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  • 1. 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
  • 2. 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.
  • 3. 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.
  • 4. iii To my parents for their love and sacrifices…
  • 5. 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.
  • 6. v Abbreviations 2D Two Dimensional 3D Three Dimensional µ-TAS Micro Total Analysis Systems CE Cell Capturing Efficiency CFD Computational Fluid Dynamics DEP Dielectrophoresis DI Deionized (Water) DM Diamagnetic DNA Deoxyribonucleic Acid EHD Electro-hydrodynamic FM Ferromagnetic LE Lyapunov Exponents LOC Lab-on-a-Chip LPCVDLow Pressure Chemical Vapour Deposition MHD Magneto-hydrodynamic MEMS Micro-Electro-Mechanical Systems N-S Navier-Stokes NdFeB Neodymium-Iron-Boron Magnet PACVDPlasma Assisted Chemical Vapour Deposition PDMS Polydimethylsiloxane PIV Particle Image Velocimetory PM Paramagnetic PTV Particle Tracking Velocimetory PZT Lead Zirconate Titanate Re Reynolds Number RNA Ribonucleic Acid SAR Split-and-Recombine SEM Scanning Electron Microscope
  • 7. vi SHM Staggered Herringbone Mixer SPM Super Paramagnetic St Strouhal Number UV Ultra-Violet
  • 8. 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
  • 9. 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
  • 10. 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
  • 11. 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
  • 12. 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
  • 13. 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
  • 14. 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
  • 15. 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
  • 16. xv References 118 Appendix A - COMSOL Multiphysics Simulation 130 Appendix B - Calculation of the largest Lyapunov exponent 142 Appendix C - PUBLICATIONS 144
  • 17. 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
  • 18. 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.
  • 19. 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.
  • 20. 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
  • 21. 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-
  • 22. 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].
  • 23. 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
  • 24. 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.
  • 25. 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.
  • 26. Chapter 2 Literature Survey Microfluidics deals with the behaviour, precise control and manipulation of micro-litre and nano-litre volumes of fluids. It is a multi-disciplinary field comprising physics, chemistry, engineering and bio-technology, with practical applications to the design of systems in which such small volumes of fluids will be used. Ascribed to the micron dimensions, microfluidics has some special characteristics such as high surface-to-volume ratio, high mass-heat transfer rate, high shear-extension rate, and low Reynolds number. Therefore, in order to understand the behaviour of micromixers, a reasonable knowledge of the theory of microfluidics is necessary. In this chapter a brief introduction to microfluidics and some of the key definitions is presented and, subsequently, a through and comprehensive literature review on micro-mixing and different reported methods is provided. 2.1. Microfluidics - basic concepts and definitions 2.1.1. Newtonian fluid A fluid is called Newtonian when the shear stress induced by the viscosity of the fluid is directly proportional to the strain gradient: du dy τ = μ (2.1) The constant of proportionality µ, is the dynamic viscosity coefficient of the fluid. Water, the fluids of interest in this research, is a Newtonian fluid.
  • 27. Chapter 2 – Literature Survey 11 2.1.2. Flow regime Laminar flow, also known as streamline flow, occurs when a fluid flows in parallel streamlines, with no disturbance between the lines. In fluid dynamics, laminar flow is a flow regime associated with high momentum diffusion, low momentum convection, and velocity and pressure independence from time. On the contrary, turbulence or turbulent flow is a flow regime characterized by chaotic, stochastic property changes. This implies lower momentum diffusion, higher momentum convection, and quick variations of velocity and pressure in time and space. Viscous forces dominate in a laminar flow regime, while inertial forces dominate in a turbulent flow regime. 2.1.3. Reynolds number In fluid mechanics, the Reynolds number is a dimensionless parameter obtained from dimensional analysis. It has an important physical meaning, since it is the ratio of inertial forces (ρv) to viscous forces (µ/L). Reynolds number is used for determining whether a flow will be laminar or turbulent and is defined as: Intertial forces vL vL Re Viscous forces ρ = = = μ ν (2.2) where: • v is the characteristic velocity of the flow • L is the characteristic length of the geometry • ρ is the fluid density • µ is the dynamic (absolute) fluid viscosity • ν is the kinematic viscosity of the fluid, ν=µ/ρ At sufficiently high Reynolds numbers, the flow becomes unstable and a turbulent regime develops. However, at lower Reynolds numbers as that in micro-channels, viscous forces dominate over inertial forces, and flow disturbance quickly gets damped. In fact, in micro-scale turbulent phenomena are practically not
  • 28. 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:
  • 29. 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.
