massoud thesis
TRANSCRIPT
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
LPCVD Low Pressure Chemical Vapour Deposition
MHD Magneto-hydrodynamic
MEMS Micro-Electro-Mechanical Systems
N-S Navier-Stokes
NdFeB Neodymium-Iron-Boron Magnet
PACVD Plasma 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.
Chapter 1 – Introduction 3
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.
Chapter 1 – Introduction 4
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
Chapter 1 – Introduction 5
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-
Chapter 1 – Introduction 6
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].
Chapter 1 – Introduction 7
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
Chapter 1 – Introduction 8
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 1 – Introduction 9
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:
dudy
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 vLReViscous 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
Chapter 2 – Literature Survey 12
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:
Chapter 2 – Literature Survey 13
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.
Chapter 2 – Literature Survey 14
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
2p ux 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
221 dp du(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
Chapter 2 – Literature Survey 15
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
Chapter 2 – Literature Survey 16
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 cdt
(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):
cJ Dx∂
= −∂
(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= =
πμT driving potentialD
6 r resis tance (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
Chapter 2 – Literature Survey 17
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 ∝2L
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
Chapter 2 – Literature Survey 18
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
Chapter 2 – Literature Survey 19
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.
Chapter 2 – Literature Survey 20
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].
Chapter 2 – Literature Survey 21
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-
Chapter 2 – Literature Survey 22
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
Chapter 2 – Literature Survey 23
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].
Chapter 2 – Literature Survey 24
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].
Chapter 2 – Literature Survey 25
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].
Chapter 2 – Literature Survey 26
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
Chapter 2 – Literature Survey 27
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.
Chapter 2 – Literature Survey 28
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.
Chapter 2 – Literature Survey 29
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
Chapter 2 – Literature Survey 30
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.
Chapter 2 – Literature Survey 31
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.
Chapter 2 – Literature Survey 32
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.
Chapter 2 – Literature Survey 33
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
Chapter 2 – Literature Survey 34
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
Chapter 2 – Literature Survey 35
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].
Chapter 2 – Literature Survey 36
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].
Chapter 2 – Literature Survey 37
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
Chapter 2 – Literature Survey 38
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.
Chapter 2 – Literature Survey 39
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.
Chapter 2 – Literature Survey 40
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
Chapter 2 – Literature Survey 41
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.
Chapter 2 – Literature Survey 42
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.
Chapter 2 – Literature Survey 43
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].
Chapter 2 – Literature Survey 44
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
Chapter 2 – Literature Survey 45
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.
Chapter 2 – Literature Survey 46
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.
Chapter 2 – Literature Survey 47
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].
Chapter 2 – Literature Survey 48
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
Chapter 2 – Literature Survey 49
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.
Chapter 2 – Literature Survey 50
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:
VmM = (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:
Chapter 3 – Magnetophoresis 53
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
Chapter 3 – Magnetophoresis 54
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
Chapter 3 – Magnetophoresis 55
magnetic field is now applied in the opposite direction it is possible to remove the
magnetization. The H-field necessary to do so is called the coercive field HC. If
the applied field is further increased it is possible to saturate the material in the
other direction. If this field is removed the magnetization will follow a new path
going through the negative remanence. If one traces the magnetic field back and
forth between the positive and negative saturation points, the hysteresis loop for
the magnetic material will be obtained. Figure 3.2 illustrates these concepts.
Figure 3.2 Magnetization (M) versus magnetic field (H) where MS is the saturation magnetization,
MR is the remanence magnetization and HC is the coercivity [5].
For a ferromagnetic material it is obvious that the magnetization is not linearly
dependent on the applied field and thus the susceptibility and relative permeability
now depends on the applied H-field and the history of the magnetic material. The
effect of ferromagnetism is very large compared to diamagnetism and
paramagnetism. Therefore, equation (3.4) does not suffice to predict the behaviour
of the material since its previous state affects the magnetization (see figure 3.2). In
these materials B>>B0.
A local order, somewhat similar to ferromagnetic materials, may also be found,
for example, in certain compounds containing ferromagnetic elements and in
some non-ferromagnetic metals. This order results in adjacent dipole moments
being equal in size and parallel but reversed in direction. These materials are
Chapter 3 – Magnetophoresis 56
called anti-ferromagnetic and have very small magnetism. Anti-ferromagnetic
materials in which adjacent dipoles are of unequal size are called ferrimagnetic
and have characteristic similar to ferromagnetic but more weakly appeared [2].
Figure 3.3 shows a summary of five different basic types of magnetic materials.
Figure 3.3 Schematic depiction of spin arrangements in different types of magnetic materials [6].
3.2.4. Super-paramagnetism and magnetic nano-particles
Super-paramagnetism is a phenomenon by which magnetic materials may exhibit
a behaviour similar to paramagnetism at temperatures below the Curie or the Neel
temperature. The width of a domain wall is a function of the magneto-crystalline
anisotropy, the exchange energy and lattice spacing of the crystal structure [5].
The domain wall is approximately a few hundred angstroms thick. As the particle
size decreases, the number of magnetic domains per particle decreases down to
the limit where it is energetically unfavourable for a domain wall to exist. In the
presence of an applied magnetic field, the spin’s orientation and subsequent
magnetic saturation is achieved with lower field strengths than with the analogous
bulk materials. When the field is decreased, demagnetization is dependent on
coherent rotation of the spins, which results in large coercive forces. The large
Chapter 3 – Magnetophoresis 57
coercive force in single domain particles is due to magneto-crystalline and shape
anisotropies for non-spherical particles [6]. The coercive force is also dependent
on particle size as shown in figure 3.4. Theoretical prediction for the single
domain size is Ds=14 nm [5].
Figure 3.4 Coercivity as a function of particle size (~diameter) .Dsp is the super-paramagnetic size
and Ds is the single domain particle size. [5].
The magnetic anisotropy, which keeps a particle magnetized in specific direction,
is generally proportional to the volume of a particle. As the size of the particle
decreases, the energy associated with the uniaxial anisotropy decreases until
thermal energy is sufficient to overcome any preferential orientation of the
moment in the particle. A single domain particle that reaches magnetization
equilibrium at experimental temperatures in short times relative to the
measurement time is commonly referred to as super-paramagnetic. Figures 3.5
and 3.6 show domain structures and magnetic response of the particles,
respectively.
Figure 3.5 Domain structures observed in magnetic particles: a) super-paramagnetic; b) single
domain particle; c) multi-domain particle [7].
Chapter 3 – Magnetophoresis 58
Figure 3.6 Magnetic response of ferromagnetic particles where response can be either multi-
domain (- - - - in FM diagram), single-domain (—— in FM diagram) or super-paramagnetic
(SPM), depending on the size of the particle [4].
It is important to recognize that observations of super-paramagnetism are
implicitly dependent not just on temperature, but also on the measurement time τm
of the experimental technique being used (see figure 3.7). If τ<<τm the flipping is
fast relative to the experimental time window and the particles appear to be
paramagnetic (PM); while if τ>>τm the flipping is slow and quasi-static properties
are observed (the so-called ‘blocked’ state of the system). A ‘blocking
temperature’ TB is defined as the mid-point between these two states, where τ=τm.
In typical experiments τm can range from the slow to medium time-scales of 100 s
for DC magnetization and 10-1 to 10-5 s for AC susceptibility [4].
Figure 3.7 Illustration of the concept of super-paramagnetism, where the circles depict three
magnetic particles and the arrows represent the net magnetization direction in those particles. In
case (a), at temperatures well below the blocking temperature TB of the particles or for relaxation
times τ (the time between moment reversals) much longer than the characteristic measurement
time τm, the net moments are quasi-static. In case (b), at temperature well above TB or for τ much
shorter than τm, the moment reversals are so rapid that in zero external field the time-averaged net
moment on the particles is zero [4].
Chapter 3 – Magnetophoresis 59
3.3. Force on a magnetized particle in a magnetic field
In order to investigate the motion of the magnetic particles, it is essential to
evaluate the exerted forces on them. Generally, there are two types of forces
generated due to the motion of a magnetized particle in the fluid advecting
influenced by a magnetic field (i.e., namely hydrodynamic and magnetic forces).
Hydrodynamic forces are induced owing to the motion of any object in the fluid.
Magnetic forces are generated due to the presence of the magnetized object(s) in a
magnetic field. While hydrodynamic forces are dealt with in chapter 5, magnetic
forces are the subject of this section.
When a magnetized particle is placed in a magnetic field, two distinct types of
magnetic forces may be identified in the domain: ‘imposed field’ and ‘mutual
particle’ interactions [8]. Imposed field interactions are formed when a single
particle, or an ensemble of non-interacting particles, is influenced by an externally
imposed field. Here, it is assumed that the particle(s) does not influence the
external field. Mutual particle interactions occur where particles are so closely
spaced that the local field of a particle influences its neighbours. This interaction
comes about in two ways; the field due to the magnetization of one particle
induces an additional magnetic moment in the neighbouring particle and also
gives rise to the in-homogeneity of the field at the position of that particle. Both
effects can exert an extra magnetic force on the subject particle.
However, in particle laden fluids when the concentration of the suspended
particles is low, assuming that the particles are uniformly distributed in the media,
the distance between neighbouring particles is large. Consequently, mutual
particle interactions are too small to be considered in force equilibrium diagram of
the particles. There is not any discrete value for the concentration below which the
particle interactions become negligible as it depends on various parameters such
as the size of the particles themselves. Nevertheless, for concentration equal to or
less than 1015 particles/m3 it is reasonably justified to assume that the only major
magnetic force on the particles is the force exerted by the external field [9]. In this
Chapter 3 – Magnetophoresis 60
research, the concentration of 1015 particles/m3 is considered for the particles in
the buffer in the course of simulations. This concentration can be provided by
most of the micro-particle suppliers. Therefore, the motion of the particles is
treated as if they are moving individually in isolation.
There are two approaches to estimate the induced magnetic force on a single
particle due to the external field: moment-energy [8] and thermodynamic [9]
methods. In this section based on the former approach, first the induced effective
magnetic moment in particles is obtained, and subsequently, the magnetic force
expression is derived.
In the beginning, we consider the dipole identification problem in general where
particle may have permanent magnetization when it is immersed in a linear media.
Then it is possible to modify extracted expressions for specific situations like
when particle is super-paramagnetic (does not include any permanent
magnetization) or when it is considered in vacuum. Let us imagine a
homogeneous sphere with radius R, permeability µ2 and net magnetic polarization
M2 which includes permanent magnetization plus any linear or non-linear
function of H0. The sphere is immersed in a magnetically linear media of
permeability µ1 and subjected to an almost uniform magnetic field H0. It is
assumed that M2 is parallel to H0 (M2H0) where H0 is the externally imposed
magnetic field vector. Figure 3.8 illustrates the sphere and direction of external
magnetic field.
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.
Chapter 3 – Magnetophoresis 61
As there is no electric current flowing in the considered space, the expression
∇×H=0 is valid everywhere and problem can be solved using a scalar potential ψ
where:
ψ= −∇H (3.6)
Assumed solutions for ψ1 and ψ2 outside and inside the sphere, respectively, have
the following forms [8]:
2
1 0ψ (r, ) H r cos Xr cos , r R−θ = − θ+ θ (3.7)
2ψ (r, ) Yr cos , r Rθ = − θ ≺ (3.8)
Here X and Y are constants which will be determined by applying two boundary
conditions. In equation (3.7) first term is the contribution of external magnetic
field and second term is the contribution of the magnetized sphere (dipole) to
magnetic potential outside the sphere. First, the magnetic potential must be
continuous across boundary between the sphere and surrounding media. Hence,
we can write:
1 2ψ (r, ) ψ (r, ) , r R,θ = θ = θ (3.9)
Second, the normal magnetic flux density must be continuous across the particle-
media interface where we can write:
1 r1 0 r2 r2H (H M ) , r Rμ = μ + = (3.10)
where 1r1
ψHr
∂= −
∂ and 2
r2ψHr
∂= −
∂ are the normal components of the magnetic
field in the media and the sphere, respectively. Using equations (3.7)-(3.10) the
constants X and Y are determined:
Chapter 3 – Magnetophoresis 62
3 30 1 00 2
0 1 0 1
X R H R M2 2
μ −μ μ= +μ + μ μ + μ
(3.11)
010 2
0 1 0 1
3Y H M2 2
μμ= +μ + μ μ + μ
(3.12)
Figure 3.9 shows the magnetic flux and magnetic field of a magnetized particle
where the contribution of externally applied field is removed.
