Massoud Thesis

Numerical Design Study of Chaotic
Mixing of Magnetic Particles
A thesis submitted for the degree of
Master of Philosophy (MPhil)
by
Massoud Zolgharni
School of Engineering & Design
Brunel University
December 2006

i
Abstract
Finding an effective mixing mechanism has been a challenge in the design
and fabrication of microfluidic platforms and so-called Lab-on-a-Chip devices due
to the fact that these devices usually operate under low Reynolds number flows
where it is not possible to create turbulence to enhance mixing. In such cases,
diffusion at the molecular scale becomes the sole mechanism of mixing. To
promote the rate of diffusion, various methods of diffusion interface stretching
have been proposed. Here, a numerical design and simulation of an active chaotic
micro-mixer for superparamagnetic particle-based mixing is presented.
Magnetophoresis force is utilized as an external force to manipulate the particles.
The aim of the design is to spread the micro-particles in order to tag the suspending
target biological entities in a bio-fluidic buffer. After the mixing process, tagged
entities can be separated from their native environment and other non-target
molecules in a separation process. Separated entities can be used in other
downstream protocols such as PCR (Polymerase Chain Reaction) or detection
algorithms in diagnostic micro-systems.
In the proposed concept, a straight channel with two embedded serpentine
conductors beneath the channel is utilized to produce the chaotic pattern in the
motion of particles and intensify the capturing of biological cells. Two flows; bio-
cells suspension and the particle laden buffer, are introduced into the channel and
manipulated by a pressure-driven flow. While the bio-cells follow exactly the
mainstream, the motion of the magnetic particles is affected by both surrounding
flow field and localized time-dependent magnetic field generated by periodic
activation of two conductor arrays.

ii
A two-dimensional numerical simulation of the mixing process is
performed in order to characterize the efficiency of the micro-mixer. In the present
study, the main focus is on the effect of two driving parameters (i.e., the fluid
velocity and frequency of magnetic activation) on the mixing quality. Two criteria,
which are dependent on the performance of the system are investigated for a wide
range of driving parameters. Lyapunov exponent (first criterion) as an index of
chaotic advection is found to be highly dependent on the driving parameters and
the maximum chaotic strength is realized corresponding to the Lyapunov exponent
of 0.36.
Moreover, it is found that capturing efficiency (second criterion) in the
mixer cannot be used as a stand alone index, which might suggest operating
conditions that are not practical. Therefore, both indices need to be taken into
account while characterizing the device. Maximum capturing efficiency is found to
be 67%, which means that more than half of the existing cells are labelled with the
magnetic particles during the mixing process.

iii
To my parents
for their love and sacrifices…

iv
Acknowledgements
First and foremost, I would like to thank my principal research advisor, Prof.
Wamadeva Balachandran. I am grateful for his frequent discussions, supports and
encouragements throughout my research which extended my abilities in research.
I would also like to thank Dr Predrag Slijepcevic for all discussions on the
biological aspects of this research.
I would also like to acknowledge Prof. Tony Anson for all his support. Throughout
my entire graduate studies at Brunel, I have been fortunate to enjoy the benefits of
working with him as my friend, supervisor and employer.
Special thanks goes to my friends. Thanks to Amir and Arash for being my best
friends who I shared the moments of happiness and sadness with. I know I can
always count on you. Thanks to Mohamad for always being by my side since the
first step of this journey. Thanks to Reza for his kind helps through all research
and life challenges over here.
I would like to express my deepest thanks to my family for their support of all my
life choices and their love, which is a constant source of strength for everything I
do. My parents, my brother and sisters, you have always been there to support me
in every endeavour I have undertaken. Your faith in me is cherished and I can
never thank you enough.
Last but not least, I would like to thank God for always hearing my voice and giving
me what I need.
Thank you.

v
Abbreviations
2D Two Dimensional
3D Three Dimensional
µ-TAS Micro Total Analysis Systems
CE Cell Capturing Efficiency
CFD Computational Fluid Dynamics
DEP Dielectrophoresis
DI Deionized (Water)
DM Diamagnetic
DNA Deoxyribonucleic Acid
EHD Electro-hydrodynamic
FM Ferromagnetic
LE Lyapunov Exponents
LOC Lab-on-a-Chip
LPCVDLow Pressure Chemical Vapour Deposition
MHD Magneto-hydrodynamic
MEMS Micro-Electro-Mechanical Systems
N-S Navier-Stokes
NdFeB Neodymium-Iron-Boron Magnet
PACVDPlasma Assisted Chemical Vapour Deposition
PDMS Polydimethylsiloxane
PIV Particle Image Velocimetory
PM Paramagnetic
PTV Particle Tracking Velocimetory
PZT Lead Zirconate Titanate
Re Reynolds Number
RNA Ribonucleic Acid
SAR Split-and-Recombine
SEM Scanning Electron Microscope

vi
SHM Staggered Herringbone Mixer
SPM Super Paramagnetic
St Strouhal Number
UV Ultra-Violet

vii
List of Figures
Chapter 1
Figure 1.1 Conceptual diagram of a typical LOC 2
Figure 1.2 (a) system level block diagram of the LOC, (b) block diagram
of sample preparation sub-system
3
Figure 1.3 Magnetic micro-particles 4
Figure 1.4 Conceptual diagram of magnetic cell sorting 5
Figure 1.6 CD4-T cell isolated with Dyna-beads 6
Chapter 2
Figure 2.1 Velocity profile of a steady pressure-driven laminar flow with
no-slip boundary conditions developed in a micro-channel
14
Figure 2.2 Basic designs in parallel lamination; (a) T-mixer, (b) Y-mixer 18
Figure 2.3 The observations of the mixing process at the junction of micro
T-mixer at different applied pressures
19
Figure 2.4 Multi-lamination 20
Figure 2.5 Vortex micro-mixers 20
Figure 2.6 Parallel lamination 21
Figure 2.7 SAR configuration 22
Figure 2.8 SAR lamination 23
Figure 2.9 Basic idea of the micro-injection mixer 24
Figure 2.10 Mixing through winding microfluidic channels shown 24
Figure 2.11 Various designs for producing chaotic advection at high
Reynolds numbers
26
Figure 2.12 Micro-mixer designs for mixing with chaotic advection at
intermediate Reynolds numbers
28
Figure 2.13 Micro-mixer designs for mixing with chaotic advection at low
Reynolds numbers
31

viii
Figure 2.14 Micro-mixer designs for mixing with chaotic advection at low
Reynolds numbers
32
Figure 2.15 Activation by moving parts 33
Figure 2.16 Active micro-mixer based on pressure field disturbance 34
Figure 2.17 Numerical simulation results 35
Figure 2.18 Schematic illustration of the micro-chamber for mixing of red
blood cells by ultrasound irradiation
36
Figure 2.19 Snapshots showing multi-bubble induced acoustic mixing in a
chamber at time
37
Figure 2.20 Active micro-mixer based on the EHD disturbance 38
Figure 2.21 Active micro-mixer based on the MHD disturbance 40
Figure 2.22 Active micro-mixer based on switching electroosmotic
disturbance
41
Figure 2.23 Electroosmotic micro-mixer 42
Figure 2.24 Active micro-mixer based on electroosmotic disturbance 43
Figure 2.25 Active micro-mixer based on dielectrophoretic disturbance 44
Figure 2.26 Active micro-mixer based on magnetophoretic disturbance 45
Figure 2.27 Mixer based on magnetophoretic disturbance 46
Figure 2.28 Schematic diagram of the realized three-dimensional
microfluidic mixer with embedded permalloy parts
47
Figure 2.29 Experimental results 47
Chapter 3
Figure 3.1 Magnetic responses associated with diamagnetic materials (left)
and paramagnetic materials (right)
54
Figure 3.2 Magnetization (M) versus magnetic field (H) 55
Figure 3.3 Schematic depiction of spin arrangements in different types of
magnetic materials
56
Figure 3.4 Coercivity as a function of particle size 57
Figure 3.5 Domain structures observed in magnetic particles: (a) super
paramagnetic; (b) single domain particle; (c) multi-domain
particle
57

ix
Figure 3.6 Magnetic response of ferromagnetic particles 58
Figure 3.7 Illustration of the concept of super-paramagnetism 58
Figure 3.8 Sphere of radius R and permeability μ2 immersed in a media of
permeability μ1 and subjected to a uniform magnetic field of
magnitude H0.
60
Figure 3.9 Magnetic flux and magnetic field for a uniformly magnetized
particle
62
Chapter 4
Figure 4.1 Channel with straight embedded conductor beneath it 66
Figure 4.2 Generated magnetic field inside the channel 67
Figure 4.3 Applied magnetic forces on the particles in the channel.
Colour-map represents the magnitude of the force and arrows
show its direction
68
Figure 4.4 Sensitive dependency on initial conditions 70
Figure 4.5 Producing stretching and folding using positive and negative
DEP forces generated by different frequencies
73
Figure 4.6 Burst-view of the proposed mixer 74
Figure 4.7 Top view of the proposed mixer illustrating one mixing unit 74
Figure 4.8 Magnetic field near the tip of one tooth in one mixing unit
during a single phase of activation
76
Figure 4.9 Magnetic forces exerted on one single particle at different
heights above the conductor in one mixing unit during a single
phase of activation
78
Figure 4.10 Conceptual diagram of one single circular tip in (a) three-
dimensional and (b) two-dimensional models
80
Figure 4.11 Variation of the force against width (H=20 μm) 81
Figure 4.12 Variation of the force against thickness (W=20 μm) 81
Figure 4.13 Concept of the model illustrating the thermal boundary
conditions
83
Figure 4.14 Temperature distribution on the boundaries 84
Figure 4.15 Maximum temperature rises in the domain for different cross-
sectional areas
84

x
Chapter 5
Figure 5.1 Schematic diagram of coupling 87
Figure 5.2 Free-body force diagram of the particle for a one-way coupled
problem
88
Figure 5.3 Boundaries of one simulated (extended) and one mixing unit 92
Figure 5.4 Phase shift control signal 93
Figure 5.5 Developed parabolic fluid field velocity inside the channel.
Arrows show the direction and magnitude of the velocity field
94
Figure 5.6 Advection of cells and particles within three and a half mixing
units when no external perturbation is applied
95
Figure 5.7 Advection of cells and particles within three and a half mixing
units with magnetic perturbation (St=0.4, V=40 μm/s)
97
Figure 5.8 Consecutive stretching and folding in trajectories which results
in chaotic advection (St=0.2, V=45 μm/s)
98
Chapter 6
Figure 6.1 Block diagram of the various techniques employed for
characterizing the micro-mixers
103
Figure 6.2 Schematic illustration showing that the λ-map 105
Figure 6.3 Schematic illustration for calculating the largest Lyapunov
exponent
105
Figure 6.4 Convergence of the largest Lyapunov exponent for one particle 107
Figure 6.5 Concept of the collision where cell is tagged by the magnetic
particle
108
Figure 6.6 Initial positions of the magnetic particle for computation of the
largest Lyapunov exponent
109
Figure 6.7 Variation of characterizing indices versus different system
operating conditions: (a) cell capturing efficiency, (b) largest
Lyapunov exponent
110
Figure 6.8 Trajectories of particles at St=0.8 and V=40 μm/s; rectangles
indicate the location of trapped particles
111

