mixing efficiency, pressure drop be-tween inlet and outlet versus number of obstacles, and (b) Mixing index versus number ofobstacles(configuration no.1∼ no.8).

125

and higher mixing efficiency. These results show that the asymmetric layout of cylindrical

obstacles were favouring mixing, but symmetric ones were not.

Figure6.7b shows the mixing index (defined in equation6.2), and the optimized con-

figuration would be one cylindrical obstacle with asymmetric layout (configuration no.3).

The criterion for designing a passive micro-mixer with cylindrical obstacles should com-

prise the capability of a micro-pump to overcome pressure drop with an adequate mixing

efficiency. Therefore, the mixing efficiency has to compromise the pressure drop to opti-

mize the design of a micro-mixer.

For a given flow rate, the total time for the flow through the channel remains the same,

because the average velocity at inlets and outlet remains the same for incompressible fluids

(conservation of mass). The local velocity, where the obstacles are, is increased to keep the

constant flow rate. The asymmetric layout of obstacles gives different resistance to the flow

in the lateral direction, so the fluids find their path through the lower resistance area, similar

to an electric circuit. This means that part of the fluid flow is distorted with the redirection

of the flow and this convective effect is shown in Figure6.8. The lateral convection created

bulk mass transport other than diffusion, and this was the reason that mixing relied largely

on the lateral convection generated by obstacles with asymmetric configurations.

Figure 6.8:Section of the channel illustrated the convective effect using velocity vectorfield

126

6.3.2 Experimental Results

To validate the simulation results by experiment, Excimer laser machined microchan-

nels with cylindrical obstacles were filled with miscible liquids to visualize the mixing

phenomena (Figure6.9). Figure6.10shows that mixing performance was poor when there

were no obstacles in the channel. Figure6.11shows that the asymmetric arrangement im-

proved mixing performance. In Figure6.11, the fluids were Ferric Ammonium Sulfate

and Ammonium Thiocyanate. The apparent diffusion was faster than it should have been

due to chemical reaction. However, there was still an obvious broadening of mixing re-

gion. Food colouring aqueous solutions had one-third diffusion coefficient compared to

the diffusion coefficient of water and ethanol. Nevertheless, Figure6.12 shows that the

experimental visualization of the mixed area (green region) using food colouring dyes is

broadened compared to that of the two configurations in Figures6.10and6.11.

15kvolts, x180, 04/03/2002200 µm

Figure 6.9:Channel with cylindrical obstacles (SEM image)

Although quantitative measurement was not possible in these experiments, by com-

paring the mass fraction distribution between simulation results and experimental results,

the visualization provided qualitative agreement with the simulation results. Therefore, the

127

Figure 6.10:Configuration no.1, channel without cylindrical obstacles (food colouring),Re = 0.4

Figure 6.11:Configuration no.5, channel with 3 cylindrical obstacles (Ferric AmmoniumSulfate and Ammonium Thiocyanate solutions), Re= 0.4

Figure 6.12:Configuration no.8, channel with 18 cylindrical obstacles (food colouring),Re = 0.4

128

MemCFD simulation software was demonstrated to be an appropriate method to evaluate

the performance of passive microfluidic mixer designs.

6.4 Summary

With viscosity dominating flow in microchannels, mixing of two fluid streams mainly

depends on diffusion. As a simplification of the ramping structure, obstacles were placed

in the microchannels, and mixing performance was investigated by simulation and experi-

ment. Obstacles in a microchannel with a low Reynolds number cannot generate eddies or

turbulence. However, the results demonstrated that obstacles could improve mixing perfor-

mance by affecting the flow pattern, and that the asymmetric arrangement of obstacles alter

the flow directions and force one fluid to flow into another to create lateral mass transport.

In this chapter, this phenomenon was termed convective effect in order to differentiate it

from pure molecular diffusion.

A very important outcome from the studies was that asymmetric layout of obstacles

could enhance microfluidic mixing, while symmetric layouts had little effect on mixing.

Placing obstacles in the microchannels was a novel method for mixing in microfluidic de-

vices, and properly designed geometric parameters, such as layout and number of obstacles,

could improve the mixing performance without significantly increasing the pressure drop.

The use of cylindrical obstacles provided only a limited number of parameters for

the study. For a more comprehensive study on obstacles, more design parameters for this

type micro-mixer needed to be investigated. Therefore, cylindrical obstacles would be

replaced by rectangular obstacles to study their effects on mixing in Chapter7. Rectangular

shaped obstacles provide more design parameters than cylindrical shaped obstacles do, and

these parameters are useful for optimizing the design of passive microfluidic mixers using

geometric variations.

129

Chapter 7

Enhancing Mixing in Microchannels

Using Square and Rectangular Obstacles

7.1 Introduction

The previous chapter revealed that although cylindrical obstacles in microchannel

could broaden the mixed region, more detailed parameters needed to be examined. To

continue the research on how structures in microchannels affect mixing, passive mixing by

rectangular obstacles was studied and reported in this chapter. One reason to use a rectan-

gular shape was the number of parameters that could be investigated, such as width, height,

gap between obstacles, position in the channel and obstacle orientation. In addition, rect-

angular structures were much easier to mesh into uniform structured elements to achieve

high simulation quality with reduced computing time.

In respect to the nature of laminar flow in a microchannel, the geometric variations

were designed to try to improve the lateral convection. By doing this, the dispersion of

one fluid into another was not only assisted by diffusion, but also, and more importantly,

130

by the convection in the lateral direction. Geometric parameters versus the mixing perfor-

mance were investigated systematically in T-type channels by applying MemCFD solvers

for microfluidics. Various obstacle shapes, sizes and layouts were studied, and the mixing

performance of microchannels with obstacles were evaluated by mass fraction.

In practice, the gap, size and other parameters associated with obstacles were not sep-

arated. In this chapter, the author provides linkages between these parameters and mixing

performance.

7.2 Numerical Modelling of Square and Rectangular Ob-

stacles

The channels were meshed into 8-nodes hexahedral elements. The simulation was

run as 3D, steady, laminar, Newtonian, with two fluids to evaluate the dispersion of one

fluid into another. The inlets were assigned flow rate (ranging from 2× 107 µm3 s−1 to 2×109 µm3 s−1 in this research) boundary condition, and all the channel walls were assigned

wall boundary condition (velocity components in Cartesian coordinates:vx = vy = vz =

0). Two fluids entered the two inlets respectively (red dye aqueous solution and blue dye

aqueous solution), with a diffusion coefficient,D = 4×10−6cm2s−1, which was calculated

according to their chemical groups (Reid et al. 1987).

For all the studies in this chapter, the T-type channels were used, with two inlets and

one outlet. In Figure7.1, the T-channel with four rectangular obstacles is illustrated. The

length, width and depth of the main channels were 5mm, 200µm and 100µm respectively.

The length of the inlet,L1, is not critical but should be sufficient for full developed flow,

which requiresL1 À 0.01wRe (Bird et al. 1960). The main channel length is divided

into three parts, whereL0 is the entry channel length for the full development of flow,Lr

is length of the section to create lateral convection andLm is the channel length for the

131

broadening of solution by diffusion.

Fluid 1

Fluid 2

Mixture

w

g g g

δ l

L1

L0 Lr Lm

a

b

Figure 7.1:T-channel with 4 rectangular obstacles, (a) 3D view; (b) layout.

7.3 Results and Discussions

The simulation results in Figure7.2show how the obstacles affected mixing. To eval-

uate the performance of a micromixer, several important factors need to be considered:

mixing efficiency (Equation6.1), pressure drop and overall mixing performance, which

was termed mixing index and defined in Equation6.2.

In this section, the effect of number, size, angle and other important parameters of

132

1 2 3 4 5 6 7 8

a

9 10 11 12 13 14 15

b

Figure 7.2:Mass fraction distributions, (a) symmetric layout of obstacles, (b) asymmetriclayout of obstacles

133

obstacles to mixing and pressure drop were evaluated using mixing efficiency, pressure

drop and mixing index respectively.

7.3.1 Number of Obstacles

In Chapter6, it was found that symmetric placement of obstacles does not improve

mixing, whereas asymmetric layout of obstacles can enhance mixing, and the mixing effi-

ciency increases with the number of obstacles. By repeating the simulations using square

obstacles, Figure7.2shows similar results. The mass fractions were extracted close to the

end of the channel and the pressure drops were calculated between the inlet and outlet.

Mass fractions and pressure drops were used to compute the mixing efficiency and mixing

index as defined in Section6.3.1. Figure7.3 shows the mixing efficiency, and it clearly

indicates that symmetric layouts have little effect on enhancing mixing and asymmetric

layouts can improve mixing efficiency.

0.00%5.00%

10.00%15.00%20.00%25.00%30.00%35.00%40.00%45.00%50.00%

0 2 4 6 8 10 12 14 16 18 20

number of obstacles

mix

ing

eff

icie

ncy

(%

)

asymmetric

symmetric

Figure 7.3:Mixing efficiency versus number of square obstacles

As explained in the previous chapter, pressure drop is a critical parameter for a mi-

crofluidic device. Figure7.4shows the pressure drop versus the number of obstacles, and

134

Figure7.5 shows the mixing efficiency measured by mixing index, which was defined as

the mixing efficiency divided by the normalized pressure drop (Section6.3.1). Although

the mixing efficiency increased with the increase of number of obstacles, the improvement

was not significant when obstacles reached a certain number, because the increase in num-

ber of obstacles also blocks the lateral convection. The pressure drop in Figures7.4 and

7.5shows that the asymmetric configuration with six obstacles gave optimized overall per-

formance. In Figure7.3, the mixing efficiency with two obstacles is better than three

0.0E+00

5.0E-06

1.0E-05

1.5E-05

2.0E-05

2.5E-05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

number of obstacles

Pre

ssu

re d

rop

(M

Pa)

asymmetric symmetric

Figure 7.4:Pressure drop versus number of square obstacles

obstacles. It does not generally agree with the view that the mixing efficiency improves

with the number of obstacles. Therefore, a further study of these two configurations was

necessary. Figure7.6uses the angle between the flow axis and obstacle axis.

The further simulation results showed that two obstacles inside a microchannel had a

better mixing efficiency than that of three obstacles (Figure7.7). This may be interpreted as

the mid obstacle actually reducing the convective effect and partially blocking the diffusion

process at the interfacial area between two fluid streams. Figure7.8shows that there is no

significant difference on pressure drop. Therefore, the mixing index shown in Figure7.9

135

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

number of obstacles

mid

x

asymmetric symmetric

Figure 7.5:Mixing index,midx, versus number of square obstacles

had the same trend as in Figure7.7, and two obstacles configuration demonstrated better

overall performance than three obstacles.

7.3.2 Size of Obstacles

The width of the obstacles was found to be a more important dimension than length,

because width was the most effective length scale to influence the flow patterns (Figure

7.10). The configuration and the simulation results are illustrated in Figure7.11. Two

streams of fluids flowed from left to right along thex-axis. All the parameters including

flow rate were fixed, except the width of the obstacles. The height of the obstacles equaled

the depth of the channel. By changing the width of the obstacles, this changed the degree

of blockage to the flow. The velocity component lateral to the flow increased, and so did

mixing efficiency, when the size of obstacles became bigger. However, at a certain width

(80% of blockage in Figure7.11) when the gaps between obstacles and wall equaled each

other, the flow pattern became symmetric. As shown previously, the mixing efficiency

136

θ

θ

Figure 7.6:Angle between flow axis and obstacle axis, two and three obstacles

137

0.00%

10.00%

20.00%

30.00%

0 20 40 60 80

Angle (degree)

Mix

ing

eff

icie

ncy

(%

)

2dots

3dots

Figure 7.7:Mixing efficiency with two and three obstacles inside a microchannel versusangle of layout

0.0E+00

2.0E-06

4.0E-06

6.0E-06

8.0E-06

0 20 40 60 80

Angle (degree)

Pre

ssu

re d

rop

(M

Pa)

2dots

3dots

Figure 7.8:Pressure drop versus two and three square obstacles

138

0.0

0.1

0.2

0.3

0.4

0.5

0 20 40 60 80

Angle (degree)

mid

x

2dots

3dots

Figure 7.9:Mixing index,midx, versus two and three square obstacles

Figure 7.10:Mass fraction distribution for two rectangular obstacle layouts with differentsize of obstacles

139

1

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

0.0% 20.0% 40.0% 60.0% 80.0% 100.0%

degree of blockage (b/w) (%)

Mix

ing

eff

icie

ncy

(%

)

Figure 7.11:Mixing efficiency versus size of obstacles

was poor for any symmetric arrangement. When the width of the obstacles continued to

increase until it reached the wall, the configuration was similar to the serpentine shaped

microchannels (Liu et al. 2000).

