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A three-dimensional (3D) defocusing-based particle tracking method and applications to inertial focusing… Winer, Michael Hubert 2014

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 ! A three-dimensional (3D) defocusing-based particle tracking method and  applications to inertial focusing in microfluidic devices   by  Michael Hubert Winer    B.A.Sc., The University of Waterloo, 2012    A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF  THE REQUIREMENTS FOR THE DEGREE OF    MASTER OF APPLIED SCIENCE  in  The Faculty of Graduate and Postdoctoral Studies  (Biomedical Engineering)    THE UNIVERSITY OF BRITISH COLUMBIA  (Vancouver)    August 2014    © Michael Hubert Winer, 2014    ! ii!Abstract   Three-dimensional analysis of particles in flows within microfluidic devices is a necessary technique in the majority of current microfluidics research. One method that allows for accurate determination of particle positions in channels is defocusing-based optical detection. This thesis investigates the use of the defocusing method for particles ranging in size from 2-18 µm without the use of a three-hole aperture. Using a calibration-based analysis motivated by previous work, we were able to relate the particle position in space to its apparent size in an image.  This defocusing method was then employed in several studies in order to validate its effectiveness in a wide range of particle/flow profiles. An initial study of gravitational effects on particles in low Reynolds number flows was conducted, showing that the method is accurate for particles with sizes equal to or greater than approximately 2 µm. We also found that the resolution of particle position accuracy was within 1 µm of expected theoretical results. Further studies were conducted in inertial focusing conditions, where viscous drag and inertial lift forces balance to create unique particle focusing positions in straight channels. Steady-state inertial studies in both rectangular and cylindrical channel geometries showed focusing of particles to positions similar to previous work, further verifying the defocusing method. A new regime of inertial focusing, coined transient flow, was also investigated with the use of the 3D defocusing method. This study established new regimes of particle focusing due to the effects of a transient flow on inertial forces. Within the transient study, the effects of fluid and particle density on particle focusing positions were also investigated. Finally, we provide recommendations for future work on the defocusing method and transient flows, including potential applications.!    ! iii!Preface    The work described in this document was conducted in the laboratories of Dr.’s Karen Cheung, Boris Stoeber and Konrad Walus. The majority of the work (Sections 4.1, 4.2, 4.3.1, and 4.3.1.1) has been published. 1. Winer M.H., Ahmadi A., Cheung K.C. (2014) Application of a three-dimensional (3D) particle tracking method to microfluidic particle focusing. Lab on a Chip, 14, 1443, DOI: 10.1039/c3lc51352a. 2. Winer M.H., Ahmadi A., Cheung K.C. (2014) Apparent size correlation: a simple method to determine vertical positions of particles using conventional microscopy. IEEE MEMS 2014, San Francisco, USA. 3. Winer M.H., Ahmadi A., Cheung K.C. (2014) Transient inertial flows: a new degree of freedom for particle focusing in microfluidic channels. IEEE MEMS 2014, San Francisco, USA. 4. Additional work (Section 4.3.1.2) included in this thesis will be presented at the MicroTAS 2014 Conference [Winer M.H., Ahmadi A., Cheung K.C. (2014) Effects of density difference between particles and fluid on inertial focusing positions in transient micro-flows. MicroTAS 2014, San Antonio, USA].  I completed all experimental work and wrote the majority of the manuscript. Dr. Ali Ahmadi provided invaluable contributions for the theoretical portions of the work. Drs. Ali Ahmadi and Karen Cheung edited the manuscript. Drs. Ahmadi and Cheung along with Jonas Flueckiger, Samantha Grist, Eugene Huang, Anna Lee and Linfen Yu aided in experimental execution. ! iv!Table of Contents  Abstract ........................................................................................................................................... ii Preface ............................................................................................................................................ iii Table of Contents ........................................................................................................................... iv List of Tables ................................................................................................................................ vii List of Figures .............................................................................................................................. viii Acknowledgements ......................................................................................................................... x Chapter 1: Introduction ................................................................................................................... 1 1.1 Particle Tracking & Positioning ......................................................................................... 5 1.1.1 Application Areas ........................................................................................................ 5 1.1.2 Major Challenges ......................................................................................................... 7 1.2 Particle Focusing ................................................................................................................ 8 1.2.1 Current Techniques ...................................................................................................... 9 1.2.2 Applications of particle focusing ............................................................................... 11 Chapter 2: Flow Focusing & Particle Tracking Theory ................................................................ 13 2.1 Flow Focusing .................................................................................................................. 13 2.1.1 Laminar and Turbulent Flows .................................................................................... 14 2.1.2 Sheath Flow Focusing ................................................................................................ 16 2.1.3 Sheathless Flow Focusing .......................................................................................... 17 2.1.4 Summary of Effects of Parameters on Inertial Focusing Positions ........................... 21 2.1.5 Steady-State and Transient Flows .............................................................................. 23 2.2 Particle Tracking .............................................................................................................. 24 2.2.1 Defocusing Principles & Approximation of Particle Diameter ................................. 26 2.2.2 Tracking Algorithms .................................................................................................. 28 Chapter 3: General Experimental Design ..................................................................................... 32 3.1 Suspension Formulations .................................................................................................... 32 3.1.1 Suspension Types ....................................................................................................... 33 3.1.2 Viscosity and Density Verification ............................................................................ 34 3.2 Optical Setup .................................................................................................................... 35 3.2.1 Camera Options ......................................................................................................... 36  ! v!3.3 Microfluidic Device Fabrication ...................................................................................... 37 3.3.1 Rectangular Channels ................................................................................................ 38 3.3.2 Cylindrical Channels .................................................................................................. 39 Chapter 4: Experimental Results & Discussion ............................................................................ 43 4.1 3D Defocusing Particle Tracking Method ........................................................................ 43 4.1.1 Calibration .................................................................................................................. 43 4.1.1.1 Calibration Assumptions ..................................................................................... 46 4.1.1.2 Experimental Error and Uncertainty for Calibration Curve ................................ 48 4.1.2 Image Acquisition ...................................................................................................... 49 4.1.3 Image Post-Processing ............................................................................................... 51 4.1.4 Particle Tracking Algorithm ...................................................................................... 53 4.1.4.1 Consideration of Peak-locking Effects in Tracking Algorithm ........................... 54 4.1.5 Application Example: Effects of Gravitational Forces on Low Reynolds Number Flows… ................................................................................................................................. 55 4.2 Steady-State Inertial Focusing Study ............................................................................... 61 4.2.1 Rectangular Channel Study ........................................................................................ 61 4.2.1.1 Discussion of Results .......................................................................................... 65 4.2.2 Cylindrical Channel Study ......................................................................................... 66 4.2.2.1 Discussion of Results .......................................................................................... 66 4.3 Transient Inertial Focusing Study .................................................................................... 68 4.3.1 Single Cycle Transient Study ..................................................................................... 68 4.3.1.1 Single Cycle Results ............................................................................................ 70 4.3.1.2 Effects of Density Difference with Transient Inertial Focusing ......................... 74 4.3.2 Multi Cycle Transient Study ...................................................................................... 77 4.3.2.1 Multi Cycle Results ............................................................................................. 79 Chapter 5: Conclusions & Future Work ....................................................................................... 83 5.1 Results Summary .............................................................................................................. 83 5.2 Future Work ..................................................................................................................... 85 5.2.1 Refinement of Tracking Method ................................................................................ 85 5.2.2 Evaluation of Transient Inertial Flows ...................................................................... 87 5.2.3 Applications ............................................................................................................... 88 Bibliography ................................................................................................................................. 90 Appendix A: DOF Calculation ................................................................................................... 104  ! vi!Appendix B: 3D-printed Cylindrical Channel Study .................................................................. 104 Appendix C: MATLAB Code for 3D Defocusing Particle Tracking Method ............................ 106 Appendix D: Arduino Code for Multi-Cycle Study ................................................................... 119 Appendix E: List of Variables .................................................................................................... 123    ! vii!List of Tables   Table 2.1. Summary of parameter effects on inertial focusing positions [92, 100]. ..................... 21 Table 2.2. Available particle tracking tools [37].. ........................................................................ 31 Table 3.1. Experimentally measured values for density and viscosity of particle solutions. ....... 35 Table 3.2. Experimental particle and solution density differences. .............................................. 35        ! viii!List of Figures  Figure 1.1. Typical PTV setup. ....................................................................................................... 6 Figure 1.2. µ-PTV example. ........................................................................................................... 7 Figure 2.1. Illustration of laminar and turbulent flow profiles within a microfluidic channel. .... 15 Figure 2.2. Schematic of symmetric hydrodynamic flow focusing (top view of a microfluidic channel). ........................................................................................................................................ 16 Figure 2.3. Lift force schematic in microfluidic channel for steady-state inertial flow. .............. 18 Figure 2.4. Particle focusing distributions in microfluidic channels ............................................ 20 Figure 2.5. Illustration of the defocusing principle.. ..................................................................... 27 Figure 3.1. Rectangular channel fabrication process. ................................................................... 39 Figure 3.2. PDMS cylindrical channel cross-section showing misalignment of two halves.. ...... 40 Figure 3.3. Illustration of cylindrical channel fabrication process. .............................................. 42 Figure 4.1. Experimental Schematic for calibration experiments ................................................. 45 Figure 4.2. General example of defocusing principle as typically seen during calibration. ......... 46 Figure 4.3. Calibration curves for various experimental designs.. ............................................... 48 Figure 4.4. Example of calibration of applied pressure to flow rate. ............................................ 51 Figure 4.5. Visualization of image post-processing algorithm. .................................................... 53 Figure 4.6. Histogram plots for peak-locking effects analysis. .................................................... 55 Figure 4.7. Illustration of the double-parabolic velocity profile. .................................................. 