xi
List of Tables
Table 2.1 Main characteristics of water at 20 ºC and 1 atm 12
Table 4.1 Properties of the reference magnetic particle 67
Table 4.2 Electrical and thermal properties of the glass and copper 82

xii
Table of Contents
Chapter 1 – Introduction
1.1. Lab-on-a-Chip Systems 1
1.2. Magnetic Cell Sorting and Isolation 3
1.3. Advantages of Magnetic Cell Sorting 5
1.4. Motivation and Objectives of the Research 7
1.5. Contribution to Knowledge 8
1.6. Outline of the Thesis 8
Chapter 2 – Literature Survey
2.1. Microfluidics-Basic Concepts and Definitions 10
2.1.1. Newtonian fluid 10
2.1.2. Flow regime 11
2.1.3. Reynolds number 11
2.1.4. Incompressible flow 12
2.1.5. Navier-Stokes equations 12
2.1.6. Steady flow 13
2.1.7. No-slip condition 13
2.1.8 Pressure driven flow (Poiseuille flow) 14
2.2. Mixing in Microfluidics 14
2.2.1. Diffusion 15
2.2.2. Chaotic advection 17
2.3. Passive Micro-Mixers 18
2.3.1. Basic T-mixer and Y-mixer 18
2.3.2. Passive micro-mixers based on multi-lamination (parallel lamination) 19
2.3.3. Passive micro-mixers based on SAR configurations (serial lamination) 21
2.3.4. Injection micro-mixers 23
2.3.5. Droplet micro-mixers 24

xiii
2.3.6. Passive micro-mixers based on chaotic advection 25
2.3.6.1. Chaotic advection at high Reynolds numbers (Re>100) 25
2.3.6.2. Chaotic advection at intermediate Reynolds numbers (10<Re<100) 26
2.3.6.3. Chaotic advection at low Reynolds numbers (Re<10) 29
2.4. Active Micro-Mixers 32
2.4.1. Micro-impellers 32
2.4.2. Pressure field disturbance 33
2.4.3. Acoustic/ultrasonic disturbance 36
2.4.4. Thermal disturbance 37
2.4.5. Electrokinetic disturbance 38
2.4.5.1. Electro-hydrodynamic (EHD) disturbance 38
2.4.5.2. Magneto-hydrodynamic (MHD) disturbance 39
2.4.5.3. Electroosmotic disturbance 40
2.4.5.4. Dielectrophoretic disturbance 43
2.4.5.5. Magnetophoretic disturbance 44
2.5. Discussion 48
2.5.1. Fabrication 48
2.5.2. Performance 48
2.5.3. Application 49
Chapter 3 – Magnetophoresis
3.1. Introduction 51
3.2. Magnetic Field and Magnetic Materials 51
3.2.1. Diamagnetic materials 53
3.2.2. Paramagnetic materials 53
3.2.3. Ferromagnetic, ferrimagnetic and anti-ferromagnetic materials 54
3.2.4. Super-paramagnetism and magnetic nano-particles 56
3.3. Force on a Magnetized Particle in a Magnetic Field 59
Chapter 4 – Basic Design of the Micro-Mixer
4.1. Sources of Magnetic Field 65
4.2. Magnetic Force due to Current Carrying Conductors 66
4.3. Chaotic Mixing 69
4.3.1. Chaos theory 69
4.3.2. Chaos in laminar flows 70

xiv
4.4. Basic Design 72
4.5. Scaling Effects 78
4.5.1. Magnetic forces 79
4.5.2. Electro-Thermal analysis (Joule heating) 82
4.5.3. Conductor size 85
Chapter 5 – Numerical Simulations and Results
5.1. Multiphase Flows 86
5.1.1. Phase coupling 86
5.1.2. Motion of a single particle in a viscous fluid 88
5.2. Numerical Simulations 90
5.2.1. Simulation procedure 90
5.2.2. Simulation parameters 92
5.3. Simulation Results 93
5.3.1. Advection of the cells and particles 94
5.3.2. Basis of chaotic advection in particles 98
Chapter 6 – Characterization of the Micro-Mixer
6.1. Mixing Assessment 100
6.1.1. Experimental techniques 101
6.1.2. Numerical techniques 101
6.2. Characterization Methods Used in this Study 103
6.2.1. Lyapunov exponents 104
6.2.2. Cell capturing efficiency 106
6.2.3. Results and discussion 108
Chapter 7 – Concluding Remarks and Future Work
7.1. Conclusions 112
7.2. Recommendations for Future Research 114
7.2.1. Modified particle properties 114
7.2.2. Three-dimensional mixing 114
7.2.3. Coupled simulations 115
7.2.4. Experiments 116

xv
References 118
Appendix A - COMSOL Multiphysics Simulation 130
Appendix B - Calculation of the largest Lyapunov exponent 142
Appendix C - PUBLICATIONS 144

Chapter 1
Introduction
1.1. Lab-on-a-Chip Systems
Over the past decade, the advent of Micro-Electro-Mechanical Systems (MEMS)
which is based on the miniaturization of mechanical components and their
integration with micro-electrical systems, has created the potential to fabricate
various structures and devices on the order of micrometers. This technology takes
advantage of almost the same fabrication techniques, equipment, and materials
that were developed by semi-conductor industries. The range of MEMS
applications is growing significantly and is mainly in the area of micro-sensors
and micro-actuators. In recent years, miniaturization and integration of bio-
chemical analysis systems to MEMS devices has been of great interest which has
led to invention of Micro Total Analysis Systems (μ-TAS) or Lab-on-a-Chip
(LOC) systems. Since the majority of chemical reactions occur in liquid
environments, the development of µ-TAS is essentially connected to the design of
liquid handling micro-devices (e.g., micro-pumps, micro-valves, micro-flow
sensors, micro-filters, micro-separators and micro-mixers). In fact, microfluidic
platforms are utilized to add an analytical functionality to the system in addition to
its electrical function(s).
New techniques to interface analytical systems with electro-mechanical
components are continuously being developed and offer the design and fabrication
of μ-TAS with a wide range of applications including drug delivery systems,
monitoring devices, nucleic acid-based analysis and automatic point-of-care
diagnostic micro-chips. In diagnostic applications, it is possible to perform all

Chapter 1 – Introduction 2
traditional bench-top protocols such as sample introduction, handling, extraction,
purification or isolation, amplification, filtering and detection.
The main advantages of μ-TAS over traditional devices lie in lower fabrication
costs, improvement of analytical performance regarding quality and operation
time, small size, disposability, precise detection, minimal human interference and
lower power consumption. Moreover, the problem of rare chemicals and samples
which restrain the application of genetic typing and other molecular analyses has
been resolved by employment of μ-TAS.
Figure 1.1 shows a conceptual diagram of a typical LOC system which includes
different microfluidic components for extraction, amplification and detection of
DNA molecules from whole human blood.
Figure 1.1 Conceptual diagram of a typical LOC. It consists of fluidic and electronic components
for human DNA analysis (© Brunel University).
Figure 1.2 illustrates block diagram of the whole system (a) and sample
preparation sub-system (b) in micro-chip under development at Brunel University.
This project focuses on the mixing process as a major step in sample preparation.

Figure 1.2 (a) system level block diagram of the LOC, (b) block diagram of sample preparation
sub-system.
1.2. Magnetic Cell Sorting and Isolation
In bio-medicine it is often essential to separate specific biological entities or cells
out of their native environment in order that concentrated samples may be
prepared for subsequent analysis in downstream or other applications [1].
Generally, there are two types of magnetic sorting when working with cells. In the
first type, cells to be isolated demonstrate adequate intrinsic magnetic property so
that magnetic separations can be performed without any modification. There are
solely two types of such cells in the nature, namely red blood cells (erythrocytes)
containing high concentrations of paramagnetic hemoglobin, and magnetotactic
bacteria containing small magnetic particles within their cells [2]. In the second
type, non-magnetic (diamagnetic) target entities have to be tagged by a magnetic
label to achieve the required contrast in magnetic susceptibility between the cell
and the solution. Through employment of a magnetic label and a proper magnetic
field, a five-order-of magnitude difference in magnetic susceptibility between a
labeled and unlabeled cell may be obtained [3]. These labels are often known as
magnetic micro-particles or micro-beads.

The tagging is made possible by modification of the surface of the particles in a
way, which leads to chemical binding between target entities and particles. In this
technique, usually polymer particles are used and their surface is chemically
functionalized through a coating process, thereby providing a link between the
particle and the target site on a cell or molecule. This coating is a specific bio-
compatible substance and can be an antibody or an m-RNA string but the
possibilities are unlimited. Figure 1.3a and 1.3b show 1 µm diameter magnetic
particles and particles with different functional groups attached to their surface,
respectively.
Figure 1.3 Magnetic micro-particles (a) 1µm Dyna-beads [4]; (b) schematic diagram of surface
functionalized magnetic particles [5].
If magnetic particles are coated with an antibody and then mixed into a solution
containing the target antigen along with other materials only the target antigens
will bind to the antibodies and thus to the magnetic particles. If the magnetic
particles can be subsequently separated from the solution the target antigens will
also be separated from the solution in this way. The separation step is made
possible through utilizing magnetic properties of the particles. The particles used
for this purpose are mostly polymer particles doped with magnetite (Fe3O4) or its
oxidized form maghemite (γ-Fe2O3) and are magnetized in an external magnetic
field. Such external field, generated by a permanent magnet or an electromagnet,
may be used to manipulate these particles through magnetophoresis phenomenon
(i.e., migration of magnetic particles in liquids).
By virtue of their small size; ranging from 100 μm down to 5 nm, particles lose
their magnetic properties when the external magnetic field is removed, exhibiting

superparamagnetic characteristics, which means they have neither coercivity nor
remanence. If the fluid mixture containing magnetically labeled cells are passed
through a region where there is magnetic field, particles and therefore tagged cells
will be immobilized while rest of the fluid is washed away. In fact, magnetic
particles are used as a label for actuation. In the next step, magnetic field is
removed and particle-cell complex is free to flow and be collected for further
analysis in downstream. Figure 1.4 illustrates the concept of magnetic cell sorting.
Figure 1.4 Conceptual diagram of magnetic cell sorting.
Prior to separation of the cell-particle complex from contaminants, magnetic
particles should be distributed throughout the bio-fluidic solution which contains
target cells. This is done by a mixing process which helps to tag the target with
particles. However, mixing remains as the main challenge throughout the whole
cell sorting process and has a significant impact on the efficiency of the protocol.
Therefore, the objective of this research is to propose a practical method for the
mixing process which is discussed in section 1.4.
1.3. Advantages of Magnetic Cell Sorting
Magnetic particle-based manipulation, in particular sorting, in microfluidic
systems is a technique, which offers to simplify and integrate the isolation and
rinsing procedures for extremely small samples of biological materials. In
molecular biology studies, this technique has been widely used for the purification
of specific target bio-molecules, e.g., cell, DNA, RNA, protein, or other macro-