Larger size obstacles block the flow more (Figure7.12), and Figure7.13shows the

overall performance was maximum around the 40% blockage, which meant the size of

obstacle was 40% of the channel width.

7.3.3 Angle of Obstacles

From the configuration described in section7.3.2, we rotate the obstacles against their

center of mass simultaneously (Figure7.14). The orientation of the obstacles causes a dif-

ferent type of disturbance to the flow (Figure7.15). The results showed that the maximum

mixing efficiency was located in the range from60o to 75o and105o to 120o.

The lateral projection area of obstacle decided the resistance to the flow, therefore

the obstacles perpendicular to the channel wall presented the highest pressure drop (Figure

140

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

0.0% 20.0% 40.0% 60.0% 80.0% 100.0%

degree of blockage (%)

Pre

ssu

re d

rop

(M

Pa)

Figure 7.12:Pressure drop versus size of obstacles

0.0

20.0

40.0

60.0

80.0

100.0

120.0

140.0

0.0% 20.0% 40.0% 60.0% 80.0% 100.0%

degree of blockage (%)

mid

x

Figure 7.13:Mixing index,midx, versus size of obstacles

7.16). Figure7.17shows the optimized overall performance is around60o and120o to 150o.

141

Figure 7.14:Mass fraction distribution, various angles of obstacle layouts

0.0% 10.0% 20.0% 30.0% 40.0% 50.0% 60.0%

0 50 100 150 200

Mix

ing

effi

cien

cy (%

)

Angle (degree)

Figure 7.15:Mixing efficiency versus angle of rectangular obstacles

142

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

0 50 100 150 200

Angle (degree)

Pre

ssu

re d

rop

(M

Pa)

Figure 7.16:Pressure drop versus angle of obstacles

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 50 100 150 200

Angle (degree)

mid

x

Figure 7.17:Mixing index,midx, versus angle of obstacles

143

7.3.4 Obstacle Offsets from Channel Walls

When two or more rectangular obstacles are placed in a microchannel, the relative

position of the obstacles to each other and the channel walls needs to be studied. In this

section, the parameter investigated was measured asa to b ratio and ten obstacles were

placed in the channel (Figure7.18). In Figure7.19, whena/b = 1, it indicated that the

obstacles lay in the center of the channel symmetrically, which did not improve the mixing

performance. Whena/b = 0, the obstacles were connected to the wall, which actually

formed a serpentine-shaped channel. However, it was preferable to have a gap between the

obstacles and the wall, as it could eliminate the dead volume created at corners between

obstacles and channel walls.

a b

Figure 7.18: Mass fraction distribution, for different offsets of obstacles from channelwalls, (a). no offset from the center; (b). with offset.

When there was no space between the obstacles and channel walls, this gave the best

mixing efficiency, however, this raised the potential for dead volume at the corners between

obstacles and channel walls. An offset of obstacles from the channel walls removes dead

volumes. Figure7.20shows there is no significant difference in pressure drop, and Figure

7.21demonstrates that obstacles closer to the wall give better overall performance of the

144

0.00% 20.00% 40.00% 60.00% 80.00%

100.00% 120.00%

0 0.2 0.4 0.6 0.8 1 1.2 a/b ratio

Mix

ing

eff

icie

ncy

(%

) b

a

a

b …

Figure 7.19:Mixing efficiency versus offset from the wall

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

0 0.2 0.4 0.6 0.8 1

a/b ratio

Pre

ssu

re d

rop

(M

Pa)

Figure 7.20:Pressure drop versus offset of obstacles from wall

145

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 0.2 0.4 0.6 0.8 1

a/b ratio

mid

x

Figure 7.21:Mixing index,midx, versus offset of obstacles from wall

device. A space between obstacle and the channel wall removes the dead volume, and the

a to b ratio between 0.1 to 0.6 gave the optimized results.

7.3.5 Gap between Obstacles

Figures7.22and7.23show the results of mixing of two fluids for two obstacle gap

arrangements. The green colour is the complete mixture. The smaller gap shows less

mixing for the same overall length. The mixing efficiency is expected to increase with the

reduction of gap between the obstacles, due to the diffusion distance being reduced by the

gap. However, a smaller gap tends to offer greater resistance to flow, and hence, reduces the

flow in the gap and increases the flow through the upper and lower spaces. Therefore, this

may mean a better mixed fluid in the smaller gap, but less of it, and the overall contribution

to mixing at the outlet by the fluid flowing through the smaller gap is then offset by reduced

flow.

Because the lateral projected area of obstacles was the same, the pressure drop re-

mained the same for the different configurations (Figure7.24). Figure7.25demonstrates

146

a b

Figure 7.22:Mass fraction distribution for different obstacle gaps, (a). 160µm gap; (b).400µm gap;

0.0% 10.0% 20.0% 30.0% 40.0% 50.0% 60.0% 70.0%

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Gap to width ratio (g/w)

Mix

ing

eff

icie

ncy

(%

)

Figure 7.23:Mixing efficiency versus gap between obstacles

147

the overall performance was improved with the increase of gap. However, the improvement

rate slowed as the gap approached the same scale as the width of the channel.

1.0E-07

1.0E-06

1.0E-05

1.0E-04

100 200 300 400

gap (µm)

Pre

ssu

re d

rop

(M

Pa)

Figure 7.24:Pressure drop versus gap between two obstacles

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

100 200 300 400

gap ( µ m)

mid

x

Figure 7.25:Mixing index,midx,versus gap,g, between two obstacles

148

7.3.6 Flow Rate

When a certain flow rate (Re ∼ 1) was reached, the mixing efficiency stopped de-

creasing (Figure7.26). For the optimized obstacle structures, which have the maximum

mixing index, i.e. the structures described in Figure7.13with 40% blockage, the mixing

efficiency increased with the flow rate:2 to 1000nl/sec (Re = 0.005 ∼ 2.5). As there was

no chaotic effects observed from the flow pattern, the interpretation of this would be that

there was an increase of convective components in the lateral direction to the flow.

The pressure drop would be expected to increase with Reynolds number (Figure7.27).

However, the slower flow improved the overall mixing performance more, which in turn

proved that slow fluid motion in a microfluidic device was preferred (Figure7.28).

40%

50%

60%

70%

80%

0 0.5 1 1.5 2 2.5 3

Reynolds number

Mix

ing

eff

icie

ncy

(%

)

Figure 7.26:Mixing efficiency versus flow rates

149

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

0 0.5 1 1.5 2 2.5 3Reynolds number

Pre

ssu

re d

rop

(M

Pa)

Figure 7.27:Pressure drop versus Reynolds number

0.1

1.0

10.0

100.0

1000.0

0 0.5 1 1.5 2 2.5 3

Reynolds number

mid

x

Figure 7.28:Mixing index,midx, versus Reynolds number,Re

150

7.3.7 Ratio of Height of Obstacles to Depth of Channel

Figure7.29 illustrated that obstacles with different heights in microchannels had an

influence on mixing performance. Higher obstacles created stronger 3D convection (Figure

7.30). The height,h, of obstacles was one variable and width,b = 280µm, was another.

Other parameters were the same as in Figure7.11. The most useful disturbance to improve

a b

Figure 7.29:Mass fraction versus obstacle height to channel depth aspect ratio,h/H, (a).h/H = 0.2, (b). h/H = 0.9

mixing is that caused by obstacles perpendicular to the interface of two fluid streams. The

full height obstacle proved to be most efficient.

When the ratio of obstacle height to channel depth was zero, the channel was with-

out any obstacles inside, and the mixing efficiency was obviously the lowest due to pure

diffusion based mixing mechanism. However, when this ratio became ’negative’, which

indicated that the obstacles became grooves, the mixing was improved again. Mixing in

microchannels having slantly patterned grooves are reported in Chapter8. Figure7.31

shows that the pressure drop increases quickly with height of the obstacles, and Figure

7.32shows that the negative height, which illustrates grooved channel, gives the best over-

151

0.0%

20.0%

40.0%

60.0%

80.0%

100.0%

-0.2 0 0.2 0.4 0.6 0.8 1

Aspect ratio (h/H)

Mix

ing

eff

icie

ncy

(%

)

Figure 7.30:Mixing efficiency versus height to depth aspect ratio of obstacles

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

-0.2 0 0.2 0.4 0.6 0.8 1

Aspect ratio (h/H)

Pre

ssu

re d

rop

(M

Pa)

Figure 7.31:Pressure drop versus obstacle height to channel depth ratio

152

0.0

100.0

200.0

300.0

400.0

500.0

-0.2 0 0.2 0.4 0.6 0.8 1

Aspect ratio (h/H)

mid

x

Figure 7.32:Mixing index,midx, versus obstacle height to channel depth ratio,h/H

all performance. This means that when pressure drop is a restriction to the design of a

micromixer, using grooved structures in the microchannel can provide an alternative, in-

stead of using obstacles. This provided a very important design guideline for micromixers.

7.4 Summary

In this chapter, square and rectangular obstacle structures in a microchannel were used

to improve mixing performance. Geometric parameters, such as number, size, gap between

obstacles, position, orientation in channel, and height ratio of obstacles, were studied for

their effect on mixing. The investigation of how flow rate affected mixing performance

illustrated promising results, and mixing efficiency remained constant or even increased

with flow rate whenRe > 1. This suggested that convective effects dominated mixing over

diffusion, and diffusion over a short distance is negligible. The results were used to identify

feasible strategies for improving mixing by placing obstacles in microchannels.

When the height of obstacles to depth of channel ratio became ‘negative’, the obstacles

153

became grooves patterned on the channel wall. These grooves provided slip-like bound-

ary conditions, because of the lower resistance to flow due to fluid to fluid friction other

than fluid to wall friction. Therefore, a lateral pressure gradient was built up and created

secondary flow. A detailed investigation of slantly patterned grooves inside a rectangular

microchannel are carried out in Chapter8 using a different numerical approach, termed

particle tracing algorithm, to trace the flow trajectories in the microchannel.

154

Chapter 8

Numerical Investigation of Mixing in

Microchannels with Patterned Grooves

8.1 Introduction

In Chapter7, square and rectangular obstacles were used to disrupt main flow to cre-

ate lateral convection, which can be also referred to as secondary flow. Stronger secondary

flow was found to improve mixing performance more. As with many other microfabri-

cated structures, the obstacles were planar, and the structure is sometimes referred to as

2.5-Dimension,which can be defined by height, width and length. The flow patterns were

mostly planar, with only two non-zero velocity components. When the height of rectan-

gular obstacles to channel depth ratio was reduced, the convective secondary flows were

not just lateral, but showed signs of of 3D convection, which indicated that all the three

velocity components were non-zero. When the height ratio became negative, which meant

the obstacles were actually decayed into grooves, 3D convection created by the grooves

could also improve mixing performance.

155

Several recently published papers investigated microfluidic mixers with slanted grooves

in microchannel walls by theoretical predictions and experimental measurement (Johnson

et al. 2002, Stroock et al. 2002). Unlike obstacles, the grooves created secondary flows

by providing local slip boundary conditions, and this type of mixer was declared a chaotic

microfluidic mixer (Stroock et al. 2002). However, the actual flow pattern in this type of

mixer was not clearly verified, and the grooves also provided splitting and recombining

effects to the main fluid streams. Therefore, grooves in microchannels needed to be further

studied to provide design guidelines for this type of micromixer.

In this chapter, the numerical model was critical for accurate simulations and is ad-

dressed first in section8.2. Then, a set of particle tracing algorithms is developed and

presented in section8.3. In section8.4, the flow patterns are analyzed using the developed

particle tracing algorithms. Streaklines and Poincare maps are computed to demonstrate

3D helical flow patterns. The splitting and folding fluid material lines by direct simulations

of two fluids diffusion-convection problems is also illustrated in this section. Section8.5

summarizes the results and concludes that the results obtained in the chapter could be used

as guidelines for designing this type of micromixers.