Figure 3.9 Magnetic flux and magnetic field for a uniformly magnetized particle.
Here Y is the magnitude of magnetic field H2 inside the sphere. In order to derive
the magnetic force on the sphere (particle) it is appropriate to assume the sphere
as a dipole where it has same effects when placed in location of the sphere. Hence,
next step is to find the magnetic moment of mentioned equal dipole. Magnetic
potential of a dipole can be estimated by [3]:
m 3 2
mr m cosψ4 r 4 r
θ= =
π π (3.13)
By comparing equation (3.13) to second term of equation (3.7), effective magnetic
moment is:
3 0 1 0
0 1 0 1
4 X 4 R2 2eff 0 2
-m ( H M )μ μ μ= π = π +
μ + μ μ + μ (3.14)
Chapter 3 – Magnetophoresis 63
First term is the contribution of the media displaced by sphere, while second term
is owing to any magnetization (which is not necessarily proportional to H0) of the
sphere itself. Now let us consider a situation when a particle is magnetically linear
where we have:
22 0 2 r
0
1 12M H , μ= χ χ = − = μ −
μ (3.15)
where χ2 and µr are susceptibility and relative permeability of the particle,
respectively. Effective magnetic moment in obtained by substituting equation
(3.15) into equation (3.14):
3 2 1 2 1
2 1 2 1
4 R 3V2 2eff 0 0m H Hμ −μ μ −μ
= π =μ + μ μ + μ
(3.16)
where V is volume of the particle. From equation (3.16) it is straightforward to
calculate the energy of the induced dipole and the force on the particle using the
standard formula for energy and forces on point-like dipoles:
3 22 10 eff 0
2 1
U m 4 R20 0m .B .H Hμ −μ
= − = −μ = πμμ + μ
(3.17)
3 22 10 0
2 1
U 2 R2m eff 0 0F (m .B) (m .H ) Hμ −μ
= −∇ = ∇ = μ ∇ = πμ ∇μ + μ
(3.18)
It is worth noting that:
• Force is a function of volume of the particle and intensity of the magnetic
field (through induced magnetic moment).
• Force is proportional to gradient of the magnetic field intensity.
• Force is directed along the gradient of the magnetic field intensity.
• If the particle is paramagnetic (µ2>µ1), it will be attracted towards higher
magnetic field regions (positive magnetophoresis). On the contrary, the
particle is repelled by the field (negative magnetophoresis). However, this
Chapter 3 – Magnetophoresis 64
is unlikely to occur where a medium with a considerable high relative
permeability is required.
Assuming that particle is in vacuum or de-ionized water where µ1=µ0, equation
(3.18) can be simplified to:
3 2r0
r
12 R2m 0F Hμ −
= πμ ∇μ +
(3.19)
Equation (3.19) is the final expression, which will be utilized to estimate the
exerted magnetic forces on the particles. Magnetic force has a major contribution
to the differential equation of motion of the particles, which must be solved in
order to obtain the Lagrangian trajectories of particles in the mixer domain. This
is dealt with in chapter five where the simulation procedure is addressed.
Chapter 4
Basic Design of the Micro-Mixer
4.1. Sources of magnetic field
Generally, there are two sources of magnetic field that can be integrated into
MEMS devices, namely permanents magnets and current-fed conductors.
Permanent magnets can generate strong fields on the order of 0.5-1 T compared to
the field generated by conductors, which is in the range of 10 mT. One advantage
in exploiting permanent magnets over conductors is that the actuation is not
involved in the Joule heating, which is a serious challenge in Bio-MEMS
applications. Various magnetic materials (e.g., NdFeB films) have been reported
by some researchers which can be fabricated in micro-scale. Fabrication
techniques comprise micro-machining of bulk magnets [1-2], screen printing/
bonding/mould injection [3-8], mechanical deformation [9-10], plasma spraying
[11-13], electro-deposition [14-18], sputtering [19-23] and pulsed laser deposition
[24-26]. However, it is not trivial to integrate the fabricated permanent magnet
with other components as it may be incompatible with the overall processing of
the micro-system. For instance, high temperature processes, such as LPCVD (low
pressure chemical vapour deposition) and thermal oxidation, are not compatible
with most magnetic materials.
On the other hand, various approaches have been taken to fabricate electromagnet
for a variety of applications such as magnetic actuators or sensors [27-30]. In
order to compensate for the reduced magnetic field strength, a current with
relatively large magnitude must be fed into the conductors. Therefore, the cross-
sectional area of the conductor has to be large to decrease the power consumption
Chapter 4 – Basic Design of the Micro-Mixer 66
and Joule heating. Common metal deposition techniques, such as sputtering and e-
beam deposition, are not the proper methods to make thick metal layer due to their
low deposition rates. In most cases, electroplating of high-conductive metals, such
as copper and gold, has been used to fabricate thick conductors utilizing
photoresist lithography as a mold [27-28]. In this way, high-aspect ratio (large
cross-sectional area) on the order of 10-100 µm thick can be fabricated.
In addition to the fabrication issue, there is one important advantage in integrating
electromagnets as the magnetic field source. The magnetic field generated by an
electromagnet is dynamic and can be used to provide time-varying and on/off
magnetic fields. This extra functionality is crucial in operation of some MEMS
devices such as mixers and separators.
4.2. Magnetic force due to current carrying conductors
In order to evaluate the magnitude and direction of magnetic forces injected on
micro-particles due to the field generated by a micro-conductor, a simple concept
is investigated as following. Let us assume a micro-channel with one straight
conductor embedded beneath the channel. The concept and dimensions are shown
in figure 4.1 (all dimensions are in microns). A uniform current density of
J=6×108 A/m2 (current of I=240 mA) is injected into the conductor. Direction of
the current is perpendicular to the xy-plane along the conductor. It is assumed that
the conductor in z-direction is unlimited; therefore a 2D model will suffice to
investigate the model.
Figure 4.1 Channel with straight embedded conductor beneath it.
Chapter 4 – Basic Design of the Micro-Mixer 67
Figure 4.2 illustrates the generated magnetic field due to the injected current. The
field inside the channel domain where particles exist is of interest; therefore the
field outside the channel is not shown. Colour-map represents the magnitude of
the field and arrows show its direction. As it can be observed, the field is
relatively stronger near the conductor where the maximum of 1800 A/m is
generated. Moving away from the conductor, the field decays quickly down to
400 A/m in the top corners of the channel.
Figure 4.2 Generated magnetic field inside the channel. Colour-map represents the intensity of the
field in A/m.
Once the magnetic field is obtained, the next step is to calculate the force applied
on the particles. The magnetic particle used in this study is M-280, Dyna-beads
(Dynal, Oslo, Norway) which is commercially available. Magnetic properties of
the reference particles are shown in table 4.1 [31].
Table 4.1 Properties of the reference magnetic particle.
Diameter, d 2.83 µm
Density, ρ 1.4 g/cm3
Relative Permeability, µr 1.76
Mass Sat. Magnetization 10.8 A.m2/kg
Saturation Magnetization 15120 A/m
Before saturation, particles are linearly magnetized with their magnetic moment
magnitude increasing in the direction of the external field. Beyond the saturation
point, magnitude of the moment tends to a constant value. Given the magnetic
Chapter 4 – Basic Design of the Micro-Mixer 68
field due the current in the conductor (see figure 4.2), particle will never reach the
saturation point and will remain in the linear area. Therefore, the equation (3.19)
in previous chapter is valid for the entire domain in the channel. By expanding
this equation in two-dimensional space, the expression for magnetic force is
obtained:
3 2 2 2r0 x y
r
y yx xx y x y
3 r0
r
12 R K H H2
H HH HK H H i H H jx x y y
1K 2 R2
μ −= πμ ∇ = ∇ +
μ +
∂ ∂∂ ∂= + + +
∂ ∂ ∂ ∂μ −
= πμμ +
mF H . ( )
.[ ( . . ) ( . . ) ] (4.1)
For the selected particle, coefficient K is 4.5×10-24. Figure 4.3 shows the magnetic
forces which would be applied on a particle if it was placed in the channel at
various positions in the domain. Colour-map represents the magnitude of
magnetic force in pN and arrows show the direction of the resultant force. As it
can be observed, the particle from any position in the channel will be attracted
towards the high strength magnetic field region close to the conductor. Also, it is
worth noting that moving away from the intense region, the force declines more
quickly than the magnetic field itself. This is due to the fact that the force is a
function of both magnetic field and its gradient and therefore, is influenced by a
double effect. A maximum force of 0.68 pN is obtained at the bottom of the
channel along its central line.
Figure 4.3 Applied magnetic forces on the particles in the channel. Colour-map represents the
magnitude of the force (pN) and arrows show its direction.
Chapter 4 – Basic Design of the Micro-Mixer 69
An appropriate concept must be adopted to use such forces to induce a chaotic
advection in the particles. In order to reach a proper design, a good understanding
of the chaos phenomenon and the mechanism to induce chaotic regimes, is
essential. Therefore, a brief introduction to chaos in given in the following
section, and subsequently, the basis of the micro-mixer design is discussed.
4.3. Chaotic mixing
4.3.1. Chaos theory
Chaos is a technical term for a particular type of irregular motion induced by a
deterministic system. Chaos is a long term phenomenon and therefore when
energy dissipation occurs, for instance in mechanical systems in the presence of
friction, a continuous input of energy is essential to maintain the chaotic response,
otherwise any observed irregularity will be transient. Chaos has now been viewed
in a wide variety of physical systems including mechanical, fluid, electronic,
chemical and even biological experiments [32].
In the context of fluid dynamics, two branches can be identified; one
corresponding to volume-contracting or dissipative systems, and the other to
Hamiltonian systems or volume-preserving systems. While chaos associated with
dissipative systems is called Eulerian or turbulent chaos, the latter is often called
Lagrangian or non-turbulent chaos, which is of interest in this study.
A state of Lagrangian chaos can be identified when the solution of the trajectory
equations has a sensitive dependency on initial conditions, and initially nearby
trajectories diverge exponentially. Since the system is frictionless or Hamiltonian,
the phase space is conservative. The possibility of having Lagrangian (laminar)
chaos without Eulerian (turbulent) chaos implies that mixing enhancement is
possible in the flows with very low Reynolds numbers.
Chapter 4 – Basic Design of the Micro-Mixer 70
Sensitivity to initial conditions means that each point in the system is arbitrarily
closely approximated by other points with considerably different future
trajectories. Therefore, an arbitrarily small disturbance in the current trajectory
may result in significantly different future behaviour. Sensitivity to initial
conditions is usually known as the "butterfly effect". This phrase refers to the idea
that a butterfly's wings might create very small changes in the atmosphere that
eventually cause a tornado to appear (or prevent a tornado from appearing). The
flapping wing represents a small change in the initial condition of the system,
which leads to a chain of events causing large-scale phenomena. Had the butterfly
not flapped its wings, the trajectory of the system might have been extensively
different. Figure 4.4 illustrates the sensitive dependency on initial condition in a
system. As it can be observed, some trajectories are evolved into different
attractors despite their close initial positions. On the contrary, trajectories with
different initial conditions can be evolved into the same attractor.
Figure 4.4 Sensitive dependency on initial conditions, blue squares represent initial state and black
squares represent equilibria.
4.3.2. Chaos in laminar flows
It has recently been generally realized that high Reynolds number turbulent flow
is not necessary for complex particle trajectories in fluid dynamics. Laminar flow,
once thought to have simple dynamics, can give rise to chaotic behaviour of
Lagrangian particle trajectories [33]. Mixing in laminar flows can be enhanced
Chapter 4 – Basic Design of the Micro-Mixer 71
through chaotic advection, the phenomenon in which particles advected by a
periodic velocity field show chaotic trajectories.