molecules, out of the heterogeneous suspension [6]. Compared to other traditional
and bench-top techniques, magnetic sorting of the cells offers several advantages.
It is relatively simple and fast and allows the target cells to be isolated directly
from crude samples such as blood, bone marrow, tissue homogenates, cultivation
media, food, water etc. Those cells isolated by magnetic method are normally pure
and viable [2].
A conventional isolation method such as centrifugation applies a large shear force
on biological entities. Use of magnetic method can eliminate mechanical force
and therefore, prevent possible damages to bio-cells. With the advent of micro-
fabrication techniques, the miniaturization and system integration of the magnetic
sorting protocol onto the chip will have a significant impact on reducing the
amount of rare samples and expensive reagents [7]. However, in such micro-
systems, the inherent problem of mixing will arise.
On the other hand, the applied magnetic field does not interfere with the
movement of ions and charged solutes in aqueous solutions (at low flow rates) as
does the electric field. Moreover, the large differences between magnetic
permeablities of the magnetic and non-magnetic materials may be exploited in
developing highly selective sorting protocols [2]. Moreover as required in some
applications, it is possible to remove the magnetic label from the isolated cells in
an elution process in order to make it ready for downstream applications and
analyses [8]. Figure 1.5 shows one isolated CD4-T cell with Dyna-beads [9].
Figure 1.5 CD4-T cell isolated with Dyna-beads [10].

1.4. Motivation and Objectives of the Research
While in bench-top protocols mixing is carried out through mechanical
phenomena, in micro-scale devices mixing remains a challenging task. Mixing
stage has a crucial effect on the efficiency of the whole sorting protocol. Without
a proper mixing, a low percentage of the target cells will be tagged by the
particles and consequently, lower number of the samples can be isolated from the
mixture later in downstream even in case of a good separation step.
However, in micro-scale devices where the Reynolds number is often less than 1,
mixing is not a trivial task due to the absence of turbulence. In this scale, laminar
pattern is a dominant feature of the micro-flows and inertial forces are dismissed.
Particularly in the case of dealing with particle laden fluids, some of the standard
micro-mixing methods such as lamination techniques are not practical due to high
probability of clogging in narrow micro-channels. Therefore, development of a
proper technique is required in order to achieve an efficient mixing and also
prevent the clogging of the channel.
The primary object of this research is to design a micro-mixer for chaotic mixing
of magnetic particles into a bio-fluidic solution containing target cells in order to
enhance the attachment of cells to particles. Moreover, a numerical model of the
mixer will be developed to characterize the design. Two indices will be utilized;
firstly calculation of Lyapunov exponents which is a common definition for
inspection of induced chaotic extent in the system, which in turn evaluates the
quantity of mixing. Secondly, the ability of the system to capture target cells will
be evaluated as a supplemental index.
Affecting parameters on the performance of the system comprise structural (i.e.,
geometrical dimensions and type of the components) and operating parameters.
Based on preliminary results obtained from the developed model, reasonably
optimized structural parameters will be adopted for the design. Subsequently, the
effect of the variation of system operating parameters will be investigated against

the above mentioned indices in order to obtain the optimum practical range of
operating parameters for the mixer.
1.5. Contribution to knowledge
The following contributions to knowledge are claimed:
• Designing a microfluidic mixer for chaotic mixing of two separate
solutions; one laden with magnetic particles and the other containing target
cells. This mixer is a part of an integrated device which will be utilized for
extraction of specific molecules out of their native environment.
• Developing a multiphysics model of the mixer combining magnetic,
fluidic and particle dynamics models. This was done using the finite
element analysis package COMSOL. Outcome of the model is the particle
velocity fields which will be used for tracing the particles.
• Developing Matlab codes for calculation of Lyapunov exponents and
therefore, inspection of chaotic extent and also evaluation of the cell
capturing efficiency in the system.
• During the course of MPhil project, the research outcome and
achievements were presented through one journal paper, two conference
papers and two posters. In addition, a design patent application was filed
for the mixer developed in this research. Details are given in Appendix C.
1.6. Outline of the thesis
This thesis comprises seven chapters and two appendices. In the first section of
every chapter, there is an opening introduction on the subject of that chapter and a
technical explanation, followed by the main body.
In the first chapter, a brief description of Lab-on-a-Chip systems and concept of
magnetic isolation is presented. An overview of the research is added at the end of
this chapter.

Chapter 2 covers a comprehensive literature review on existing techniques of
micro-mixing. The review process is based on the reported mechanism of mixing
and includes two major divisions; passive and active types. Also main
characteristics of microfluidic regimes are introduced in this chapter.
Chapter 3 provides a brief description of technical background of magnetic
materials and fields. Underlying physics of super-paramagnetic micro-particles
and the origin of magnetophoretic forces are also addressed in details.
Chapter 4 describes the design of the micro-mixer and proposed mechanism to
create chaotic advection in the motion of the particles. Geometrical structure and
dimensions of the micro-channel and micro-conductors are introduced. An
estimation of generated magnetic field by micro-conductors and injected
magnetophoretic forces on the particles is presented.
Chapter 5 presents detailed simulation procedure of the system. Utilized software
packages and mathematical methods are explained. Simulation results for
different operating parameters are given subsequently, together with a through
discussion.
In chapter 6 the mixing process is evaluated by inspection of chaotic regimes.
Moreover, ability of the system to capture target cells, which is the main goal of
the design, is used as a supplemental index to characterize the device.
Finally in chapter 7, main conclusions are drawn and some suggestion for the
future research in order to further investigate the performance of the mixer and
optimize the design, are presented.

Chapter 2
Literature Survey
Microfluidics deals with the behaviour, precise control and manipulation of
micro-litre and nano-litre volumes of fluids. It is a multi-disciplinary field
comprising physics, chemistry, engineering and bio-technology, with practical
applications to the design of systems in which such small volumes of fluids will
be used. Ascribed to the micron dimensions, microfluidics has some special
characteristics such as high surface-to-volume ratio, high mass-heat transfer rate,
high shear-extension rate, and low Reynolds number. Therefore, in order to
understand the behaviour of micromixers, a reasonable knowledge of the theory of
microfluidics is necessary. In this chapter a brief introduction to microfluidics and
some of the key definitions is presented and, subsequently, a through and
comprehensive literature review on micro-mixing and different reported methods
is provided.
2.1. Microfluidics - basic concepts and definitions
2.1.1. Newtonian fluid
A fluid is called Newtonian when the shear stress induced by the viscosity of the
fluid is directly proportional to the strain gradient:
du
dy
τ = μ (2.1)
The constant of proportionality µ, is the dynamic viscosity coefficient of the fluid.
Water, the fluids of interest in this research, is a Newtonian fluid.

Chapter 2 – Literature Survey 11
2.1.2. Flow regime
Laminar flow, also known as streamline flow, occurs when a fluid flows in
parallel streamlines, with no disturbance between the lines. In fluid dynamics,
laminar flow is a flow regime associated with high momentum diffusion, low
momentum convection, and velocity and pressure independence from time. On the
contrary, turbulence or turbulent flow is a flow regime characterized by chaotic,
stochastic property changes. This implies lower momentum diffusion, higher
momentum convection, and quick variations of velocity and pressure in time and
space. Viscous forces dominate in a laminar flow regime, while inertial forces
dominate in a turbulent flow regime.
2.1.3. Reynolds number
In fluid mechanics, the Reynolds number is a dimensionless parameter obtained
from dimensional analysis. It has an important physical meaning, since it is the
ratio of inertial forces (ρv) to viscous forces (µ/L). Reynolds number is used for
determining whether a flow will be laminar or turbulent and is defined as:
Intertial forces vL vL
Re
Viscous forces
ρ
= = =
μ ν
(2.2)
where:
• v is the characteristic velocity of the flow
• L is the characteristic length of the geometry
• ρ is the fluid density
• µ is the dynamic (absolute) fluid viscosity
• ν is the kinematic viscosity of the fluid, ν=µ/ρ
At sufficiently high Reynolds numbers, the flow becomes unstable and a turbulent
regime develops. However, at lower Reynolds numbers as that in micro-channels,
viscous forces dominate over inertial forces, and flow disturbance quickly gets
damped. In fact, in micro-scale turbulent phenomena are practically not

encountered. As will be discussed, such characteristics make the mixing process
difficult in microfluidic devices where the goal of the mixing is to distribute the
molecules by a random, non-reversible process. At the macroscopic scale where
inertial forces dominate, local instability in the form of turbulence can be easily
created to enhance the mixing [1]. On the contrary, when the Reynolds number is
small, any assay to induce turbulent disturbance is quickly damped by viscous
forces. Thus diffusion becomes the only viable process of the mixing for low
Reynolds number flow.
2.1.4. Incompressible flow
Certain fluids undergo very little change in density despite the existence of large
pressures. In such circumstances when density variation in a problem is
inconsequential, the fluid is called incompressible and the density is treated as a
constant value in computations. Water is an incompressible fluid and table 2.1
lists the main characteristics of water in standard conditions of pressure and
temperature.
Table 2.1 Main characteristics of water at 20 ºC and 1 atm.
Density ρ
Dynamic viscosity
µ
Kinematic
viscosity ν
998 kg/m3
1.0 ×10-3
kg/(m.s) 1.01 ×10-6
m2
/s
2.1.5. Navier-Stokes equations
The Navier-Stokes (N-S) equations are a set of fundamental differential equations
that explain the motion of the fluid substances such as liquids and gases. These
equations are derived from conservation principles (i.e., conservation of mass,
momentum and energy) and are the governing constitutive equations of
conventional flows. The vectorial form of the N-S equations for an incompressible
Newtonian flow is:

2d
( P )
dt
ρ = ρ + −∇ + μ∇
V
B V (2.3)
where:
• V is the velocity
• B is the body force per unit mass
• P is the pressure
In Computational Fluid Dynamics (CFD), the N-S equations are the most
common equations to solve fluid dynamics problems with finite elements analysis
methods.
2.1.6. Steady flow
A flow is called steady when flow characteristics (e.g., velocity components) and
thermodynamic properties at each position in space are invariant with time.
Individual fluid particles may move, but at any particular position in domain, such
particle behaves just like as any other particle when it was at that point. There is
no time dependency in parameters for steady flow equations (d/dt=0).
2.1.7. No-slip condition
When a fluid flow is bounded by a solid surface, molecular interactions cause the
fluid in contact with the surface to seek momentum and energy equilibrium with
that surface. All liquids essentially are in equilibrium with the surface they contact
[2]. Then, all fluids at a point of contact with a solid take on the velocity of that
surface which means the fluid relative velocity at all liquid-solid boundaries is
zero (Vfluid=Vwall). In other words, the outermost molecule of a fluid sticks to
surfaces past which it flows. This is called the no-slip condition and serves as the
boundary condition for analysis of the fluid flow past a solid surface. The no-slip
condition gives rise to the velocity profiles of a flowing fluid as it is shown in the
following section.