8.2 Numerical Setup

The performance of microfluidic mixers with patterned grooves was evaluated using

several numerical approaches. A Computational Fluid Dynamics (CFD) package for Mi-

crofluidic applications was used to simulate the 3D velocity field as well as two fluids

mixing, and algorithms were developed to compute rate of shear and particle trajectories to

study the stretching and folding of fluids.

The 3D geometric models (Figure8.1) were kept consistent with a paper by Stroock et

al. (2002a). In order to obtain high quality structured mesh elements, a CFD preprocessing

156

z

x

y

W

Outlet

inlet -1

inlet -2 Lr

Note: not to scale

H

Ltotal

Lm L0

Figure 8.1:Fluid volume in microchannel with grooved bottom surface.L0 is the entrychannel length,Lr is the grooved channel length andLm is the channel length to completethe mixing by diffusion

package, Gambitr v2.04, was used to build 3D models and 8-node Hexahedral elements

(Figure8.2). It should be emphasized that high quality uniform mesh elements are crit-

ical to achieving accurate simulation results, especially for simulating two fluids mixing.

The uniformity of the mesh can be achieved by applying advanced mesh mapping or sub-

mapping techniques (covered in section3.3). The density of the mesh elements was first

calibrated by comparing the simulation results with analytical solutions of viscous flow in

a rectangular channel. By this approach, the size of the element could be decided for mod-

elling channels with patterned grooves. Then, by mesh sensitivity study, the mesh density

can be increased or decreased until a stable solution is reached, which either can be de-

termined by comparing with an analytical solution or when the simulation reaches a point

where accuracy is insensitive to mesh density. Because the size of the numerical model

is directly related to mesh density, sometimes it is necessary to reduce the unnecessary

density of mesh. MemCFD v2001.3d from Coventor was used to solve the Navier-Stokes

equations and the diffusion-convection problem.

The simulations were run as steady, laminar, Newtonian, and with one fluid (or two

fluids in the case of solving diffusion-convection problems). In Figure8.1, both inlet1

and inlet2 were assigned the same volumetric flow rate boundary conditions. In the case

of simulations using two fluids, 100% fluid1 was assigned to inlet1 and 100% fluid2 was

157

z

x

y

Figure 8.2:Meshed fluid volume

assign to inlet2 with the same volumetric flow rate value. In this section, the mean ve-

locities used ranged from 100µm s−1 to 50000µm s−1, which could be worked out

from the flow rates. All the channel walls were assigned “WALL” boundary condition

(vi = 0, i = 1, 2, 3, represented three orthogonal velocity components). The diffusion co-

efficient,D, was4×10−6cm2 s−1 for simulating two fluids diffusion-convection flow. A set

of particle tracing algorithms was developed and coded in Fortran77 to compute the particle

trajectories and Poincare maps using the 3D velocity field exported from the CFD simu-

lations (AppendixB.2). The length, width and depth of the channels were 5mm, 200µm

and 100µm respectively. Simulations were performed on Win NT4.0 with PentiumIII

800MHz CPU and 512MB memory. The number of mesh elements (cells) in the models

ranged from 250,000 to 300,000.

158

8.3 A Particle Tracing Algorithm

To evaluate the performance of a mixer, one can place non-diffusive particles in the

velocity fields. To observe the advection of these particles, the convection of mass transport

can be evaluated. In experiments, measurement of velocity can be carried out by injecting

very fine coloured ink streams in the flow fields or by particle image velocimetry (PIV).

A numerical approach was developed that places virtual particles, which have no physical

properties, in the velocity fields. The trajectories of these particles can be computed by the

Lagrangian method, which records the spatial positions of the particles at each time step.

In a steady flow, particle trajectories can be integrated with the system of equations (8.1):

dp(t)

dt= v(p(t), t) (8.1)

wherep(t) is the particle position at timet, andv is the velocity field. Integrating equation

(8.1) yields:

p(t + δt) = p(t) +

∫ t+δt

t

v(p(t), t)dt (8.2)

whereδt is the time step.

The integral term on the right hand side can be evaluated numerically using a multi-

stage method or a multi-step method. Regardless of how it is solved, the end result is

a displacement, which when added to the current position,p(t), gives the new particle

location at timet + δt.

In a discrete velocity field, the velocity of a particle at certain position is interpolated

by the element containing this position. Hence, this element needs to be located first.

After the element is located, the particle velocity can be interpolated. Then, a time step is

assigned to integrate its new position. The procedure is repeated until the particle leaves

the flow domain. The algorithm is described in words below:

159

find cell containing initial position (point location)while ( particle in grid )

determine velocity at current position(interpolation)calculate new position (integration)find cell containing new position (point location)

endwhile

8.3.1 Point location

Point location is required to find the cell that contains a specified point. For the com-

plex mesh of microchannels with grooved surface, it is difficult to locate particle position

in the 8-node hexahedral elements. So it is necessary to split hexahedral elements into 5 or

6 tetrahedral elements, and 5 tetrahedral elements are used commonly (in Figure8.3). For

each tetrahedron, assume that we have a pointp with coordinatesp(x, y, z). The simplest

way to find the element into which pointp falls is to perform a loop over all the tetrahe-

drons, evaluating their shape-functions,Ni, with respect top(x, y, z) using Equation8.3.

x =∑

i

Nixi; y =∑

i

Niyi; z =∑

i

Nizi; 1 =∑

i

Ni (8.3)

Tetrahedrization is only performed in the cells along the path of the line and the results

do not have to be stored. Each tetrahedral element has 4 shape-functions (i = 4). The above

equations can be written into a linear system (Equation (8.4)), and the shape functions can

be evaluated by the standard Gaussian elimination method.

x = X · N,−→ N = X−1 · xp (8.4)

min(Ni, 1−Ni) > 0,∀i (8.5)

The criterion set forth in Equation (8.5) is used to determine whether a point lies within

160

1

8

76

5

3

4

2

Figure 8.3:Hexahedral cell decomposed into 5-Tetrahedrons

the confines of an element. All the shape functions must have positive values but less than

unity to determine a point within an element. Negative values or values larger than unity

mean that the point is outside the element. This can be interpreted geometrically as the

shape functions are the ratios of four volumes divided by pointp(x, y, z) to the volume

of the tetrahedron (in Figure8.4) (Li et al. 1999). The sub-volumes cannot be negative

or larger than the volume of the element, therefore, any negative value ofNi or 1 − Ni

means that the pointp(x, y, z) is outside this tetrahedral element. Then, the point location

proceeds by advancing to, and crossing the respective face into the adjoining tetrahedron.

The worst violator of the four conditions is used to predict which tetrahedron to try next.

Even if the bounding tetrahedron is not found in the immediate neighbor, then by always

moving in the direction of the worst violator the search technique will converge upon the

correct cell.

8.3.2 Interpolate velocity

One of three techniques may be used for the spatial interpolation of velocity: phys-

ical space linear interpolation, volume weighted interpolation, and linear shape function

interpolation. All three are mathematically equivalent and produce identical interpolation

161

p(x,y,z)

N4

N1 N3

N2

Figure 8.4:Tetrahedron geometry.Ni, i = 1, 4. are the shape functions

functions (Kenwright & Lane 1996). The linear shape function was the most efficient

technique for this application because it reused the shape functionNi, i = 1, 4 computed

during point location. The linear shape function for spatial velocity interpolation is:

v =4∑

i=1

Ni × vi (8.6)

where,vi are the velocity vectors at the 4 vertices of the tetrahedron.

Integration

Many integration methods are shown in the literature, ranging from the simple first-

order Euler scheme to the fourth-order Runge-Kutta scheme or even higher-order methods,

162

applied with fixed or variable time steps. The fourth-order Runge-Kutta scheme becomes:

p(t + δt) = p(t) + (k1 + 2k2 + 2k3 + k4)/6

k1 = δt · v(p(t), t)

k2 = δt · v(p(t) + k1/2, t + δt/2)

k3 = δt · v(p(t) + k2/2, t + δt/2)

k4 = δt · v(p(t) + k3/2, t + δt/2)

(8.7)

whereki, i = 1 ∼ 4 are the parameters decided by particle positionp, velocity v, time t

and time stepδt. Thus, using equation (8.7), a particle can be numerically integrated and

traced through the flow field. In this section, an adaptive variable time step fourth-order

Runge-Kutta scheme was used.

8.4 Results and Discussions

With the algorithms developed in the section (8.3), one arbitrary periodical flow sec-

tion (Figure8.5) of the exported velocity field for a single fluid flow was extracted to be

used to compute the Poincare map. Mixing of two fluid streams was also presented to

demonstrate the twisting and stretching of the interfacial area between them.

8.4.1 Flow patterns

One periodical flow was illustrated for vector planes in Figure8.6, which corre-

sponded to surface of sections (π2i, i = 0 to 4) in Figure8.5. The void areas in the vector

planes indicated the cross section of the solid parts of the grooves. The aspect ratio of

grooves (α) can be defined as half of the groove depth to channel depth (H). The flow pat-

terns for high aspect ratio (α = 0.30) grooves were more complex than that of low aspect

163

H

w

αH

200

0 π / 2 π 3 π / 2 2 π s e c t i o n p l a n e s

Figure 8.5:One periodic section of patterned grooves, with section planes labelled from0 ∼ 2π, and particles advected from plane 0 to plane2π

ratio (α = 0.05). The rotating like flow patterns can be called secondary flow. Therefore, a

higher aspect ratio created stronger secondary flow.

8.4.2 Streaklines

A streakline is defined as a line formed by the particles, which pass through a given

location in the flow field. A streakline can be made visible by injecting a dye into the fluid

at the given location. In a steady flow, the streaklines coincide with the particle traces, and

the streamlines are lines to which the velocity vectors are tangent at all points.

To visualize the effects of the mixing numerically, a particle may be traced along the

streakline. The algorithms to determine streaklines are given in AppendixB.3. In Figure

8.7, the set of streaklines are twisting like a helical shape. This indicates folding and

stretching of fluids, which favours mixing. For the patterned grooves with the periodic

configuration, the velocity field can be used repeatedly. CFD simulations could simulate

only a section of the channel length due to the limitation of computer speed and memory.

However, by reusing the periodic velocity field repeatedly, streaklines may, in principle, be

164

0

105

0 200

130

0 200

π/2

105

0 200

130

0 200

π

105

0 200

130

0 200

3π/2

105

0 200

130

0 200

2π

105

0 200

130

0 200

α = 0.05 α = 0.30

Figure 8.6:Cross section velocity vector planes,0 ∼ 2π, Re = 5

165

0 1000

2000 3000

4000 5000

0

50

100

150

200 0

60

120

Z (

µm)

y (µm)

x (µm)

Fluid 1

Direction of fluid

Fluid 2

Start points

Figure 8.7:Streaklines of two neighboring fluid particles from two fluid streams respec-tively, α = 0.20, Re = 5

computed for a longer channel length.

The distances between neighboring fluid elements’ traces (Figure8.7) were calculated

along the flow direction. A plot of the normal distance between the two traces along the

length of the channel showed that the distance between the traces varied randomly (Figure

8.8). The plot indicates that stretching and folding of the fluid material lines occurs.

8.4.3 Poincare maps

The same algorithms to compute streaklines can be used to record Poincare maps,

which are useful for evaluating mixing performance. To generate a Poincare map in a

spatially periodic system, one or a series of passive particles are advected by the periodic

velocity field and pass through a series of periodic planes in the system. For the mixer

considered here the plane is located at the exit of each mixing segment, and each position of

the particle which hits this plane is recorded. The plane is called the Poincare map. Regular

patterns in the Poincare map indicate integrable (non-chaotic) behavior, while jumbled

166

0.0

20.0

40.0

60.0

80.0

3000 3500 4000 4500 5000 5500 6000

Distance in flow direction (µm)

Gap

bet

wee

n t

wo

str

eakl

ines

( µm

)

Figure 8.8:Gap between the two Streaklines in Figure 8.7

patterns may indicate the existence of chaotic behavior. In Figure8.5, a spatially periodic

system between section plane 0 and plane2π was clipped from the exported velocity field,

and the Poincare maps were actually recorded on plane2π.