58 Figure 4.8. Cubic relationship between change in x-position and change in z-position of a particle in the flow due to settling. ................................................................................................ 59 Figure 4.9. Theoretical and experimental settling study results. .................................................. 60  ! ix!Figure 4.10. Conjectured and experimental focusing trends for particles in the rectangular channel study. ............................................................................................................................... 63 Figure 4.11. Comparison of results between previous work [5] and steady-state study results. .. 64 Figure 4.12. Steady-state cylindrical study experimental data PDF contour plots. ...................... 67 Figure 4.13. Experimental schematic. ........................................................................................... 70 Figure 4.14. Relative velocity of fluid (as tracked using 2 µm beads) and of 15.5 µm beads vs. time following the time at which the outlet pressure was matched via the Fluigent to the inlet pressure. ........................................................................................................................................ 71 Figure 4.15. Transient study results.. ............................................................................................ 73 Figure 4.16. Particle Distribution Histogram Illustration. ............................................................ 75 Figure 4.17. Particle Distribution Peak Width and Position Plots, comparing effects of density mismatch due to variation in particle density. .............................................................................. 76 Figure 4.18. Particle Distribution Peak Width and Position Plots, comparing effects of density mismatch due to variation in fluid density. ................................................................................... 76 Figure 4.19. Diagram of multi-cycle study setup. ........................................................................ 79 Figure 4.20. Reynolds number profile for Remax = 87.02 ............................................................. 80 Figure 4.21. Rectangular channel transient multi-cycle results .................................................... 81 Figure 4.22. Rectangular channel histogram for multi-cycle transient particle focusing. ............ 82 Figure B-1. 400x400um channel. ................................................................................................ 105 Figure B-2. Height/Radius measurements for several cylindrical channel samples. .................. 105 Figure B-3. Optical profilometry screen-capture for 250 µm diameter channel ........................ 106    ! x!Acknowledgements     I would like to thank Dr. Karen Cheung for her advice and support throughout my time at the University of British Columbia. Her expertise and quick, creative approaches to solving a variety of problems has been both humbling and extremely educational. My time spent with the other members of her research group has been just as valuable, and I would like to extend thanks to all of them: Jonas, Sam, Nomin, Eric, Daljeet, Solmaz, Soroush, Selim, Daniel, Riley, Cynthia, Anna, Eugene, and Bernard. All of your advice and friendship taught me a lot about myself and provided new perspectives on how to work and learn with others to reach a greater goal than any individual could.  I would like to also extend my sincerest appreciation to Dr. Ali Ahmadi, who acted as a mentor and guide throughout my research. Both his helpful and generous demeanour and his deep knowledge of theoretical fluid mechanics were extremely important to the success of the work described here. I wish him the greatest of success in his future endeavours. Similarly, I would like to thank Dr. Linfen Yu for her daily help with laboratory work, setups, and general advice for experimental design. As an aside, it was great fun teaching her son Kevin piano for a year. I would also like to thank Dr. Boris Stoeber and Dr. Dana Grecov for serving on my thesis committee.    Finally, I would like to thank my family, especially my parents Ute and Philip Winer, as well as my brothers Marc and Derek and their families. I love all of you and your guidance is my deepest inspiration. To the beginning of something new. I don’t want to be smart, because being smart makes you depressed. ! 1!Chapter 1: Introduction   Since the early 1980s, microfluidics has become a dominant analysis technique in a variety of fields. Considering its origins in materials engineering, microelectronics, molecular biology and chemistry, it is no surprise that microfluidics has become a hallmark of the modern technology age of interdisciplinary research and scale-down.  Microfluidics involves the design of systems in which small volumes (typically 10-18 to 10-9 litres) of fluids are handled for control, manipulation and behavioural study. Typically these fluids are moved, mixed, separated or somehow processed in numerous applications, using both passive and active control mechanisms. By taking advantage of micro domain phenomena such as laminar flows, surface tension, diffusion, and electrowetting, new technologies can be developed to reduce energy consumption and allow for substantial scale-down of various laboratory techniques. Perhaps the most revolutionary technologies are related to molecular biology procedures such as enzymatic assays, DNA analysis, and other biochemical studies such as disease detection.  As many microfluidic devices and techniques involve particle analysis, particle tracking methods have become an integral part of this technology. Most research has been conducted in single particle tracking, which involves the observation of the motion of individual particles within the fluid. Trajectories of particles can be determined by collecting data on the physical coordinates in space of each particle over time. Particle tracking velocimetry (PTV) is perhaps the most well-known examples of applying particle tracking to determining fluid profiles by tracking the positions of a high concentration of particles over time in a fluid. Particle tracking can be conducted in numerous ways, but optical detection is the most common. Choices for optical detection generally fall within bright field or fluorescence microscopy categories. Aside  ! 2!from the light source, many image acquisition setups exist, including single-camera and multi-camera approaches.  As a means of particle manipulation within microfluidic systems, particle focusing has garnered a great deal of attention. By controlling the positions of particles within a microfluidic flow, applications such as separation, filtration, sorting, or mixing are achievable. Two main techniques – sheath flow (e.g. hydrodynamic focusing) and sheathless flow (e.g. inertial focusing) – are used in various experimental designs depending on the outcome required. One of the most important applications of flow focusing is flow cytometry. Flow cytometry is a chemical and/or physical technique used in cell counting, cell sorting, or general particle detection and uses optical, electrical, or biochemical methods to determine information about individual cells in a large sample. Many flow cytometers employ microfluidic technologies for advantages such as high throughput by using particle focusing in a continuous flow environment. As a major method for diagnosis of health disorders, most notably blood cancers or other related issues, the potential for microfluidics and particle focusing techniques used in flow cytometry fields is significant. Despite the remarkable achievements of previous work in the areas of particle tracking and analysis, there remain many challenges. This is especially the case in high-throughput, continuous flow applications such as flow cytometry. Due to the high flow rates of these environments, sophisticated optical and/or electrical setups must be utilized to capture images of particles accurately. For example, many previous works require high-magnification lenses, with laser fluorescence sources to illuminate marked particles effectively [1-4]. Simple ways of particle tracking within particle focusing applications is limited. Past research has explored both two-dimensional (2D) and three-dimensional (3D) particle tracking methods. However, 3D  ! 3!methods typically require expensive multiple camera setups, or sophisticated optics designs, while 2D setups limit the resulting data to particle position analysis in two dimensions. As an example, one of the most common 2D techniques in recent years, particle streak velocimetry, takes averaged information of particles across an image and normalizes the results without maintaining the integrity of each particle in the dataset and therefore eliminating any particle trajectory or velocity data [5].  This thesis presents a simple method for tracking particles of sizes ranging in the micrometer scale in three dimensions. The tracking method is based on optical defocusing principles and is calibrated by comparing the apparent size of the particles to their vertical position in space. The application of this 3D technique focuses on inertial focusing phenomena including both steady-state and transient flow conditions. The method was verified using a gravitational force experiment and several steady-state particle focusing experiments, including both rectangular and cylindrical channel geometries. It was then used to determine the effects of a rapidly changing (transient) flow rate on particle focusing positions through the manipulation of the relative velocity between the particles and the fluid flow. Results found using the 3D particle tracking technique indicate that particles can focus in differing equilibrium positions depending on the use of a constant or changing flow rate over time. Transient inertial flow rates were used to determine the effects of a relative velocity between the particle and fluid on particle focusing positions. Both a single-cycle and multi-cycle study was conducted, using rectangular channels. Results indicate that a cyclic transient flow creates new equilibrium focusing positions, especially in high-Reynolds number flows (Re > 80). Lastly, transient flow rates were used to examine the density difference between the solution and the particle as a means of further manipulating the relative velocity between the particles and the flow, with evidence that an  ! 4!increased density difference causes a shift of particle focusing peak positions in microfluidic flows.  This thesis consists of the following five chapters: • Chapter 1: This chapter includes an introduction to microfluidics, particle tracking and positioning research, an exploration of popular particle focusing techniques in microfluidic devices, and the current applications and limitations of particle tracking for focusing experiments. • Chapter 2: This chapter presents a detailed theoretical background of flow focusing, including a comparison of two main focusing types: sheath flows and sheathless flows. Further theoretical explanation of sheathless flow, specifically inertial flows, is presented. A single-particle force analysis within a microfluidic channel is included for both rectangular and cylindrical channels. A discussion of transient flow effects and relative velocity is also included. Finally, the main principles of particle tracking are presented, referencing previous work in this area. • Chapter 3: This chapter describes general experimental setup and design for the majority of the work discussed in this paper. This includes particle suspension preparations, optical setup, camera choices, and microfluidic device fabrication. • Chapter 4: This chapter presents the experimental work. The first section presents the development of the calibration-based 3D particle tracking technique, including image acquisition, image post-processing, and the calibration of the method. Verification of the method is presented via a gravitational force study. Steady-state flow experiments are described as further verification of the tracking technique, including results from rectangular and cylindrical channels. Transient flow experiments are also described,  ! 5!including a density-variance study and a multi-cycle transient study in rectangular channels. • Chapter 5: This chapter summarizes the experimental results and general findings of the thesis and highlights remaining challenges. Recommendations for future work are also included.  1.1 Particle Tracking & Positioning  Particle tracking is an integral part of experimental design for thousands of applications in the modern engineering world. It is known as perhaps the simplest and most powerful method of quantifying flow regimes in a variety of environments. Its power comes in its instantaneous depiction of magnitudes, primarily velocity or vorticity, over large physical areas. Particle tracking has been used in several decades of research in order to quantitatively evaluate flow patterns or regions of interest, especially in liquids [6-12]. In its modern context, tracking typically involves seeding the flow with particles in order to either follow the flow or distinguish differences between fluid and particle characteristics within a system. Traces are determined typically via optical detection. Resulting information is manifested in pictures and includes particle position, velocities, and trajectories of motion. This data was processed manually [13-15] in earlier work but is compiled and analyzed digitally today [16, 17].  1.1.1 Application Areas Particle tracking and position data collection is a fundamental approach to solving numerous fluidic problems. Most work in this area has been conducted using liquids, although some research in gaseous flows supplements this effort [18, 19]. Scalability of particle tracking also  ! 6!allows for flow mapping of systems both large (such as fuel engines [20]) and small (protein membrane dynamics [21]). Within liquid flow applications, both single-phase [22] and multi-phase [23] systems have been analyzed. Micro-scale applications are centered on confining flows to unique physical geometries and building understanding on the mechanical principles which guide the flow. Therefore, an extremely well-known and common technique within particle tracking is PTV [24-29]. Using particles that follow the flow, researchers are able to conduct complex fluid mechanics experiments within micro-scale regions and map magnitude trajectories over time. Figure 1.1 shows a typical PTV setup.   Figure 1.1. Typical PTV setup. As the particles flow through the channel, the optical setup takes several images of the particles moving through space. Each image is time-stamped and then velocity trajectories are built based on the differences in particle position between time t and time t’ [30].   ! 7! In the case of microfluidics, PTV is known as micro-PTV (µ-PTV). An example of a common application for µ-PTV techniques is cellular dynamics [31-35]. Figure 1.2 below shows an example of µ-PTV.  