Laminar flow can be one-, two-, or three-dimensional, and may be steady (time-
independent) or unsteady (time-dependent) [33]. However, chaos cannot occur in
steady two-dimensional flows, but only in three-dimensional and two-dimensional
time-dependent flows. In a two-dimensional flow, time-dependency may be
considered as an added third dimension. Since a two-dimensional simulation is
carried out in this study (this is discussed in the next section), the kinematics of
Lagrangian chaos in such flows is investigated as follows:
The starting point for the chaotic advection is the Lagrangian representation of the
fluid as:
x y zdx dy dzu (x,y,z, t) , u (x,y,z, t) , u (x,y,z, t)dt dt dt
= = = (4.2)
where ux, uy and uz are the Cartesian components of the velocity field u(ux,uy,uz).
Here we are dealing with laminar flows where the fluid is incompressible (i.e.,
∇·u=0). In the case of a two-dimensional flow, the incompressibility of the flow
implies that:
∂∂
− =∂ ∂
yx uu 0x y
(4.3)
Therefore, there exists an exact differential dψ so that:
dx dy,dt y dt x
∂ψ ∂ψ= =∂ ∂
(4.4)
where ψ is known as stream function and is equivalent to Hamilton’s equations of
motion in classical mechanics. In fact, two-dimensional kinematics of advection
by an incompressible flow is equivalent to the Hamiltonian dynamics of a one-
degree-of-freedom system (ψ=ψ(x,y)). It can be proven that the motion in two-
Chapter 4 – Basic Design of the Micro-Mixer 72
dimensional static flow is always regular and integrable [35-36]. In this case, the
fluid elements (or particles) flow along the streamlines and cannot produce
chaotic regimes. Two-dimensional unsteady incompressible flows, on the other
hand, have the dynamics of one-degree-of-freedom time-dependent Hamiltonians
(ψ=ψ(x,y,t)), which are generically non-integrable. In this case, it is possible for
the system to exhibit chaotic particle trajectories as it is non-integrable. The time-
dependency of ψ can be caused by some simple external variation of the flow
system. The two space dimensions and time together provide the minimum of
three dimensions required for chaos to take place.
Although time-dependency as the third dimension is usually referred to the fluid
velocity components, the same strategy can be utilized for particles’ velocity
components. Solid particles suspended in a fluid, follow the flow streamlines and
therefore, will have the same velocity field. If an oscillating component is added
to their motion through using external forces, chaotic advection can be produced.
Relative to integrable advection, chaotic advection enhances stretching and
folding of material interfaces. This deformation of fluid–fluid boundaries
increases the interfacial area across which diffusion occurs, which typically leads
to significantly more rapid mixing [34]. In the case of mixing of the suspended
particles which here is of interest, this consecutive deformation leads to an
increase in attachment of biological cells onto the particles. Therefore, the key to
effective mixing lies in producing stretching and folding of lines in two-
dimensional (surfaces in three-dimensional) flows which can be equated with
chaos. In rough terms, a necessary condition for chaos is the crossing of
streamlines, which must occur at different times.
4.4. Basic design
Now the question arises on how to employ magnetic forces generated by current-
fed conductors to produce chaotic regimes in motion of the magnetic particles. To
answer this question, let us start with investigating the dielectrophoretic chaotic
Chapter 4 – Basic Design of the Micro-Mixer 73
mixer mentioned in chapter 2 [37]. In this mixer attractive or repulsive forces on
polystyrene particles in aqueous medium are induced by changing the frequency
of the signal. High frequencies (10 MHz) generate a negative DEP force, where
particles are repelled from electrode edges, while at lower frequencies (100 KHz)
positive DEP forces move the particles to the electrode edges, as they correspond
to highest field gradients (see figure 4.5). The combination of the two opposite
motions generates stretching and folding required for chaotic advection.
Figure 4.5 Producing stretching and folding using positive and negative DEP forces generated by
different frequencies [37].
However, this concept cannot be used for magnetophoretic forces due to the major
limitation of these forces, i.e., merely attractive forces can be generated by
magnetophoresis. Therefore, opposite force, which is required to repel the
particles attracted to low fluid velocity field region cannot be generated. Even if
the particles are exposed to an AC magnetic field, they will be attracted towards
the source. This is due to this fact that an alternating magnetic field can only
change the direction of polarity in the particles considering the relaxation time
(time between moment reversals) of the particles. Hence, another strategy must be
adopted to utilize magnetophoretic forces.
In this investigation, a straight channel with two embedded serpentine conductors
beneath the channel is used to produce chaotic patterns in the motion of particles
Chapter 4 – Basic Design of the Micro-Mixer 74
and intensify the capturing of bio-cells. The burst-view of the mixer is depicted in
figure 4.6. Two flows; target cells suspension and the particle laden buffer, are
introduced into the channel and manipulated by pressure-driven flow (see figure
4.7). While the cells follow the mainstream in upper half section of the channel
(they are transported by convection of the suspending bio-fluid), the motion of
magnetic particles is affected by both the surrounding flow field and the localized
time-dependent magnetic field generated by sequential activation of two
serpentine conductors (here we call the advection due to bulk flow field passive
and due to magnetophoretic forces active for the sake of distinction of these
phenomena).
Figure 4.6 Burst-view of the proposed mixer.
Figure 4.7 Top view of the proposed mixer illustrating one mixing unit (all dimensions are in
microns).
Chapter 4 – Basic Design of the Micro-Mixer 75
As discussed in section 4.2, magnetic field declines quickly as we move away
from the conductors. To overcome this problem and inject particles from various
positions across the channel, geometry of the conductors must be adopted in a
way that generates larger forces. Circular form of the tip of each tooth in
conductors provides an intensified magnetic field in the centre where it acts as a
sink for particles. Particles from various positions in the upstream and
downstream sides move across the streamlines and are attracted towards the centre
of nearest activated tip where the maximum magnetic field exists. By using a
proper periodical current density and structural geometry, chaotic patterns can be
produced in the particles’ motion, which leads to their better mixing into the bio-
fluid suspension.
Dimensions are shown in figure 4.7. Channel is 150 µm wide and 50 µm deep.
Conductors are 50 µm high and 25 µm wide in the section and distances between
centre of circular tips of the conductors are 100 µm and 65 µm in x and y
directions, respectively. Each row of upper and lower conductors is connected to
the power supply alternately. Employment of two rows of conductors at both sides
of the channel allows pulling particles from each side, therefore, compensating for
the limitation of magnetic forces not being bi-directional. The mixing operation
cycle consists of two phases. In the first half cycle, one of the conductor arrays in
switched on while the other one is off. In the next half cycle, the status of
conductor arrays is reversed. Each mixing unit consists of two adjacent teeth from
opposite conductor arrays and the mixer is composed of a series of such mixing
units which are connected together. Boundaries of one mixing unit are illustrated
in figure 4.7.
Figure 4.8a shows one mixing unit with its magnetic field generated near the
circular tip of the conductor when a current of 750 mA is injected into one
conductor array and is turned off in the opposite array during a half cycle of
activation. The colour-map represents variations in the magnetic field intensity at
a plane 10 µm above the surface of the conductor where the maximum magnitude
of the field is about 6000 A/m at the centre of the circular tip (point P). Figures
Chapter 4 – Basic Design of the Micro-Mixer 76
4.8b and 4.8c show the magnitude of the total magnetic field (H=(Hx2+Hy
2)1/2) in
x-y plane along two lines A-A and B-B, respectively. Graphs show the field at
different heights above the conductor and as expected, the closer to the conductor,
the stronger the magnetic field can be observed.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x 10
-4
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
L (m)
Mag
netic
Fie
ld (A
/m)
5 µm20 µm2 µm15 µm10 µm
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x 10
-4
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
L (m)
Mag
netic
Fie
ld (A
/m)
15 µm10 µm2 µm20 µm5 µm
(a)
(b)
(c)
Figure 4.8 Magnetic field near the tip of one tooth in one mixing unit during a single phase of
activation, (a) colour-map of the field (A/m), (b) & (c) magnitude of the field at different heights
above the conductor along lines A-A and B-B, respectively.
Chapter 4 – Basic Design of the Micro-Mixer 77
Magnetic forces exerted on one single particle along lines A-A and B-B are
plotted in figure 4.9 showing the x and y components of the force and also the
total resultant force at different heights above the conductor. As expected, the
maximum force (~6 pN) is applied on particles near the conductor and inside the
circle of its tip where the intensity of magnetic field is at its maximum value.
Although the magnetic field is maxima at the centre point P, the force on particles
is relatively small at this point. This is due to the fact that the magnetic force is
proportional to the gradient of the field which is almost constant in the
neighbourhood of the point P (see figures 4.8b and 4.8c). In moving away from
the conductor, the force drops significantly due to a dramatic decrease in the
magnetic field, which in turn affects the magnetic moment.
It is worth noting that the magnetic force is three-dimensional and the z
component of the force is downward, which together with gravity, pull the
particles towards the bottom of the channel and restrict their motion to a two-
dimensional pattern. In fact, this component has no contribution to the chaotic
motion of the particles and is assumed not to be influential on the process of
mixing. Therefore, a two-dimensional simulation is conducted in this study and
planar forces at 10 µm above the surface of the conductors are of interest as closer
layers might not be practical due to fabrication restrictions.
Chapter 4 – Basic Design of the Micro-Mixer 78
0 1 2 3 4 5 6 7 8 9x 10
-5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 x 10-11
L (m)
F (N
)
10 µm15 µm2 µm20 µm5 µm
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x 10
-4
-8
-6
-4
-2
0
2
4
6
8 x 10-12
L (m)
Fx (N
)
10 µm15 µm2 µm20 µm5 µm
1 2 3 4 5 6 7 8 9x 10
-5
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1 x 10-11
L (m)
Fx (N
)
10 µm15 µm2 µm20 µm5 µm
0 1 2 3 4 5 6 7 8 9 10x 10
-5
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1 x 10-11
L (m)
Fy (N
)
10 µm15 µm2 µm20 µm5 µm
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x 10
-4
-8
-6
-4
-2
0
2
4
6
8 x 10-12
L (m)
Fy (N
)
10 µm15 µm2 µm20 µm5 µm
1 2 3 4 5 6 7 8 9x 10
-5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 x 10-11
L (m)
F (N
)
2 µm20 µm5 µm10 µm15 µm
(a) (b)
(c) (d)
(e) (f)
Figure 4.9 Magnetic forces exerted on one single particle along lines A-A and B-B at different
heights above the conductor in one mixing unit during a single phase of activation, (a) & (b) x-
components, (c) & (d) y-components, (e) & (f) resultant force.
4.5. Scaling effects
One way to produce larger magnetic forces is to generate stronger magnetic fields
by increasing the current in the conductors. However, this approach is limited as
Chapter 4 – Basic Design of the Micro-Mixer 79
the maximum current density cannot exceed certain values. Otherwise, problem of
Joule heating arises. Current density J, is a function of current I, and the cross-
sectional area of the conductor A (J=I/A). Therefore, one solution is to use a
conductor with large cross-sectional area which in turn allows increasing the
current while the current density is held in the permissible range. Nevertheless, as
discussed in section 4.1, utilization of thick conductors is restricted by fabrication
techniques. On the other hand, employment of large conductors can diminish the
magnitude field gradient which in turn declines the force at points of interest.
Another solution is to optimize the conductor geometry through modifying its
cross-sectional area. In this way, it is possible to generate intensified fields with
high gradients at points of interest. In this study, conductors with rectangle
sections are adopted and the process of optimizing the cross-sectional area is
carried out through determining the thickness and width of the conductor and their
ratio. The scaling effects are investigated considering two crucial issues, namely
magnitude of the magnetic forces and Joule heating phenomenon.