In Figure8.9a, a small aspect ratio (α = 0.05) had a regular recirculation pattern,

which implied that it was not chaotic. With high aspect ratio (α = 0.30), the flow pattern

illustrated in Figures8.9b and8.9c became more irregular. However, there was still not

sufficient evidence to verify it was chaotic advection. Nevertheless, the particles trajectories

were circling around the flow axis. Hence, by simply counting the dots per circle (illustrated

with the dark lines in Figure8.9) in the Poincare map, the necessary length of the grooved

channel to complete one recirculation can be calculated. The folding and stretching of

material lines is closely related to the recirculations. In this aspect, the necessary length

for one circulation may be used as a criterion to evaluate the performance of micromixers

with patterned grooves. In Figure8.10, these lengths had an exponential relation with the

groove aspect ratio. Whenα = 0.3, the necessary lengths to complete one circulation were

also calculated for different Reynolds numbers,Re = 5 and0.01 (Figures8.9b and8.9c).

The results were the same and this indicated that the recirculation had little to do with flow

velocity, but mostly was a function of geometric parameters, especially the aspect ratio of

167

a

b

c

Figure 8.9:Poincare map,45o patterned grooves: (a)α = 0.05, Re = 5; (b)α = 0.30, Re =5; (c)α = 0.30, Re = 0.01.

168

grooves.

102

0.1

101

100

10-1

0.2 0.3

Len

gth

of

gro

ov

ed

chan

nel

(m

m)

Groove Aspect Ratio ( )

Figure 8.10:Length of grooved channel to complete one circulation, forH/w = 0.5

Moreover, the recirculations can reorient fluids layers, hence, the reduction of the

diffusion path can be more effective in the depth direction, because most of microchannels

have low depth to width ratios due to the planar nature of micro-devices. In this aspect,

shallow channels with patterned grooves can achieve rapid mixing. In addition, the length

of channel required to achieve complete mixing is a criterion to evaluate the performance

of a passive microfluidic mixer. In Figure8.1, Lr is a function of groove aspect ratio, and

Lm is proportional to the mean flow velocity. Therefore, when designing a micromixer,

instead of increasing the velocity to create chaotic advection to improve mixing, the use

of wide shallow channels with patterned grooves can offer an alternative solution to mix

fluids in a constrained channel length.

169

8.4.4 Simulation of Mixing in Microchannels with Patterned Grooves

In the previous sections, the particle tracing technique was used to optimize the design

of micromixers with patterned grooves. In this section, MemCFD was used as an alternative

approach to evaluate mixing performance.

To initiate the simulations, the geometric model and boundary conditions were set

in accordance with the description in section8.2. Two fluids were assigned to two inlets

respectively to simulate diffusion-convection problems, withu = 50000µm/s (Re = 5),

α = 0.1 andD = 4 × 10−6cm2/s. Due to anisotropic patterned grooves, the interfacial

area between two fluid streams was shifted from the flow axis and stretched. The shift

of interfacial area and relative mass fraction (percentage of fluid1, or fluid2, distributed

spatially in the channel) are shown in Figure8.11. In the figure, both patterns were derived

from the same simulation result. In Figure8.11(a), only a thin interfacial layer between

two fluids is illustrated in a 3D view, which shows that the area of this layer is enlarged by

stretching and twisting. Figure8.11b shows the mass fraction pattern. The two fluids were

represented by two colours, the green area (or bright area in a grey print) indicated where

the fluids became mixed. Figure8.12 shows a similar simulation run for groove aspect

α = 0.30 at Re = 5. The twisting effect was much stronger and the interfacial regions

between two fluid streams were enlarged by the stretching, and even torn apart.

The shifting of fluid streams could be analyzed by calculating the rate of shear just

below the flat plate (Stroock et al. 2002). A measure of the helicity of the flow can be

made by comparing the transverse and longitudinal rate of shear. In this section, the rate of

shear was computed from the 3D velocity field exported from the CFD simulations. The

measurement of helicity can be provided by the angleΩ between the channel axisx and

interfacial line of two fluid streams, which is shifted by the helical flow pattern (Figure

8.11).

In Figure8.13, the mean helicity computed numerically in this research is compared

170

a

b

100% fluid 1 100% fluid 2Mixed

Ω

Ω

Figure 8.11:Two fluid streams flow into a T-type channel with patterned grooves,α = 0.10andRe = 5. (a). The interfacial line is indicated by the bright mixed region. The helicitywas measured by the angleΩ between this interfacial line and the channel axis, (b). Thevisualization of mixing

171

Figure 8.12:Interfacial regions whenα = 0.30 andRe = 5

0.0

4.0

8.0

12.0

16.0

0 0.1 0.2 0.3

Aspect ratio, α

Rat

e o

f S

hea

r (t

an(

)x1

00)

analytical prediction numerical prediction measured

Ω

Figure 8.13:Mean helicity measured by the angleΩ for H/w = 0.5

172

with published analytical predictions and three experimental measured points from Stroock’s

paper (Stroock et al. 2002). Whenα > 0.2, the numerical prediction does not agree with

the analytical one due to the finite channel depth to width ratio (H/w = 0.5) used in this

study.

Stroock’s analytical prediction gave the mean helicity, and the periodic terms were not

counted in the calculation. In Figure8.14, the helicity was computed numerically along

the channel longitudinal direction for different Reynolds numbers. The amplitudes of the

waves were proportional to the Reynolds number, however, the waves vibrated about the

same value, which was the mean used in Figure8.13. The number of peaks in wave form

coincide with the number of periodic grooves in the channel. We can conclude that the

mean value of helicity only relies on groove aspect ratioα, width of the grooves and depth

of channel, and is independent of Reynolds number. Aspect ratio,α, contributes the most

to the helicity, and deeper grooves improve the mixing performance.

0

0.5

1

1.5

2

2.5

3

3.5

4

1200 1700 2200 2700 3200

Distance in the x direction (µm)

Rat

e o

f S

hea

r (t

an( Ω

)x10

0)

Re = 0.25 Re = 0.5 Re = 2.5 Re = 5

200 µm

Figure 8.14:Helicity at different Reynolds numbers, numerical estimations forα = 0.10

173

8.5 Summary

A set of particle tracing algorithms was developed and used together with CFD sim-

ulations to evaluate the performance of micromixers with patterned grooves. The particle

tracing technique was used to construct Poincare maps, which showed at low groove aspect

ratio (α = 0.05), the flow pattern was regular, and became less regular when the groove

aspect ratio became larger (α = 0.3). There was insufficient evidence to indicate chaotic

advection existed in such systems. However, the ability to create recirculation was analyzed

by calculating the necessary channel length to generate one recirculation. It was shown that

this length had an exponential relationship with the groove aspect ratio and was not signif-

icantly affected by the flow velocity. The diffusion path in the micromixer can be reduced

by the folding and stretching, and is directly related to the number of recirculations.

Simulation of diffusion-convection problems using a CFD package provides an under-

standing of mass transport in the micromixers with patterned grooves. Although the length

of channel that can be simulated is limited by the computer memory and speed, it is possi-

ble to demonstrate the stretching and folding of processes of material lines. Also, the rate of

shear computed from the simulations can be used to evaluate the performance of this type

of mixer. The mean value of helicity calculated numerically is independent of Reynolds

number (forRe ≤ 5), which agrees with the published results (Stroock et al. 2002).

Passive mixers with patterned grooves in microchannels have the drawback of creating

dead volumes, however, deeper grooves improve the mixing efficiency and reduce the chan-

nel length required for complete mixing. Also, these mixers work at relatively lower flow

velocity (Re ≤ 5), which reduces pressure drop, and are compatible with microfabrication

processes.

The investigation of micromixers with patterned grooves concluded the systematic

research of geometric variations on microfluidic mixing. The next chapter will summarize

174

the results and contributions to the knowledge, and make recommendations for further

research.

175

Chapter 9

Conclusions and Further Research

9.1 Overview

This study has investigated the ability of various geometric configurations to improve

the performance of microfluidic mixing, by using computational fluidic dynamic (CFD)

simulations, experimental investigations and developed particle tracing algorithms for so-

phisticated geometric configurations. This chapter presents the findings, contributions and

limitations of this research on passive mixing in microchannels. The last section makes

recommendations to extend this study on microfluidic mixing.

9.2 Contributions of the Research

9.2.1 Clarification of Related Theories to Microfluidic Mixing

The clarification of theories that could be applied to microfluidics is necessary and

helpful to this research. Therefore, the first contribution of this research was the identifica-

176

tion of applicable theories to micro-flow and microfluidic mixing.

This research started from the review of existing mixing techniques in a micro-device.

In microfluidic devices, it is very important to mix two or more reagents together. However,

in a typical microfluidic device, viscosity dominates flow, and as a result, mixing of fluids in

microfluidic devices is by virtue of diffusion, which is a slow process. This attracted many

researchers attention to study methods to improve microfluidic mixing. These methods

can be categorized in either the active or the passive mixing methods. Most active micro-

mixers are complex to fabricate and require an external power source. Passive mechanisms

have also been studied extensively, however, previous researchers have concentrated on

typically one of geometric factors, and an overall view of geometric configurations and their

effect on mixing was not clear. Therefore, it was the intention of this research to provide a

comprehensive research on passive mixing with various geometric configurations.

The influence of various geometric configurations of microchannels to mixing per-

formance could be studied experimentally or numerically. Because of the various com-

binations of design parameters of a microfluidic mixer, it is impractical to apply solely

experimental approaches. In this study, advanced CFD packages were used as the pri-

mary research method. Experimental approaches were also used as the complementary

method to prove the numerical approaches faithfully represented the physical principles of

microfluidic mixing. Work was undertaken to correctly and accurately carry out numeri-

cal simulations. In chapter3, several methods were introduced to improve the quality of

numerical simulation, such as mesh density and sensitivity study. Then, CFD simulations

results were calibrated by comparing with the analytical results of Hele-Shaw flow in a

finite aspect ratio rectangular micro-channel. It was found that velocity field predicted by

CFD simulation could be validated by this method. However, in the case of simulating a

convection-diffusion problem, even though the velocity field was accurately predicted by

using sufficient mesh density, the simulation of mass transport was very sensitive to the

mesh quality and could not accurately predict fluids concentration distributions. It was

177

found that a uniform high quality mesh needs to be applied with sufficient mesh density.

In the case of modelling fluids in a complex configured microchannel, it is difficult to

achieve a uniform mesh. With the help of an advanced mesh generator, and mesh conver-

sion program developed in this research (AppendixB.1), high quality uniform mesh could

be translated into the MemCFD solvers. By comparing with the published experimental re-

sults, the numerical result using improved mesh could simulate a real convection-diffusion

problem. A simple experimental arrangement was also setup for carrying out tests. The

equipment included an optical microscope, a CCD camera and an image capture computer.

The simple laboratory configuration could give qualitative comparisons between experi-

ment visualizations and numerical simulations. A fabrication technique to produce simple

micro-mixers in microchannels was also described.

After the methodology was decided, several characteristics of microfluidic flow and

mixing were clarified. Three aspects were investigated, the influence of viscosity, the

convection-diffusion in a micro-device and the geometric characteristics of a common mi-

crochannel. It is widely accepted that the flow in a microchannel is slow viscous incom-

pressible flow. The investigation of micro-flow is to study the flow at very low Reynolds

number. Because the majority of microchannels are structured in a rectangular or similar

cross-section shape, viscous flow in a rectangular channel was studied. Because there was

a lack of theory to study micro-flow, it was the intention of this research to link microflu-

idics to the study of viscous flow in a duct, such as Hele-Shaw flow. Because the theory for

viscous flow in a duct has been well-developed, this linkage is important to provide theory

to the study of microfluidics. The linkage between Taylor-Aris dispersion and microflu-

idic mixing was also provided. The theory of Taylor-Aris dispersion in a small channel is

well-developed in analytical chemistry. By theoretical analysis and experimental proof, the

conditions of microfluidic mixing are very much similar to that of the Taylor-Aris disper-

sion. The Taylor dispersivity is normally a few orders greater than pure molecular diffusion

coefficient, therefore, this provided a guide to this research to introduce a convective effect

178

to improve microfluidic mixing.

To evaluate the convective effect, the Peclet number, which is the bulk mass transport

to diffusion mass transport ratio, was introduced. For a microfluidic device, it was found

that only a range of Peclet numbers need to be investigated. At very low Peclet numbers,

mixing is in the pure molecular diffusion zone, and mixing method other than diffusion is

not required. Also at very high Peclet numbers, a high Peclet number induces high pressure

drop and can make it difficult to drive the fluids through the microchannel. Therefore, the

Peclet number is an important parameter for the design of a micromixer, and was also used

to form the scope of this research.