2.1.8. Pressure-driven flow (Poiseuille flow)
Pressure-driven flow is commonly found in fluid handling systems, including
microfluidic devices. For a pressure-driven flow in rectangular cross-sectional
channels with no-slip boundary conditions, velocity profiles are parabolic. The N-
S equations with restriction of two-dimensional steady flow simplifies to:
2
2
p u
x y
∂ ∂
= μ
∂ ∂
(2.4)
The solution to this differential equation can be found with the boundary
conditions (the velocity is zero at the channel wall (y=d/2)):
= −
μ
2
21 dp d
u(y) ( y )
2 dx 4
(2.5)
where u is the velocity, µ is the dynamic viscosity of the fluid and p is the
pressure. Figure 2.1 illustrates the developed parabolic velocity profile with the
highest velocity along central streamline.
Figure 2.1 Velocity profile of a steady pressure-driven laminar flow with no-slip boundary
conditions developed in a micro-channel.
2.2. Mixing in microfluidics
Mixing is a fundamental step in most of the microfluidic systems used in
biochemistry analysis where biological processes such as enzyme reactions often
engage reactions that require mixing of reactants. Mixing is also essential in LOC

platforms for tagging of specific entities by some labels such as magnetic particles
which are used for actuation. Micro-mixers can be integrated in a microfluidic
platform or utilized as a stand-alone device. However, mixing several fluids at the
micro-scale is not as easy as it might seem at first glance. As discussed earlier, the
Reynolds number at these dimensions is usually quite small and no turbulence
takes place. Therefore, flow streamlines do not interfere with each other which
results in zero mixing. Nevertheless, over small distances mixing can be
performed by diffusion phenomenon. Alternatively, mixing may be enhanced by
chaotic patterns, which can be induced by various schemes.
Micro-mixers can be generally categorized as passive and active mixers. In
passive micro-mixers where no external energy is required, the mixing process
can rely on diffusion or chaotic advection. Passive mixers can be further
categorized by their arrangement for the mixed phases such as lamination,
injection, chaotic advection and droplet. In active micro-mixers an external field is
used to generate disturbance to enhance the mixing process. Therefore, active
mixers can be categorized by their type of external sources such as pressure,
temperature, electrokinetics, and acoustics. Almost in all active mixers the basis
of mixing is the chaotic advection of the flows. In the following sections, a brief
introduction on the diffusion and chaotic advection phenomena is given and,
subsequently, the review considers various types of passive and active micro-
mixers.
2.2.1. Diffusion
Diffusion is the instinctive spreading of matter (particles), heat, or momentum and
represents one type of transport phenomenon. It is the movement of entities from
regions with higher chemical potential to lower chemical potential. One type of
diffusion is the molecular diffusion (Brownian motion) in which we are dealing
with transfer of the matter. Here, chemical potential can be interpreted as the
concentration of molecules or particles. In fact, Brownian motion is an entropy-
minimizing process occurring in the presence of a non-uniform distribution of

molecules. In microfluidic systems, the molecular diffusion is the dominant
mechanism of mixing of mass species unless some external perturbation is
applied. It is, however, mostly too slow and thus impractical in many cases,
especially for large molecules. Let us estimate the characteristic time of diffusion.
The reason for the diffusion is the large gradient of the concentration of the fluid
molecules (or suspended particles) which exists when two different liquids have a
common interface. The mathematical model of diffusion can be described by
Fick’s second law [3]:
= ⋅ = ⋅∇2dc
D div grad c D c
dt
(2.6)
where c is the concentration for a particular fluid molecule type and D is the
solute diffusion constant. For steady state diffusion (when the concentration
within the diffusion volume does not change with respect to time) the equation
(2.6) is reduced to Fick’s first law, which gives the flux of the diffusing species as
a function of the change in concentration in space (distance):
c
J D
x
∂
= −
∂
(2.7)
where J is the diffusive mass flux per unit of area (area perpendicular to x) and x
is the position. D, diffusion coefficient or diffusivity, is defined as:
κ
= =
πμ
T driving potential
D
6 r resistance
(2.8)
where
• κ is the Boltzmann’s constant (=1.35054×10-23
[J/K])
• T is the absolute temperature of the fluid
• r is the molecular radius of the solute
• µ is the dynamic viscosity of the fluid
Temperature dependency of the diffusion coefficient is associated with this fact
that the Brownian motion of the particles is due to the applied forces from small

liquid molecules which are excited by the temperature. The average time for the
suspended entity to diffuse over a given distance is directly proportional to the
square of the distance:
τ ∝
2
L
D
(2.9)
where L is the characteristic mixing length (e.g., channel width) and τ is the time
of mixing. τ can be up to the order of 105
seconds for particles with 1 μm diameter
dispersed in water solution diffusing a distance of 100 μm. Obviously, such a
diffusion time is not realistic and microfluidic devices that employ natural
diffusion as their sole mixing mechanism will not be able to satisfy the rapid-
mixing requirement in bio-chemical analyses. Therefore, an innovative method of
mixing is essential to enhance the process. As equation (2.9) suggests, the rate of
diffusion is dependent on diffusion coefficient, and the mixing length. Both
viscosity and diameter are intrinsic properties of the solution and the chosen
species, and thus the only remaining possibility of enhancing diffusion is to
increase the contact surface and decrease the diffusion path (see equation (2.7)).
2.2.2. Chaotic advection
In addition to diffusion, advection is another important form of mass transfer in
flows. Advection is normally parallel to the main flow direction, and is not
functional for the transversal mixing process. However, the so-called chaotic
advection can enhance the mixing in microfluidic devices significantly. Mixing in
these devices generally involves two steps; at first, a heterogeneous mixture of
homogeneous domains of the two fluids is created by advection and,
subsequently, diffusion between adjacent domains leads to a homogeneous
mixture at the molecular level [4]. In the context of micro-mixers, the question
arises on how the principle of chaotic advection can be implemented, as macro-
scale techniques such as employment of stirrers are not available. Chaotic
advection can generally be produced by special geometries and three-dimensional
structures in the mixing channel or induced by an external force in passive and

active micro-mixers, respectively. The basis of chaotic mixing will be addressed
in details later in chapter 4.
2.3. Passive micro-mixers
Because of their simple concept, passive mixers were one of the first microfluidic
devices reported. Here we review the passive mixers based on their arrangement
for the mixed phases.
2.3.1. Basic T-mixer and Y-mixer
As discussed earlier, fast diffusion mixing can be accomplished by decreasing the
mixing path and increasing the contact surface between two liquid phases.
Lamination separates the inlet streams into “n” sub-streams and then joins them
into one stream. The most simple design is a channel with merely two inlets (n =
2); known as the T-mixer or the Y-mixer [5,6]. Figures 2.2a and 2.2b illustrate the
design of a typical T-mixer and Y-mixer, respectively [7].
Figure 2.2 Basic designs in parallel lamination; (a) T-mixer, (b) Y-mixer.
Since the basic T-mixer depends solely on molecular diffusion, a long mixing
channel is required to accomplish the process. Nevertheless, efficient mixing may
be achieved in a short mixing length at the expense of increasing the Reynolds
number [8,9]. A chaotic regime can be induced at these high Reynolds numbers.
Wong et al [9] reported a T-mixer which utilizes Reynolds numbers up to 500,
where flow velocity is as high as 7.60 m/s at a pressure of up to 7 bar (figure 2.3).
However, in such micro-mixers, the high velocities on the order of 1 m/s or even

higher require high supply pressures. The high pressure may be a crucial
challenge for bonding and inter-connection techniques.
Figure 2.3 The observations of the mixing process at the junction of micro T-mixer at different
applied pressures: (a) 1.12 bar; (b) 1.88 bar; (c) 2.11 bar; (d) 2.48 bar; (e) 2.77 bar, (f) 4.27 bar [9].
The design of a T-mixer may be enhanced by roughening [10] or throttling [11]
the channel wall and entrance, respectively. At rather high Reynolds numbers the
basic T-mixer can be further modified by implementation of some obstacles in the
channel, which generate vortices and chaotic advection. These types are discussed
in section 2.3.6.
2.3.2. Passive micro-mixers based on multi-lamination (parallel lamination)
When the number of sub-streams is greater than 2, the concept of multi-
lamination is realized. Multi-laminating flow configurations can be realized by
different types of feed arrangements. As explained, lamination is based on the
concept of reducing the mixing path by making narrow channels [12-15]. Another
method to make narrow paths is implementation of interdigital structures in the
channel [16]. The flow is usually driven by pressure, but can also be generated by
electrokinetic forces [17-19]. Figure 2.4 illustrates the concept of multi-lamination
and three reported mixers based on this principle.

Figure 2.4 Multi-lamination; (a) concept, (b) principle of the lateral micro-mixer, (c) mixing of red
and green ink at p= 7.8 kPa [15], (d) photograph of a micro-mixer consisting of a mixing device
with an interdigital channel structure [16].
Vortex (cyclone) mixers are a further type of multi-laminating mixers where fast
vortices are generated to enhance mixing with multiple inlet streams focused in a
circular chamber [20-21]. In the work by Hardt et al [22], a numerical model was
used to analyse the streamline distributions of the three-dimensional vortex and to
predict the mixing performance of the micromixer. It was found that when the
Reynolds number is higher than a critical value of 2.32, a self-rotation effect is
induced in the circular micro-chamber, which in turn generates a three-
dimensional vortex. Respective flow patterns were confirmed by microscopy
analysis and resemble the prediction made by CFD analysis (figure 2.5).
Figure 2.5 Vortex micro-mixers; (a) concept of the mixer, (b) cross-section of vortex chamber
showing mixing process (green indicates complete mixing) [20].