Figure 1.2. µ-PTV example. Stereoscopic velocity field of a T-mixer operating at Re = 120 [36].  1.1.2 Major Challenges Tracking particles consistently and accurately over time presents several major challenges. Tracking can be separated into two main events: detection and linking. Detection involves distinguishing the particles from the background image by some criteria and labeling their location relative to a known physical origin. Of course, manual detection of large numbers of particles over time would be extremely burdensome. Thankfully, modern computational ability typically allows for particle detection within large sample sizes [37]. However, detection of the particles from the images, especially establishing intensity thresholds and subtracting background noise can prove difficult and require sophisticated image processing algorithms.  ! 8!Linking, the second step of tracking, is the temporal aspect of the process, and involves locating the same particle in two or more images with varying positions. This step can be extremely difficult depending on the frequency of images, number of particles per image, and expected trajectories. Several researchers have built positioning algorithms in order to discern the particles from the image backgrounds, as well as differentiate ‘real’ particle traces from false positives or other anomalies [38-43]. However, as noted in previous summary articles, there is no universally accepted algorithm or particle tracking method that is superior to all others across all applications [44]. Typically, one method may be superior to others depending on variables including particle sizes, fluidic geometry constraints, particle concentrations, and experimental setup.    1.2 Particle Focusing Microfluidic devices have become a standard in recent years as tools to manipulate particles. The many advantages of microfluidics, including enhanced efficiency, reduced sample and reagent use, and shorter processing times have already made many common laboratory procedures much easier [45-48]. For many applications, focusing particle positions into a uniform, continuous stream is a necessary step prior to detecting, counting, and manipulation. As an example of its usefulness, focusing some biological particles toward the center of a channel prevents them from being adsorbed by the channel walls [46]. Particles can be focused in two or three dimensions. Typically, two-dimensional focusing pushes particles towards a uniform horizontal plane, away from side walls but still spread vertically. This is usually sufficient for many counting applications, but sorting in the vertical plane can sometimes prove difficult [49-52]. Another substantial issue with two-dimensional focusing is the detection of multiple  ! 9!particles overlapping one another in the imaging plane [53-56]. 3D particle focusing avoids these issues by manipulating particle positions in both horizontal and vertical planes.   1.2.1 Current Techniques The most simplistic form of particle focusing involves constricting the physical space in which a particle can travel to the extent that only a single particle can flow through a channel at any given time. This in essence is a form of three-dimensional focusing, and ensures particles will travel in a single file across the imaging area [57]. However, this method requires low concentrations of particles, and a limited throughput is expected. The majority of particle focusing techniques can fall under two main types: sheath flow and sheathless flow. Sheath flow uses one or more fluids to pinch the particle suspension fluid and therefore focuses particles. Sheathless flows require pressure-driven or electrokinetic systems in which particles are focused by externally applied fields or internally induced flow fields [54, 55].  Sheath Flow Focusing Techniques  Sheath flow focusing is perhaps the most commonly used method in microfluidic devices. A sheath flow is typically a stream of fluid which surrounds the sample fluid which contains the particles or cells. This type of focusing requires the use of either pressure-driven [58-60] or electrically induced [61, 62] particle-free flows to focus the particle-rich fluid. The sample stream is the inner stream, with a concentric flow of sheath fluid. This setup is widely used in commercial flow cytometers to narrow the central flow, creating a single file of particles through hydrodynamic focusing. Multiple sheath flows surrounding the particle flow are used to focus particles in two or three dimensions. Earlier work in this area has been summarized in multiple articles [53, 54, 56, 63]. Most recent work has been diverse in physical channel  ! 10!geometries and sheath flow choices. As an example, Howell et al. [64] proposed a 3D sheath flow method by using grooved microchannels. This method uses chevrons or stripes fabricated into the channel to focus the particles vertically, and accomplishes 3D sheathing using as few as one sheath input. In some applications, particles are required to be focused towards the bottom or top of the channel, rather than the centre. In these cases, unique sheath flows have been investigated [65]. A unique attribute of sheath flow focusing is its independency from particle size. This focusing technique is mostly dependent on the flow-rate ratio between the sheath flows and the particle-rich fluid. The focusing particle stream can also be tuned to a variety of positions easily in the channel geometry. However, the addition of multiple sheath flows sometimes complicates the fabrication of the channels, adding cost.   Sheathless Flow Focusing Techniques  Sheathless methods of focusing typically rely on external or internal forces to manipulate the particle suspension. This has been accomplished with external forces such as acoustic [66], magnetic [67], dielectrophoretic (DEP) [68], and optical [69], or internally by hydrophoretic [70], inertial [71] or dielectrophoretic [72] means. Therefore, sheathless methods are typically sub-categorized into active or passive by the nature of the forces involved. Active force methods include acoustic [73-75], axisymmetric [76], and electrode-based alternating current (AC) DEP [77]. Passive techniques include the common hydrodynamic focusing method [78], hydrophoretic focusing [79], insulator-based DEP focusing [80], and inertial focusing [81-83]. One major advantage of sheathless flow is the avoidance of diluting the sample via introduction of accompanying sheath flows in the channel. The particle tracking method described in this work was used for a variety of inertial focusing experiments, mainly in straight channels of  ! 11!varying geometries. Inertial focusing has become a hallmark for high-throughput particle focusing in recent years [5, 84, 85].  1.2.2 Applications of particle focusing Particle focusing can be used for a wide variety of applications. The specific focusing method is typically tailored to the application required, and many focusing methods have been created specifically for a given application or to overcome certain challenges.  Sheath flows are highly versatile for manipulating particle positions to very specific locations within a channel. For example, the chevron-based technique has been used to focus polystyrene beads and integrated into a microflow cytometer for multiplexed detection of bacteria and toxins that were thought to coat bead surfaces [86]. As another sheath flow example, a micro-weir structure was used to separate polystyrene beads of various sizes in the vertical direction [87]. Finally, vertically confined flows have been used to push particles close to electrodes embedded in microfluidic devices used in detection [88].  In sheathless focusing, applications vary from ultra-rapid particle focusing for flow cytometry [89] to blood cell analysis [90] to chemical treatment of cell [91] to DNA filtration [79] to high-throughput flow cytometry [34].  For either major particle focusing technique, commercial flow cytometry is perhaps the largest application. As flow cytometry is used to count and/or characterize cells and other particles, focusing is usually required to help discern different types of particles from one another. Focusing also helps to keep particles in a specific order, or for filtering particles based on size, morphology, or bio-chemical or electro-chemical characteristics. Finally, many flow cytometers are aided by particle focusing to ensure the position of the particles is aligned with  ! 12!the various tools used to verify their existence or other attributes. An example of this is in impedance flow cytometry, in which particles must be properly aligned with electrodes outside the channel in order to register a resistance signature related to the particles in the flow. Each method of particle focusing has several pros and cons, but in general particle flow focusing opportunities are vast and offer the user the ability to manipulate particles within microfluidic devices.   !   ! 13!Chapter 2: Flow Focusing & Particle Tracking Theory The main parts of this work are related to two major fields of research in microfluidics: flow and particle focusing, and particle tracking. These two areas overlap significantly. In order to develop sophisticated methods of particle filtration, separation, counting, sorting, or mixing, a manipulation of the fluid flow or particle flow trajectories must take place. As a means of verifying the location of particles in a flow, particle tracking is used. This chapter thoroughly describes the background and key principles found in both flow focusing and particle tracking work, including well-developed theory (such as cylindrical channel focusing and PIV analysis) and the newest areas of research (such as transient flows and defocusing-based tracking methods). It serves as the motivation for the experimental work described in chapters 3 and 4.  2.1 Flow Focusing Flow focusing, and focusing of particles within flows, is a well-established area of research. Within microfluidics, the hallmark of flow focusing is the understanding of velocity profiles within the physical region under investigation. Once some observation of these profiles is undertaken, manipulation of the profiles can be done to significantly alter the positions and directions of motion of particles found within the fluid suspension. Many sub-categories of flow focusing therefore exist in order to fully utilize the fluid mechanics and dynamics principles available to create unique or useful focusing positions for particles to be analyzed or otherwise influenced by external forces.  !! ! 14!2.1.1 Laminar and Turbulent Flows The fundamentals of fluid flow dynamics relate to whether the flow direction and trajectory is uniform or random. This is especially true in microfluidics, where small volumes allow for precise control of fluidic movement in space and time. Two distinct regimes of flow exist – laminar and turbulent. In general laminar flow is generally associated with low velocity or high viscosity fluid motion. To the contrary, turbulent flows are associated with higher velocity, or low viscosity fluids that are generally ‘forced’ against physical barriers that result in rapid trajectory changes. These two flow regimes can be characterized and quantified using the Reynolds number. Named after the Irish-born fluid mechanics savant Osborne Reynolds, this dimensionless number relates several physical quantities of fluid motion to one another:  Re = !!!!!!" !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(1) where DH is the hydraulic diameter of the channel (m), Q is the volumetric flow rate (m3/s), A is the channel cross-sectional area (m2), ρ is the density of the fluid (kg/m3), and µ is the dynamic viscosity of the fluid (kg/m·s).  The hydraulic diameter is defined as: !! = ! 2!"! +! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2) where H is the channel height (m) and W is the channel width (m). This Reynolds number is also sometimes referred to as the channel Reynolds number (Rec). Other work reports the Reynolds number as a particle Reynolds number (Rep = Re(a/Dh)2 where a is the particle diameter) [92]. In essence, the dimensionless Reynolds number relates the inertial and viscous forces involved in a flow, and determines if a flow is laminar or turbulent. In general, it is known that if Re < 2300, the flow is laminar, while for higher Re, the flow is turbulent. Figure 2.1 illustrates laminar and turbulent flows within a channel.   ! 15!  Figure 2.1. Illustration of laminar and turbulent flow profiles within a microfluidic channel.  For all experiments discussed in this paper, Re < 120 and therefore laminar flow was expected. This is especially true as channel geometries did not change significantly within each microfluidic device, as described in Chapter 3. For microfluidics experiments, laminar flow is a typically suitable condition that allows for reproducible particle manipulation within the fluid. When pressure-driven laminar flow is present in a closed microfluidic channel, the velocity profile takes on an approximately parabolic shape. This is due to the shear force along the walls of the channel (no-slip condition) that lower the fluid velocity while the streamlines in the centre of the channel (zero shear condition) have a higher fluid velocity (as seen in Figure 2.1 above). By using various types of laminar flow, including multiple parallel flows, fluid mechanics theory can be used to accomplish many different particle focusing tasks. As noted in Chapter 1, two main types of laminar flow particle focusing exist in microfluidics: sheath flow and sheathless flow focusing.    ! 16!2.1.2 Sheath Flow Focusing In sheath flow focusing, multiple laminar flows are used to direct a main particle suspension flow into a specific position within the channel. This concept is most commonly applied in what is known as hydrodynamic focusing. In hydrodynamic focusing, one or more flows are fed via side channels into a main channel in which a pre-existing central particle suspension flow exists. By adjusting the density, viscosity, and flow rate of the side (sheath) flows with respect to the central flow, the central flow can be manipulated into a desired region.  Figure 2.2 illustrates this concept [58]. This particle focusing technique is very commonly used in flow cytometry applications in order to align the suspended cells or particles through the detection region.    Figure 2.2. Schematic of symmetric hydrodynamic flow focusing (top view of a microfluidic channel). Qi refers to the inlet (central) volumetric flow rate, Qs refers to the side flow rates, Qo refers to the outlet flow rate, and wi, ws, wo and wf refer to the inlet, side, outlet, and final central channel and flow sizes respectively. vf refers to the final flow velocity direction of the central flow [58].   ! 17!Of course, there are many other methods of sheath flow focusing, but many rely on similar theoretical background to establish specific central flow profiles manipulated by side flows.   