Another important factor is the geometry of a micromixer. Because many micromix-

ers are structured with rectangular shaped cross-sections, the depth to width aspect ratio

of a microchannel was also investigated. The influences of channel depth to width aspect

ratio on pressure drop in the channel and mixing performance were investigated. It was

found that mixing benefited from increasing of the channel aspect ratio, and pressure drop

increased if the aspect ratio is either larger or less than unity. However, because of the dif-

ficulties to fabricate high aspect ratio channels, and also high channel depth to width aspect

ratio creates optical distortion for analyzing mixing results, therefore, shallower channels

are normally used in microfluidic applications. A small channel aspect ratio increases the

distance for the fluids to travel, and hence, it has poor mixing performance. Therefore,

the intention of this research was to find methods to enhance mixing performance in a

microfluidic analytical device with low aspect ratio microchannels.

9.2.2 Modifications to Serpentine-Shaped Microchannels

Three modifications of serpentine structures or serpentine-like structures were inves-

tigated in Chapter5. In these designs, transverse convection was brought into the main

stream in the longitudinal direction. Hence, the convection could disturb the main flow to

179

induce bulk mass dispersion.

Some microfluidic applications do not require a complete mixing to achieve homoge-

nous solution. A concept of micromixer could generate different concentration gradients

was proposed in this chapter. The mixer was divided into several mixing zones by parti-

tion walls. The mixing simulation results showed that different concentration zones were

created in these divided zones.

The second modification to the serpentine micromixer design added by-passing chan-

nels. Partial flows were induced in the by-passing channels. The partial flow reduced the

amount of fluid entering the main channel from one side of the channel, therefore, one

fluid stream became thinner and closer to the wall. The slower flow motion at the wall and

thinner diffusion distance improves the mixing performance.

The last approach was to use 3D ramping structure to enhance mixing. It was inspired

by the complex 3D micro-structures created by Excimer Laser micromachining. At the

beginning of this research, the effects of these 3D structures on microfluidic mixing were

not clear. Using CFD modelling technique, the diffusion-convection simulation results

showed the positive influence of these structures on microfluidic mixing. At high Reynolds

numbers (Re = 50), irregular flow pattern in a 3D ramped channel was also shown. The 3D

ramping structure was novel and could be used to enhance mixing in a microchannel.

9.2.3 Design Guidelines for Obstacle Structures

3D ramping structures were difficult to parameterize for further investigation using

CFD. Therefore, the 3D structures were simplified to obstacles. The study of obstacle

structures were carried out in Chapters6 and7. In Chapter6, it was found that cylindrical

obstacles could provide disturbances to the flow that was similar to ramping structures,

and resistance and disruption could be controlled by number and position of obstacles. By

180

studying distribution of mass fraction and velocity field affected by the obstacles in the

channels, the effect on mixing could be revealed. Obstacles in a microchannel with a low

Reynolds number cannot generate eddies or turbulence. However, the results demonstrated

that obstacles could improve mixing performance by affecting the flow pattern, and that the

asymmetric arrangement of obstacles alter the flow directions and force one fluid to flow

into another to create lateral mass transport. In Chapter6, this phenomenon was termed

convective effect in order to differentiate it from the pure molecular diffusion.

A very important outcome from the studies was that asymmetric layout of obstacles

could enhance microfluidic mixing, while symmetric layouts had little effect on mixing.

Placing obstacles in the microchannels was a novel method for mixing in microfluidic de-

vices, and properly designed geometric parameters, such as layout and number of obstacles,

could improve the mixing performance without significantly increasing the pressure drop.

In Chapter7, rectangular shaped obstacles provided more design parameters to be

studied, and these parameters are useful for optimizing the design of passive microfluidic

mixers using geometric variations. These parameters, such as number, size, gap between

obstacles, position, orientation in channel, and height ratio of obstacles, were studied for

their effect on mixing. The investigation of how flow rate affected mixing performance

illustrated promising results, and mixing efficiency remained constant or even increased

with flow rate whenRe > 1. This suggested that convective effects dominated mixing over

diffusion, and diffusion over a short distance is negligible. It was found the greater the

lateral convection created by obstacles, the better the mixing performance. However, pres-

sure drop is also a critical performance criteria. A mixing index was defined to combine

the mixing performance and pressure drop for the design of a microfluidic mixer. Using the

mixing index, it was found that grooved patterns gave better overall mixing performance.

The results were used to identify feasible strategies for improving mixing by placing ob-

stacles in microchannels.

181

9.2.4 Design of Microfluidic Mixers with Patterned Grooves

When the height of obstacles to depth of channel ratio became ‘negative’, the ob-

stacles became grooves patterned on the channel wall. These grooves provided slip-like

boundary conditions, because of the lower resistance to flow due to fluid to fluid friction

rather than fluid to wall friction. Therefore, a lateral pressure gradient was built up to create

secondary flow. A detailed investigation of slantly patterned grooves inside a rectangular

microchannel was carried out in Chapter8 using a numerical approach, termed particle

tracing algorithm, to trace the flow trajectories in the micro-channel.

A set of particle tracing algorithms was developed in this research and used together

with CFD simulations to evaluate the performance of micromixers with patterned grooves.

The particle tracing technique was used to construct Poincare maps, which showed at low

groove aspect ratio (α = 0.05), that the flow pattern was regular, and became less regular

when the groove aspect ratio became larger (α ∼ 0.3). There was insufficient evidence

to indicate chaotic advection existed in such systems. However, the ability to create re-

circulation was analyzed by calculating the necessary channel length to generate one re-

circulation. It was shown that this length had an exponential relationship with the groove

aspect ratio and was not significantly affected by the flow velocity. The diffusion path in

the micromixer can be reduced by the folding and stretching, and is directly related to the

number of recirculations.

Simulation of diffusion-convection problems using a CFD package provides an under-

standing of mass transport in the micromixers with patterned grooves. Although the length

of channel that can be simulated is limited by the computer memory and speed, it is possi-

ble to demonstrate the stretching and folding processes of material lines. Also, the rate of

shear computed from the simulations can be used to evaluate the performance of this type

of mixer. The mean value of helicity calculated numerically is independent of Reynolds

number (forRe ≤ 5), which agrees with the published results (Stroock et al. 2002).

182

Passive mixers with patterned grooves in microchannels have the drawback of creating

dead volumes, however, deeper grooves result in better mixing efficiency and reduce the

channel length required for complete mixing. Also, these mixers work at relatively lower

flow velocity (Re ≤ 5), which reduces pressure drop, and are compatible with microfabri-

cation processes.

9.3 Limitations of the Research

The scope of this research was restricted within a limited range of Reynolds num-

bers. Mixing two or more fluid streams with very slow motion does not need assistance

other than by pure molecular diffusion, and fluids with very high velocity can be mixed

by conventional methods. However, microfluidic mixing provides a uniform mixture that

other technologies cannot achieve, and the study of high output micromixers for chemical

synthesis is another important aspect of research on microfluidic mixing.

One limitation of the research was the computational resources. It was impractical

to simulate the complete length of the microchannel in many cases. Instead, most of the

numerical models were built to simulate a section of the device. Although this did not

affect the evaluation of mixing performance of individual mixers, it can limit its application

to complex designs. In addition, because the size of the numerical model is limited by the

computer memory and main frequency of the computer central processing unit (CPU), the

quality of computer simulations is also affected. This is another reason why large models

could not be simulated.

Experimental measurements of mixing efficiency can provide overall information and

is useful for evaluating prototype micromixers. This research designed a simple testing

setup, which could provide qualitative measurement, in comparison to sophisticated mi-

crofluidic diagnostic technology used by some researchers (Santiago et al. 1998), including

183

µPIV and other advanced high power laser optical diagnostic systems.

9.4 Recommendations for Further Research

9.4.1 Experimental Methods

Numerical simulation provides a versatile method to study flow and mixing in mi-

crochannels. Numerical simulations can represent many practical problems when the mod-

els are correctly setup. Numerical approaches omit many less important conditions as in a

real environment, such as temperature, humidity and other environmental noise, and focus

on principal parameters. Numerical modelling and simulation may reveal more informa-

tion about the problem, especially when the incompressible viscous flows can be predicted

by numerical simulations. However, numerical modelling and simulation can represent the

reality only when the theories are well-developed and validated, such as in the case of in-

compressible viscous flow studied in this research. However, because the governing equa-

tions for µCFD involving electrohydrodynamics (EHD), magnetohydrodynamics (MHD)

and driving forces other than pressure, have not been fully developed, CFD was restricted

to be widely applied to microfluidic application. For example, the instability in an EHD

flow under high voltage has not built its theoretical foundation (Oddy et al. 2001), hence,

its governing equations have not been developed. In such cases, direct experimental mea-

surements of flow field are desirable. Nevertheless, even for the flow that can be predicted,

experimental measurements are required to validate prediction and build up confidence in

the simulations.

A personal communication in January 2003, with Professor Juan Santiago of Stanford

University suggested that the micro particle image velocimetry (µPIV) was now a mature

technique and could provide rich information in micro flows. Hence, 2D or even 3D veloc-

ity field of micro-flow can be measured and recorded.µPIV can be a complementary tool

184

to study microfluidic mixing.

There are still noµPIV algorithms for unsteady micro-flows, because it is difficult, if

not impossible, to direct a laser sheet beam into the micro-flow field. Instead, the depth-

of-field of the microscope has been used in currentµPIV systems and this also limits its

application to measure unsteady or transient flows. Intensified CCD cameras also put re-

striction onto the developmentµPIV systems for its high cost. However, relatively low

cost CCD cameras still can measure flow velocity up to several hundred micrometers per

second, and this is sufficient for many microfluidic devices.

9.4.2 Complex 3D Geometry

3D Ramping

Although it is difficult for other microfabrication techniques to build very complex

3D structures, it is relatively simple for Excimer Laser micromachining. Applying mask

dragging and rotation techniques, periodic or non-periodic 3D structures in microchannels

can be rapidly fabricated. In Chapter5, a 3D ramping structure was briefly introduced

and it was shown that it was possible to create 3D convection for high Reynolds number.

A microfluidic mixer with grooved surfaces has dead volume, which restricts its applica-

tion, and 3D ramping can provide a solution to the dead volume problem. However, more

research needs to be done to reveal the parameters of 3D structures that could create 3D

helical-shape secondary flows. This research has shown that periodic geometric structures

were the necessary conditions to create periodic secondary flows, and further research of

3D structures may provide another solution to microfluidic mixing.

185

Mixer with Patterned Grooves

The results in Chapter8 have shown that the analysis of Poincare maps to be a useful

approach for the study of mixing in a system that involves periodic recirculations. However,

for Re ≤ 5, the use of the Poincare mapping technique did not provide a sound justification

for labelling micromixers reported in this research as chaotic or not. Chaotic mixing can

be verified by demonstrating that at least one positive Lyapunov exponent exists in the

system (Wolf, Swift, Swinney & Vastano 1985), and it is recommended as a topic of further

work. Chaotic mixing, which defines a flow region intermediate between turbulent and

laminar advection, has been recently introduced into microfluidic mixing literature.

Moreover, Figure8.10in Section8.4.3provides the relation between length of channel

to complete one recirculation and groove aspect ratio, for the channel depth to width ratio

H/w = 0.5. A chart of this relation for a range ofH/w ratios, ie. 0.1 to 1, would be a

useful guideline for the design of the T-type micromixer reported here, in a field where at

present only few design guidelines have been established.

The periodic structures investigated in Chapter8 were compared with the results in

one of Stroock’s paper (Stroock et al. 2002). In another Stroock’s similar paper (Stroock

et al. 2002), the patterned grooves changed orientation several times down the channel.

However, whether these variants are better for mixing than pure periodical structures or

worse is not clear, and is an interesting topic for further research.

9.5 Summary

This research developed feasible methods to enhance microfluidic mixing by using

geometric configurations in microchannels of microfluidic devices.