An alternative concept to reduce the mixing path for multi-lamination micro-
mixers is hydrodynamic focusing. The basic design for hydrodynamic focusing is
a relatively long channel with three inlets. The middle inlet is dedicated to the
sample flow, while the solvent streams join through two encompassing inlets and
act as the sheath flows (figure 2.6a).
Hydrodynamic focusing technique was first developed to enable a fast mixing
process in less than one second. It reduces the stream width and, consequently, the
mixing path. Knight et al [23] reported a prototype with a narrow mixing channel
of 10 μm×10 μm in section. The sample fluid may be focused to a specific width
by adjusting the pressure ratio between the sample flow and the sheath flows. In
this way, diffusion distances are significantly reduced by compressing the fluid
layer to a few micrometers, resulting in a mixing in the milliseconds range [24].
Walker et al [25] reported a micro-mixer based on hydrodynamic focusing used
for cell infection (figure 2.6b).
Figure 2.6 Parallel lamination; (a) concept of the hydrodynamic focusing, (b) an image of the blue
food colouring stream in the middle and water streams on either side [25].
2.3.3. Passive micro-mixers based on Split-and-Recombine configurations
(serial lamination)
Split-and-Recombine (SAR) micro-mixers can improve the mixing by splitting
and later joining the streams, creating sequentially multi-laminating patterns
(figure 2.7a). For instance, the inlet streams may be first joined horizontally and
then in the next stage vertically. SAR mixing commonly relies on a multi-step
procedure. The basic operations are: splitting of a bi- or multi-layered stream
perpendicular to the main orientation into sub-streams, re-direction or re-

alignment of the sub-streams, and the recombination of these. These basic steps
are usually accompanied by one or more re-shaping steps [26]. After m splitting
and joining stages, 2m
liquid layers can be laminated. The process leads to a 4m-1
times improvement in the mixing time [7].
Figure 2.7 SAR configuration; (a) concept of join-split-join [7], (b)-(d) cross-sectional views of
layer configurations within a SAR step: b) splitting; c) re-arrangement of the sub-streams and
recombination; d) reshaping [27].
Branebjerg et al. [28] and Schwesinger et al. [29] were among the first who
considered a micron-sized implementation of the SAR approach. Since then,
several kinds of micro-mixers have been realized utilizing some kind of multi-step
SAR approach. The designs of SAR mixers differ in the exact geometry by which
they actually achieve the fluidic arrangement. Ramp-like [28], fork-like [29], and
curved architectures [27] with and without splitting plane were reported. In
context with micro-technological applications, the SAR concept is especially
appealing, since it allows achieving fine multi-lamination with moderate pressure
drops and without severe fabrication constraints.
The concept of the SAR lamination micro-mixing may also be utilized for
electrokinetic flows [30]. Using electroosmosis flows between the multiple
intersecting channels, mixing was considerably enhanced (see figure 2.8). Figure
2.8c shows the flow configuration in the mixer. As it can be seen, diffusive
mixing is enhanced and convective mixing also takes place, which would not
occur in open channels. Melin et al [31] later reported a similar design for a

pressure-driven flow. However, this design works only for discrete liquid
samples.
Figure 2.8 SAR lamination; (a) concept, (b) fabricated mixer, (c) how mixing might occur [30].
2.3.4. Injection micro-mixers
The basis of the injection mixer is similar to the SAR lamination mixer. However,
instead of splitting both inlet flows, the mixer solely splits the solute flow into
many sub-streams and injects them into the solvent flow. On top of one stream is
an array of nozzles, which create a number of micro-plumes of the solute. These
plumes enlarge the contact surface and decrease the mixing path, thereby
improving the mixing efficiency [7].
Miyake et al [32] presented an injection micro-mixer with 400 nozzles arranged in
a square array. The mixer has an area for mixing, which is very flat and thin with
micro-nozzles provided at the bottom of the mixing chamber. First, the mixing
area is filled with one liquid, and the other liquid is injected into the area through
the micro-nozzles, making many micro-plumes. The nozzles are positioned very
closely in rows, 10-100 µm apart, in order that the plumes may quickly diffuse for
this distance. Thus, effective mixing will be performed without any additional
driving. Figure 2.9 shows the concept of the injection mixro-mixer. Similar
technique for the mixing with different nozzle shapes was reported by other
researchers [33-35].

Figure 2.9 Basic idea of the micro-injection mixer [32].
2.3.5. Droplet micro-mixers
An alternative method for reducing the mixing path is to form droplets of the
mixed liquids. The movement of a droplet may lead to an internal flow field and
make the mixing inside the droplet feasible. If mixing is achieved by droplet
movement only, this is passive mixing due to convection. Droplets may be
generated and manipulated individually using pressure [36] or capillary effects
such as thermo-capillary [37] and electro-wetting [38]. Moreover, droplets may be
generated by virtue of the large difference of surface forces in a narrow channel
with multiple immiscible phases such as oil-water or water-gas [39]. In this type,
by using a carrier liquid such as oil, droplets of the aqueous samples can be
formed. While moving through the channel, the shear force between the carrier
liquid and the sample accelerates the mixing process in the droplet (figure 2.10).
Figure 2.10 Mixing by winding microfluidic channels shown, (a) experimentally (left: a scheme of
the microfluidic network. right: photograph of plugs) and (b) schematically [39].

2.3.6. Passive micro-mixers based on chaotic advection
Chaos cannot occur in steady two-dimensional flows, but only in three-
dimensional and two-dimensional time-dependent flows. In two-dimensional
flows, time-dependency may be considered as an added third dimension. Time-
dependency may be induced by external forces, which is the principle of active
mixing class and will be dealt with in section 2.4. In passive micro-mixers the
basic idea is to modify the configuration and shape of the channel in a way that
leads to splitting, stretching, and folding of the flow. Here, we classify the passive
chaotic mixers based on the range of flow Reynolds number; high, intermediate
and low. However, it is not always possible to dedicate a particular design to a
specific range of Reynolds number.
2.3.6.1. Chaotic advection at high Reynolds numbers (Re>100)
A simple method is to insert obstacle structures in the mixing micro-channel in
order to induce the chaotic advection. Various configurations and arrangements
have been reported. Lin et al [40] used seven cylinders of 10 μm diameter placed
in a narrow channel (50 μm × 100 μm × 100 μm) to enhance mixing. The mixing
was performed with Reynolds numbers ranging from 200 to 2000 and a reaction
time of 50 μs. Wang et al [41] reported a mixer using the same type of obstacles
with different arrangements and carried out a numerical investigation of the
mixing at high Reynolds numbers. The mixing channel was 300 μm in width, 100
μm in depth and 1.2-2 mm in length, and the diameter of the obstacle was 60 μm
(see figures 2.11a and 2.11d). It was revealed that obstacles in a channel at low
Reynolds numbers cannot generate eddies or re-circulations. However, simulation
results showed that obstacles could enhance the mixing performance at high
Reynolds numbers. As shown in figures 2.11b and 2.11e, obstacles may also be
placed on the channel’s walls [7,42].

Figure 2.11 Various designs for producing chaotic advection at high Reynolds numbers; (a)&(d)
obstacles placed in the channel [41], (b)&(e) obstacles placed on the wall [42], (c)&(f) zigzag
channel [43].
An alternative method to generate chaotic advection is utilizing zigzag channels to
produce re-circulation around the turns in the channel. Mengeaud et al [43] used a
micro-channel with a width of 100 μm, a depth of 48 μm and a length of 2 mm
(see figures 2.11c and 2.11f). In conducting a numerical investigation, they
adopted the periodic steps of the zigzag shape as the main optimization parameter.
Reynolds number was varied ranging from 0.26 to 267 and a critical Reynolds
number of 80 was found. Below this number the mixing process relied entirely on
diffusion whereas as at higher Reynolds numbers, mixing was performed by the
generated re-circulations at the turns along the channel. The re-circulations could
induce a transversal component of the velocity, which enhances the mixing
process.
2.3.6.2. Chaotic advection at intermediate Reynolds numbers (10<Re<100)
Most of the micro-mixers in this category are based on the modified three-
dimensional twisted channels, but there may be some exceptions as well. For
instance, Hong et al [44] presented an in-plane micro-mixer with two-dimensional
modified Tesla structures (figures 2.12a to 2.12d). The Coanda effect in this

structure leads to chaotic advection and enhances mixing noticeably. The mixer
performs well at Reynolds numbers higher than 5.
Liu et al [45] reported a three-dimensional serpentine mixing channel comprised
of a series of C-shaped segments placed in perpendicular planes (figures 2.12e to
2.12g). The micro-mixer has two inlet channels joined in a T-junction and a
sequence of six mixing segments. It was observed that the mixer is that the mixing
time is short at higher Reynolds numbers; chaotic advection only occurred at
Reynolds numbers ranging from 25 to 70.
Chen and Meiners [46] reported a complex mixing unit which consists of two
connected out-of-plane L-shapes (figures 2.12h to 2.12j). A single mixing unit
measures about 400 μm × 300 μm and the mixer is composed of a series of such
mixing units. Effective mixing could be obtained on short length scales with a
purely laminar flow with the Reynolds number of Re= 0.1-2. This concept was
called as ‘flow-folding topological structure’ by the authors.
Park et al [47] presented the results for mixing two fluids in a three-dimensional
passive rotation micro-mixers using the break-up process (figures 2.12k to
2.12m). The complex channel rotates and separates the two fluids by partitioning
walls, and consequently, generates smaller blobs exponentially. In practical
experiments, over 70% mixing was achieved at Re=1, 10 and 50, only after
passing through a 4 mm long channel.
Vijayendran et al [48] reported a three-dimensional serpentine mixing channel
where the channel was designed as a series of L-shaped segments in perpendicular
planes (see figure 2.12n). The mixer was experimentally tested at Reynolds
numbers of 1, 5 and 20. The results indicated that better mixing was achieved at
higher Reynolds numbers. Jen et al [49] proposed various designs of twisted
micro-channel providing a third degree of freedom for chaotic advection. Mixing
of methanol and oxygen was numerically investigated at different velocities (0.5-
2.5 m/s). Figures 2.12o to 2.12q illustrate the concept of the twisted channels.

Figure 2.12 Micro-mixer designs for mixing with chaotic advection at intermediate Reynolds
numbers: (a) modified Tesla structure, (b)-(d) experimental results for the Tesla structure, (e) C-
shape concept, (f)-(g) photographs of side-by-side experiments at Reynolds numbers of 12 and 70,
respectively, (h) connected out-of-plane L-shapes, (i) topologic structure, (j) mixing of two
fluorescently labelled protein solutions in a six-stage mixer at Reynolds number of 0.1. (top).
mixing of the same dyes in an aqueous 54% glycerol solution with ten fold higher viscosity at
Reynolds number of 0.1 (bottom), (k) twisted micro-channel concept, (l) schematic diagrams of
twisted mixer segment and flows in the channel re-circulating along the walls, (m) images of
mixer obtained using a confocal scanning microscope; one fluid is propagating into the other fluid
along the 4 mm long channel, (n) L-shape concept, (o)-(q) other designs of the twisted channel.