2.1.3 Sheathless Flow Focusing As described in Chapter 1, there are many methods of sheathless flow focusing in microfluidic devices. These include acoustic, DEP, magnetic or hydrophoretic means. Each method has an extensive theoretical background for externally manipulating the flow of particles within a suspension. The sheathless focusing method most closely related to its common sheath flow cousin hydrodynamic focusing is inertial focusing. Inertial migration of particles was first discovered in the early 1960s, and since has become a major field of research [93, 94]. As a passive technique, inertial focusing manipulates particles without any externally applied forces beyond those found in common Poiseuille flow.  Inertial focusing, as is evident in its title, is caused by focusing based on inertial forces found in Reynolds number flows larger than Stokes flows (Re < 1) but found in a relatively low Reynolds number range (Re < 150). Inertial focusing has been most commonly examined in straight channels. In this situation, a balance arises between two major forces: viscous drag (FD), which keeps particles along a specific fluid streamline, and the inertial lift force (FL) that leads to lateral migration across streamlines. In general, there are two well-understood components to the steady-state lift force: a wall-induced lift force (FW) that acts up the velocity gradient and away from the wall towards the channel centre, and a shear-induced lift force (FS) that acts down the velocity gradient and towards the walls of the channel [95, 96]. Equations for these forces are shown [97]:  ! 18! !! = 6!"#!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(3) !! = !!!!"!"#! !!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(4) !! = !!!!!!"#! !!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(5) where Cw and Cs are constants affiliated with each force, ρ and µ are the fluid density and viscosity, Umax and Uf are the maximum fluid velocity and relative velocity between the particle and the fluid traveling through the centre of the particle, a is the particle diameter, and Dh is the hydraulic diameter. It is the net lift force that is responsible for the equilibrium focusing positions of particles seen in inertial focusing experiments. Figure 2.3 illustrates the effect of these lift forces on a single particle positioned in a channel.   Figure 2.3. Lift force schematic in microfluidic channel for steady-state inertial flow. The two lift forces (wall-induced and shear-induced) act together to create equilibrium positions in channel flows. The particle diameter and velocity are a and U, respectively [83].  These forces are the main contributors to determining the final focusing positions of particles in microfluidic channels. However, depending on the specific geometry of the channel  ! 19!walls, another force can become significant. While the lift forces described take into account focusing of particles towards the centre of the channel or towards the walls, it has been noted in previous work that equilibrium of particles near the walls of rectangular channels is asymmetrical [98]. Specifically, when particles reach close to the walls, they focus further into clusters near the midpoints of the walls, rather than the corner regions of the channels [5, 82, 99, 100]. This secondary particle focusing can be attributed to a rotation-induced lift force, FΩ. When particles are far from the walls, this rotation force is minimal, especially in comparison to the shear-induced lift force [98, 101-104]. However, it has been shown that this rotation becomes more significant when shear is high and particles are close to the wall [104]. Scaling of the rotational force has been related to the particle diameter, a, as FΩ ∝ a3 [105], directed towards the midpoint of each side by acting against the parabolic flow velocity gradient found in microfluidic channels. Since particles can migrate towards either midpoint from a corner region due to the shear-dependency of the rotational force [104], the height/width aspect ratio of the channel plays a role in the final equilibrium positions of particles. Therefore, in rectangular/square channels, particles reach one state of positional equilibrium related to the wall-induced and shear-induced lift forces, and then a second equilibrium positions towards the midpoints of the walls related to the rotation-induced lift force [83]. In cylindrical channels, the rotation-induced lift force is negligible compared to the other forces, and therefore particle equilibrium positions are only determined via the balance of the shear-induced and wall-induced lift forces. It is well-documented that particles focused in a cylindrical channel reach a stable annulus at approximately 0.6 times the radius of the channel [93, 106]. Figure 2.4a and 2.4b show the particle equilibrium focusing distributions for cylindrical and rectangular channels, respectively.    ! 20!  Figure 2.4. Particle focusing distributions in microfluidic channels. (a) Cylindrical channel. Note the particle focusing equilibrium at the Segre-Silberberg annulus (0.6R). (b) Rectangular channel. As noted, the particles will first focus towards the walls based on the effects of the wall-induced and shear-induced lift forces (stage I). In stage II, the particles are more influenced by the rotation-induced lift force and push towards the middle of each wall, depending on the height/width aspect ratio of the channel [71, 83].     ! 21!2.1.4 Summary of Effects of Parameters on Inertial Focusing Positions In summary, several parameters affect the inertial focusing positions of particles in microfluidic devices. A recent summary article has provided details for each parameter and how it affects the overall focusing positions [92]. These findings are summarized in Table 2.1 below.  Table 2.1. Summary of parameter effects on inertial focusing positions [92, 100]. Parameter Effects Reynolds number Both the wall and shear lift forces increase with increasing flow velocity and Re, with the shear lift having a greater increase. Therefore, as Re increases particle focusing positions move towards the wall. However, as the particles reach towards the walls, the velocity decreases due to the parabolic gradient, thereby increasing the effect of the wall force and pushing the particles back towards the center of the channel. Particle size (a/Dh) For a/Dh << 1, the positions approach 0.6(Dh/2). As a/Dh approaches 1, equilibrium positions shift towards the channel center. Particle concentration For particle concentrations where the fraction of particle diameters per channel length is > 75%, multiple streams are observed. In these cases, there is a large increase in particle-particle hydrodynamic interactions. As a result, a portion of the particles are pushed out of the focusing streams and form new nearby focusing streams.  ! 22!Parameter Effects Entry Length In general, if the entry length is too short, particles will not reach their final equilibrium positions at the centre of the walls of square/rectangular channels at high Re. In this case, they will only partially, if at all, focus to these secondary positions, remaining in their primary focusing positions towards the corners or in a general annulus shape between the walls and the center of the channel. Particle Density An increase in particle density relative to the fluid density causes the particles to spread wider from their equilibrium positions and in general, towards the walls of the channel. This is due to the response time of the particle relative to the fluid velocity surrounding it. Fluid Density An increase in fluid density relative to particle density has a similar effect as particle density changes, however it is caused by changes in the magnitude of the lift force acting on the particles, since FL∝ ρf. Fluid Viscosity For higher viscosity, shear gradient is decreased due to lower flow rate at a fixed pressure change. Therefore, particles focus closer to the center of the channel.     ! 23!2.1.5 Steady-State and Transient Flows  As previously discussed, manipulation of cells and particles in microfluidic systems using inertial forces has shown enormous potential for numerous applications including cell/particle filtration, separation and flow cytometry [63, 107]. In essence, the inertial focusing of particles is based on the manipulation of particle positions in the directions perpendicular to the main flow direction by lateral forces.  In recent years, numerous system designs have been proposed to achieve higher throughput size-based separation using the inertial focusing method [82, 84, 108, 109]. In many of these studies, the lateral forces on the particle resulted from a steady-state flow in the channels, and depending on the geometry and scale of the system, different equilibrium positions and focusing regimes were achieved.  It is known that these lateral forces strongly depend on the relative velocity of the particle and the main flow [105]. Therefore, by manipulating the relative velocity between the particles and flow, new focusing paradigms can be established [110]. In order to experimentally determine the effects of transient flows on particle focusing, it is important to understand the fundamental theories surrounding this phenomenon.  The equation of motion of spheres moving in a transient flow in the direction of motion is [105]: ! !!!!" = !! + !!! + !!" + !!"                                               (6) where !! and ! are the particle velocity and mass respectively, !! is the quasi-steady-state drag force, !!! is the lift force due to the fluid shear and particle rotation (also known as the slip-shear lift force), !!" and !!" !are the added mass and stress gradient forces, respectively. More specifically, !!" accounts for the work required to change the momentum of the surrounding  ! 24!fluid as the particles accelerate, and !!"  is the fluid stress gradient force which accounts for forces that would exist in the absence of the particles due to the hydrostatic pressure gradient. The equilibrium positions are obtained when the lateral forces balance each other, and lateral velocity of the particles become zero. The slip-shear lift force contains a dominant relative velocity term: !!! = !! !!!!!! !"# ∙ !!"#!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(7)           where !!, !!, !! and !!"# are the fluid density, lift coefficient, particle diameter and the relative velocity, respectively. According to equation 7, by changing the flow velocity as a function of time, the balance of forces in equation 6 changes, and therefore the velocity of the particle and consequently the relative velocity will change. This will lead to a new balance of forces and therefore different lateral equilibrium positions of focused particles. As further investigation, it can be shown by evaluation of the forces from equation 6 that this relative velocity is related to both the fluid and particle densities, including a ratio of the two: !!!!" = ! !!!!!!!!! !!!!" − ! !"!!!!!!! !! !! − !! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(8)!where Vp and Vf  are particle and fluid velocity, ρp and ρf are particle and fluid density, µ is fluid viscosity and a is particle diameter. Therefore, the relative density between the fluid and particle in a suspension plays a significant role in particle focusing in transient flows.  2.2 Particle Tracking In order to fully evaluate inertial particle focusing positions within microfluidic channels, it is essential to create a three-dimensional analysis system. Recently, many 2D and 3D tracking methods have been developed to obtain the particle position and therefore trajectory and flow  ! 25!field information in microfluidic systems. 2D particle tracking typically consists of capturing data on individual fluorophores, as is the case with micro-PTV [24, 35, 36, 111]. Previous work has shown the ability to track individual fluorophores in two dimensions to sub-nanometre accuracy [112]. Two-dimensional techniques are simpler to implement than their 3D counterparts, however the significant advantage of analysis in three dimensions is clear, especially in microfluidic applications including characterization of particle focusing and mixing devices. In order to address the limitations of 2D approaches, several 3D techniques have been introduced. These are typically divided into single camera (including confocal scanning microscopy anamorphic/astigmatic imaging, digital holographic microscopy and deconvolution microscopy) and multi-camera (including stereoscopic imaging and tomographic imaging) approaches [113-121]. Advantages and disadvantages of each method have been recorded in recent summary articles [3, 122, 123]. Several of these techniques have been used successfully to track the three-dimensional path of particles as small as 100 nm with high accuracy [1, 2, 124-126]. However, many of these methods also require intricate optical setups that are typically expensive, including numerous lenses, cameras, light sources, and aperture designs. Of the methods described, astigmatic particle tracking velocimetry (APTV) is perhaps the simplest in experimental design, and has been used for particle detection in applications including acoustophoretic devices [127, 128]. However, work with APTV has not included high Reynolds number flow profiles such as with inertial microfluidics [129]. Published experimental analysis of inertial microfluidics has almost exclusively been done using 2D methods, the most common of which is microparticle streak velocimetry (µ-PSV) [82]. The limited work in 3D inertial focusing analysis has used confocal or stereoscopic microscopy [99, 109]. Therefore, our main goal with this work is to find a simple method of 3D analysis of micron-sized particle  ! 26!distributions in focusing applications of varied Reynolds number. Particle defocusing methods, in which particles are imaged at varying vertical positions relative to the focal plane of the microscope so that their z-position is correlated to their apparent size, lends itself well to a simple experimental setup. Previously, resolutions of several hundred nanometres were attainable using defocusing methods with a three-hole aperture design [130-132]. However, the three-hole design must be custom-made and requires sophisticated understanding of optics. Other calibration-based defocusing methods have been used without an aperture mask, but typically require high magnification or high numerical aperture (NA) objective lens setups as the methods are reliant upon fitting the experimentally measured intensity profile to the point-spread function (PSF) of a single fluorescent particle [125, 133]. Despite these restrictions on experimental setup, diffraction ring defocusing methods with nano-scale particles have become relatively commonplace, with little work shown for larger particles (1-50 µm in size, equivalent to the size of most cells) [134].  2.2.1 Defocusing Principles & Approximation of Particle Diameter Defocusing-based particle tracking methods have garnered significant attention in recent years [2, 135, 136]. In general, it is known through simple optics analysis that a particle that is not on the focal plane of an objective lens appears larger than its real size as viewed through the objective. Figure 2.5 illustrates this concept.   ! 27! Figure 2.5. Illustration of the defocusing principle. A typical one-hole aperture within the lens shows a solid angle of light rays emanating from two point sources, A and B. Point A is located on the reference/focal plane, while point B is located at some distance from this plane. Point A appears focused at the image plane in A’, while B is projected as a blurred image B’ of larger diameter instead of its expected size B’’.  L is the distance from the reference/focal plane to the aperture [136].  The most common of these is a holographic approach known as micro-3D-defocusing digital particle image velocimetry (µ-3D-DDPIV). Using a three-hole aperture, a single particle illumination source can be converted into three sources of equal intensity in a triangular pattern [130]. The diameter formed by a circle intersecting the centre of each of the three points is linearly related to the vertical position of the particle in space [131]. This work has been thoroughly investigated for a variety of applications, and includes equation-based and calibration-based methods of relating the triangulated diameter to the vertical position [132]. However, no work has been done using a simpler one-hole aperture approach. For reasons unrelated to defocusing-based particle tracking (namely providing a more thorough investigation of the depth of correlation parameter vital to PIV analysis), recent work has been conducted to determine an equation relating a particle’s intensity diameter in an image to its vertical position. Using approximations of a Gaussian particle intensity distribution and a  ! 28!single-lens system as in previous work [137], an equation was derived to approximate the relationship between the image particle diameter, and the vertical position of the particle in space relative to the focal plane [138]: !! = ! !!! + 1.49!! !!!!"! − 1 + 4!! !!!!"! − 1 !! !! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!(9) where de is the expected/observed diameter of the particle, M is the magnification of the objective lens, λ is the wavelength of the light emitted from the particle, NA is the numerical aperture of the objective lens, z is the z-position of the particle with z = 0 as the origin (bottom of the channel), and no is the refractive index of the materials between the objective and the particle.  Refraction is an important parameter to consider in these circumstances, as typical experimental setups involve several materials placed between the objective and the particles to be imaged. When the refractive index of the immersion medium of the lens ni is different from the working fluid nw, refraction occurs. This is also the case when materials such as PDMS, glass, and air are included in the immersion medium. By proposing the assumption of a planar focal plane, then the new z-position can be determined with relation to the ratio of the two refractive indices [138]: !! = !!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(10) In this way, a defocusing-based method without the use of a three-hole aperture can be created to determine vertical positions of particles within a 3D micro-scale flow.   2.2.2 Tracking Algorithms Apart from imaging particles in three dimensions, tracking is a useful and sometimes necessary tool in determining velocity trajectories in flows. Digital image analysis is therefore a  ! 29!key aspect of many applications, as particles must be found and tracked consistently and accurately within large data sets. The time evolution of the distribution of particles in space and time can be modelled with the following equation: ! !, ! = ! ! !− !! ! ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(11)!!!!!!!  Where ri (t) is the location of the ith particle in the field of N particles at time t. Most software that has been developed to extract ! !, !  from a sequence of digital images consists of several steps, including: imperfection correction of individual images, particle position location, position location refinement, removal of “false” or erroneous particle data, and linking of particle positions between images through time t. In a more general sense, there are two parts of tracking particles: (1) the distinction of particles within an image and their separation from the background, and (2) the association of particles from frame to frame to make connections and therefore determine particle trajectories across the data set. The first step, known as segregation, can be done in both bright-field and fluorescence conditions. In either case, the intensity distribution of the particle in question is discerned using a threshold, and the centroid position of each particle can be found by using a theoretical or experimentally acquired model of the point spread function (PSF) of this distribution [40].  In the second step, association can be determined in many ways. The most straightforward strategy is to apply local nearest-neighbour linking. However, particles in three dimensions are sometimes difficult to track, as they disappear or reappear in the focal plane, or overlap with other particles in the flow. More sophisticated algorithms have therefore been developed to address these issues, including spatiotemporal tracing [139] and graph-based optimization [140, 141]. As another alternative, Bayesian estimation approaches have also been  ! 30!explored [142, 143]. Many tools already exist for particle tracking, with various pros and cons. Table 2.2 below lists several of these.  !   ! 31!Table 2.2. Available particle tracking tools [37]. The columns indicate (from left to right) the name of the tool, availability (Free = freeware, Paid = paid license code required or available as a paid service only, Request = freely available from the developers on request), the platform on which the tool runs (Lin = distribution for Linux, Mac = distribution for Mac OS X, Win = distribution for Microsoft Windows), whether source code is available, whether it was developed primarily for cell tracking or for particle tracking, whether it can track multiple objects.  ! 32!Chapter 3: General Experimental Design This chapter provides a detailed explanation of experimental design developed for the majority of the work. Although several studies were conducted for various microfluidic systems, solution/particle variations, or pressure and flow conditions, many similarities in design were chosen to provide some comparison as well as ease of setup across the entirety of the work. These similarities include particle/solution choices for suspensions, optical setups chosen including microscopes and camera choices, and microfabrication techniques for the microfluidic devices. This chapter illustrates these choices and the reasons for each, and acts as a framework for the results and discussion of chapter 4. Some details in experimental setup are further noted in chapter 4, depending on specific differences in setup between studies.  3.1 Suspension Formulations  Particle suspension formulation is a critical aspect of flow analysis and particle focusing techniques. Fluidic parameters must be taken into consideration for each study in order to create the optimal environment for particle focusing. Most importantly, as previously described, is the Reynolds number of the given fluidic system. More specifically, a key variable associated with inertial flow focusing is the fluid Reynolds number, Re. Within particle focusing techniques, the Reynolds number is used to compare expected particle positions within a new experimental design when previous experimental work has shown specific particle focusing positions. Each variable within the Reynolds number equation, including density, viscosity, and channel geometry, are essential to determining expected particle focusing positions. Therefore, for each study conducted in inertial microfluidics, these parameters must be considered. ! 33! This chapter details the formulations used in all experimental work described later, specifically discussing the variables associated with particle focusing and combinations of materials used in formulations in order to reach pre-defined variable ranges within the Reynolds number for focusing particles.    3.1.1 Suspension Types In order to investigate several aspects of the Reynolds number dependency on inertial flows and particle focusing positions within these flows, several suspension formulations were made. Options included varying both the particle and solution choices in order to match with previous work as well as compare characteristics such as density and viscosity.   Particle Types The majority of the experimental work was conducted using 15.5 µm (± 1.52 µm)  polystyrene (PS) beads (Bangs Laboratories, CC FS07F, IN, USA). The beads were green fluorescent (emission wavelength = 502 nm) and the size was chosen to mimic the size of many cells typically found in microfluidic assays such as white blood cells or circulating tumour cells (CTCs). In order to determine the velocity of the fluid flow at any given time, some experiments were conducted using 2 µm (± 0.06 µm) red (emission wavelength = 612 nm) fluorescent PS beads (Thermo Scientific, R0200, USA) in a PIV technique as in previous work [144]. As a method of comparison of the effect of particle density on focusing characteristics, 15 µm (± 2.78 µm) poly(lactic-co-glycolic acid) (PLGA) red (emission wavelength = 668 nm) fluorescent beads (Phosphorex, LGFR15K, MA, USA)  were also used.  ! 34!Solution Types  Several choices of liquids were made in order to emphasize various components of the Reynolds number effect on particle focusing positions. Most experiments used a solution comprised of de-ionized (DI) water and 1% by volume of Tween-20 surfactant (Sigma-Aldrich, P1379, ON, Canada). The surfactant was added to avoid particle aggregation in the solutions. Due to the low percentage of Tween-20, there was little effect on density or viscosity of the water (as shown in Section 3.2). In order to create a variance of density difference between the solution and the particles in some experiments, a mixture of water and ethanol (1:4 parts water:ethanol)  was used with the PS beads. Finally, a solution of dimethyl sulfoxide (DMSO) mixed with methanol (1:1 ratio) was used in cylindrical channels in order to create a refractive index matched solution to the polydimethylsiloxane (PDMS) used to form the channel itself while keeping density and viscosity similar to water.   3.1.2 Viscosity and Density Verification To quantitatively determine density and viscosity of each solution type, two simple methods were used: a scale and volumetric pipette method for density, and a rheometer (Anton Paar Physica MCR 301) for viscosity. To measure density, 5 mL of each solution was measured using a volumetric pipette (precision of +/- 0.01 mL), and then weighed to 4 decimal places using a high-accuracy balance. To measure viscosity, 1 mL of each solution was placed on the disk of the rheometer, and data was taken across 10 time steps using a circular disk setup. Three trails were done for each solution. Experimental values are tabulated below in Table 3.1.    ! 35!Table 3.1. Experimentally measured values for density and viscosity of particle solutions. Solution Type Density (g/mL) Viscosity (mPa.s) H2O + 1%vol Tween-20 0.9972 0.97 H2O + Ethanol (1:4) 0.8430 0.97 DMSO + Methanol (1:1) 0.9805 0.98  A portion of the work described in this thesis is related to the use of varying solutions and/or particles in order to affect the overall density difference between the particles and the solution. In the case of our experiments, the PS beads used have a density of 1.05 g/mL, while the PLGA beads used have a density of 1.3 g/mL. Therefore, the density difference can be exaggerated by either changing the particle type or solution type, or both. Table 3.2 below shows the variations of changing both particles and solutions for the experiments conducted in this work.  Table 3.2. Experimental particle and solution density differences. Particle/Solution Choice Δρ = ρp – ρs (g/mL) PS/H2O + 1%vol Tween-20 0.0528 PLGA/H2O + 1%vol Tween-20 0.3028 PS/H2O + Ethanol (1:4) 0.2070 PS/DMSO + Methanol (1:1) 0.0695  3.2 Optical Setup A major priority for the work described here is the simplicity of the experimental setup. As described in Chapter 1, many experimental setups have been used in particle tracking techniques, both 2D and 3D as well as including many optical setup variations. However, the majority of these systems include custom-made aperture masks, multi-camera systems, and expensive objective lens choices. Our system was chosen as a very simple option for 3D particle tracking. In essence, our experimental optical setup is what would be considered a basic system found in many laboratory settings. A Nikon Eclipse TE2000-U inverted microscope was used, ! 36!with a standard 10X objective lens (Nikon Plan Fluor 10X/0.30, WD = 16 mm, DOF (depth-of-field) = 293 µm, NA = 0.3, see Appendix A for DOF calculation). No major alterations to the microscope were made, with standard stages including micrometer adjustment. A sCMOS camera (LAVision GmBH Imager sCMOS, MI, USA) or a high-speed CMOS camera (Phantom Miro eX4, Vision Research, USA) were used to acquire images/information about particles in the flow.  3.2.1 Camera Options The two camera options chosen were used in all experiments described in our work. The first camera, a sCMOS design, is similar to those used in many 2D particle tracking in high-throughput flows [5]. This camera is a relatively inexpensive choice, providing adequate resolution (2560 x 2160 pixels) and an easily variable exposure. The resolution was chosen to allow for particles to be composed of many pixels (for example, with a 10X magnification lens, a 15.5 µm particle is typically composed of 55-75 pixels with an average diameter of approximately 20-40 pixels). For all experiments using this camera, an exposure of 100 ms was chosen. This exposure time seemed to provide a balance between adequate light intensity collected from the particles while avoiding ‘streaking’ of particles moving in the flow. The frame rate chosen for this camera was 30 fps, which is the highest available considering the resolution chosen. The second choice of camera was the high-speed camera. This choice allowed for an increased frame rate (7150 fps) with a reduced resolution (320 x 240 pixels). This choice was used in higher Re conditions, in which streaking of particles was clearly evident using the sCMOS camera. This camera was also chosen for transient flow studies, as the resulting shifts in particle positions due to applied relative velocity occurred within an extremely short time-frame ! 37!(on the order of ms).  This camera option used a variety of exposure times, ranging from 1-140 ms depending on the specific application. Considering the settings chosen, the high-speed camera was able to capture data sets of approximately 1.3 seconds (using the maximum memory of the camera for any single set). The high-speed camera was also used in bright-field imaging results, in order to test if the particle tracking method developed in this work was successful in determining particle sizes accurately without the use of fluorescence (providing a more robust design).  