4.5.1. Magnetic forces
Since the circular tips of the conductors generate the major part of the forces, a
circle is assumed to resmble a single tip of the conductor in order to evaluate the
effect of the conductor size. This assumption allows conducting an axisymmetric
two-dimensional simulation which reduces the consumption of computer
resources and increases the accuracy of the model significantly. The concept is
shown in figure 4.10a. The cross-section of the conductor and channel in plane A-
A is depicted in figure 4.10b illustrating all geometrical parameters. The inner
radius (Ri) of the conductor and distance between the channel and conductor (h)
are 15 µm and 10 µm, respectively. Thickness (H) and width (W) are varied
between 20 µm and 50 µm with 5 µm step. A current of 750 mA is fed into the
conductor and the radial component of the generated force is evaluated along line
B-B which represents the resultant of the x and y components of the force in
Cartesian system in three-dimensional model.
Chapter 4 – Basic Design of the Micro-Mixer 80
Figure 4.10 Conceptual diagram of one single circular tip in (a) three-dimensional and (b) two-
dimensional models.
Figure 4.11 shows the variation of the force against width where the thickness is
20 µm. As it can be observed, increasing the conductor width decreases the
maximum magnitude of the force (here it occurs at about 20 µm from the central
axis). However, the force generated by wider conductors extends more which
means a wider area is influenced by the force. In narrow conductors, the force is
stronger in the vicinity of the centre and diminishes hastily outside the tip.
Chapter 4 – Basic Design of the Micro-Mixer 81
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x 10-4
-2
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0 x 10-12
L (m)
Fr (N
)
20253035404550
Figure 4.11 Variation of the force against width (H=20 µm).
Figure 4.12 shows the variation of the force against the thickness where the width
is 20 µm. The thinner the conductor, the stronger the magnetic force. The
maximum generated force in H=20 µm is decreased down to almost 25% when
the thickness of H=50 µm is used. In fact, thick conductors extend the field along
the y direction below the channel which is not useful for the mixing. However, the
extension of the force in x direction is not affected by the thickness of the
conductor. Therefore, use of sections with small thickness down to the point
where Joule heating is not an issue, is of interest.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x 10-4
-2
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0 x 10-12
L (m)
Fr (N
)
20253035404550
Figure 4.12 Variation of the force against thickness (W=20 µm).
Chapter 4 – Basic Design of the Micro-Mixer 82
4.5.2. Electro-Thermal analysis (Joule heating)
Joule heating (also known as ohmic heating or resistive heating) refers to the
increase in temperature of a conductor as a result of resistance to an electrical
current flowing through it. At an atomic level, Joule heating is the result of
moving electrons colliding with atoms in a conductor, where momentum is
transferred to the atom, increasing its kinetic energy. This phenomenon is
governed by the Joule's law, relating the amount of heat released from an
electrical resistor to its resistance and the charge passed through it:
2 LQ I R t , RA⋅ ρ
= ⋅ ⋅ = (4.5)
where Q is the heat generated by a constant current I flowing through a conductor
of electrical resistance R, for a time t. Terms L, ρ and A are length, electrical
resistivity of the material and cross-sectional area of the conductor, respectively.
Here, the effect of the conductor geometry on Joule heating is through its cross-
sectional size which is inversely related to the electrical resistance. Copper is
considered as the bulk material for the conductors.
In order to evaluate the temperature rise in the mixer, a simplified model of one
conductor tip with surrounding substrate (here is the Glass) which represents half
a mixing unit is considered as shown in figure 4.13. Three cross-sectional area
(highlighted in the figure) are considered; small (25 µm × 25 µm), medium (35
µm × 35 µm) and large (45 µm × 45 µm). A constant current of 120 mA is passed
through the conductor and the generated heat (temperature rise) is investigated
against the size. Thermal boundary conditions for the model are shown in figure
where free convective cooling is considered for the sides exposed to air with a
temperature of Ta=290 K and heat transfer coefficient of h=50 W/m2K. For the
two lateral faces where the other mixing units are attached, thermal insulation is
considered assuming that the same quantity of heat is generated in lateral mixing
units and therefore, no conduction occurs through mentioned boundaries.
Chapter 4 – Basic Design of the Micro-Mixer 83
Figure 4.13 Concept of the model illustrating the thermal boundary conditions. Surrounding area is
air with h=50 W/m2K at 290 K.
Electrical resistivity ρ for copper and glass is the reciprocal of the temperature-
dependent electrical conductivity σ:
σ =ρ + α −0 0
1(T)(1 (T T ))
(4.6)
where ρ0 is the resistivity at the reference temperature and α is the temperature
compensation slope of the material. Electrical and thermal properties and density
for the both materials are given in table 4.2.
Table 4.2 Electrical and thermal properties of the glass and copper.
a Resistivity at reference temperature of Ta=293 K.
Figure 4.14 shows the temperature rise in a conductor with cross-sectional area of
A=35 µm× 35 µm=1225 µm2 where the colour-map represents the temperature
Property / Material Glass Copper
Thermal conductivity K (W/m.K) 1.38 400
Heat capacity Cp (J/kg.K) 703 385
Resistivitya ρ0 (Ωm) 1010 1.72×10-8
Temperature coefficient α (1/K) 0.55×10-6 17×10-6
Density (kg/m3) 2203 8700
Chapter 4 – Basic Design of the Micro-Mixer 84
distribution on the boundaries. The maximum temperature of 300.18 K is deduced
on the copper face while the lowest temperature (300 K) is observed on outside
boundaries of the substrate where the material is directly exposed to convective
cooling of the air. Shown in figure 4.15, a graph of the maximum temperature rise
versus the cross-sectional area of the conductor; the larger the area, the less the
temperature rise. Temperature rise in smallest area (~19 K) is almost twice as that
in the largest area (~8 K). This is due to this fact that as the area increases, the
electrical resistance, and consequently, the generated heat deceases. Therefore
from the heat generation point of view, it is preferable to employ wider and
thicker conductors.
Figure 4.14 Temperature distribution on the boundaries (cross-sectional area A=1225 µm2).
400 600 800 1000 1200 1400 1600 1800 2000 2200
296
298
300
302
304
306
308
310
312
Cross-sectional area (micron2)
Tem
pera
ture
(K)
A=25*25
A=35*35
A=45*45
Figure 4.15 Maximum temperature rises in the domain for different cross-sectional areas. The
current is constant in all models.
Chapter 4 – Basic Design of the Micro-Mixer 85
4.5.3. Conductor size
Based on the results obtained from evaluations of the effect of conductor size on
the magnetic force and temperature rise, it is deduced that a compromise between
both issues must be reached. Therefore, a thickness and width of 35 µm is
considered for preliminary simulations. The adopted width allows applying
required forces on the particle in a relatively large area. Moreover, the moderate
thickness can compensate for the Joule heating owing to the larger current
densities.
Chapter 5
Numerical Simulations and Results
Processes for the separation and mixing of particles often depend on the behaviour
of the particles when they are subjected to the action of a moving fluid and
external forces. Most of the methods for the determination of particles’ path in the
domain involve relative motion between the particles and a fluid. In this chapter, a
brief introduction on the multiphase flows and phase coupling is presented first,
followed by a discussion of the flow of a spherical particle relative to the fluid.
Subsequently, the simulation procedure and parameters are explained and finally,
the results are presented.
5.1. Multiphase flows
A phase refers to the solid, liquid, or vapour state of matter. A multiphase flow is
the flow of a mixture of phases such as gases (bubbles) in a liquid, or liquid
(droplets) in gases, and so on. As in the case of this study, liquid-solid flows
consist of flows in which solid particles are carried by the liquid.
5.1.1. Phase coupling
An important concept in the analysis of multiphase flows is coupling. If the flow
of one phase affects the other while there is no reverse effect, the flow is said to
be one-way-coupled. If there is a mutual effect between the flows of both phases,
then the flow is two-way-coupled. Coupling can take place through mass,
momentum, and energy transfer between phases. Mass coupling is the addition of
Chapter 5 – Numerical Simulations and Results 87
mass through evaporation or the removal of mass from the carrier stream by
condensation. Momentum coupling is the result of an interaction force, such as a
drag force, between the dispersed and continuous phase. Energy coupling occurs
through heat transfer between phases [1]. Here, the momentum coupling which
explains the hydrodynamic interactions is of interest and a schematic diagram of
this type of coupling is shown in Figure 5.1.
Figure 5.1 Schematic diagram of coupling [1].
A further definition of coupling includes four-way coupling in which
hydrodynamic interactions extends beyond the two-phase interactions. Four-way
coupling addresses the situation where, in addition to ‘discrete phase’-‘carrier
phase’ (particle-liquid) interaction, particle-particle collisions also affect the
multiphase motion. However, two-way and four-way couplings effects become
important when particle fraction exceeds certain values which is the situation in
dense flows. Strictly speaking, mutual couplings do not apply to particle laden
flows with a concentration lower than 1015 particles/m3 [2]. Therefore,
considering the mentioned concentration, one-way coupling is assumed to be
valid in this study and as in the case of magnetic interaction which was discussed
in chapter 3, motion of the particles is treated as if they are moving individually.
Chapter 5 – Numerical Simulations and Results 88
5.1.2. Motion of a single particle in a viscous fluid
Consider the motion of a spherical particle inside a viscous fluid which flows
slowly enough for the condition Re<1 to be satisfied. No-slip condition applies on
the surface of the sphere and the centre of coordinates is coincident with the
centre of the sphere. The flow domain is much larger than the diameter of the
sphere (this is referred to sometimes as an “infinite domain”). The multiphase
flow in one-way coupled and there is no magnetic interaction between particles.
Hydrodynamic and magnetic forces that may act on the particle are shown in
figure 5.2 illustrating the free-body force diagram of the particle. Net force acting
on the particle will be used to extract the velocity expression which can yield the
Lagrangian equation of motion.
Figure 5.2 Free-body force diagram of the particle for a one-way-coupled problem.
Term Fm is the magnetic force which was obtained in chapter 3. Term Fd refers to
the parasitic drag force which is the force that resists the movement of the particle
through surrounding fluid. The drag force is generated in parallel to the relative
motion of the particle to the flow and in opposite direction. The magnitude of the
drag force depends on both particle and flow properties. Generally, conditions of
flow relative to a spherical particle are characterised by the particle Reynolds
number:
pudRe ρ
=μ
(5.1)
where
ρ is the density of the fluid
µ is the viscosity of the fluid
d is the diameter of the sphere
u is the velocity of the particle relative to the fluid
Chapter 5 – Numerical Simulations and Results 89
For the case of creeping flow (i.e., flow at very low velocities relative to the
particle), the drag force Fd on the particle was obtained by Stokes [3] who solved
the hydrodynamic equations of motion, the Navier–Stokes equations, to give:
dF 3 du= πμ (5.2)
Drag is made up of friction forces, which act in a direction parallel to the particle's
surface (primarily along its sides, as friction forces at the front and back cancel
themselves out), plus pressure forces, which act in a direction perpendicular to the
particle's surface (primarily at the front and back, as pressure forces at the sides
cancel themselves out). Skin friction constitutes two-thirds of the total drag on the
particle as given by equation (5.2). Therefore, the total force Fd is made up of two
components:
(i) skin friction: 2πμdu
(ii) form drag: πμdu
While several textbooks make this distinction between the two parts of the drag
force, here drag force is considered as a single entity that arises from the
interactions between the fluid and the particle and not two different forces.
Equation (5.2), which is known as Stokes’ law is applicable only at very low
values of the particle Reynolds number where Rep<0.2 which is encountered in
micro-flows [4]. The velocity of the particle due to the magnetic and drag forces
can be described by Newton’s second law:
p mum F 3 dut
∂= − πμ
∂ (5.3)
where mp is the particle mass. At steady state, the two opposing forces are equal
in magnitude and the spheres move at constant velocity. When magnetic force is
exerted, particle accelerates and reaches the terminal velocity Vm as follows:
Chapter 5 – Numerical Simulations and Results 90
mm
FV3 d
=πμ
(5.4)
The particle reaches this velocity in a very short time which is known as particle
relaxation time and is estimated by:
2
p d18ρ
τ =μ
(5.5)
where ρp is the particle density. Given the particle properties (table 4.1) and
viscosity of 0.001 kg/ms (characteristic of water at room temperature), particle
relaxation time is less than 100 ns. During this time, particle moves a minute
distance which can be neglected in tracing the particle position. Hence, ignoring
the acceleration phase, we assume particles react to magnetic forces with no delay
and total velocity of the particle at each moment (Vp) is the sum of velocity due to
fluid field (Vf; passive advection) and velocity due to the magnetic field (Vm;
active advection):
mp f m f
FV V V V3 d
= + = +πμ
(5.6)
Equation (5.6) will be used in the following section to obtain the Lagrangian
trajectories of the particles.