2.3.6.3. Chaotic advection at low Reynolds numbers (Re<10)
One of the most promising types of the passive micro-mixers falls in this category
which works based on the idea of placing micro-structured objects within the flow
passage on one side of the channels. Johnson et al [50] and Stroock et al [51] were
the first to investigate this concept and since then, much effort has been dedicated
to improve their proposed mixers. In Johnson et al [50], a T-mixer was modified
with a pulsed UV excimer laser to ablate a series of slanted wells at the junction
(figure 2.13a). This structure allowed inducing a high degree of lateral transport
across the channel in either electroosmotic or pressure-driven flows. The
performance of the slanted-well design was evaluated over a range of effective
electroosmotic flow rates. The captured fluorescence microscopy images for two
flow rates are shown in figures 2.13c and 2.13d for slanted-well design (left) and
T-channel without the ablated wells (right).
Stroock and his colleagues [51-53] pointed out another way of creating secondary
re-circulating flows in a channel. They considered geometries with grooved
channel walls, such that at least one of the walls contains ridges standing at a
tilted angle with the main flow direction. Two different groove patterns were
considered; obliquely oriented and staggered ridges. Corresponding schematic
designs are shown in figures 2.13e and 2.13i, respectively. They referred to later
one as the staggered herringbone mixer (SHM).
One way to induce a chaotic pattern is to subject volumes of fluid to a repeated
sequence of rotational and extensional local flows. This sequence of local flows in
the SHM may be obtained by varying the shape of the grooves as a function of
axial position in the channel: The alteration in the orientation of the herringbones
between half cycles exchanges the positions of the centres of rotation and the up-
and down-wellings in the transverse flow. When a pressure-driven fluid flows
over such a surface, the grooves can be viewed as if they induce a slip flow in a
particular direction. Confined to a channel, the flow develops re-circulation
patterns, which leads to an exponential increase of specific interface, therefore to

fast mixing. The SHM mixing is superior to similar channels without inserted
structures or with straight ridges only. SHM can work well at a Reynolds numbers
ranging from 1 to 100. Confocal micrographs of vertical cross-sections of the
channel in both designs are shown in figures 2.13h and 2.13k.
The effect of chaotic advection in a channel with grooves was numerically
investigated by Wang et al [54] and Schonfeld and Hardt [55] for channels with
straight ridges and Aubin et al. [56] for both patterns using CFD methods. They
showed that an exponential stretching of the fluid interface occurs where with
simple linear grooves (straight ridges), the interface area increases more slowly.
Also ablation of grooves on the PDMS substrate by a laser was investigated by
Lim et al [57].
Kim et al [58] improved the design of straight ridges with embedded barriers
parallel to the flow direction. The embedded barrier changes the original elliptic
mixing pattern (developed in SHM) to a hyperbolic pattern as shown in figure
2.14a. Cross-sectional confocal microscope images of the mixing patterns at
several positions are shown in figure 2.14b.
A miniaturized version of a conventional large-scale static mixer with helical
elements was presented by Bertsch et al [59]. This concept modifies the three-
dimensional inner surface of a cylindrical mixing channel. Two kinds of
geometries were studied. The first type was composed of a series of stationary
rigid elements that form intersecting channels to split, rearrange and combine
component streams. The second type is comprised of a series of short helix
elements arranged in pairs, each pair composed of a right-handed and left-handed
element arranged alternately in a pipe (figure 2.14c). To numerically characterize
the efficiency of the mixer, they injected 65000 evenly distributed particles in the
half section of the pipe just before the mixer inlet. Figure 2.14d shows the location
of the particles at regularly spaced axial locations along both types of the mixers.
Moreover, experimental results revealed that the mixing efficiency of the mixer
made of intersecting channels is better than the mixer made of helical elements.

Figure 2.13 Micro-mixer designs for mixing with chaotic advection at low Reynolds numbers: (a)
concept of slanted ribs, (b)-(d) fluorescence images of experiments with and without ablated wells,
(e) slanted grooves, (f) schematic diagram of the channel with ridges, (g) optical micrograph
showing a top view of a red stream and a green stream flowing on either side of a clear stream in
the channel, (h) fluorescent confocal micrographs of vertical cross-sections of the channel, (i)
staggered herringbone grooves (SHM), (j) schematic diagram of one-and-a-half cycles of the
SHM, (k) confocal micrographs of vertical cross-sections of the channel.

Figure 2.14 Micro-mixer designs for mixing with chaotic advection at low Reynolds numbers: (a)
schematic view of the barrier embedded micro-mixer, (b) cross-sectional confocal microscope
images of the mixing pattern, (c) cut-of view of the micro-mixer structures made of intersecting
channels and helical elements, (d) plots of the locations of particles at regularly spaced axial
locations along the micro-mixers (from left to right); left plot corresponds to the beginning of the
mixers. The top line corresponds to the mixer made of intersecting channels, whereas the bottom
line corresponds to the mixer made of helical elements.
2.4. Active micro-mixers
As discussed earlier, in active micro-mixers an external field is used to generate
disturbance to enhance the mixing process. Most of the active mixers rely on the
chaotic regime induced by virtue of the induced periodic perturbation. In the
following, various active mixers classified by the type of employed external
sources are presented.

2.4.1. Micro-impellers
Traditionally, stirring with impellers is the most common way to perform mixing
of large volumes. However, several miniaturised stirrers have been developed for
mixing of the liquids in micro-scale [60-62]. In macroscopic stirrers, the stir-bar
or propeller rotation causes turbulence by increasing the local velocity. In micro-
scale, the stir-bar helps mixing by providing more interfacial area rather than
inducing turbulence. Claimed advantages of such mixers are the possibility to
match the impeller diameter to the mixing volume, carry out large-area mixing,
undergo mixing on-demand (switch on/off), and the flexibility of the mixing
approach regarding the choice of liquids [63]. A micro-stir-bar with a span of 400
µm was fabricated and placed at the interface between two liquids in a PDMS
channel by Ryu et al [61]. An external magnetic field provided by a rotating
magnet in a hotplate/stirrer drives the stirrer remotely (figures 2.15a and 2.15b).
Experimental results proved that nearly complete mixing is achieved instantly.
Figure 2.15c shows the sequential shots of the mixing operation of a micro stir-bar
at different time lapses.
Figure 2.15 Activation by moving parts: (a) schematic view, (b) fabricated magnetic stir-bar, (c)
sequential shots of the mixing operation of a micro stir-bar at different time lapses [61].
2.4.2. Pressure field disturbance
Pressure disturbance was one of the earliest methods used in active micro-mixers.
Deshmukh et al [64] presented a T-mixer with pressure disturbance where an

integrated micropump drives and stops the flow in the channel to divide the mixed
liquids into multiple serial segments and make the mixing process independent of
convection. The performance of this mixer was later evaluated and mixing was
found to proceed quickly in the mixing channel [65]. Figure 2.16 shows the
mixing process at different stages. The pressure disturbance may also be
generated using external embedded micropumps [66].
Figure 2.16 Active micro-mixer based on pressure field disturbance: (a) schematic view, (b) steady
flow, (c) the top stream is stopped for 1/6 seconds, (d) the bottom stream is stopped, (e) mixed
fluid after many pulses [64].
Another method to achieve pressure disturbance is the generation of pulsing
velocity by alternating switches of the flows from a high to a low flow rate,
periodically. In this way, a pulsation of the whole stream is achieved promoting
axial mixing. Glasgow and Aubry [67] reported a simple T-mixer and detailed
CFD simulations with a pulsed side flow at a small Reynolds number of about 0.3.
When both inlets have constant flow rates, the mixing zone is confined to a
narrow band around the horizontal interface (figure2.17a). Time pulsing of one
inlet flow rate distorts the interface to an asymmetrically curved shape which
changes with time. Therefore, liquid transport is promoted and mixing is
improved (figure 2.17b). The degree of mixing was 22%, being 79% larger than
for constant flows. The periodicity and the number of pulsing streams have a
significant effect on the mixing efficiency. The best results were obtained for two
pulsed inlet flows having a phase difference of 180º with the same amplitude and
frequency. CFD simulations showed the bending of the fluid interface along the

channel cross-section and associated stretching and folding in the direction of the
flow. The corresponding degree of mixing was considerably increased to 59%.
Figures 2.17c and 2.17d show the simulation results for two pulsed inlet flows
with phase differences of 90º and 180º, respectively.
Figure 2.17 Numerical simulation results, (a) constant mean velocity in both inlets, (b) , pulsed
flow from the perpendicular inlet (c) and (d) two inlet flows pulsed at a 90 and 180 degree phase
difference, respectively [67].

Same concept was extended to the multiple pulsing injection of flows into one
channel, thereby generating chaotic advection [68]. However, such devices
require a complex computer controlled source-sink system. A further modelling
work on pressure disturbance was reported by Okkels and Tabeling [69].
2.4.3. Acoustic/ultrasonic disturbance
Acoustic (ultrasonic) actuation may be utilized to stir the fluids in active micro-
mixers [70-72]. However, ultrasonic mixing may be a challenging issue in
applications for biological analysis owing to the temperature rise due to acoustic
energy. Many biological fluids are sensitive to high temperatures. Moreover,
ultrasonic waves around 50 kHz are harmful to biological samples by virtue of the
possible cavitations. The non-destructive ultrasonic mixer reported by Yasuda
[73] used loosely focused acoustic waves to induce stirring movements where the
wave was generated by a piezoelectric zinc oxide thin film (figure 2.18a). The
actuator was driven by a programmable function generator providing a 500
kHz/3.5 MHz sine waves and programmed waveforms corresponding to the
thickness-mode resonance of the piezoelectric film. The mixer performed without
any consequential temperature increase and could be used for fluids sensitive to
the temperature. Figures 2.18b to 2.18d show the mixing of red blood cells by
ultrasound irradiation.
Figure 2.18 Schematic illustration of the micro-chamber for mixing of red blood cells by
ultrasound irradiation: (a) longitudinal section (b) area A in the chamber, (c) micrograph before
ultrasound irradiation, (d) micrograph during ultrasound irradiation [73].