3.3 Microfluidic Device Fabrication Microfluidic devices were fabricated using soft lithography techniques, as described in previous work [145]. Lithography masters were fabricated in the Advanced Materials and Process Engineering Laboratory (AMPEL) Advanced Nanofabrication Facility (ANF), which includes a class 1000 lithography room and two class 10000 thin film rooms. 4-inch silicon wafers were procured (University Wafer, MA, USA) and used as the substrate for the master. SU-8 3050 photoresist (MicroChem, MA, USA) was spin-coated onto the silicon wafers to a thickness of 85 µm. Each wafer was subsequently soft-baked at 95 ºC for 40 minutes. Following the soft bake, the wafers were exposed to UV light (0.684 J cm-2 at 400 nm) through a negative film mask (Qingyi Precision Maskmaking, Shenzhen, China). Each mask was made using a design created in CleWin (Version 5, Phoenix Software, Holland). After UV exposure, the wafers were baked again at 65 ºC for 1 minute and then 95 ºC for 5 minutes. Following the post-exposure bake, each wafer was developed using SU-8 developer (MicroChem, MA, USA) for approximately 5 minutes.  ! 38!3.3.1 Rectangular Channels The rectangular channel design was created as a simple example for evaluating the particle focusing properties of inertial flows. As channel geometry remained constant for the majority of the experimental work, it became simpler to determine and compare the effects of density, viscosity, particle size, and flow rate on the particle focusing positions. The rectangular channel design used was a 85 µm x 100 µm x 4 cm (H x W x L) shape.  PDMS devices were fabricated with a mixture of 10:1 PDMS base to curing agent (Sylgard 184, Dow Corning, MI, USA). The mixture was poured onto the silicon wafer masters and cured in a 70 ºC oven for approximately 1 hour. The PDMS cast was cooled and then peeled away from the silicon wafer substrate. Each mould was then trimmed and holes were punched into the structure at the inlet and outlet of each channel on the design. 75 x 50 x 1 mm No. 1 glass slides (VWR Scientific, PA, USA) were then cleaned with acetone, dried with N2 gas and placed with the PDMS moulds into a plasma chamber (Harrick Plasma, NY, USA). An oxygen plasma exposure was then applied for 1 minute, and the glass slides and moulds were then removed and bonded together carefully. Each completed ‘chip’ was then post-baked for approximately 1 hour in the 70 ºC oven. Syringe tips (0.020 x 0.5 mm, NordsonEFD, OH, USA) were used as inlet and outlet feeds, secured and sealed to the PDMS using epoxy (ITW Devcon, MA, USA). Figure 3.1 shows the rectangular channel fabrication process.  ! 39! Figure 3.1. Rectangular channel fabrication process. (a-f) are as described in the above figure.  3.3.2 Cylindrical Channels A cylindrical channel design was also investigated as a focus on the channel geometry parameter within inertial fluidics phenomena. Experimental micro-particle focusing work in cylindrical channels has been limited in the past [146]. This is mostly due to the difficulty of fabricating a cylindrical shaped channel using conventional soft lithography or even other more complex lithography techniques. Most methods proposed require fabrication of two semicircular halves that are then bonded together [147]. One difficulty in this design lies in the alignment requirements of the two halves to avoid overlap of the sides, especially over a long channel length. Our work (headed by undergraduate summer student Anna Lee) attempted this strategy using a 3D printed mould as the master for each half of the cylinder, however alignment proved to be inconsistent, as did the shape of the cylinder (more elliptical than circular in cross-section ! 40!with ‘pinching curves’ at the regions where two halves meet, possibly causing issues with focusing due to the pseudo-circular geometry). Appendix B gives further details from the 3D printed cylindrical channel work. Figure 3.2 below shows a typical result from this work.   Figure 3.2. PDMS cylindrical channel cross-section showing misalignment of two halves. Molded from 3D-printed master structure. The channel geometry is not quite circular, with pinched curves at the points where the two halves should meet (located in the yellow circles in the figure).  Another strategy for cylindrical channel fabrication involves thin-film PDMS and back-pressures to bend and mould the films into semi-circular shapes [148]. Along with the previous issue of misalignment, this technique requires several extra steps in the fabrication process, including the creation of a PDMS base and thin films and control of pressure during the casting step. Finally, in order to avoid the semicircular designs and alignment issues, some work has been done in ! 41!using a sacrificial inner cylindrical shape that, once moulding around it has taken place, the sacrificial layer can be removed [149]. A variation on this technique was used in this work, by using thin wire as the sacrificial layer. First, a layer of 10:1 PDMS was poured into a petri dish and cured. Following this, a wire of length ~5 cm and diameter 150 µm (checked using a micrometer caliper) was placed straight on the PDMS layer and taped at the ends into place. Additional PDMS was then poured over the wire and cured. Following these steps, the two PDMS layers were separated along the cured ‘grain’ that bonded them together. This is relatively easy to do, as the first PDMS layer was cured previously to the second. By doing so, the wire between them can be delicately removed by hand. Each side is then left to re-join the other. Since only one side of the two PDMS layers were spread from one another, the two can re-join without alignment issues. Following the removal of the wire, each PDMS chip is allowed to further bake with several pounds of weight on it via a petri dish/metal block weight in order to ensure the two PDMS layers re-bond properly. Epoxy resin (ITW Devcon, MA, USA) is then applied to both ends of the chip where the wire was previously in order to close off the channel. Finally, inlet and outlet holes are punched, and the PDMS chip is bonded to a glass slide to avoid leakage from the punched holes in a similar fashion to the rectangular channels. Figure 3.3 below illustrates the entire process.  ! 42! Figure 3.3. Illustration of cylindrical channel fabrication process. (a-f) are as described in the above figure.  Our method of cylindrical channel fabrication is cheap and relatively simple in its fabrication process, as it only requires PDMS and standard electrical wire of a specific/known diameter. The major drawback to this method is the restriction on channel design. Although our straight channel was relatively easy to produce, more sophisticated design including spirals, loopback structures or general curves in the channel would be much more difficult to produce, especially to micrometer precision. !  ! 43!Chapter 4: Experimental Results & Discussion This chapter describes the experimental work conducted for this thesis, and represents the majority of the effort beyond theoretical understanding and experimental design. Experimental discussion of the three-dimensional defocusing particle tracking method developed. This includes calibration of the system, image acquisition and post-processing methods, and particle tracking algorithm considerations. This tracking method is then used in several verification studies, including a gravitational force effect study and steady-state inertial flow studies in rectangular and cylindrical channels. As a novel technique, transient inertial flows are discussed and experimental results of particle focusing for both single- and multi-cycle flows are shown. Sources of error within the particle tracking method are also discussed.  4.1 3D Defocusing Particle Tracking Method This chapter section describes the application of a defocusing-based 3D particle position tracking technique for microfluidic particle focusing. The technique is calibration-based and used for typical microfluidic channel dimensions and finite Reynolds number (Re < 120). Although our primary application is inertial focusing, this technique is also implemented to determine gravitational force effects to show scaling thresholds of the technique and also illustrate its use for a variety of experimental designs.  4.1.1 Calibration The main component of the defocusing-based particle position tracking method developed is the calibration of the z-positioning with the defocused size of the particles. In order ! 44!to correlate the z-position to the particle’s apparent diameter, an experimental curve validating the relationship between these variables must be found. To accomplish this task, particles must be held in a known location in space and their apparent diameters must be recorded. For each particle type (PS, PLGA), an agarose gel suspension was used to hold the particles in a randomly oriented distribution within the microfluidic channels used. This suspension was made using DI water, 3-5 wt% agarose powder (Sigma-Aldrich, ON, Canada) depending on the density of the particles, and 0.1 wt% of the particles of choice. Prior to adding the particles, the agarose mixture was left at 85 ºC overnight (approx. 10 hours) to mix and dissolve the gelling agent. Upon adding the particles, the suspension was rapidly syringed into the channels while still liquid, quickly solidifying and ensuring the particles were suspended at varying z-positions in space. In the case of the cylindrical channels, DI water was substituted for a DMSO:methanol (n = 1.405 [150]) solution in order to match the refractive index of the solution to PDMS (n = 1.41 [151]). This is important in order to remove optical aberrations or scattering irregularities from the calibration process. Figure 4.1 is a schematic of the experimental setup used in the calibration.   ! 45! Figure 4.1. Experimental Schematic for calibration experiments  The particles were imaged using fluorescence and/or bright field microscopy. The Nikon Eclipse TE2000-U microscope was used, with a standard 10X optical lens (Nikon Plan Fluor 10X/0.30, WD = 16 mm), NA = 0.3. The sCMOS and high-speed cameras were used for different calibration curves, depending on the final application of the particle/channel dataset to be acquired. The data from the calibration was acquired in three steps. First, the bottom of the channel was found and set as the z = 0 position in relation to the focal dial on the microscope. Then, the z-position from the bottom of the channel to each particle was determined with relation to the original position of the microscope’s focal dial. Finally, each particle was imaged with the focal plane of the microscope set to z = 0, thereby acquiring an apparent size value relative to its known z-position in space. Figure 4.2 shows a general example of the defocusing principle on particle diameter as seen during calibration.  ! 46! Figure 4.2. General example of defocusing principle as typically seen during calibration. Focused particles maintain a diameter close to the real diameter (15.5 ± 1.52 µm) while defocused particles have an intensity distribution spreading wider as they are positioned further from the focal plane. With the focal plane set at the bottom of the channel (z = 0), particles positioned closer to the top appear to be larger than those at the bottom. Two experimental examples are shown [152].  Z-position and apparent size data for each particle in the suspension was collected and a calibration curve of z-position and apparent particle diameter was made. As a simplified theoretical check on the experimental data, equation 9 was used as an adaptation of previously described work [138]. Refraction of the objective light occurs through the microfluidic channel (including the glass and PDMS substrates) and the solution medium before reaching the particle. Therefore, zw = nw/ni zo, where ni = nsolids/nair (nair = 1), nsolids = nPDMS/nglass (nPDMS = 1.41, nglass = 1.52) and nw varies depending on the solution; magnification M = 10 as in previous work [153].   4.1.1.1 Calibration Assumptions Equation 9 above was derived under several assumptions: (1) uniform illumination in space, (2) a single thin-lens system and (3) particle image intensity distribution modelled as a Gaussian function [137]. We can include assumption (1) as inherent within of our experimental ! 47!setup, however during data analysis, we avoided using data collected from particles too close to the edges of the image frame due to possible fringing effects from the illumination source at these locations. Considering assumption (2), a single-lens approximation does introduce error since our real microscope system is more complex. As identified in previous work, a more complex microscope system would cause high error in this simplified equation when a large magnification or large NA lens is used (M ≥ 20X and NA ≥ 0.4) [138]. Manifestations of this error include a shift in the rate of defocusing (i.e. the slope of de related to z), an asymmetric defocusing pattern when considering objects above and below the focal plane, and possible optical distortions at the outer regions of the sCMOS image acquisition sensor. Our NA (0.3) and magnification (10X) are both below the values leading to these error sources. Furthermore, because of the positioning of our focal plane at the bottom of the channel (z = 0), any asymmetric effects can be neglected since all of our objects will be physically above this plane (z > 0). As pertaining to assumption (3), although particles used in the justification of this single-lens equation ranged from dp = 1 to 5 µm and our particles are slightly larger at dp = 15.5 µm, a Gaussian approximation of the intensity distribution remains valid [137]. Other defocusing methods use much smaller particles to experimentally determine the point-spread function (PSF) via diffraction ring patterns found in high magnification, high-NA images (M = 20 to 60X, NA ≥ 1) [133, 134, 154, 155]. These diffraction patterns most closely resemble Bessel functions in shape. However, the intensity profile of larger particles can be approximated as a Gaussian function when viewed under relatively low magnification and NA [137, 156]. Considering all of the optical effects described, our experimental setup fits the assumptions used to derive equation 9. The calibration curves developed under these assumptions were used in all further experiments to relate apparent diameter of particles in flows to their vertical position in the channel. The ! 48!experimental results from several calibration curves were matched with the theoretical predictions from equation 9 and are shown in Figure 4.3.   Figure 4.3. Calibration curves for various experimental designs. 