5.2. Numerical simulations
5.2.1. Simulation procedure
A two-dimensional numerical simulation is carried out assuming that particles are
neutrally buoyant and their motion in z direction is either zero or negligible as
discussed in chapter 4. Procedure consists of two steps: first, the steady-state
velocity field of an incompressible Newtonian fluid (water) and time-dependent
magnetic field are computed using commercial multiphysics finite element
package Comsol (COMSOL, UK) and velocities of the particles due to the fluid
Chapter 5 – Numerical Simulations and Results 91
and magnetic fields are extracted (see Appendix A). Then trajectories of particles
are evaluated by integrating the sum of velocities using Euler integration method
and through developed Matlab codes:
mp p f
Fr V dt (V ) dt3 d
= ⋅ = + ⋅πμ∫ ∫ (5.7)
This method of integration is adopted because the equation of motion of the
particles is highly stiff due to quick changes in the magnetic field and,
consequently, in magnetic forces when the signal phase change occurs. Therefore,
some commonly used algorithms, such as Runge-Kutta, for computing the
solution to differential equations method can take a relatively long time to solve
such stiff problems. In order to obtain accurate results, a small discrete time-step
(10 ms) is considered for particle tracking procedure and, where necessary, the
fluid velocity and the magnetic intensity is linearly interpolated between two
adjacent grids. Neutral diffusion (Brownian motion) of magnetic particles of this
size is insignificant. Diffusion rate for such a particle in water would be in the
order of 10-13 m/s. Advection due to magnetic forces is approximately 105-106
times greater than the diffusion fluxes. Therefore, diffusion is neglected. In fact,
particles are small enough not to agitate the flow, but large enough not to get
involved with Brownian motion, moving only with the surrounding flow itself.
Although the trajectories of the particles will suffice to numerically evaluate the
induced chaotic regimes in the mixer, trajectories of the biological cells are also
obtained. These trajectories will be later utilized to quantify the efficiency of the
mixer. Trajectories of the cells are obtained using the same method as for
magnetic particles, with the exception that cells are magnetically inactive and
simply follow the mainstream in the fluid flow field. The size of biological
entities may vary from a few nano-meters (such as proteins) to several micro-
meters (such as cells). In this study, cells are considered to be spheres of 1 μm
diameter.
Chapter 5 – Numerical Simulations and Results 92
For reduction of the computational domain, the smallest possible mixing unit with
periodic boundary conditions must be used. For a periodic mixer like the one
proposed, the flow field solution is also periodic and remains invariant in each
mixing unit. This indicates that every single mixing unit contains all the
information of the flow in the whole mixer system. However, the same cannot be
said for the magnetic field as the generated field by neighbouring teeth affects the
field in each unit. In other words, one mixer unit as shown in figure 5.3 cannot
comprise the field of outer teeth if it is simulated as a stand-alone domain. Hence,
one extended unit which includes adjacent teeth at both sides is used (see figure
5.3 for the boundaries of this extended unit) and field solution is extracted for
inner unit. Other teeth are too far to influence the field in the inner unit and their
effect is not taken into account.
Figure 5.3 Boundaries of one simulated (extended) and one mixing unit.
5.2.2. Simulation parameters
In order to characterize the mixing efficiency and to optimize the design, effects
of a large number of different parameters can be investigated. These parameters
include geometry (e.g., channel dimensions, size of the conductors and spacing
between them), bulk flow rate, particle characteristics, particle concentration, and
magnitude and frequency of the current. Nevertheless, it seems impractical to
consider the effect of variation of all influential parameters simultaneously due to
computational restrictions. Therefore, based on preliminary calculations, a
reasonably optimized geometry for the channel and conductors (as discussed in
the previous chapter) and a current magnitude of 750 mA, are adopted and the
Chapter 5 – Numerical Simulations and Results 93
effect of variation of two driving parameters; namely the bulk flow velocity and
frequency of the current, has been investigated against mixing efficiency. The
frequency is the reciprocal of the period which consists of two half-cycles of
pulsing signal as shown in figure 5.4. T is the length of period and Tp is the length
of a half-cycle or phase.
p
1 1fT 2T
= = (5.8)
Figure 5.4 Phase shift control signal.
The ratio of these driving parameters is defined as a dimensionless number St
(Strouhal number):
f LStV
= (5.9)
where f is the frequency, L is the characteristic length (here, distance between two
adjacent teeth), and V is the bulk mean velocity of the fluid.
5.3. Simulation results
Figure 5.5 illustrates the developed parabolic fluid field velocity inside the
channel for a steady flow with a bulk velocity of 40 µm/s. Colour map represent
the pressure and arrows show the direction and magnitude of the velocity. As it
can be observed, the velocity profile is uniform along the channel length and
Chapter 5 – Numerical Simulations and Results 94
merely pressure is changing which is not of interest in investigating the mixing
process. The pressure drop along a 1 mm long channel is about 3.5×10-3 Pa.
Density and viscosity of the liquid remain almost constant for such range of
pressure variation. Therefore, the premise that each mixing unit contains all
necessary information for the velocity field is valid.
The width of the channel is 150 µm and considering the fluidic properties of water
(ρ=103 kg/m3, μ=10-3 kg/ms), the Reynolds number would be in order of 10-3
emphasizing the flow is absolutely laminar.
Figure 5.5 Developed parabolic fluid field velocity inside the channel. Arrows show the direction
and magnitude of the velocity field.
5.3.1. Advection of the cells and particles
Figure 5.6 illustrates the position of the particles and cells while advecting within
three and a half mixing units. Bio-cells (red dots) and magnetic particles (blue
dots) enter the first mixing unit (across line A-A) from the left in upper and lower
halves of the section, respectively, and with the same concentration. When there is
no magnetic actuation, both cells and particles remain in their initial section and
simply follow the streamlines of the parabolic velocity profile in Poiseuille flow.
In this situation, tagging might occur only in the middle of the channel along the
interface between two halves. All dimensions in the figure are normalized to the
characteristic length (200 µm).
Chapter 5 – Numerical Simulations and Results 95
Figure 5.6 Advection of cells and particles within three and a half mixing units when no external
perturbation is applied. Dimensions are normalized to the characteristic length.
Fig. 5.7 illustrates a typical effect of magnetic actuation (St=0.4, V=40 μm/s)
within the same mixing units at different snapshots. When the external field is
applied, particles travel across the streamlines as well as the interface. Therefore,
they find the opportunity to spread in upper section where they can meet and
collect cells. Magnetically inactive cells will have the same behaviour as previous
situation when no perturbation was applied. As it can be observed, some particles
far from the central line of the channel remain in the lower section as the magnetic
forces in these regions are not strong enough to attract them during the lower
array activated half-cycle. In order to agitate more particles, one solution is to
inject higher magnitudes of the current into the conductors. An alternative
solution can be driving the flow at lower velocities and use of longer activation
periods, whereupon these particles will have the chance to be advected towards
the central region before any phase change occurs. The effect of operating
parameters on the advection of the particles is examined in next chapter.
Chapter 5 – Numerical Simulations and Results 96
(for the caption see next page)
Chapter 5 – Numerical Simulations and Results 97
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). Dimensions are normalized to the characteristic length.
Chapter 5 – Numerical Simulations and Results 98
5.3.2. Basis of chaotic advection in particles
In order to explain the basis for chaotic advection in the proposed micro-mixer,
trajectories of four particles (particles 1-4 in figure 5.8) are considered as typical
trajectories in the mixer. Particles are released in the first mixing unit uniformly
with the spacing of 10 µm when St=0.2 and V=45 µm/s. During the first half-
cycle, first array (conductor I) is on and second array (conductor II) is off. Particle
1 feels a strong magnetic force in y direction and tends to move in this direction
while it is advected by the mainstream in x direction. Note that depending on its
location in the channel which determines both drag force in the Poiseuille flow
and magnetic force, particle 1 can have a positive or negative velocity in x
direction according to equation (5.6). Particle 2 is further from the conductor I and
does not find any chance to be attracted upwards completely during the first half-
cycle. Therefore, two initially nearby particles diverge inducing the mechanism of
stretching, which is marked with a rectangle. In this phase particle 1 is exposed to
the target cells across different streamlines and captures them in case on any
collision.
Figure 5.8 Consecutive stretching and folding in trajectories which results in chaotic advection
(St=0.2, V=45 μm/s).
In the following half cycle, electric current is injected into the conductor II and
turned off in conductor I. In this phase, particle 1 is free to move from the
previous location and is further advected by the mainstream until it approaches a
region of strong enough magnetic force and, consequently, is pulled towards the
centre of conductor II. Particle 2 is subject to a small magnitude of magnetic force
in y direction (see figure 4.9c in chapter 4) but tends to move faster than the
Chapter 5 – Numerical Simulations and Results 99
mainstream by virtue of magnetic force in x direction. In this phase, particle 2
approaches and tags the target cells, if any, along one streamline. Folding is
achieved where two distant trajectories converge and even in some operating
conditions cross each other. Consecutive stretching and folding can be produced
in this way which is the basis of chaos.
Particles 3 and 4 which are too far from the conductor I to be attracted, are
dragged downstream by the fluid and gradually move towards the upper half
section of the channel. After passing a few mixing units, almost all particles
penetrate to cells’ region and fluctuate in a chaotic regime confined to the tips of
two conductor arrays.
Chapter 6
Characterization of the Micro-Mixer
Characterization of the micro-mixers can be regarded as evaluation of several
aspects of the device, including mixing time, mixing length and quality of the
mixing process. The quantification of the extent of mixing (which is the subject of
this chapter) is crucial for evaluation of performance as well as design
optimization of micro-mixers. Despite the numerous recent works on micro-
mixers, characterization of micro-mixers still remains a challenging issue. In the
following, a brief description on different techniques for the characterizing the
mixers is presented, followed by detailed assessment of the proposed concept in
this study.
6.1. Mixing assessment
While the performance of some micro-components such as pumps, or extractors
can be characterized by comparison of standard parameters (e.g. pump frequency,
or volumetric mass-transfer coefficient), a similar protocol for the analysis of
mixing is more complicated. The best solution for mixing characterization would
be a locally in-line measurement of the concentration profiles along the flow axis.
However, sensors of such small size and fast response time are not available at
present [1]. Therefore, the quality of today’s micro-mixers has to be characterized
by indirect means. Here, various techniques are classified into two categories;
namely experimental and numerical approaches. Since this study is a numerical
design study, the main focus will be on the later approach.
Chapter 6 – Characterization of the Micro-Mixer 101
6.1.1. Experimental techniques
Experimental techniques primarily rely on optical inspection of the fluids
transported through the mixer. The most common quantification technique is
using dilution of a tracer dye to determine the extent of mixing (dilution-type
experiments) which is done by the aid of microscopic-, photo- or video- cameras.
Usually, fluorescent dye streams are observed, followed by the evaluation of the
corresponding recorded intensity image. Since the concentration of the dye is
proportional to the intensity of the recorded image, the uniformity of the
concentration image can be quantified by determining the standard deviation of
the pixel intensity values [2]. In cases where the imaging direction is
perpendicular to the fluid layers, the two layers, even at the channel entrance,
appear to be completely mixed. In such cases, an imaging system with a confocal
microscope is required for the three-dimensional spatial distribution of the
concentration field [3-4].
Another quantification method is measuring the fluorescent product of a chemical
reaction (reaction-type experiments). The simplest outcome of a reaction is the
formation of a coloured species and therefore, the intensity of the product is a
direct measure of the extent of mixing. Typically, this process is an acid–base
reaction with a dye having a fluorescence quantum yield that is pH-sensitive [5].