In a typical microchannel, viscosity dominates flow, and as a result, the Reynolds

186

number is low and the flow is laminar. Therefore, mixing of two or more fluid streams

in microfluidic devices is by virtue of diffusion, which is a slow process. On the other

hand, mixing two or more reagents together is a critical part of sample preparation in a

micro Total Analysis System (µTAS), and mixing elements need to be integrated on the

microchip. Therefore, microfluidic mixing is challenging and attracts the attention of many

researchers. Based on the mixing mechanism, micromixers have pure molecular diffusion

at one end (Kamholz & Yager 2001) and chaotic mixing on the other (Liu et al. 2000,

Stroock et al. 2002). Based on structures, micromixers can be categorized in either the

active or the passive mixing methods. Most active micromixers enhance mixing by stirring

flow. The stirring can be done mechanically (Lu et al. 2002) or by the means of magneto-

hydrodynamic (MHD) (Lemoff & Lee 2000), electro-hydrodynamic (EHD) (Choi & Ahn

2000) or by acoustic streaming (Zhu & Kim 1998, Yang et al. 2000) to create secondary

flows. The secondary flows can stretch and fold material lines to reduce the diffusion path

between fluid streams, and hence, enhance mixing. Active micromixers are particularly

suitable for chamber mixing, however, most active micromixers are complex to fabricate

and require an external power source.

Some passive mixers reduce diffusion path between fluid streams by splitting and re-

combining (Schwesinger et al. 1996, Koch et al. 1998, Ehrfeld et al. 1999). Recently, pas-

sive chaotic micromixers were investigated by using 3D serpentine-type channels (Beebe

et al. 2001, Liu et al. 2000). The flow in 3D serpentine channels demonstrated chaotic

advection at high flow rates (Re ∼ 70) and good mixing performance. However, the chal-

lenges of serpentine mixers are the micro-fabrication of complex 3D structures and the

need of high Reynolds number to stir the fluids to generate chaotic advection.

In this thesis, the author started the study by understanding passive microfluidic mix-

ing in a rectangular microchannel to provide the necessary background theory. Because

most of the effort in this research was on the study of various geometric structures, the

geometric variations were then defined as the channel geometric topologies and structures

187

inside a microchannel. Inspired by the serpentine shaped micromixers, modified serpentine

or serpentine-like structures were applied to improve their mixing performance. The 3D

ramping structures indicated promising mixing performance. However, 3D ramping struc-

tures had limited and arbitrary parameters, and were difficult to model, therefore, the 3D

ramping structures were simplified into cylindrical obstacles inside a microchannel. The

study on obstacles was extended to square and rectangular obstacles, which provided a rich

source of parameters for the study. During the research, overlapping studies by other re-

searchers were found that investigated grooved microchannels. Because of the relevance to

the present study of geometric structures of this research, extended research on mixers with

slantly grooved surfaces was carried out with particle tracing algorithms developed by the

author.

The vast parameters to be studied meant that there was a lack of experimental tools,

and therefore, the study was carried out principally by computational methods. To comple-

ment the numerical approach, selected samples were verified by experiments qualitatively.

The experiments conducted in this research were visualization of mixing in channels with

cylindrical obstacles and the visualization of Taylor-Aris dispersion in a straight rectangu-

lar channel. The numerical computations of mixers with grooved surfaces were compared

with the published analytical and measured data.

In conclusion, the principal objective of this research was to systematically investi-

gate the relation between various geometric configurations and their stirring mechanisms

towards microfluidic mixing by using computational fluid dynamic (CFD) simulations and

experimental validations. The results of this research provide a link between individual

geometric configuration of a passive mixer, and useful guidelines for the design of mi-

crofluidic passive mixers.

188

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200

Appendix A

List of Publications

A.1 Journal Publications

Wang, H., Iovenitti, P., Harvey, E. C. & Masood, S. (2003). Numerical investigation of

mixing in microchannels with grooved surfaces,Journal of Micromechanics and Micro-

engineering13(6):801-8.

Wang, H., Iovenitti, P., Harvey, E. & Masood, S. (2002). Optimizing layout of obstacles

for enhanced mixing in microchannels,Smart Materials and Structures11(5): 662-7.

201

A.2 International Conference Publications

Wang, H., Iovenitti, P., Harvey, E. & Masood, S. (2003). Passive mixing in microchannels

by applying geometric variations,SPIE, Micromachining and Microfabrication, San Jose,

USA, 4982: 282-9.

Wang, H., Iovenitti, P., Harvey, E. & Masood, S. (2002). Mixing of two fluid streams in a

microchannel using Taylor-Aris dispersion effect,Proceedings of SPIE, Smart Electronics

and MEMS, Melbourne, Australia,4937: 158-163.

Wang, H., Iovenitti, P., Harvey, E. & Masood, S. (2001). Mixing of liquids using obsta-

cles in microchannels,Proceedings SPIE, BioMEMS and Smart Nanostructures,Adelaide,

Australia,4590: 204-212.

Wang, H., Masood, S., Iovenitti, P. & Harvey, E. (2001). Application of fused deposition

modelling rapid prototyping system to the development of microchannels,Proceedings of

SPIE, Smart Electronics and MEMS, Adelaide, Australia,4590: 213-220.

Wang, H., Iovenitti, P., Harvey, E. & Masood, S. (2000). A simple approach for modelling

flow in a microchannel,SPIE, Smart Electronics and MEMS, Melbourne, Australia,4236:

99-106.

202

A.3 Local Publications and Presentations

Wang, H., Iovenitti, P., Harvey, E. C. & Masood, S. (2002). Passive mixing in a microchan-

nel, in D. Toncich (ed.),Profiles in Industrial Research Knowledge and Innovation 2002,

Industrial Research Institute Swinburne, Hawthorn, pp. 261-268, ISBN 1876 567 04X

Wang, H., Iovenitti, P., Harvey, E. C. & Masood, S. (2001). Mixing of fluids in microchan-

nels, in D. Toncich (ed.),Profiles in Industrial Research Knowledge and Innovation 2001,

Industrial Research Institute Swinburne, Hawthorn, pp. 225-236, ISBN 1876 567 03 1

Wang, H., Iovenitti, P., Harvey, E. C. & Masood, S. (2000). Flow in micro-domain, in D.

Toncich (ed.),Profiles in Industrial Research Knowledge and Innovation 2000, Industrial

Research Institute Swinburne, Hawthorn, ISBN 1876 567 023

203

Appendix B

Selected Fortran Codes Developed to

Study Microfluidic Mixing

In this appendix, programs were developed for the purpose of numerical simulations.

These programs included Fortran code for converting neutral mesh to an I-Deas universal

mesh format (AppendixB.1), that was used to improve mesh quality through simulations

carried out in this research, a program developed in Section4.2 to calculate viscous flow

velocity distribution in a rectangular duct (AppendixB.2) and a main program for particle

tracing developed in Section8.3(AppendixB.3).

B.1 Fortran Codes for Mesh Conversion

For complex geometric structures, MemCFDTM

could not generate uniform structured

mesh. Another finite element mesh generation package, Gambitr 2.04, was used. The

following Fortran code was developed to import Gambitr neutral mesh into MemCFDTM

.

This mesh converting program had been used throughout the whole research project pre-

204

sented in this thesis. The flow chart is shown in FigureB.1

C*******************************************************************C** *C** convert Gambit neutral file to ideas universal neutral file *C** for import mesh file to CoventorWare *C** *C** by Hengzi Wang *C** *C** 10-July-2002 *C** *C** hengzi_wang@yahoo.com.au *C** *C** IRIS, SWINBURNE UNIVERSITY OF TECHNOLOGY *C*******************************************************************C** *C** PROGRAM GAMBIT2UNV *C** *C*******************************************************************C**C**

PROGRAM GAMBIT2UNVUSE DFPORTIMPLICIT NONECHARACTER*80 NEUFILE,UNVFILE,CONTRINFO,GFILE,HED,PROG,REC5,

& ENDOFSECTION,MTHYEAR,GROUP,ELEMENT,MATERIAL,NSFLAGS,& ELMAT,TODAY,MATNAM

INTEGER NUMNP,NELEM,NGRPS,NBSETS,NDFCD,NDFVL,ND,NE,I,& NTYPE,NDP,NODE(30),NGP,NELGP,MTYP,NFLAGS,ISOLVE(10),& NELT(1000000),REDATE(8),MATNUM,NODET

REAL(8) REVL,X,Y,ZC**C** Give the name for both input and output mesh filesC**

WRITE(*, ’(A,$)’) ’ Enter the Input mesh file, .neu: ==>’READ(*, ’ (A) ’) NEUFILEWRITE(*, ’(A,$)’) ’ Enter the OUTPUT mesh file: .unv==>’READ(*, ’ (A) ’) UNVFILE

C**C** Open Gambit neutral file, and ideas universal neutral fileC**

OPEN(UNIT=10,FILE=NEUFILE,FORM=’FORMATTED’,STATUS=’OLD’)OPEN(UNIT=20,FILE=UNVFILE)

C**

205

Read gambit neutral mesh

Write universal datasets 1710, 164, 2400, 2420, 2411, and 2412.

End of writing?

yes

no

Covert mesh topology to universal dataset 2412

yes

End of converting? no

Export mesh to a universal file.

Figure B.1:Flow chart to convert gambit neutral mesh to universal mesh

206

C** Write IDEAS universal dataset 1710C**

WRITE(20,’(I6)’) -1WRITE(20,’(I6)’) 1710WRITE(20,*)’=================================================

&=====================’WRITE(20,*) ’MATERIAL’WRITE(20,*)’=================================================

&=====================’MATNUM=2

WRITE(*,’(A,$)’) ’INPUT MATERIAL NAME: ’READ(*,’(A) ’) MATNAMMATNAM=’WATER’WRITE(20,10) MATNUM,MATNAM

10 FORMAT(I10,2X,A40)WRITE(20,’(I10,A)’) 0,’ LINE(S) OF TEXT’WRITE(20,’(I10,A)’) 0,’ MATERIAL CLASS(ES)’WRITE(20,’(I10,A)’) 0,’ MATERIAL ATTRIBUTE(S)’WRITE(20,’(I10,A)’) 0,’ MATERIAL COMPONENT(S)’WRITE(20,’(I10,A)’) 0,’ MATERIAL SPECIFICATION(S)’

WRITE(20,*) ’-------------------------------------------------&---------------------’

WRITE(20,’(I10,A)’) 0,’ MATERIAL VARIABLE(S)’WRITE(20,*) ’-------------------------------------------------

&---------------------’WRITE(20,’(I10,A)’) 0,’ MATERIAL PROPERT(IES)’

WRITE(20,*) ’-------------------------------------------------&---------------------’

WRITE(20,’(I10,A)’) 1,’ REFERENCE ENTITIES’WRITE(20,’(I10,A)’) 1,’ MATERIAL TYPES’

WRITE(20,*) ’FEM ISOTROPIC& MATERIALS’

WRITE(20,*)’=================================================&=====================’

WRITE(*,*) ’end of writing dataset 1710...’WRITE(*,*) fdate()WRITE(*,*) ’start writing dataset 164...’WRITE(20,’(I6)’) -1

C**C** Write IDEAS universal dataset 164C**

WRITE(20,’(I6)’) -1WRITE(20,’(I6)’) 164WRITE(20,’(I10,A20,I10)’) 5,’mm (milli-newton) ’,2

207

WRITE(20,’(3D25.17)’) 1000.0,1000.0,1.0,273.149999999999977WRITE(20,’(I6)’) -1

C**C** Write IDEAS universal dataset 2400C**

WRITE(20,’(I6)’) -1WRITE(20,’(I6)’) 2400WRITE(20,’(I12,2I6,I12)’) 1001,7,1,0WRITE(20,’(A)’) ’FEM_1’WRITE(20,*)WRITE(20,’(32I2)’) 0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,

&0,0,0,0,0,0,0,0,0,0TODAY=FDATE()TODAY=TODAY(5:)PRINT *, TODAYWRITE(20,’(A20,3I12)’) TODAY,30,30,17WRITE(20,’(I12)’) 0WRITE(20,’(I6)’) -1

C**C** Write IDEAS universal dataset 2420C**

WRITE(20,’(I6)’) -1WRITE(20,’(I6)’) 2420WRITE(20,’(I10)’) 1002WRITE(20,’(A)’) ’PART_1’WRITE(20,’(3I10)’) 1,0,8WRITE(20,’(A)’) ’CS1’WRITE(20,’(1P3D25.16)’) 1.0,0.0,0.0,0.0,1.0,0.0,0.0,0.0,1.0,