15.5 µm PS beads in water, 150 µm diameter cylindrical channel (green), 15.5 µm PS beads in 1:1 DMSO:methanol, 150 µm diameter cylindrical channel (blue), 15.5 µm PS beads in water, 100 x 85 µm rectangular channel (orange). PLGA beads have the same mean size as PS beads, therefore the same calibration curve can be used. Solid lines are theoretical predictions based on equation (9).  4.1.1.2 Experimental Error and Uncertainty for Calibration Curve Previous work describes a linear trend between apparent diameter and z-position, whereas others have found a closer fit to equation 9 [125, 131, 138]. After completing an R2 regression fit for a linear function and equation 9 for each calibration curve, R2linear ranges from 0.8973 to 0.9244 and R2eq.6 ranges from 0.9090 to 0.9312. We have decided to use equation 9 as a theoretical fit for our experimental data as it has a more thoroughly understood theoretical basis ! 49!as discussed in Section 4.2.1.1 [138]. As verified by the R2 calculations, the experimental data has an error ranging from 7.8 to 9.1% from equation 9. Coefficient of variation (CV = σ/µ; σ is the standard deviation, µ is the mean) was used to compare our experimental error with several hypothesized sources of uncertainty. Uncertainty is most likely attributable to several experimental steps related to human uncertainty. Each particle was deemed in focus based on observation through the microscope by eye, where a particle appears to be the smallest diameter compared to its size at any other focal plane. To determine a quantitative measure of this uncertainty, the same particle was observed and brought into focus 10 times, with this dataset collected for each calibration curve setup. Standard deviation of this observed in-focus position was 1.3 µm (CV = 8.4%) for 15.5 µm PS particles and 1.1 µm (CV = 7.3%) for 15 µm PLGA particles. Another source of uncertainty could be related to the standard deviation of the particle size (± 1.52 µm, CV = 9.8% for PS particles; ± 2.78 µm, CV = 18.5% for PLGA particles). Since each particle has a slightly different size, two particles at the same z-position can have differing apparent diameters. Notably, particles with the approximately equivalent z-position across all studies have CV ranging from 6.6 to 7.9%, which falls within the uncertainty described above. While the manufacture of particles could be altered based on more stringent production specifications, the human source of uncertainty is unavoidable and restricts the use of this experimental technique.  4.1.2 Image Acquisition Data for all studies described were collected using the same optical setup as the calibration curve with a standardized procedure. The microfluidic devices described in Section 3.3 are fitted with tubing of 0.5 mm inner diameter on the inlet and outlet. The inlet tubing was ! 50!fit to a flow pressure control system (Fluigent MFCS-8C, Villejuif, France). The outlet was fed to a 2 mL collection vial. The bead solution described in Section 3.1 was fed through a T-junction into the inlet of the microfluidic channel at a known pressure dictated by the user through the Maesflo (v2.1.3) software accompanying the Fluigent system. As shown in Figure 4.4, a simple study was done to correlate pressure from the Fluigent to flow rate using a flow meter (Fluigent Flowell, Villejuif, France). The flow resistance of the Flowell was found to be approximately 1000 times lower than the entire system ( Applied pressure can then be correlated to a given Reynolds number based on the following equation: !! !"#$ = 4.7088×10!"(!"/!! ∙ s)!!"! !!!! !!!!! !!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(12) where Re is the Reynolds number, µf and ρf are the fluid viscosity and density, and H and W are the height and width of the channel. Beads then travel through the channel, where the sCMOS camera was positioned along its length (1 mm from the inlet for the gravitational force study (Section 4.2.5) and 1.5 cm from the inlet for all other studies (Sections 4.3 and 4.4)) to acquire sets of images over time. The focal plane of the objective was always kept at the bottom plane of the channel (z = 0) to validate the use of the calibration curve results later. The sCMOS camera captured 300 images at 30 Hz, for sets of 10 s time intervals. Alternatively, the high-speed camera captures 9000 images at 7150 Hz, for sets of 1.26 s. A number of data sets were collected in order to have at least several thousand (typically 5000-10000) individual particle trajectories for each flow rate. This was done to reduce error based on initial position. Data sets were taken in several different channels with the same dimensions and averaged. Data was acquired for 0.246 ≤ Re ≤ 75.4. Particle image density ranged from 0-50 particles/image, with estimated displacement between frames of 50-2000 pixels depending on the chosen Reynolds number. ! 51!Therefore, total number of data sets ranged from 5 – 50 depending on the flow rate (Re) and study, to have a total number of particles (data points) of at least 3000 for each flow rate.   Figure 4.4. Example of calibration of applied pressure to flow rate. The experiment was conducted using our setup for a range of applied pressures (0-20 mbar). Values were collected only for Q ≤ 7 µL/min as this was the limit of our flow meter. This linear relationship was determined for our specific microfluidic geometries and solutions. The linear coefficient (m = 0.32 µL/min·mbar) was used to correlate P (mbar) to Q (µL/min) and therefore correlating pressure to Reynolds number [152].  4.1.3 Image Post-Processing Once the data sets are acquired using the simple optical setup, each set was analysed using the free image processing software ImageJ and a custom MATLAB code to remove background. Specifically, a bright-field channel edge image was taken from each data set to determine the location of the channel edges within the frame. Then a background image was taken from each data set and is subtracted from each image containing particles. A spatial bandpass filter using a convolution of Gaussian and Boxcar functions was applied to each image ! 52!to remove background noise and smooth the individual particle intensities. The bandpass filter works by convolving with the appropriate kernels (i.e. particles) within an image. It is a two-step process. First, a low-passed image is produced by convolving the original image with a Gaussian. Next, a second low-passed image is produced by convolving the original image with a Boxcar function. By subtracting the boxcar version from the Gaussian version, we use the Boxcar to perform a high-pass filter operation. This code, including the MATLAB script implemented, was developed prior to this work by John C. Crocker and David G. Grier [157]. In this work, particle tracking is used to eliminate multiple examples of the same particle in different frames, in order to remove duplicity in the statistical significance of the data. The bandpass filter was supplemented with the MATLAB dilation and erosion functions to further resolve the edges of the each particle from the background. Dilation involves applying an operator to a binary image to gradually enlarge the boundaries of regions of interest. Thus, areas of interest grow in size while holes within the regions become smaller. Similarly, erosion reduces the boundaries of regions of interest. Parameters related to the bandpass, dilation and erosion functions were optimized for our systems. Finally, the MATLAB regionprops function was used to approximate the centroid location and equivalent diameter of each particle. A final overlay of the original image with the result of the image post-processing algorithm was used to visually confirm the accuracy of the method before a full image set was processed. Appendix C shows examples of the MATLAB code developed from previous work and adapted and optimized for this 3D defocusing particle tracking method. Figure 4.5 is a visual explanation of the entire image post-processing algorithm. The centroid and diameter data was then saved and compiled using Excel for each experimentally tested flow rate.  ! 53!  Figure 4.5. Visualization of image post-processing algorithm. (a) An original image from the sCMOS camera is imported into MATLAB. The following images (b-e) are of a zoomed-in portion of the total image to emphasize the effect of the algorithm. (b) A zoomed-in portion of the total image, no post-processing conducted at this step. (c) A bandpass filter, including a convolution of Gaussian and Boxcar (a dual-sided Heaviside) functions, is used to remove background noise and better resolve the edges of the beads. (d) Dilation and erosion functions embedded in MATLAB are used to approximate the finite edges of the beads (an octahedral geometry is used rather than a higher-sided shape to reduce computational time). (e) The results of the dilation/erosion functions are mapped using regionprops and overlaid over the original image to verify accuracy of the algorithm. One image is checked in this manner for each image set before the algorithm is run for all images in the set [152].  4.1.4 Particle Tracking Algorithm The tracking algorithm used was originally developed by John C. Crocker and David G. Grier [157]. Given positions of n number of particles at a time t(i), and m possible new positions of the particles at a time t(i+1), the tracking algorithm considers all possible trajectories of the n old positions with the m new positions and chooses the trajectory which results in the minimal total squared displacement. Particles leaving or entering the images were treated as distinct, new particles. Tracking individual particles was used to determine individual particle velocity vectors or to more precisely understand conditions for particle displacement within the channel due to gravitational settling or other forces including inertial or fluid viscous drag.  a" b" c " d" e"! 54! 4.1.4.1 Consideration of Peak-locking Effects in Tracking Algorithm It has been noted in several PIV and PTV works that error can be caused by a sub-pixel resolution issue known as peak-locking [158-160]. In many image processing algorithms, particle diameter and centroid positions are calculated based on an approximation to the closest pixel along the edge (in the case of diameter) or the closest pixel to the centre of the particle (in the case of centroid determination). This error is especially prevalent in cases where the particles are close to the size of individual pixels in the image. Due to the large size of the particles and high resolution of the images, each particle is approximately 35-40 pixels in diameter when using the sCMOS camera, or 20-25 pixels using the high-speed camera. Therefore, error associated with peak-locking would be on the order of 2-5%. As a thorough confirmation for all Re considered in our experiments, a 2D histogram plot (Figure 4.6a) of the probability density function (PDF) of the sub-pixel centroid positions, as well as a 1D histogram plot (Figure 4.6b) of the sub-pixel part of the diameter were made. Notably, there is no discernible peak-locking effects in these analyses, which confirms that this image post-processing algorithm is viable for our technique.    ! 55!  Figure 4.6. Histogram plots for peak-locking effects analysis. (a, left figure) 2D histogram plot of probability density function (PDF) of the sub-pixel centroid positions of particles. Data was accumulated from both gravitational and inertial focusing studies. Bin size set to 0.03 (3% of total pixel size). X and Y axes refer to the sub-pixel (decimal) centroid value for each particle detected via the defocusing method and image post-processing algorithm described in the accompanying paper. Colour bar refers to the number of particle positions within each sub-pixel region. Blue points represent raw data. There is no discernible pattern in this plot in either the x- or y-axis. (b, right figure) 1D histogram plot of probability density function (PDF) of the sub-pixel parts of the diameter of particles from gravitational and inertial focusing studies. Bin size set to 0.02. Similarly to Figure 4.6a, there is no discernible pattern in the particle counts across the sub-pixel diameter parts. Therefore, it can be concluded that peak-locking effects are negligible [152].  4.1.5 Application Example: Effects of Gravitational Forces on Low Reynolds Number Flows A large body of work has investigated the effects of inertial forces on particle positioning in channels [5, 34, 71, 84, 85]. However, most studies were conducted using non-density-matched particles without taking into account the possible effects of gravitational forces on their positions. Therefore this study acts as a validation of our method of vertical position while developing a more thorough understanding of the forces affecting particle positioning. ! 56! Centroid and diameter data was collected for particles for 0.246 ≤ Re ≤ 1.237 using the sCMOS camera and Fluigent system. Particles that have been imaged at both the far-left (within the first 50 pixels, excluding the first 10 pixels following assumption (2) for eq. 6) and far right (within the last 50 pixels, excluding the last 10 pixels) of the frame were tallied and their change in diameter across the frame was monitored. The channel walls were carefully oriented perpendicular to the camera frame to avoid particle initial position affecting particle travel length across the channel. The overall change across the frame length (1.35 mm) was then determined for each particle and results were averaged over the total number of particles observed.  In order to complete the validation, a theoretical analysis of the settling effects of gravitational forces on particles in the flow must be prepared and compared to the experimental results. Three forces are considered: gravitational (Fg), buoyancy (Fb) and Stokes drag force of the particles relative to the medium flow (FD). !! − !! − !! = !! !!!!" !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(13) with, !! = !!!"!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(14) !! = !!!"!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(15) !! = 3!"!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(16) Integration of equation 13 leads to the following result: !!!!!!!!!!!!!!!!!!!!!!!!! ! = !! 1− !!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(17) !! = !! !! − !! !!!18! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(18) ! = !!!!!18! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(19) ! 57!where ρp is the particle density, ρf is the fluid density, µ is the fluid viscosity, g is the gravitational acceleration, dp (equivalent to a previously) is the particle diameter, V is the particle volume, u∞ is the steady-state sedimentation velocity in the z-direction, uz is the sedimentation velocity in the z-direction, and τ is a time constant related to the particle’s motion in the fluid. For our particular rectangular channel geometry, τ = 16.84 µs and u∞ = 10.60 µm/s. Therefore, within approximately 17 µs, up ≈ u∞. The above result can be used to relate the particle’s initial (x,y,z) coordinates in space to its final position 1.35 mm downstream, with the sedimentation velocity approximately equal to u∞. Using the first approximation of a first-order Ritz velocity profile, the x-component velocity distribution in non-dimensional form is obtained as [161]: !! ∆! = !"