6.1.2. Numerical techniques
Most numerical methods are focused on solving a mass transport equation in order
to visualize and evaluate the mixing performance of micro-mixers. For this
purpose, massless non-interacting virtual particles or tracers are placed in the flow
field and their trajectories are computed using a Lagrangian method. This particle
tracing algorithm can be used to both visualize and evaluate the mixing
performance of the mixers. If the virtual tracers are distinguished by different
Chapter 6 – Characterization of the Micro-Mixer 102
colours, interpreting the results of particle advection can lead to formation of
Poincare maps or used in other methods, which are based on plotting the position
of the tracers at various snapshots or downstream distances revealing the
evolution of the mixing patterns. Poincare sections, which relate information on
the chaotic nature of the flow, are generated by tracking the tracers through the
flow and recording their cross-sectional positions at different downstream
coordinates [5]. The cross-sectional positions are then superimposed to form a
two-dimensional plot. The disposition of the points in the Poincare map can be
used to study the chaotic nature of the flow. Figure 5.7 in the previous chapter,
which illustrated the advection of the particles and cells with two different
assigned colours, falls in this category as a simple example.
Quantifying the performance of the mixer, trajectories of the particles can also be
used to calculate Lyapunov exponents. In this case, the underlying assumption in
the interpretation of the results is that mixing can be achieved efficiently only in
chaotic flows and therefore, this method is appropriate for mixers based on the
chaotic advection. In fact Lyapunov exponents are used primarily to evaluate the
chaotic behaviour of the system and the strength of chaos can be related to the
mixing performance. Computation of the Lyapunov exponents is one of the
employed techniques in this study, which is dealt with in the next section.
An alternative approach is to measure the mixing using statistical analyses of the
concentration samples in the mixture such as information entropy or Shannon
entropy. The Shannon entropy (S) is the rigorous measure of the mixing and it has
been used in many different scientific areas [6]. The Shannon entropy is
determined from statistical properties where: (I) it depends on the probability
distribution p only; (II) the lowest entropy (S = 0) corresponds to one of the p
being 1 and the rest being zero (i.e., perfect order, complete segregation); (III) the
largest value for the entropy is achieved when all ps are equal to each other (i.e.,
complete disorder, perfect mixing). However, in order to obtain more accurate
results, a large number of particles needs to be tracked, which would require an
Chapter 6 – Characterization of the Micro-Mixer 103
extremely long computational time. Figure 6.1 shows a diagram of various
methods including both experimental and numerical methods.
Figure 6.1 Block diagram of the various techniques employed for characterizing the micro-mixers.
6.2. Characterization methods used in this study
In order to quantitatively evaluate the degree of mixing, two criteria are computed
for the investigated range of simulation parameters. A common definition of
mixing quality is based on the inspection of chaotic regimes developed in the
mixer and calculating the Lyapunov exponent is a standard method of
investigating chaos. However, as discussed earlier, Lyapunov exponents are used
primarily to evaluate the chaotic behaviour of the system, but not the mixing
performance directly. Therefore, as a supplemental index, the ability of the system
to capture the target cells is introduced, which can represent the performance of
Chapter 6 – Characterization of the Micro-Mixer 104
the mixer. Since the buffer is intrinsically particle laden and in fact, this is the
mixing of these particles which is of interest, the trajectories of magnetic particles
are used for calculation of characterizing indices.
6.2.1. Lyapunov exponents
As explained in chapter 4, sensitivity to initial conditions is an indication of chaos
related to fluid mixing. In chaotic systems, time evolution of two initially nearby
particles shows exponential divergence. Lyapunov exponent (or Lyapunov
characteristic exponent) is the average exponential rate of divergence or
convergence of initially neighbouring orbits in the phase space and is defined as:
t
d(t)1lim lnt d(0)→∞
λ = (6.1)
where d(t) and d(0) are distance between two orbits at time t and initial time,
respectively. Calculation of the Lyapunov exponent can be used to detect the
incidence of chaos, measure its extent, and investigate the relationships between
various affecting parameters and chaos. For chaotic mixing problems, Lyapunov
exponent reflects the dispersion rate of the fluid particles and in this study, it is
used to quantify the chaotic advection of magnetic particles.
Lyapunov exponents are defined as a spectrum with n components in an n-
dimensional phase space (the rate of divergence or convergence can be different
for different orientations of initial separation vector). However, normally it
suffices to consider its largest component to describe the system because it
determines the predictability of a dynamical system. A positive value is the
signature of chaos, while zero indicates stable properties. A negative value of λ is
an indication of dissipative systems. Whereas the (global) Lyapunov exponent
gives a measure for the total mixing of a system, it is sometimes interesting to
estimate the local behaviour around a specific point in the domain (these values
are usually called local Lyapunov exponents). When λ is positive in part of flow
Chapter 6 – Characterization of the Micro-Mixer 105
regions (labelled as region ‘a’ in figure 6.2), while in other regions it is zero/near
zero (region ‘b’), the mixer is only partially chaotic and the mixing is incomplete.
Figure 6.2 Schematic illustration showing that the λ-map.
Generally the calculation of Lyapunov exponents cannot be carried out
analytically, and in most cases one must resort to numerical techniques. Here
Sprott’s method [7] is used to calculate the largest Lyapunov exponent (hereafter
λl). This method utilizes the general idea of tracking two initially close particles,
and calculates average logarithmic rate of separation of the two particles. Further
description of the Sprott’s method is given in Appendix B.
Figure 6.3 shows the convergence of λl versus time for one particle with two
different driving parameters and also in the absence of the external field. Without
magnetic perturbation, λl convergences to zero indicating a steady flow. At
Strouhal number of St=0.2 and bulk fluid velocity of V=35 μm/s, it convergences
to a constant value of about 0.45 whereas at velocity of 40 μm/s and the same
Strouhal number, this value is about 0.19. Hence, it can be deduced that in the
former operating condition, the system exhibits a stronger chaotic behaviour.
Chapter 6 – Characterization of the Micro-Mixer 106
Close observation of the graphs reveals that there are some points where λl
declines quickly (marked with circles in the figure). Considering the time of these
incidents, it can be inferred that near the end of each phase, some particles enter
the centre of tips and become trapped in these regions yielding a zero value for λl
(for that specific time) and reducing the overall λl. When the driving current is
switched and system is in next half-cycle, particles are free to flow with the
mainstream up to the point where they are re-attracted by nearby active tooth
leading to an increase in λl. Examination of λl for various particles reveals that
regardless of system operating parameters, λl approaches its converged value
generally after a period of 20s of activation.
0 5 10 15 20 25 30 35 40 45 50-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time (s)
Lyap
unov
Exp
onen
t
St=0.2 , V=35 µm/sSt=0.2 , V=40 µm/sNo magnetic forceFrequent re-injection
Figure 6.3 Convergence of the largest Lyapunov exponent for one particle.
6.2.2. Cell capturing efficiency
In this section, a new supplemental index for characterization is introduced, which
can be implemented for this particular concept. Up to now, solely the chaotic
behaviour of magnetic particles has been discussed. Since the ultimate goal of the
system is to improve the attachment of cells to magnetic particles, efficiency of
Chapter 6 – Characterization of the Micro-Mixer 107
capturing of the target cells can be another criterion for characterizing the mixer.
It offers the ability to examine the efficiency of the mixer versus chaotic advection
and also analyze the effect of driving parameters on the cell capturing ratio. This
method uses the idea of monitoring the trajectories of cells and particles to predict
their collision (if any) in the mixer domain.
Both particles and cells with a uniform distribution and the same concentration
(1015 particles/m3) enter the lower and upper halves of the channel, respectively. It
is assumed that collision happens when the distance between the centre of circular
particle and cell becomes smaller than the sum of their radii (as shown in figure
6.4) and then cell is attached to the particle due to chemical binding. Low flow
velocities in the order of 10 µm/s allow the cells to be in the proximity of the
particles long enough for the chemical binding to occur. Since reference particles
are larger than the cells in this study, more than one cell may be attached to a
single particle. In such occasions, once a cell is attached, it will be counted and
then assumed as removed, leaving the particle ready for another collision.
However, after every collision the trajectories of the particles must be re-
calculated using new free-body force diagram. Although the driving force is the
same for the cell/particle complex (magnetic force is applied merely on the
particles), the drag coefficients need to be modified according to the number of
the attached cells. Subsequently, Capturing Efficiency (CE), i.e., ratio of the
captured cells to the total number of entered cells, is calculated after a period of
20s of mixing.
Figure 6.4 Concept of the collision where cell is tagged by the magnetic particle. Rp and Rc are the
radius of the particle and cell, respectively.
Chapter 6 – Characterization of the Micro-Mixer 108
6.2.3. Results and discussion
Both criteria are calculated for a wide range of simulation parameters. For
computation of the largest Lyapunov exponent, 21 particles are uniformly
distributed in upper half of the first mixing unit as the initial positions and λl is
calculated for each individual particle using the method explained in section 6.2.1.
The initial positions of the particles are shown in figure 6.5. The time period is
20s when the particles approach their constant value of λl. In order to quantify the
extent of chaos over the entire domain in the upper section (where cells exist), the
average of λls of 21 particles is taken.
Figure 6.5 Initial positions of the magnetic particle for computation of the largest Lyapunov
exponent.
Figure 6.6a illustrates variation of CE (capturing efficiency) for different driving
parameters (St=0.2-1) where each graph represents the values of CE for a constant
fluid velocity (V=30-50 μm/s). Results for λl calculated over the same range of
driving parameters are shown in figure 6.6b. When no external agitation is
applied, no penetration of particles into the cells’ region occurs and collision
happens merely at the interface between upper and lower halves of the flows.
Application of external forces leads to an increase in both indices. The global
Chapter 6 – Characterization of the Micro-Mixer 109
variations of λl is almost identical for different bulk flow velocities; the maximum
chaos happens at St=0.4, while the minimum occurs at St=0.8. CE exhibits a
similar behaviour at Strouhal numbers less than 0.6, which means that an increase
in chaos leads to an increase in captured cells. Maximum values for λl and CE are
realized at St=0.4, which are 0.36 and 67%, respectively.
Figure 6.6 Variation of characterizing indices versus different system operating conditions: (a) cell
capturing efficiency, (b) largest Lyapunov exponent.
At higher Strouhal numbers (namely 0.8), two indices show different variations.
Although at high bulk flow velocities (larger than 40 μm/s) a good agreement
between two indices can still be observed, in the case of lower velocities they
show contradicting behaviours. At low velocities of the bulk flow, some particles
Chapter 6 – Characterization of the Micro-Mixer 110
are advected until they are attracted to the centre of one tip in the upper conductor.
In the vicinity of the channel wall, flow velocity is much less than the central
region as the parabolic velocity profile in Poiseuille flow is developed in the
channel. Since the magnetic forces are significantly large in the centre of the
conductor, these particles will be retained in this area. Even after the current is
switched to the opposite array, due to low fluid velocity particles will not have the
opportunity to escape from the previous conductor and come close enough to the
opposite conductor. Therefore, in the next period of activation, particles are again
pulled back towards the same region quickly and become trapped again. For such
particles, mixer is partially acting like an asymptotically stable system which
results in a decrease in the Lyapunov exponent of the whole domain.
However, trapped particles play the role of nearly fixed posts which collect
multiple cells when they meet them, thereby increasing the value of CE. Although
the efficiency is relatively high, in practice, it is a challenging issue where trapped
particles can clog the channel. Figure 6.7 illustrates the trajectories of such
particles at Strouhal number of St=0.8 and bulk velocity of V=40 μm/s. Five
particles are released in the first unit and after flowing along three and a half units,
three particles (particles 1-3) are trapped and never exit the channel. Trajectories
of these particles are plotted with dotted lines and locations of traps are marked
with rectangles. In such operating conditions, the mixer is only partially chaotic,
and the mixing is incomplete. However, when the Strouhal number is low, i.e., in
case of longer time periods, particles have the chance to move away from these
attractors, even though the velocity is low.
Figure 6.7 Trajectories of particles at St=0.8 and V=40 µm/s; rectangles indicate the location of
trapped particles.