& 0.0,0.0,0.0WRITE(20,’(I6)’) -1

C**C** Read Gambit neutral file, and Write Ideas dataset 2411C**C**C** Read the header

READ(10,100) CONTRINFOREAD(10,100) GFILEWRITE(*,100) GFILEREAD(10,110) HEDWRITE(*,110) HEDREAD(10,110) PROGWRITE(*,110) PROGREAD(10,110) MTHYEAR

208

READ(10,110) REC5C** Write the header of 2411

WRITE(20,’(I6)’) -1WRITE(20,’(I6)’) 2411READ(10,130) NUMNP,NELEM,NGRPS,NBSETS,NDFCD,NDFVLWRITE(*,130) NUMNP,NELEM,NGRPS,NBSETS,NDFCD,NDFVLREAD(10,100) ENDOFSECTIONWRITE(*,100) ENDOFSECTION

100 FORMAT(A40)110 FORMAT(A80)120 FORMAT(A44,F5.2)130 FORMAT(6I10)140 FORMAT(5X,’NUMNP’,5X,’NELEM’,5X,’NGRPS’,4X,’NBSETS’,5X,’NDFCD’,

& 5X,’NDFVL’)C**C** Read and write nodal coordinatesC**

READ(10,’(A30)’) CONTRINFOWRITE(*,’(A30)’) CONTRINFOND=1DO WHILE(ND.LT.NUMNP)READ(10,150) ND,X,Y,ZWRITE(20,’(4I10)’) ND,1,1,11WRITE(20,’(1P3D25.16)’) X,Y,ZEND DO

150 FORMAT(I10,1X,3E20.12)READ(10,’(A40)’) ENDOFSECTIONIF (ENDOFSECTION.EQ.’ENDOFSECTION’) THENWRITE(20,’(I6)’) -1ELSEWRITE(*,*) ’Reading error, exit 2411’ENDIF

C**C** Read and write elements (Ideas 2412)C**

WRITE(20,’(I6)’) -1WRITE(20,’(I6)’) 2412READ(10,’(A30)’) CONTRINFONE=1DO WHILE(NE.LT.NELEM)READ(10,160) NE,NTYPE,NDP,(NODE(I),I=1,NDP)

C** 115 for linear elements, and 116 for parabolic elementsC** always use 115 transfer mesh from Gambit to CoventorWareC** unless, FEMTOOL is called!

209

WRITE(20,’(6I10)’) NE,115,1,2,7,NDPC**C** Convert gambit topology to ideas universal topologyC**C** Exchange no.3 & 4 elements

NODET=NODE(3)NODE(3)=NODE(4)NODE(4)=NODET

C** Exchange no.7 & 8 elementsNODET=NODE(7)NODE(7)=NODE(8)NODE(8)=NODETWRITE(20,’(8I10)’) (NODE(I),I=1,NDP)END DO

160 Format(I8,1X,I2,1X,I2,1X,7I8:/(15X,7I8:))C**

READ(10,’(A30)’) ENDOFSECTIONIF (ENDOFSECTION.EQ.’ENDOFSECTION’) THENWRITE(20,’(I6)’) -1ELSEWRITE(*,*) ’Reading error, exit 2412’ENDIF

C**C** Read and write element group control information recordC**

READ(10,’(A30)’) CONTRINFOWRITE(*,’(I6)’) -1READ(10,170) GROUP,NGP,ELEMENT,NELGP,MATERIAL,MTYP,NSFLAGS,NFLAGSWRITE(*,180) GROUP,NGP,ELEMENT,NELGP,MATERIAL,MTYP,NSFLAGS,NFLAGS

170 FORMAT(A8,I10,A11,I10,A10,I10,A8,I10)180 FORMAT(A8,I10,A11,I10,A10,I10,A8,I10)

READ(10,’(A40)’) ELMATWRITE(*,’(A40)’) ELMATREAD(10,190) (ISOLVE(I),I=1,NFLAGS)WRITE(*,190) (ISOLVE(I),I=1,NFLAGS)

190 FORMAT(10I8)READ(10,190) (NELT(I),I=1,NELGP)WRITE(*,190) (NELT(I),I=1,NELGP)

C**READ(10,’(A30)’) ENDOFSECTIONIF (ENDOFSECTION.EQ.’ENDOFSECTION’) THENWRITE(*,’(I6)’) -1ENDIFCLOSE(10)

210

CLOSE(20)STOPEND

211

B.2 Fortran Codes for Velocity Field in a Rectangular Duct

The analytical prediction of viscous flow in a rectangular channel comprises Fourier

series. Because these Fourier series converge very quickly, a finite number of Fourier

series should be calculated (FigureB.2). This program gives the analytical solution of flow

field in a rectangular microchannel. This program was developed and used in Section4.2

to calculate the velocity field in a rectangular microchannel, and was used throughout this

research to find out proper mesh uniformity and density to achieve high quality simulations.

program heleshawUSE DFPORT

cc This function was used to calculate the velocity distribution inc a microchannel with finite aspect ratio - you know I am talkingc about the velocity field a microchannel.c

real*8 muu,ymin,ymax,zmin,zmax,dy,dz,z,W,b,L,G,u(100,100),qfr,PI1,u0,u1(100,100),u2(100,100),za(100),ya(100),kucharacter*40 uyz,TODAYinteger*4 npts,n,ny,nz

C**C** Read flow (qfr) and the output file name.C**

WRITE(*,’(A,$)’)’Enter the flow rate in (mu3/sec): ==>’READ(*, *) qfrWRITE(*, ’(A,$)’) ’ Enter the OUTPUT velocity field: .txt==>’READ(*, ’(A) ’) uyz

OPEN(UNIT=20,FILE=uyz)

c qfr = 2e6muu = 1.002e-9ymin = -200ymax = 200

c width of the channelzmin = -50zmax = 50

c thickness of the channelnpts = 40dy = (ymax-ymin)/nptsdz = (zmax-zmin)/npts

212

Input channel geometry

Calculate pressure gradient at certain channel width to depth aspect ratio

Calculate velocity field, with the above pressure gradient

Export velocity field

End

Figure B.2:Flow chart for viscous flow in a rectangular channel

213

a = (ymax-ymin)/2b = (zmax-zmin)/2L = 5000PI = 3.14159265359G1 = 0.0

c aspect ratio = 4ku = 14.2

c input of flow rate to calculate pressure gradient GG = ku*muu*qfr/(ymax-ymin)/(zmax-zmin)/(zmax-zmin)**2write(20,’(1p1e16.6)’) Gn = 1000ny = int((ymax-ymin)/dy)nz = int((zmax-zmin)/dz)

do i = 1,ny+1do j = 1,nz+1

u (i,j) = 0.0enddoenddo

do i=1,nz+1za(i) = zmin+(i-1)*dz

enddo

do i=1,ny+1ya(i) = ymin+(i-1)*dy

enddo

do i = 1,ny+1do j=1,nz+1

do k = 1,800u(i,j) = u(i,j)+(16*G*a**2)/(muu*PI**3)*((-1)**(k-1)*

1 (1-cosh((2*k-1)*PI*za(j)/2/a)/cosh((2*k-1)*1 PI*b/2/a))*cos((2*k-1)*PI*ya(i)/(2*a))/(2*k-1)**3)

enddoenddo

enddowrite(*,*) nz+1,ny+1write(20,’(1P1E20.6)’ ) (za(i),i=1,nz+1)write(20,*)write(20,*)write(20,’(1P1E20.6)’ ) (ya(i),i=1,ny+1)write(20,*)

214

write(20,*)write(20,’(1P41E20.6)’ ) ((u(i,j),i=1,ny+1),j=1,nz+1)

write(20,*)

C**TODAY=FDATE()PRINT *, TODAYWRITE(20,*) TODAY

C**CLOSE(20)

stopend

215

B.3 Fortran Codes for Particle Tracing - Main Program

This program was developed to compute the particle trajectories in a micromixer. One

periodic flow field was clipped from the exported simulation results and used repeatedly to

record streaklines or Poincare map. It was a lengthy program, and only the main program

is included here to describe the logical flow of the algorithms. The detailed subroutines are

not included in this thesis but can be worked out from Section8.3, where the algorithms

were described. This program was developed to calculate Poincare maps and streaklines

so that mixing performance of micromixers with grooved surface could be evaluated. The

flow chart is shown in FigureB.3

C*******************************************************************C** *C** extract velocity vector field from CoventorWare exported *C** result text file(s). The extracted velocity field is used to *C** calculate the POINCARE MAP TO EVALUATE PARTICLE CONVECTION. *C** *C** by Hengzi Wang *C** *C** 02-Sept-2002 *C** *C** hengzi_wang@yahoo.com.au *C** *C** IRIS, SWINBURNE UNIVERSITY OF TECHNOLOGY *C*******************************************************************C** *C** PROGRAM POINCARE version 3.0 BETA *C** *C*******************************************************************C** version 3, change the interpolation strategy. Use eight nodesC** to interpolate the velocity.C**

PROGRAM POINCAREUSE DFPORTUSE IMSLF77IMPLICIT NONECHARACTER*80 TECFILE,XSECFILE,CONTRINFO,NODES,ELEMS,RESTS,TODAYCHARACTER*150 GFILEINTEGER NNODE,NELEM,ND,FNODE(100000),NODEX,NANG,NSTR,I,J,K,L,M,N,

216

Read simulation result

Determine and clip one periodic grooved section

End of writing?

yes

no

Assigning initial points locations

yes

In the flow domain? no

Locate the mesh element containing this particle

Integrating new position

Interpolating velocity of this particle

Record the exit point to Poincare map

Stop when cycles are completed

Initial point <= exit point

Figure B.3:Flow chart to compute particle trajectories

217

& YI,POIN,L1,L2,L3,NDD,NEIDX,NIIDX,NJIDX,NKIDX,NE,NL,3 NIMAX,NJMAX,NKMAX,NS,IDX(8),NMAX,NP,N5TET(5,4),NEWNS,4 NI,NJ,NK,NEIN(8),CELLIDX(2,2,2),IELE(500000,8),NNODEUP

LOGICAL FLAGDOUBLE PRECISION CORD(500000,3),V(500000,3),PRESURE(500000),

& MF1(500000),MF2(500000),VISO(500000),XMIN,XMAX,XRES,& YMAX,YMIN,ZMAX,ZMIN,XMID,YMID,ZMID,XINC,XM,XS,& DELTZ,DELTA,FLORAT,GRIDX,AREAO,STRP(20,3),XN1(3),DX,& XSTART,XSTOP,VN(3),VP(3),STRLIN(2000,3),POICAR(10000,3),& K1(3),K2(3),K3(3),K4(3),PV(100,100,3),& P1(5,100,100,3),& P1V(100,100,3),P2V(100,100,3),P3V(100,100,3),P4V(100,100,3),& T,DT,P,XP(0:2),YP(0:2),ZP(0:2),EPSILON,& P5(100,100,3),P5V(100,100,3),PT(3),XI(5),ETA(5),ZETA(5)

COMMON /CO/CORD,V/EL/IELE1 /MAXS/XMAX,XMIN,YMAX,YMIN,ZMAX,ZMIN,XSTART,XINC

C**C** Give the name for both input and output results filesC**

WRITE(*,’(A,$)’)’Enter the ASCII tecplot result file, .TXT: ==>’READ(*, ’ (A) ’) TECFILEWRITE(*, ’(A,$)’) ’ Enter the OUTPUT velocity field: .txt==>’READ(*, ’ (A) ’) XSECFILEXINC=200.0

C**C** Open TECPLOT file, and export velocity vector fileC**

OPEN(UNIT=10,FILE=TECFILE,FORM=’FORMATTED’,STATUS=’OLD’)OPEN(UNIT=20,FILE=XSECFILE)OPEN(UNIT=30,FILE=’POIMAP.TXT’)

C**C** Start to write data into vecfileC**

TODAY=FDATE()PRINT *, TODAYWRITE(30,*) TODAY

C**C** Read the headerC**

READ(10,’(A)’) CONTRINFOREAD(10,’(A)’) GFILEWRITE(*,’(A)’) GFILEREAD(10,’(A)’)READ(10,110) NODES,NNODE,ELEMS,NELEM,RESTS

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WRITE(*,110) NODES,NNODE,ELEMS,NELEM,RESTS110 FORMAT(A7,I6,A4,I6,A30)C**C** Read and write nodal coordinatesC** Read and write velocity fieldC**