  • 30. 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 2 p u x y ∂ ∂ = μ ∂ ∂ (2.4) The solution to this differential equation can be found with the boundary conditions (the velocity is zero at the channel wall (y=d/2)): = − μ 2 21 dp d u(y) ( y ) 2 dx 4 (2.5) where u is the velocity, µ is the dynamic viscosity of the fluid and p is the pressure. Figure 2.1 illustrates the developed parabolic velocity profile with the highest velocity along central streamline. Figure 2.1 Velocity profile of a steady pressure-driven laminar flow with no-slip boundary conditions developed in a micro-channel. 2.2. Mixing in microfluidics Mixing is a fundamental step in most of the microfluidic systems used in biochemistry analysis where biological processes such as enzyme reactions often engage reactions that require mixing of reactants. Mixing is also essential in LOC
  • 31. 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
  • 32. 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 c dt (2.6) where c is the concentration for a particular fluid molecule type and D is the solute diffusion constant. For steady state diffusion (when the concentration within the diffusion volume does not change with respect to time) the equation (2.6) is reduced to Fick’s first law, which gives the flux of the diffusing species as a function of the change in concentration in space (distance): c J D x ∂ = − ∂ (2.7) where J is the diffusive mass flux per unit of area (area perpendicular to x) and x is the position. D, diffusion coefficient or diffusivity, is defined as: κ = = πμ T driving potential D 6 r resistance (2.8) where • κ is the Boltzmann’s constant (=1.35054×10-23 [J/K]) • T is the absolute temperature of the fluid • r is the molecular radius of the solute • µ is the dynamic viscosity of the fluid Temperature dependency of the diffusion coefficient is associated with this fact that the Brownian motion of the particles is due to the applied forces from small
  • 33. 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 ∝ 2 L D (2.9) where L is the characteristic mixing length (e.g., channel width) and τ is the time of mixing. τ can be up to the order of 105 seconds for particles with 1 μm diameter dispersed in water solution diffusing a distance of 100 μm. Obviously, such a diffusion time is not realistic and microfluidic devices that employ natural diffusion as their sole mixing mechanism will not be able to satisfy the rapid- mixing requirement in bio-chemical analyses. Therefore, an innovative method of mixing is essential to enhance the process. As equation (2.9) suggests, the rate of diffusion is dependent on diffusion coefficient, and the mixing length. Both viscosity and diameter are intrinsic properties of the solution and the chosen species, and thus the only remaining possibility of enhancing diffusion is to increase the contact surface and decrease the diffusion path (see equation (2.7)). 2.2.2. Chaotic advection In addition to diffusion, advection is another important form of mass transfer in flows. Advection is normally parallel to the main flow direction, and is not functional for the transversal mixing process. However, the so-called chaotic advection can enhance the mixing in microfluidic devices significantly. Mixing in these devices generally involves two steps; at first, a heterogeneous mixture of homogeneous domains of the two fluids is created by advection and, subsequently, diffusion between adjacent domains leads to a homogeneous mixture at the molecular level [4]. In the context of micro-mixers, the question arises on how the principle of chaotic advection can be implemented, as macro- scale techniques such as employment of stirrers are not available. Chaotic advection can generally be produced by special geometries and three-dimensional structures in the mixing channel or induced by an external force in passive and
  • 34. 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
  • 35. 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.
  • 36. 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].
  • 37. 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-
  • 38. 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
  • 39. 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].
  • 40. 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].
  • 41. 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].
  • 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
  • 43. 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.
  • 44. 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.
  • 45. 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
  • 46. 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.
  • 47. 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.
  • 48. 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.
  • 49. 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
  • 50. 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
  • 51. 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].
  • 52. 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].
  • 53. 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
  • 54. 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.
  • 55. 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.
  • 56. 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
  • 57. 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.
  • 58. 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.
  • 59. 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].
  • 60. 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
  • 61. 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.
  • 62. 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.
  • 63. 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].
  • 64. 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
  • 65. 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.
  • 66. 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.
  • 67. 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:
  • 68. Chapter 3 – Magnetophoresis 52 0= μB H (3.1) where constant µ0 (=4π×10-7 [H/m]) is the permeability of free space. In other media B is no longer a linear function of H. Nevertheless, they are still related by the permeability of the medium which is not necessarily a constant. When a specimen is placed in a magnetic field arising from an external source, there is a field inside the specimen and its atoms or molecules are magnetized. The field B is made up of two contributions. One is the original field B0 present when the specimen was absent (here it is assumed that the external field will remain unaffected by magnetization of the specimen) and other contribution is the field Bm owing to the magnetization of the specimen. The total field B is the sum of the fields from two sources: 00 mB = B + B = (H+ M)μ (3.2) Magnetization M is defined as the magnetic moment per unit volume and m is the magnetic moment on a volume V of the material: V m M = (3.3) All materials are magnetic to some extent, with their response depending on their atomic structure and temperature [4]. They may be conveniently classified in terms of their volumetric magnetic susceptibility, χ, where = χM H (3.4) describes the magnetization induced in a material by H. By combining equations (3.2) and (3.4) we can write:
  • 69. 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
  • 70. 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
  • 71. 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
  • 72. 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
  • 73. 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].
  • 74. 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].
  • 75. 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
  • 76. 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 (M2║H0) 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.