An air bubble in a liquid can perform as an actuator, when it is energised by an
acoustic field. The bubble surface behaves like a vibrating membrane and this
type of actuation is mainly dependent on the bubble resonance characteristics.
Bubble vibration due to a sound field generates friction forces at the air/liquid
interface which leads to a bulk fluid flow around the air bubble (known as
cavitation or acoustic micro-streaming). Liu et al [75-75] used acoustic streaming
around an air bubble for mixing where streaming was induced by the field
generated by an integrated PZT actuator. Fluidic movements led to the global
convection flows with “Tornado” pattern in the vicinity of the bubbles. The time
required to fully mix the whole chamber was approximately 45 s. Figure 2.19
shows snapshots of multi-bubble induced acoustic mixing in a chamber at
different time stages. Further acoustic devices for mixing water and ethanol [76]
as well as water and uranine [77] were reported. Yaralioglu et al [78] also used
acoustic streaming to perturb the flow in a conventional Y-mixer.
Figure 2.19 Snapshots showing multi-bubble induced acoustic mixing in a chamber at time (a) 0 s;
(b) 28 s; (c) 1 min 7 s; (d) 1 min 46 s. [74].
2.4.4. Thermal disturbance
According to equation (2.8), diffusion coefficient is highly dependent on
temperature. Therefore, thermal energy may also be utilized to enhance the
mixing. Mao et al [79] generated a linear temperature gradient across a number of

parallel channels in order to examine the temperature dependence of fluorescent
dyes. Also a micro-mixer with a gas bubble filter activated by a thermal bubble
actuated micropump was successfully demonstrated by Tsai and Lin [80]. The
generated oscillatory flow could induce disturbance and wavy interface to
increase the contact area of fluids and accelerate the mixing process.
2.4.5. Electrokinetic disturbance
Electrokinetics is the study of the motion of bulk fluids or selected particles
embedded in fluids when they are subjected to electric or magnetic fields.
Electrokinetic forces can be utilized to manipulate liquid and/or particles in
micro-mixers as an alternative to pressure-driven flow. In the following, various
active mixers classified regarding the employed electrokinetic forces are
presented.
2.4.5.1. Electro-hydrodynamic (EHD) disturbance
Electro-hydrodynamic effect has been used to generate chaotic flows in micro-
mixers [81-82]. A simple geometry mixer was proposed, which works based on
the EHD force when the fluids to be mixed have different electrical properties and
are subjected to an electric field [81]. The electrodes are arranged so that the
electric field is perpendicular to the interface between the two fluids, creating a
transversal flow. Figure 2.20a illustrates the concept of the EHD mixer.
Figure 2.20 Active micro-mixer based on the EHD disturbance: (a) schematic view of the mixer,
(b)-(d) visualization of the flow.

Two fluids of identical viscosity and density, but with different electrical
conductivities and permittivities were used for experiments. Each fluid enters the
microfluidic chamber in its own inlet channel. As soon as they meet, a jump in
electrical conductivity and/or permittivity is generated at the interface between the
two fluids, which has no effect as long as the electric field is absent. However, as
the fluids enter the electric field influence zone close to a pair of facing electrodes,
they are subjected to an electrical force, which creates a transversal secondary
flow across the interface between the two fluids, therefore destabilizing the
interface and enhancing the mixing process. By alternating the voltage and
frequency on the electrodes, efficient mixing was obtained in less than 0.1s at a
low Reynolds number of 0.02. Figures 2.20b to 2.20d show the photographs of the
experiment.
2.4.5.2. Magneto-hydrodynamic (MHD) disturbance
The magneto-hydrodynamic force has been utilized in an active micro-mixers
reported by Bau et al [83]. This mixer uses the arrays of electrodes deposited on a
conduit’s wall as shown in figure 2.21a. By applying alternating potential
differences across pairs of electrodes, currents are induced in various directions in
the solution. In the presence of a magnetic field, the coupling between the
magnetic and electric fields induces body (Lorentz) forces in the fluid which in
turn produce mixing movement in the chamber. The Lorentz force can roll and
fold the liquids in a mixing chamber.
Figures 2.21b to 2.21f illustrate the deformation of a line of dye resulting from the
application of the Lorentz forces. After each time unit (a few seconds), the
polarity of the electrodes and the direction of the Lorentz force are reverse and the
dye returns to its previous initial position. After several reversals, dye continues to
deform in opposite directions and eddies are formed. These concepts work only
with an electrolyte solution. Since the electrodes can be patterned in various ways;
relatively complex flow fields can be generated.

Figure 2.21 Active micro-mixer based on the MHD disturbance: (a) schematic (top-view)
depiction of the mixer, (b) top-view of the fabricated mixer at beginning of the experiment when a
thin line of dye is laid across the cavity, (c)-(f) deformation of the dye line.
2.4.5.3. Electroosmotic disturbance
Lin et al [84-85] reported a T-form micro-mixer using alternatively switching
electroosmotic flow. A switching DC field is utilized to generate an
electroosmotic force which concurrently drives and mixes the electrolytic fluid
samples (figure 2.22a). It was shown that a mixing performance as high as 97%
can be obtained within a mixing distance of 1 mm downstream from the T-
junction when a 6 kV/m driving voltage and a 2 Hz switching frequency are
applied. Figure 2.22b presents the flow contours for optimized operating
conditions in the cases of low and high driving voltages.
Design and fabrication of a ring electroosmotic chaotic micro-mixer with
integrated electrodes was reported by Zhang et al [86] and numerical investigation
of the same mixer was later carried out by Chen et al [87]. Figure 2.22c shows the
SEM picture of the mixer. It takes two fluids from different inlets and combines
them into a single channel where the fluids enter the central loop in downstream.
Four microelectrodes are positioned on the outer wall of the central loop with an
angular distance of 45º. These microelectrodes impose a spatially varying electric
field, and the fluids are manipulated via the electroosmotic slip boundary
condition before they enter the outlet channel. Electric potentials on the

microelectrodes are time-dependent, which adds the third dimension necessary for
chaotic mixing. Generated electroosmosis agitates the low Reynolds number flow.
Figure 2.22d shows the streamlines at t=25 s obtained from simulations and figure
2.22e illustrates the induced stretching and folding of a small volume of fluid. Red
and blue curves are particle trajectories starting from the upper and lower half of
the inlet, respectively.
Figure 2.22 Active micro-mixer based on switching electroosmotic disturbance: (a) schematic
representation of the mixer, (b) flow contours for two optimized operating conditions at low and
high driving voltages, (c) SEM picture of the ring micro-mixer, (d) streamlines at t=25 s, (e)
stretching and folding of a small volume of fluid.
Sasaki et al [88] presented a mixer based on AC electroosmotic flow, which is
induced by applying an AC voltage to a pair of coplanar meandering electrodes
configured in parallel to the channel. The mixing time was 0.18 s, which was 20-
fold faster than that of diffusional mixing without an additional mixing
mechanism.
Tang et al [89] also utilized an electroosmotic flow to improve mixing where
switching on or off the voltage supplied to the flow generates fluid segments in
the mixing channel. This flow modulation scheme was capable of injecting
reproducible and stable fluid segments into microchannels at a frequency between
0.01 Hz and 1 Hz.

A mixer based on periodical field-effect control to dynamically manipulate local
flow field in the micro-channel was demonstrated by Wu and Liu [90]. The
proposed mixing mechanism combines temporal modulation (periodical out-of-
phase AC radial voltage control) with spatial modulation (asymmetric
herringbone-electrode feature) on the ζ-potential of the channel walls to induce
complex flow field for mixing enhancement (figure 2.23a). Numerical and
experimental results showed that good mixing efficiency of over 90% can be
achieved within a 5 mm long micro-channel (figures 2.23b and 2.23c).
Figure 2.23 Electroosmotic micro-mixer: (a) T-shape mixer with embedded electrodes, (b)
simulation results for the transverse velocity vectors at different cross-sections (y–z planes) along
the channel, (c) photograph of an electroosmotic flow with uranine dye.
In another case, oscillating electroosmotic flow in a mixing channel/chamber is
caused by an AC voltage [91]. The pressure-driven flow becomes unstable in a
mixing channel and the rapid stretching and folding of material lines associated
with this instability can be used to stir fluid streams with Reynolds numbers of
order of unity. Figure 2.24 shows schematic concepts of the mixers and also time-
stamped images showing an initially stable interface and its development after the
onset of the instability in both channel and chamber concepts.

Figure 2.24 Active micro-mixer based on electroosmotic disturbance: (a)-(d) schematic view of
mixing channel and chamber, (e) & (f) time-stamped images obtained from mixer with channel
and chamber configurations, respectively.
2.4.5.4. Dielectrophoretic disturbance
When a polarizable particle is exposed to an electric field, a dipole is induced in
the particle. If the electric field is non-uniform, the particle experiences a force
that can move it towards the high or low-electric field region, depending on the
particle polarizability compared with the surrounding medium. This phenomenon
is known as dielectrophoresis (DEP). If the polarizability of the particle is higher
than the solution, the force is towards the high field strength region (positive
DEP). Otherwise, the force is towards the lower field region (negative DEP) [92].

DEP has been utilized in active chaotic micro-mixers by Deval et al [93] and Lee
et al [94]. Figure 2.25a and 2.25b show a schematic view and the fabricated
mixer, respectively. Chaotic advection was generated by embedded polystyrene
particles with a combination of electrical actuation and local geometry channel
variation. Figure 2.25c shows the evolution of an interface as it advects through
the chamber. Where the electric field is constantly set to zero, the interface
remains flat as it travels across the chamber. However, when it is periodically
switched on and off, stretching and folding can take place, resulting in a
favourable situation for mixing. The yellow line indicates the evolution of the
interface between particle solution (lower part) and DI water (upper part).
Figure 2.25 Active micro-mixer based on dielectrophoretic disturbance: (a) schematic view, (b)
fabricated mixer, (c) stretching and folding as dielectrophoretic force is applied.
2.4.5.5. Magnetophoretic disturbance
The magnetic field-induced migration of particles in liquids is known as
Magnetophoresis which is dealt with in details in next chapter. Recently, in
addition to separation, magnetophoretic forces are exploited to enhance the
mixing of the particles in a solution in micro-scale devices.
A magnetic force driven chaotic micro-mixer was reported, in which magnetic
particles are stirred by the local time-dependent magnetic field to enhance the
attachment of magnetic particles onto biological molecules suspended in the
medium [95-98]. A serpentine channel geometry with the perpendicular electrodes

arrangement was used to create the stretching and folding of material lines. It is
claimed that good mixing was achieved in a short time (convective time of less
than 10 s) and distance (mixer length of 1.3 mm). Figure 2.26a shows the
fabricated magnetic mixer with a serpentine shaped channel. Magnetic particles
do not mix without an external disturbance. In figure 2.26b, particles are dispersed
over the entire channel at downstream due to magnetophoretic perturbation.
Figure 2.26 Active micro-mixer based on magnetophoretic disturbance: (a) fabricated mixer, (b)
dispersed particles over the entire channel at downstream.
Another micro-mixer is presented by Rong et al. [99] using magnetic micro-tips
for active mixing of magnetic particles or bio-cells. Mixing is achieved by a
combined rotational/vibrational force exerted on the particles as the magnetic tips
are sequentially excited to produce a rotating magnetic field. Mixer is driven by
three magnetic pole pairs excited with electromagnets coupled to magnetic pole
tips using through-hole vias as shown in Figure 2.27a and 2.27b.
Before excitation, particles are randomly distributed in the centre junction region
(figure 2.27c). When a sequential driving signal is applied, the magnetic particles
move around in the junction region of the channels with the applied magnetic
field. Figure 2.27d shows the mixing action before and after applying drive signal
to the three pair tips. Two liquids with and without magnetic particles are
introduced via separate channels. In the absence of excitation, two fluids will flow
separately according to laminar flow theory. If sequential actuation is applied to
the pole tips, particles in the junction region will be agitated by both rotating and
vibrating motion, which will then produce a rapid mixing action. However,
manufacturing the proposed mixer requires the utilization of complex micro-
fabrication techniques.