Chapter 6 – Characterization of the Micro-Mixer 111
Some particles (here particles 4-5) seem to have nearly similar and close
trajectories. There are two possible scenarios. First, two particles may flow along
the same Lagrangian path but at different times. In this case these particles will
capture different cells if any collision happens. The other scenario occurs when
operating conditions force the particles to take almost the same trajectories at the
same time, despite their different initial positions.
Finally, cell capturing ratio is calculated for a mixing time of 30s which improves
the maximum efficiency up to 75%. However, longer mixing times require longer
channels. In fact, mixing length is a function of bulk fluid velocity as well as
mixing time. For instance, 30s of mixing process at bulk velocity of 45 μm/s can
take place through 5.5 mixing units (corresponding to CE=66%) while solely
three units are required for 20s of mixing at bulk velocity of 35 μm/s
(corresponding to CE=57%). Therefore, one needs to reach the best compromise
between efficiency and size of the design.
Chapter 7
Concluding Remarks and Future Work
7.1. Conclusions
In this thesis, a micro-mixing device for magnetic particles is designed and it is
shown that a straight channel with two embedded serpentine conductors beneath
the channel can be utilized to produce the chaotic pattern in the motion of
particles and intensify the capturing of biological cells. Two flows; bio-
cells/molecules suspension and the particle laden buffer, are introduced into the
channel and manipulated by pressure-driven flow. While the cells follow the
mainstream, the motion of magnetic particles is affected by both the surrounding
flow field and the localized time-dependent magnetic field generated by
sequential activation of two serpentine conductors.
Prior to numerical simulations, preliminary modelling is carried out to reach a
reasonably optimized geometrical structure for the conductors. It was found that a
compromise between the magnitude of the applied forces and the amount of
generated heat must be reached in determining the dimensions of the cross-
sectional area in the conductors. Subsequently, a two-dimensional numerical
study of the mixing process is performed in order to characterize the efficiency of
the micro-mixer. Although employed simulation techniques and developed codes
allow the evaluation of the effect of various geometry configurations and particle
characteristics, this study focuses on the effect of two driving parameters (i.e., the
fluid velocity and frequency of magnetic activation) on the mixing quality.
Chapter 7 – Concluding Remarks and Future Work 113
Outline of the simulation procedure is as follows:
Steady-state velocity field of an incompressible Newtonian fluid (water)
and time-dependent magnetic field were computed using commercial
multiphysics finite element package COMSOL.
Passive and active advection of the particles and cells (cells have only
passive advection) were extracted from the model in order to investigate
the motion of them in the mixing domain.
Trajectories of particles were evaluated using developed codes in Matlab.
Two indices which are highly dependent on the performance of the system
were investigated for a wide range of driving parameters (namely the bulk
flow velocity and frequency of the current), thereby characterizing the
mixer. Trajectories of the particles were used in order to:
o Detect chaos in their motion and quantify its extent by calculating
the Lyapunov exponents.
o Examine the capability of the system to capture target bio-cells (as
a supplemental index).
Based on the simulation results, optimum driving parameters were
concluded.
In the present configuration, the stretching and folding mechanism which
consecutively arises along the mixing channel is considered to be a major cause of
the chaotic behaviour. Lyapunov exponent as an index of the chaotic advection is
found to be highly dependent on the Strouhal number where the maximum chaotic
strength is realized in Strouhal numbers close to 0.4, which corresponds to the
Lyapunov exponent of 0.36. It is shown that capturing efficiency 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 can be separated out of the
Chapter 7 – Concluding Remarks and Future Work 114
medium; although this could be further increased at the cost of longer mixing time
and channel length.
7.2. Recommendations for future research
This project was originally initiated with the interest whether the magnetic
particles can be arbitrary manipulated and mixed in a two-dimensional micro-flow
channel. Therefore, the main interest has been focused on the physical phenomena
and behaviour of magnetic particles. However, when one tries to use the present
design in the real applications, there are some issues which might be worth
considering.
7.2.1. Modified particle properties
One serious challenge in the proposed mixer can be the problem of Joule heating
which is directly influenced by the magnitude of the current injection. In order to
diminish the generated heat, polymeric particles with higher concentration of the
magnetic materials as the core can be used (i.e., particles with higher
permeability). This way, larger magnetic moments and subsequently, stronger
magnetic forces may be generated. Therefore keeping the force at the same level,
magnetic field can be minimized which in turn, reduces the required current
density. Moreover, it is an important issue from power consumption point of
view.
7.2.2. Three-dimensional mixing
This study investigated the mixing of the magnetic particles in a two-dimensional
design. In order to increase the output of the device, a three-dimensional mixing
can be a great step forward. However, as explained in the basic design section,
merely attractive forces can be applied through magnetic field and a repulsive
force which is required to move the particles in z-direction is lacking. One
Chapter 7 – Concluding Remarks and Future Work 115
possible option is to embed identical conductors in the top layer (cover layer) in
order to drag particles from top as well as bottom. Nevertheless, it is likely that
magnetic forces generated by the top conductors not to be strong enough to attract
the particles from a large distance at the bottom of the channel. In that case, one
can use soft magnetic materials such as permalloy which can be embedded in the
same way as the conductors. Utilizing a soft magnetic material outside the mixing
domain, it performs as a shield which can protect the field from being lost.
Therefore, magnetic field can be concentrated in the required space with a
significantly increased magnitude. Some basic numerical examinations reveal a 5-
fold increase in the field in case of an added permalloy.
7.2.3. Coupled simulations
As discussed in chapter 5 (simulation procedure), the investigation is based on the
premise that there is no hydrodynamic interactions between particles and
surrounding fluid and the problem is faced with by using one-way coupled
analysis. In fact, in this study the mixing of particles into the bio-fluid has been of
interest. However, if the buffers contain a high concentration of the particles, the
mutual interactions must be taken into account. Use of higher concentrations may
have two advantages. First, increased number of the particles in the same volume
can improve the ratio of the tagged target entities and therefore, enhance the
efficiency of the mixer. Second, in case of two- and four-coupled circumstances
the mixing of the buffers can also be attained. In fact, it would be possible to
utilize the magnetic particles as a mediator element to perform liquid-liquid
mixing whether there is any target cell to be collected or not. It is worth noting
that in some applications such extracting DNA molecules from the whole human
blood, the process of the liquid-liquid mixing is essential. In bench-top protocols,
a lysis buffer suspended with magnetic particles is mixed with the blood. As the
first step, the mixed lysis buffer will lyse the white blood cells and release DNA
molecules. In the following step, released DNA molecules can be tagged with the
magnetic particles.
Chapter 7 – Concluding Remarks and Future Work 116
As mentioned earlier, even if collection of the cells is not the aim of the mixer,
concentrated particles can be used for liquid-liquid mixing solely. Prior to mixing,
they can be loaded in any buffer easily and separated using simple magnetic
forces later in downstream. However, simulation of such coupled problems calls
for implementation of sophisticated mathematical methods and computational
facilities.
7.2.4. Experiments
Needless to say, the best method for the evaluation of the proposed mixer is
fabricating and conducting practical experiments. Standard micro-fabrication
techniques can be utilized to manufacture different layers of the device. The
fabrication involves micro-patterning of the serpentine conductors in thin films
and microfluidic channels, surface treatment to facilitate flow of particle laden
buffers in the micro-channels, packaging of the microfluidic chip and introduction
of fluid inlet/outlet. Conductors can be fabricated using micro-photolithography
and electroplating as other film deposition processes such as sputtering are
employed for deposition of thin films. For insulation of the conductors, PACVD
(Plasma Assisted Chemical Vapour Deposition) can be used to cover the surface
with silicon oxide. The initial materials for the microfluidic chip can be silicon
and glass so that anodic bonding can be employed for chip packaging. However,
the cover layer needs to be made of a transparent material, thereby allowing the
optical experiments to be conducted. Other materials such as plastic and PDMS
may be investigated using soft lithography and hot embossing techniques as
microfluidic chips based on polymer materials are low cost and potential for
disposable devices.
Once fabricated, the device can be evaluated through various techniques discussed
in the previous chapter for a reasonable range of operating parameters. One
interesting experiment may be recording the motion of the particles using PIV
(Particle Image Velocimetory) or PTV (Particle Tracking Velocimetory)
techniques. Particularly in chaotic systems, these experimentally obtained
Chapter 7 – Concluding Remarks and Future Work 117
trajectories of the particles can be used for computing analytical indices such as
Lyapunov exponents and performing visualization methods such as Poincare
maps in order to investigate the behaviour of the system. This way, the validity of
the models can be evaluated by comparing the numerical and experimental results.
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Appendix A
COMSOL Multiphysics Simulation
This appendix explains the COMSOL Multiphysics package for simulations and
how to use it for both electromagnetic and fluid field problems. When creating a
model in COMSOL Multiphysics, the typical modelling steps include:
1. Creating or importing the geometry
2. Meshing the geometry
3. Defining the physics on the domains and at the boundaries
4. Solving the model
5. Post-processing the solution
Among the above list, solving the model is one of the most crucial steps.
Therefore, the issue of solving a model in the COMSOL is explained in the
following (the main part of this appendix is based on the COMSOL 3.2
documentation).
A.1. Solver Overview
The COMSOL Multiphysics Solvers
COMSOL Multiphysics includes a set of solvers for PDE-based problems and
depending on the type of the problem, the proper set must be chosen. Table A.1
summarizes the available types:
Appendix A- COMSOL Multiphysics Simulation 131
Table A.1 Available types of solvers in COMSOL.
SOLVER USAGE
Stationary linear solver
For linear or linearized stationary PDE problems
Stationary nonlinear solver
For nonlinear stationary PDE problems
Time-dependent solver
For time-dependent PDE problems (linear or nonlinear)
Eigenvalue solver For eigenvalue PDE problems
Parametric linear solver
For linear stationary PDE problems depending on a parameter
Parametric nonlinear solver
For nonlinear stationary PDE problems depending on a parameter
Adaptive solver For stationary (linear or nonlinear) or eigenvalue PDE problems using adaptive mesh refinement
Selecting an Analysis Type
Many application modes suggest a set of analysis types as an application mode
property. The possible analysis types vary with the application area. Typical types
you can expect to see include stationary, eigenfrequency, transient, time-
dependent, and parametric analyses.
You can choose the analysis type when selecting most application modes in the
Model Navigator. To change the analysis type later on, choose Properties from the
Physics menu to open the Application Mode Properties dialog box. The analysis
types often set up various equations in the application mode, and the type also
suggests a default solver. To override this default, go to the Solver Parameters
dialog box.
For multiphysics models, the ruling application mode determines the analysis type
and it also suggests a default solver. When you change the ruling application
mode or its analysis type in the Model Navigator, the program changes certain
solver settings accordingly.
Appendix A- COMSOL Multiphysics Simulation 132
In addition to the analysis type, other application mode properties include the
default element type, and weak constraints. The weak constraint property enables
more-accurate reaction-force computations by introducing an extra equation,
which in turn means extra work for the solver.
Selecting a Solver
The analysis type generally selects an appropriate solver, so normally it is not
necessary to select one yourself. If you do prefer to make a selection, the first
question to ask is whether the problem is stationary or time-dependent. Most real-
world phenomena develop in time, but you might know that a given solution
approaches a stationary value.
A.2. Modelling Electromagnetics
This section explains the application modes in COMSOL Multiphysics for
electromagnetics and how to use them for electromagnetic field simulations.
Fundamentals of Electromagnetics
The problem of electromagnetic analysis is that of solving Maxwell’s equations
subject to certain boundary conditions. Maxwell’s equations are a set of
equations, written in differential or integral form, stating the relationships between
the fundamental electromagnetic quantities. These quantities are:
• The electric field intensity, E
• The electric displacement or electric flux density, D
• The magnetic field intensity, H
• The magnetic flux density, B
• The current density, J
• The electric charge density, ρ
Appendix A- COMSOL Multiphysics Simulation 133
You can formulate the equations in differential or integral form. This discussion
presents them in differential form because it leads to differential equations that the
finite element method can be handle. For general time-varying fields, Maxwell’s
equations are:
The first two equations are also referred to as Maxwell-Ampère’s law and
Faraday’s law, respectively. The last two are forms of Gauss’ law in the electric
and magnetic form, respectively. Another fundamental relationship is the equation
of continuity:
Out of these five equations only three are independent. The first two combined
with either the electric form of Gauss’ law or the equation of continuity form an
independent system.