  • 77. 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 1 r1 ψ H r ∂ = − ∂ and 2 r2 ψ H r ∂ = − ∂ 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:
  • 78. Chapter 3 – Magnetophoresis 62 3 30 1 0 0 2 0 1 0 1 X R H R M 2 2 μ −μ μ = + μ + μ μ + μ (3.11) 01 0 2 0 1 0 1 3 Y H M 2 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 mcos ψ 4 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 R 2 2 eff 0 2 - m ( H M ) μ μ μ = π = π + μ + μ μ + μ (3.14)
  • 79. 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: 2 2 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 3V 2 2 eff 0 0m H H μ −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 1 0 eff 0 2 1 U m 4 R 2 0 0m.B .H H μ −μ = − = −μ = πμ μ + μ (3.17) 3 22 1 0 0 2 1 U 2 R 2 m eff 0 0F (m.B) (m .H ) H μ −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
  • 80. 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 2r 0 r 1 2 R 2 m 0F H μ − = 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.
  • 81. 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
  • 82. 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.
  • 83. 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
  • 84. 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 2r 0 x y r y yx x x y x y 3 r 0 r 1 2 R K H H 2 H HH H K H H i H H j x x y y 1 K 2 R 2 μ − = πμ ∇ = ∇ + μ + ∂ ∂∂ ∂ = + + + ∂ ∂ ∂ ∂ μ − = πμ μ + 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.
  • 85. 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.
  • 86. 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
  • 87. 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 z dx dy dz u (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 0 x 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-
  • 88. 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
  • 89. 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
  • 90. 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).
  • 91. 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
  • 92. Chapter 4 – Basic Design of the Micro-Mixer 76 4.8b and 4.8c show the magnitude of the total magnetic field (H=(Hx 2 +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 1 x 10 -4 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 L (m) MagneticField(A/m) 5 µm 20 µm 2 µm 15 µm 10 µm 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 10 -4 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 L (m) MagneticField(A/m) 15 µm 10 µm 2 µm 20 µm 5 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.
  • 93. 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.
  • 94. Chapter 4 – Basic Design of the Micro-Mixer 78 0 1 2 3 4 5 6 7 8 9 x 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 µm 15 µm 2 µm 20 µm 5 µm 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 10 -4 -8 -6 -4 -2 0 2 4 6 8 x 10 -12 L (m) Fx(N) 10 µm 15 µm 2 µm 20 µm 5 µm 1 2 3 4 5 6 7 8 9 x 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 µm 15 µm 2 µm 20 µm 5 µm 0 1 2 3 4 5 6 7 8 9 10 x 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 µm 15 µm 2 µm 20 µm 5 µm 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 10 -4 -8 -6 -4 -2 0 2 4 6 8 x 10 -12 L (m) Fy(N) 10 µm 15 µm 2 µm 20 µm 5 µm 1 2 3 4 5 6 7 8 9 x 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 µm 20 µm 5 µm 10 µm 15 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
  • 95. 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.
  • 96. 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.
  • 97. 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 1 x 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) 20 25 30 35 40 45 50 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 1 x 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) 20 25 30 35 40 45 50 Figure 4.12 Variation of the force against thickness (W=20 µm).
  • 98. 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 L Q I R t , R A ⋅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/m2 K. 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.
  • 99. 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/m2 K 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
  • 100. 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 ) Temperature(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.
  • 101. 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.
  • 102. 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
  • 103. 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.
  • 104. 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: p ud Re 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
  • 105. 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 m u m F 3 du t ∂ = − 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:
  • 106. Chapter 5 – Numerical Simulations and Results 90 m m F V 3 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 d 18 ρ 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): m p f m f F V V V V 3 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
  • 107. 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: m p p f F r V dt (V ) dt 3 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.
  • 108. 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
  • 109. 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 1 f T 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 L St V = (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
  • 110. 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).
  • 111. 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.
  • 112. Chapter 5 – Numerical Simulations and Results 96 (for the caption see next page)
  • 113. 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.
  • 114. 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
  • 115. 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.
  • 116. 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.
  • 117. 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
  • 118. 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
  • 119. 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
  • 120. 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)1 lim ln t 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
  • 121. 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.
  • 122. 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) LyapunovExponent St=0.2 , V=35 µm/s St=0.2 , V=40 µm/s No 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
  • 123. 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.
  • 124. 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
  • 125. 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
  • 126. 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.
  • 127. 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.
  • 128. 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.
  • 129. 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
  • 130. 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
  • 131. 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.
  • 132. 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
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  • 146. 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:
  • 147. 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.
  • 148. 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
  • 149. 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.
  • 150. 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
  • 151. 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
  • 152. 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
  • 153. 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,
  • 154. 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.
  • 155. 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
  • 156. 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:
  • 157. 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.
  • 158. 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 i 1 n i 1 d (t)1 lim ln n 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
  • 159. 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.
  • 160. 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)