WRITE(*,*) ’Now reading the file, please wait...’DO ND=1,NNODEREAD(10,’(10E16.6)’) CORD(ND,1),CORD(ND,2),CORD(ND,3),V(ND,1),

& V(ND,2),V(ND,3),PRESURE(ND),MF1(ND),MF2(ND),VISO(ND)C & V(ND,2),V(ND,3),PRESURE(ND)

IF (ND.EQ.NNODE) THENWRITE(*,’(A)’) ’END OF READING NODES’READ(10,*)ENDIFEND DO

CDO N=1,NELEMREAD(10,’(8I8)’) (IELE(N,J),J=1,8)IF (N.EQ.NELEM) THENWRITE(*,’(A)’) ’END OF READING ELEMENTS’ENDIFEND DO

CWRITE(*,’(A)’) ’CLOSING THE INPUT FILE’CLOSE(10)DO I=1,10WRITE(*,’(8I8)’) (IELE(I,J),J=1,8)ENDDOWRITE(*,’(8I8)’) (IELE(NELEM,J),J=1,8)

C**C** FIND THE BOUNDRARY (XMIN,XMAX,YMIN,ZMIN,YMAX ZMAX)C**

NNODEUP=1DO I=1,NELEM

DO J=1,8IF (IELE(I,J).GE.NNODEUP) NNODEUP=IELE(I,J)

ENDDOENDDOWRITE(*,’(I8)’) NNODEUP

XMIN=CORD(1,1)XMAX=CORD(1,1)DO I=1,NNODEUP

IF (CORD(I,1).GT.XMAX) THEN

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XMAX=CORD(I,1)ELSE IF (CORD(I,1).LT.XMIN) THEN

XMIN=CORD(I,1)ENDIF

END DOWRITE(*,*) ’XMIN = ’,XMIN,’XMAX = ’,XMAX

CYMIN=10000.0YMAX=0.0DO I=1,NNODEUP

IF (ABS(CORD(I,1)-XMAX).LE.0.01) THENIF (CORD(I,2).GT.YMAX) THEN

YMAX=CORD(I,2)ELSE IF (CORD(I,2).LT.YMIN) THEN

YMIN=CORD(I,2)ENDIF

ENDIFEND DOWRITE(*,*) ’YMIN = ’,YMIN,’YMAX = ’,YMAX

CZMIN=CORD(1,3)ZMAX=CORD(1,3)DO I=1,NNODEUPIF (CORD(I,3).GT.ZMAX) THEN

ZMAX=CORD(I,3)ELSE IF (CORD(I,3).LT.ZMIN) THEN

ZMIN=CORD(I,3)ENDIFEND DOWRITE(*,*) ’ZMIN = ’,ZMIN,’ZMAX = ’,ZMAXXMID=(XMIN+XMAX)/2YMID=(YMAX+YMIN)/2ZMID=(ZMAX+ZMIN)/2

C**NANG=INT(XMAX/XINC)

C**XSTART=0.0XSTOP=0.0DO L=1,NANG

IF(XSTART.LT.XMID) THENXSTART=XSTART+XINC

ENDIFEND DOXSTOP=XSTART+XINC

220

CXS=XMIN

DO I=1,NNODEUPIF(ABS(CORD(I,2)-YMIN).LE.0.1.AND.ABS(CORD(I,3)-

& ZMIN).LE.0.1) THENIF ((XSTART-CORD(I,1)).GE.0.0.AND.CORD(I,1).GE.XS) THEN

XS=CORD(I,1)NS=I

ENDIFENDIF

END DOCALL CS(NELEM,NS,NMAX,IDX,NEIN)DO I=1,NMAXIF (IDX(I).EQ.1) THEN

NS=IELE(NEIN(I),4)ENDIFENDDOXSTART=CORD(NS,1)XSTOP=XSTART+XINC

c write(20,*) ’main program’c write(20,*) XMAX,XMIN,YMAX,YMIN,ZMAX,ZMIN,XSTART,XINC

CC*** SEARCH THE DOMAIN FOR ALL NODES GREAT THEN XSTART AND LESSC*** THAN XSTOPCC SERACH THE ELEMENTS CONTAINING NODE (NS)C

NODEX=0DO I=1,NNODEUP

IF((CORD(I,1)-XSTART).GE.-20.0.AND.(CORD(I,1)-XSTOP)1 .LE.20.0) THEN

NODEX=NODEX+1FNODE(NODEX)=I

ENDIFEND DO

CFLORAT=1E9GRIDX=8.0AREAO=ABS((ZMAX-ZMIN)*(YMAX-YMIN))

WRITE(30,’(A)’) ’Write the Poincare map to a file:’WRITE(30,’(2A20)’) ’CORD Y’,’CORD Z’

DO I=1,10STRP(I,1)=XSTART

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STRP(I,2)=YMIN+(YMAX-YMIN)/11*Ic STRP(I,2)=YMID+ABS(YMAX-YMID)/10*(I-1)c STRP(I,2)=6.75E+02c STRP(I,3)=ZMID

STRP(I,3)=50.0END DO

WRITE(20,*) ’ START POINT’WRITE(20,’(3F16.6)’) (STRP(I,1),STRP(I,2),STRP(I,3),I=1,10)

CC STREAKLINE dye injected from fixed position for period of timeC - tracer of dye show the fluid flow.C initial position (x0,y0,z0),t=0, find the path (x(t),y(t),z(t))C motion of a particle is give by dx/dt=vx; dy/dt=vy;dz/dt=vzCC==================================================================C*** START TO GET PARTICLE TRACKS ***C==================================================================

POIN=0EPSILON = 2.0

C GIVET DELTA T (TIME STEP)C

DO 200 I=1,10C CALL INTERPOLATION ROUTINE TO CALCULATE THE VLOCITYC TO COMPUTE THE VELOCITY FOR NEXT POINTCC Define the time step for x=vx*dtCC INITIALIZING START POINT OF STREAKLINE

DO L=1,3STRLIN(1,L)=STRP(I,L)

END DOC

DO 300 J=1,200NSTR=1DT=GRIDX/(FLORAT/AREAO)

CC GET THE VELOCTIY OF THE START POINT

DO K=1,3PT(K)=STRLIN(NSTR,K)

ENDDOCALL PS(NODEX,FNODE,NS,PT)CALL CS(NELEM,NS,NMAX,IDX,NEIN)CALL CELIDX(NELEM,NMAX,IDX,NEIN,CELLIDX)CALL ITP(NMAX,CELLIDX,VN,PT,FLAG)

222

IF (FLAG) GOTO 333IF (NMAX.EQ.8) THENIF (CORD(NS,3).GT.PT(3)) THEN

NEWNS = IELE(CELLIDX(1,1,1),2)CALL CS(NELEM,NEWNS,NMAX,IDX,NEIN)CALL CELIDX(NELEM,NMAX,IDX,NEIN,CELLIDX)CALL ITP(NMAX,CELLIDX,VN,PT,FLAG)

ELSENEWNS = IELE(CELLIDX(1,1,2),6)CALL CS(NELEM,NEWNS,NMAX,IDX,NEIN)CALL CELIDX(NELEM,NMAX,IDX,NEIN,CELLIDX)CALL ITP(NMAX,CELLIDX,VN,PT,FLAG)

ENDIFENDIF

333 if(abs(vn(1))+abs(vn(2))+abs(vn(3)).eq.0.0) GOTO 330CC*** GET THE PARTICLE TRACKC RESTORE DELTA T

DO 350 WHILE (STRLIN(NSTR,1).LE.XSTOP)CC*** USE OF 4TH-ORDER RUNGE-KUTTA TO INTEGRATE VELOCITYC

NSTR = NSTR+1C210 CALL RK4V(NODEX,FNODE,NSTR,STRLIN,VN,DT)

DO L=1,3XN1(L)=STRLIN(NSTR,L)

ENDDODT = DT/2

CC CALL RK4V(NDD,NSTR,STRLIN,VN,DT)

CALL RK4V(NODEX,FNODE,NSTR,STRLIN,VN,DT)C

DX = 0.0DO L=1,3

DX=DX+(XN1(L)-STRLIN(NSTR,L))**2ENDDO

DX=SQRT(DX)IF (DX.GT.0.0.AND.DX.LT.EPSILON/2.0) THEN

DT=DT*4GOTO 210

ENDIFC

DO 400 WHILE (DX.GE.EPSILON)

223

CCALL RK4V(NODEX,FNODE,NSTR,STRLIN,VN,DT)

CDO L=1,3

XN1(L)=STRLIN(NSTR,L)ENDDO

CDT = DT/2CALL RK4V(NODEX,FNODE,NSTR,STRLIN,VN,DT)DX = 0.0DO L=1,3

DX = DX + (XN1(L)-STRLIN(NSTR,L))**2ENDDODX=SQRT(DX)

400 CONTINUECC CALL VVEC(NODEX,FNODE,NSTR,STRLIN,VN)

DO K=1,3PT(K)=STRLIN(NSTR,K)

ENDDOCALL PS(NODEX,FNODE,NS,PT)CALL CS(NELEM,NS,NMAX,IDX,NEIN)CALL CELIDX(NELEM,NMAX,IDX,NEIN,CELLIDX)CALL ITP(NMAX,CELLIDX,VN,PT,FLAG)IF (FLAG) GOTO 334

IF (NMAX.EQ.8) THENIF (CORD(NS,3).GT.PT(3)) THEN

NEWNS = IELE(CELLIDX(1,1,1),2)CALL CS(NELEM,NEWNS,NMAX,IDX,NEIN)CALL CELIDX(NELEM,NMAX,IDX,NEIN,CELLIDX)CALL ITP(NMAX,CELLIDX,VN,PT,FLAG)

ELSENEWNS = IELE(CELLIDX(1,1,2),6)CALL CS(NELEM,NEWNS,NMAX,IDX,NEIN)CALL CELIDX(NELEM,NMAX,IDX,NEIN,CELLIDX)CALL ITP(NMAX,CELLIDX,VN,PT,FLAG)

ENDIFENDIF

C334 if(abs(vn(1))+abs(vn(2))+abs(vn(3)).eq.0.0) GOTO 330C350 CONTINUEC

IF (ABS(STRLIN(NSTR,1)-XSTOP).LE.0.01) THEN

224

ELSEDO L=0,2

XP(L)=STRLIN(NSTR-3+L,1)YP(L)=STRLIN(NSTR-3+L,2)ZP(L)=STRLIN(NSTR-3+L,3)

END DOSTRLIN(NSTR,1)=XSTOPSTRLIN(NSTR,2)=P(XP,YP,2,XSTOP)STRLIN(NSTR,3)=P(XP,ZP,2,XSTOP)

END IFC

WRITE(*,’(A,$)’) ’ CYCLING...’WRITE(*,’(4X, I, 4X, I)’) J,NSTR

CC WRTIE TO POINCARE MAPC

POIN=POIN+1POICAR(POIN,1)=STRLIN(NSTR,2)POICAR(POIN,2)=STRLIN(NSTR,3)

WRITE(20,’(A,$)’) ’Write the streakline for point (x,y,z):’WRITE(20,’(1P3E16.6)’) (STRP(I,L),L=1,3)WRITE(20,’(3A20)’) ’CORD X’,’CORD Y’,’CORD Z’

C+++ WRITE THE STREAKLINE TO FILEC DO L1=1,NSTRC WRITE(20,’(1P3E20.6)’) (STRLIN(L1,L2),L2=1,3)C END DOC RESAMPLE THE START POINT

STRLIN(1,1)=XSTARTSTRLIN(1,2)=STRLIN(NSTR,2)STRLIN(1,3)=STRLIN(NSTR,3)

C+++C300 CONTINUEC+++ WRITE THE STREAKLINE TO FILE330 WRITE (30,’(A,$)’) ’POINCARE MAP, POINT: ’

WRITE (30,’(2X,I8)’) IDO L1=1,POIN

WRITE(30,’(1P2E20.6)’) (POICAR(L1,L2),L2=1,2)END DO

C+++POIN=0

200 CONTINUEC**

TODAY=FDATE()

225

PRINT *, TODAYWRITE(30,*) TODAY

C**CLOSE(20)CLOSE(30)

C**STOPEND