Figure 2.27 Mixer based on magnetophoretic disturbance: (a) & (b) schematic view and fabricated
mixer, repectively, (c) working principle, (d) mixing action before and after applying derive signal.
In another report, active fluid mixing was demonstrated in micro-channels where
mixing was based on the manipulation by a local alternating magnetic field of
self-assembled porous structures of magnetic micro-particles that are placed over
the section of the channel [100]. The mixing is the result of the chaotic splitting of
the fluid streams by the structures. In fact, they have followed the approach of
placing obstacles in the micro-channel in order to create stirring (convective)
effects by forcing one fluid stream into another. Another factor is the possibility to
induce a rotational motion of the magnetic micro-particles by using an AC
magnetic field.
Figure 2.28 is a schematic diagram of the realized three-dimensional and
monolithic microfluidic chip with embedded permalloy parts. Magnetic field is
generated by an external electromagnet, brought in mechanical contact with the
permalloy parts. When placed in the field, particles start interacting by means of
the magnetic dipole interaction. This interaction induces a spontaneous clustering
of the particles into larger structures. Using a sinusoidally varying magnetic field
(1 Hz<f<100 Hz), a rotational motion of the particles was induced, thereby
enhancing the fluid perfusion by the structure that behaved as a dynamic random
porous medium.

Figure 2.28 Schematic diagram of the realized three-dimensional microfluidic mixer with
embedded permalloy parts [100-101].
Fig. 2.29 shows the experimental results where according to the authors, a 70%
(static field) and 95% (AC field) mixing efficiency over a channel length as small
as the channel width (200 µm) and at velocity of 5×10-3
m/s was obtained.
However, the most important drawback of the mentioned system is its complexity
from the fabrication point of view.
Figure 2.29 Experimental results [101].

2.5. Discussion
Each of the investigated mixers has its own specific advantages and drawbacks
and there is not any particular type as the best general candidate for the mixing
process in micro-scale. Therefore, one must decide on an appropriate mixer type
considering various parameters such as desired functionality, fabrication costs,
disposability, and operating conditions.
2.5.1. Fabrication
Generally speaking, passive micro-mixers are more preferable as no external
source is required to drive these devices. Integrating actuation mechanisms such
as heaters, micro-conductors, power generators and controllers to provide the
required external energy in active mixers, calls for employment of sophisticated
fabrication techniques, which in turn adds an extra cost to the manufacturing
process. This may be a challenging issue particularly for disposable devices.
However, there are some exceptions in passive mixers where fabrication of micro-
channels with three-dimensional configurations such as Tesla structure, staggered
herringbone parts and obstacles is as complex as active mixers. Perhaps, most
convenient mixers from fabrication point of view are passive mixers, which rely
on lamination techniques and no complex structure or component is required to
operate them.
2.5.2. Performance
Performance of the micro-mixer can be a crucial factor in determining the proper
type of mixing mechanism for a particular application. Extent of the mixing of
micro-particles in bio-fluid, for instance, has a significant effect on the quality of
whole magnetic isolation process. Therefore, a mixing technique with sufficient
capability must be adopted for this protocol. Efficiency may also be interpreted as
the mixing time or the space required (e.g., channel length) to achieve the full
extent of the mixing as in most of the integrated systems, a considerable effort is

dedicated to minimizing these factors. In fact, one often needs to reach a
compromise between different parameters regarded as the efficiency of the mixer.
Moreover, controllability of the mixer must be factored in. While active mixers
can be activated on-demand (switch on/ff), in a passive mixer there is not any
chance to operate the device in particular ranges of time or space.
2.5.3. Application
Micro-mixers are widely used in chemical, biological and medical analysis
applications where one deals with variety of fluidic environments. Each type of
fluids has its own intrinsic properties such as viscosity, density, electrical
properties, etc. Therefore, based on the working fluid, a proper type of the mixing
technique must be adopted as some of the mixers are designed to work with
particular liquids. For instance, in most active mixers where the driving force is
electrokinetic, the possibilities for two mixing phases are limited; MHD mixers
work solely with electrolyte solutions, in EHD mixers two fluids are expected to
have distinct different electrical properties such as conductivity and permittivity,
electroosmotic mixers are highly dependent upon pH and the concentration of the
different ion species in the solution, and finally in dielectrophoretic and
magnetophoretic mixers, presence of some polarisable elements in mixing phases
is essential.
On the other hand, another major limiting factor for mixing phases must be taken
into account for almost all passive mixers, which rely on lamination methods; if a
particle laden fluid is passed through narrow channels the probability of clogging
is very high. Moreover, in those mixers where embedded conductors are utilized
to supply necessary electric or magnetic field for actuation, heat generation can be
a challenging issue for buffers sensitive to high temperatures. The same problem
is observed in acoustic micro-mixers.

In addition to the type of mixing liquids, operating conditions such as pressure
and bulk fluid velocity (Reynolds number) may be a crucial parameter in choosing
the suitable micro-mixing mechanism. For instance, as discussed earlier, a passive
micro-mixer with inserted obstacles which relies on the chaotic advection is not
an appropriate candidate for mixing of flows with low velocities.
Having considered the properties of buffer containing magnetic particles and the
presence of particles themselves, in this research it was decided to employ
magnetophoretic forces to perform the mixing as the same type of force is used
for separation stage. Besides this view to ultimately integrate the mixer to the
magnetic isolation chip as its particular application, it was intended to propose a
mixer with flexibility of the mixing approach regarding the choice of liquids.
Magnetic particles can be loaded into most fluids and be utilized as a label for
actuation. After the mixing, particles can be easily separated in downstream.

Chapter 3
Magnetophoresis
3.1. Introduction
Migration of magnetic particles in a fluid due to an inhomogeneous magnetic field
is known as magnetophoresis, which is the magnetic analogue of
dielectrophoresis. Magnetophoresis finds its application in separation and mixing
processes where some entity of interest is either magnetic itself or attached to, for
example, a magnetic bead. Since most materials show negligible para or
diamagnetism, it is usually necessary to introduce a magnetic ‘handle’ or ‘label’
such as the mentioned magnetic beads. In the following sections, general
principles of the magnetophoresis will be addressed.
3.2. Magnetic field and magnetic materials
In order to investigate the behaviour of magnetic particles under the influence of
magnetic forces, it is essential to know the theory of magnetic field and magnetic
materials, which is the origin of desired forces for manipulation of particles. This
section reviews the magnetic field theory in brief. Further details can be found in
one of many textbooks on magnetism (e.g. [1-3]).
A magnetic field intensity H is produced whenever there is electrical charge in
motion (e.g., an electrical current flowing in a conductor). Magnetic field intensity
is measured in the unit of amperes per meter [A/m]. In free space, magnetic flux
density B in tesla [T] is a linear function of H and we can write:

Chapter 3 – Magnetophoresis 52
0= μB H (3.1)
where constant µ0 (=4π×10-7
[H/m]) is the permeability of free space. In other
media B is no longer a linear function of H. Nevertheless, they are still related by
the permeability of the medium which is not necessarily a constant.
When a specimen is placed in a magnetic field arising from an external source,
there is a field inside the specimen and its atoms or molecules are magnetized.
The field B is made up of two contributions. One is the original field B0 present
when the specimen was absent (here it is assumed that the external field will
remain unaffected by magnetization of the specimen) and other contribution is the
field Bm owing to the magnetization of the specimen. The total field B is the sum
of the fields from two sources:
00 mB = B + B = (H+ M)μ (3.2)
Magnetization M is defined as the magnetic moment per unit volume and m is the
magnetic moment on a volume V of the material:
V
m
M = (3.3)
All materials are magnetic to some extent, with their response depending on their
atomic structure and temperature [4]. They may be conveniently classified in
terms of their volumetric magnetic susceptibility, χ, where
= χM H (3.4)
describes the magnetization induced in a material by H. By combining equations
(3.2) and (3.4) we can write:

0 0 r o rμ χ μ (1+ χ μ μ μ 1+ χ= + = = =B (H H) )H H , (3.5)
where µr is relative permeability and is related to susceptibility of the material.
The difference between B and B0 in equation (3.2) is dependent on the magnetized
material. Most of the magnetic materials can be classified into four categories
which are discussed hereafter.
3.2.1. Diamagnetic Materials
When a magnetic field is applied to a piece of material the orbital motion of the
electrons in the atoms will be affected, and a very small magnetic moment is
induced, which is opposite in direction and proportional to the applied magnetic
field. This phenomenon is termed diamagnetism, and materials in which this is the
dominant magnetic effect are termed diamagnetic materials. A diamagnetic
material is repelled towards field-free regions, but so weakly that sensitive
apparatus is required to measure the repulsive force. For diamagnetic materials, χ
is negative and falls in the range -10-6
to -10-3
. M is opposite the H, hence B<B0.
3.2.2. Paramagnetic Materials
If the atoms of the material possess a permanent dipole moment due to unpaired
electron spins (the orientation of the dipole moments of individual molecules is
random in the absence of a magnetic field), these dipole moments will tend to
align themselves to an externally applied magnetic field and thus they will
enhance the field. Such materials are called paramagnetic materials. The
molecules also acquire induced magnetic dipole moments, but this diamagnetic
effect is usually smaller than the paramagnetism due to the permanent moments.
A paramagnetic material is pulled into the field. Although the attraction is usually
weak, it can sometimes be strong enough to observe in a simple way. For
paramagnetic materials, χ is positive and falls in the range 10-6
to 10-1
. M is in the

same direction as H, hence B>B0. Figure 3.1 shows the magnetic response of a
typical diamagnetic and paramagnetic material subjected to an external field.
Figure 3.1 Magnetic responses associated with diamagnetic materials (left) and paramagnetic
materials (right) [4].
3.2.3. Ferromagnetic, Ferrimagnetic and Anti-ferromagnetic Materials
In some materials the quantum mechanical exchange energy of the atoms is so
large that they interact with the surrounding atoms. Below a certain temperature
called the Curie temperature, magnetic moments will tend to align to each other
even in the absence of a magnetic field. This also means that these materials can
support a permanent magnetization when no externally applied field is present. A
volume of a ferromagnetic material in which all the atomic moments are aligned
to each other is called a magnetic domain. For an un-magnetized material these
domains will cancel each other and the net magnetization will be zero. When a
magnetic field is applied, the domains that are parallel to the applied field will
grow and the others will shrink thus giving rise to a net magnetization. When the
parallel domains have grown to fill the entire piece of material the material is said
to be saturated, and the material is said to have reached its saturation
magnetization.
However, when the field is removed the parallel domains will not shrink enough
to remove the magnetization completely. There will be a remanent magnetic field
which is known as the remanence BR due to the remanent magnetization MR. If a

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