Constitutive Relationships
To obtain a closed system, you need the constitutive relationships describing the
macroscopic properties of the medium. They are:
where ε0 is the permittivity of vacuum, μ0 is the permeability of vacuum, and σ is
the electrical conductivity. In the SI system the permeability of a vacuum is
4π×10-7 H/m.
Appendix A- COMSOL Multiphysics Simulation 134
Potentials
Under certain circumstances it can be helpful to formulate a problem in terms of
the electric scalar potential V and magnetic vector potential A. They are given by
the equalities:
which are direct consequences of the magnetic case of Gauss’ law and Faraday’s
law, respectively.
Material Properties
This discussion has so far only formally introduced the constitutive relationships.
These seemingly simple relationships can be quite complicated at times. In fact,
these relationships require some special considerations when working with four
main groups of materials:
Inhomogeneous materials
Anisotropic materials
Nonlinear materials
A material can belong to one or more of these groups. Inhomogeneous materials
are the least complicated. An inhomogeneous medium is one in which the
constitutive parameters vary with the space coordinates so that different field
properties prevail at different parts of the material structure.
For anisotropic materials the field relationships at any point differ for different
directions of propagation. This means that a 3x3 tensor is necessary to properly
define the constitutive relationships. If this tensor is symmetric, the material is
often referred to as reciprocal. In these cases you can rotate the coordinate system
Appendix A- COMSOL Multiphysics Simulation 135
such that a diagonal matrix results. If two of the diagonal entries are equal, the
material is uni-axially anisotropic; if none of the elements have the same value,
the material is bi-axially anisotropic.
In some nonlinear materials the permittivity or permeability depend on the
intensity of the electromagnetic field. Nonlinearity also includes hysteresis effects
where not only the existing field intensities influence a material’s physical
properties but the history of the field distribution also plays a role.
Boundary and Interface Conditions
To get a full description of an electromagnetic problem, you must also specify
boundary conditions at material interfaces and physical boundaries. At interfaces
between two media, you can mathematically express the boundary conditions as:
where and denote the surface charge density and surface current density,
respectively, and n2 is the outward normal from medium 2. Of these four
equations, only two are independent. This is an overdetermined system of
equations, so you would like to reduce it. First select either equation one or
equation four. The select either equation two or equation three. Together these
selections form a set of two independent conditions. From these relationships, you
can derive the interface condition for the current density,
Electromagnetic Quantities
The table below shows the symbol and SI unit for most of the physical quantities
that appear in the Electromagnetics Module. Although COMSOL Multiphysics
Appendix A- COMSOL Multiphysics Simulation 136
supports other unit systems, the equations in the Electromagnetics Module are
written for SI units.
Table A.2 Electromagnetic Quantities in COMSOL.
QUANTITY SYMBOL UNIT ABBREVIATION
Angular frequency ω radian/second rad/s
Attenuation constant α meter-1 m-1
Capacitance C farad F
Charge q coulomb C
Charge density (surface) ρs coulomb/meter2 C/m2
Charge density (volume) ρ coulomb/meter3 C/m3
Current I ampere A
Current density (surface) Js ampere/meter A/m
Current density (volume) J ampere/meter2 A/m2
Electric displacement D coulomb/meter2 C/m2
Electric field E volt/meter V/m
Electric potential V volt V
Electric susceptibility χe (dimensionless) -
Electrical conductivity σ siemens/meter S/m
Energy density W joule/meter3 J/m3
Force F newton N
Frequency ν hertz Hz
Impedance Z, η ohm Ω Inductance L henry H
Magnetic field H ampere/meter A/m
Magnetic flux F weber Wb
Magnetic flux density B tesla T
Magnetic potential (scalar) Vm ampere A
Magnetic potential (vector) A weber/meter Wb/m
Magnetic susceptibility χm (dimensionless) - Magnetization M ampere/meter A/m
Permeability μ henry/meter H/m
Permittivity ε farad/meter F/m
Polarization P coulomb/meter2 C/m2
Poynting vector S watt/meter2 W/m2
Appendix A- COMSOL Multiphysics Simulation 137
Propagation constant β radian/meter rad/m
Reactance X ohm Ω
Relative permeability μr (dimensionless) -
Relative permittivity εr (dimensionless) -
Resistance R ohm Ω Resistive loss Q watt/meter3 W/m3
Torque T newton-meter N·m
Velocity v meter/second m/s
Wavelength λ meter m
Wave number k radian/meter rad/m
A.3. Fluid Mechanics
This section explains how to use the Incompressible Navier-Stokes application
mode for the modeling and simulation of fluid mechanics and fluid statics.
Navier-Stokes Application Mode
When studying liquid flows, it is often safe to assume that the material’s density is
constant or almost constant. You then have an incompressible fluid flow. Using
the Incompressible Navier-Stokes application mode you can solve transient and
steady-state models of incompressible fluid dynamics.
Variables and Space Dimension
The Incompressible Navier-Stokes application mode solves for the pressure p and
the velocity vector components. It is available for 2D, 2D axisymmetric, and 3D
geometries.
PDE Formulation and Equations
Use the Incompressible Navier-Stokes application mode to model incompressible
flow in fluids. The Navier-Stokes equations for fluid flow,
Appendix A- COMSOL Multiphysics Simulation 138
are deduced for incompressible Newtonian flow. However, for both Cartesian and
axisymmetric coordinates, COMSOL Multiphysics uses a generalized version of
the Navier-Stokes equations to allow for variable viscosity (non-Newtonian
fluids). Starting with the momentum balance in terms of stresses, the generalized
equations in terms of transport properties and velocity gradients are:
The first equation is the momentum balance, and the second is the equation of
continuity for incompressible fluids. The following variables and parameters
appear in the equations:
• η is the dynamic viscosity.
• ρ is the density.
• u is the velocity field.
• p is the pressure.
• F is a volume force field such as gravity.
These application modes are general enough to account for all types of
incompressible flow. In practice, though, successful analysis of turbulent flows
requires simplifications of the description of transport of momentum.
Subdomain Settings
The subdomain quantities are listed in table A.3.
Appendix A- COMSOL Multiphysics Simulation 139
Table A.3 Subdomain quantities.
PARAMETER VARIABLE DESCRIPTION
ρ rho Density
η eta Dynamic viscosity
F F Volume force
Density
This material property specifies the fluid density.
Dynamic Viscosity
This term describes the relationship between the shear stresses in a fluid and the
shear rate. Intuitively, water and air have a low viscosity, and substances often
described as thick, such as oil, have a higher viscosity. You can describe a non-
Newtonian fluid by defining a shear-rate dependent viscosity.
Boundary Conditions
The boundary conditions for the Incompressible Navier-Stokes application mode
are:
Table A.4 Boundary conditions, Navier-Stokes equations..
BOUNDARY CONDITION DESCRIPTION
u = u0 = (u0, v0, w0) Inflow/Outflow velocity
T = -p0n Outflow/Pressure (total stress tensor)
p = p0, K = 0 Outflow/Pressure (viscous stress tensor)
Slip/Symmetry, 2D
Slip/Symmetry, 3D
u = 0 No slip
Appendix A- COMSOL Multiphysics Simulation 140
Normal flow/Pressure, 2D (total stress tensor)
Normal flow/Pressure, 2D (viscous stress tensor)
Normal flow/Pressure, 3D (total stress tensor)
Normal flow/Pressure, 3D (viscous stress tensor)
T = 0 Neutral (total stress tensor)
K = 0 Neutral (viscous stress tensor)
u = 0, K = 0 Axial symmetry
Inflow/Outflow velocity
At an inflow or outflow boundary you can specify the fluid’s velocity field as
in 3D; in the 2D case, drop the last component.
Outflow or Pressure
Using the total stress tensor form, this boundary condition states that the total
force on the boundary is a pressure force defined by p0:
Using the viscous stress tensor form, the pressure is set to p0, and the viscous
force is zero:
Appendix A- COMSOL Multiphysics Simulation 141
This means that the implementation of the outflow condition for the total stress
sensor form uses a Neumann boundary condition, which provides a better-posed
problem than the Dirichlet boundary condition in the viscous stress tensor form.
Appendix B
Calculation of the largest Lyapunov exponent
Sprott’s method utilizes the general idea of tracking two initially close particles,
and calculates average logarithmic rate of separation of the two particles. The
numerical procedure is shown in figure.
Schematic illustration for calculating the largest Lyapunov exponent.
For any arbitrary particle, a virtual particle is considered with a minute distance of
d(0) from the chosen particle and trajectories of these particles are tracked. At the
end of each time-step, the new distance, d(t), between real and virtual particles
and also the value of ln⎪d(t)/d(0)⎪ are calculated. The virtual particle is then
placed at distance d(0) along its connecting line to the real particle (procedure
known as Gram-Schmit Reorthonormalization, GSR). After repeating this process
for several time-steps, λl will be converged and is evaluated by:
→∞ =λ =
Δ ∑n
i1
n i 1
d (t)1lim lnn t d(0)
where Δt is the duration of one time-step and n is the number of steps. By using
this algorithm, the orientation of the orbit is kept along the direction of maximum
Appendix A- Calculation of the largest Lyapunov exponent 143
expansion and therefore, the largest Lyapunov exponent is achieved. An
appropriate choice of d(0) is one that is about 1000 times larger than the precision
of the floating point numbers that are being used. Therefore, a value of 0.01 μm
will suffice for initial distance. This algorithm, however, is shown to be robust to
any choice of d(0) and the frequency of normalization.
Appendix C
PUBLICATIONS
1. M. Zolgharni, S. M. Azimi, M. R. Bahmanyar, W. Balachandran
A numerical design study of chaotic mixing of magnetic particles in a microfluidic bio-separator, Journal of Microfluidics and Nanofluidics (2006) in-press
2. M. Zolgharni, S. M. Azimi, H. Ayers, W. Balachandran
Labelling of Biological Cells with Magnetic Particles in a Chaotic Microfluidic Mixer, 2nd Annual IEEE Int. Conf. on Nano/Micro Engineered and Molecular Systems (IEEE-NEMS), Bangkok, Thailand, January 16-19 (2007) in-press
3. M. Zolgharni, B. J. Jones, R. Bulpett, A. W. Anson, J. Franks
Tribological behaviour and surface properties of diamond-like carbon for efficiency improvements of coated drill bits, International Journal of Machine Tools and Manufacture (2006) submitted
4. M. Zolgharni, S. M. Azimi, M. R. Bahmanyar, W. Balachandran
A Microfluidic Mixer for Chaotic Mixing of Magnetic Particles, 10th Annual NSTI Nanotechnology Conference, Santa Clara, California, May 20-24 (2007) accepted
5. S. M. Azimi, M. R. Bahmanyar, M. Zolgharni, W. Balachandran
Numerical Investigation of a Magnetic Sensor for DNA Hybridization Detection Using Planar Transformers, Electronics Letters (IEE) (2006) submitted
6. S. M. Azimi, M. R. Bahmanyar, M. Zolgharni, W. Balachandran
An Inductance-based Sensor for DNA Hybridization Detection, 2nd Annual IEEE Int. Conf. on Nano/Micro Engineered and Molecular Systems (IEEE-NEMS), Bangkok, Thailand, January 16-19 (2007) in-press
7. S. M. Azimi, M. R. Bahmanyar, M. Zolgharni, W. Balachandran
Using Spiral Inductors for Detecting Hybridization of DNAs Labeled with Magnetic Beads, 10th Annual NSTI Nanotechnology Conference, Santa Clara, California, May 20-24 (2007) accepted
PATENT
M. Zolgharni, W. Balachandran “ Microfluidic device for extraction of biological cells” U.S Patent Disclosure, Brunel University, Jan. 2007. (Patent Pending)