!" ∆! = !94 !!" 1− 4 !!! ! 1− 4 !! + ∆!! ! !!!!!!!!!!!!!!!(20) The y-position is considered constant (uy ≈ 0) as there are no forces acting along the y-axis during flow to shift particles along this direction. Experimental results confirmed this assumption, as particles tend to shift within 1 µm along the y-axis over the length of the frame. However, particles that begin at a y-position closer to the walls of the channel will have a final position closer to z = 0 than particles with an initial y-position closer to the centre of the channel (Figure 4.7). This is due to the double-parabolic velocity profile within a rectangular microfluidic channel.   ! 58! Figure 4.7. Illustration of the double-parabolic velocity profile. The fluid velocity profile in a rectangular channel can be approximated as a double-parabola for two axes (y,z). Since the highest fluid velocity is found in the centre of the channel, the particle velocity will also be highest at this position. Therefore, particle settling will be lowest for a particle initial position (x,y,z) = (x,0,H/2), and increase as particles are initially positioned closer to the channel walls. The initial and final positions of the two particles are illustrated. x1 = 0, x2 = L = 1.35 mm downstream (the width of a capture frame) [152].  Integration of dx/dt over t and application of height and width constants (de-normalization) for the y and z terms leads to this final relationship between a particle’s x and z positions within the channel: ! = ! 81!"2! !! − !! !"!!! 1− 4 !!! ! ∆! − 43!! !! + ∆! ! − !!! !!!!!!!!!!(21)  where Q is the input flow rate decided by the user and H and W are the height and width of the channel, respectively. The input height, zo was maintained at a certain height (at approximately zo = H/2) for all data to ensure consistency of gravitational effects across all data. Figure 4.8 ! 59!shows this relationship for a selection of the Reynolds numbers chosen from the experimental procedure.    Figure 4.8. Cubic relationship between change in x-position and change in z-position of a particle in the flow due to settling. These are results analytically derived from equation 21. The relationship is indeed cubic, however, for small changes in x (as in our case, where Δxmax = 1.35 mm), the relationship can be approximated linearly. In general, the higher the Reynolds number, the higher the flow rate, and therefore the lower the change in z across a change in x [152].   Although the theoretical approximation is a third-order polynomial, the (x, Δz) relationship approximates linearly. This is because for x ≤ 1.35 mm, Δz reaches a maximum of approximately 5 µm, while H = 85 µm. Therefore, (Δz/H)3 << (Δz/H). Figure 4.9 combines the theoretical and experimental results for gravitational settling.   ! 60! Figure 4.9. Theoretical and experimental settling study results. Results are illustrated as simple lines and error bars are illustrated using semi-transparent zones as results are averaged across many particles (100-1000 depending on the pressure chosen).  As expected, an increase in Reynolds number causes a decrease in overall settling. Most importantly however, experimental results are within 1-2 µm of the theoretical prediction for each Reynolds number (i.e. applied pressure) chosen. Uncertainty (shown as coloured bands) ranges from ± 0.2 µm to ± 0.39 µm from the average slopes, increasing with increasing Reynolds number [152].  In general, the experimental results match well with predicted theoretical trends of settling across the channel. Experiments show a change in z-position of the particles of 0.5-3.5 µm across the frame length. This is associated with a 2-5 µm diameter change (~3-8 pixels) depending on the particle’s initial position. Experimental value uncertainty (i.e. error) increases in general with increasing Reynolds number. This is reasonable considering that increased initial flow rate pushes particles towards the walls and bottom of the channel at the inlet, creating a more random initial distribution. ! 61!The effect of gravity on particle positions becomes relatively small as Re increases above 1 for our system (Δz ≤ 0.05(dp)). Therefore, most previous work is sound in that flow rates are typically much higher and therefore the lift forces related to inertial effects become predominant. However, the regime in which gravitational forces are of an equivalent order to the other forces acting on particles in microfluidic flows could be used to create unique focusing positions. Primarily, however, this gravitational settling study was used to verify that defocusing can be used for 3D tracking of moving particles in a microfluidic channel, and as determined using the calibration curve method, has a particle diameter resolution limit of approximately 2 µm.  4.2 Steady-State Inertial Focusing Study A simple steady-state inertial focusing study was conducted in order to determine experimental focusing positions of particles due to inertial effects, further validating the 3D defocusing-based tracking method developed. Steady-state inertial focusing is an important application of our tracking method, as many researchers working in this area wish to have a complete 3D understanding of the particle positioning without a complex experimental setup or post-processing algorithm. The study was conducted using both rectangular and cylindrical channels, in order to validate our method with comparison to previous results as well as show any new trends in focusing at the specific Reynolds numbers chosen.   4.2.1 Rectangular Channel Study  Our first study to be conducted used the rectangular channel geometry. If the geometry is a rectangular channel, high flow rates cause a 6-point symmetrical convergence of particles close ! 62!to the centre of each of the two longer walls of the channel as well as the four corners, as shown in Figure 4.10a [5, 82, 162-165]. The particles then tend to migrate to a 2-point equilibrium depending on how far down the channel the particles have travelled [83].This study was used to verify this theoretical expectation using the 3D defocusing method, and also determine if any new focusing trends exist in Reynolds number regimes that were not previously investigated.  Following a procedure similar to that outlined in Section 4.1.2, image sets were collected using the sCMOS camera in several rectangular channels of the same dimensions at a position 1.5 cm from the channel inlet and particle trajectories were found using the tracking algorithm. 2D plots of the cross-section of the channel were developed using MATLAB to map particle positions at varying flow rates. Due to the density of particles in specific regions in these plots, the data was converted to a probability density plot using a probability density function (PDF). The PDF was derived by determining the number of particles found in partitioned bins of the 85 x 100 µm channel cross-section. These values were then normalized by the bin with the highest number of particles. Figure 4.10b shows the results from the seven flow rates chosen for this study, including a simplified diagram of the expected focusing regimes. These flow rates (corresponding to specific Re) were chosen as they have been previously used to identify unique focusing positions of particles in square or rectangular channels [5, 109].  ! 63! Figure 4.10. Conjectured and experimental focusing trends for particles in the rectangular channel study. (a) A schematic of the expected trend for inertial focusing of PS particles in our specific rectangular geometry. Schematic aspect ratio of channel height/width is to scale. Particles begin in the centre of the channel and move outwards towards the walls as Reynolds number of the flow increases. Due to the 85:100 height/width ratio, particles are expected to focus towards 4-6 positions at the top and bottom of the channel as Re increases. (b) Probability density contour plots of experimental data for steady-state inertial focusing using the three-dimensional defocusing method. Particles are initially found close to the bottom of the channel due to low velocity and the effect of gravitational settling. As Reynolds number increases, particles form a circular cross-section in the channel, slowly shifting away from the center and spreading towards the walls. Eventually (at Re = 75.4), particles appear to be primarily positioned in the central vertical plane of the channel, with spreading towards the upper and lower walls. Each cross-section was split into 100 x 100 bins (bin size of 1 x 0.85 µm). The color bar values refer to a normalization of the number of particles found within each bin (0 indicating no beads found, 1 indicating the maximum number of beads found across the entire cross-section) [152].  ! 64! . Figure 4.11. Comparison of results between previous work [5] and steady-state study results. Our results (the bottom figure) show similar trends to past work (the top figure). The main difference is the broadening of the distribution in previous work across the channel width compared to our results – however this could be attributed to the differing a/Dh ratio (0.09 from previous work and 0.16 from our work). ! 65!4.2.1.1 Discussion of Results Figure 4.11 illustrates the comparable particle distribution profiles along the channel width from our steady-state work and previous work [5]. Overall, the trends are quite similar, confirming our setup works effectively, including the 3D tracking method. As shown in Figure 4.10, particles tended to migrate away from the center of the channel towards the walls along the y-axis, also spreading towards the top and bottom walls at higher flow rates. This generally follows trends indicated from previous work; however, particles tend to more frequently stay along the central vertical plane rather than migrating towards the top and bottom walls. Also, there is an oval-shaped particle distribution between 10 ≤ Re ≤ 35 which has not been previously described [5, 71, 82, 109] and may represent an intermediate focusing regime between unfocused particles and particles focused towards the walls as indicated in Figure 4.10a. This is most likely due to the relatively high particle size/channel size ratio (a/Dh = 0.167) compared to previous work (ranging from 0.05 to 0.1) [5, 82, 166]. Finally, we compare our results and method to those previously used in inertial microfluidics. As previously noted, typically 2D results are used to validate inertial effects experimentally, the most common of which is µ-PSV [5, 83]. However, 2D methods such as this give little understanding of individual particle trajectories, or vertical positioning information. Furthermore, 2D methods are difficult to normalize, as they are reliant on average intensity of many hundreds of particles viewed in a channel. Our simple 3D method demonstrated here allows for individual particle tracking to create an overall cross-section of particle positioning in the channel as seen in Figure 4.10b.    ! 66!4.2.2 Cylindrical Channel Study Similarly to the rectangular channel study conducted above, a steady-state cylindrical channel study was conducted to verify the use of the 3D tracking method in various channel geometries. The cylindrical channel used was fabricated as described in Section 3.3.2. The channel is approximately 150 µm in diameter, and the high-speed camera was used instead of the sCMOS. Three data sets were collected for each Reynolds number, each including several thousand (5000-15000) data points. One major difference was the use of a DMSO:methanol 1:1 ratio solution replacing the water solution used previously. This was done in order to match the refractive index of the solution to the surrounding PDMS channel structure, thereby eliminating optical aberrations and scattering effects due to refraction along the curved channel surface. Also, a new calibration curve was made specifically for the cylindrical channel, as shown in Figure 4.3.   4.2.2.1 Discussion of Results Figure 4.12 shows the results from the steady-state cylindrical study. 2D cross-sectional PDF plots were constructed in the same way as the rectangular channel study.    ! 67!    Figure 4.12. Steady-state cylindrical study experimental data PDF contour plots. Particles are initially found close to the centre of the channel due to low applied pressure with some spreading towards the walls due to the pseudo-focused nature of the low Reynolds number flow. As Reynolds number increases, particles form a circular cross-section in the channel, moving away from the center and spreading towards the walls. At Re = 22.20, particles appear to be primarily positioned in a ring shape in the channel, with the central radius of the ring at approximately R = 0.63. Each cross-section was split into 100 x 100 bins (bin size of 1.5 x 1.5 µm). The color bar values refer to a normalization of the number of particles found within each bin (0 indicating no beads found, 1 indicating the maximum number of beads found across the entire cross-section). The yellow borders represent the approximate edges of the channel walls. Data recorded 1.5 cm away from the channel inlet.  ! 68!In general, the data shows a similar trend to previously shown results, specifically the resulting 0.6R focusing position trend at high Re [93, 94]. For Re = 22.20, the average R found for the particles from the centre of the channel ((y,z) = (75,75)) is 0.6328.   4.3 Transient Inertial Focusing Study Along with the steady-state inertial focusing study, a transient study was conducted. This study is the first of its kind, and uses a transient flow profile to manipulate particle positioning based on the effects of rapidly changing the relative velocity of the particle with respect to the fluid. By changing the relative velocity, the balance between the inertial lift forces as well as the viscous drag force shifts. This is due to a shift in the viscous drag which then alters particle position due to change in the parabolic velocity profile of the channel, as well as the other transient lift forces, namely the slip-shear lift. The experiment was designed using specific Reynolds numbers so that the results are directly comparable to the steady-state results previously shown.  4.3.1 Single Cycle Transient Study For the initial transient studies and as a proof-of-concept, a simple single transient cycle setup was used. To achieve the desired transient effect, the Fluigent system was set up to control flow from both the inlet and outlet of the channel. In this way, the flow was quickly stopped by applying a reverse pressure from the outlet against the inlet applied pressure. The inlet pressure was set to the chosen pressure related to Reynolds number (from equation 12). After reaching the steady-state (i.e. constant) flow regime, the outlet pressure was increased to match the inlet pressure, effectively stopping the flow in less than 100 ms (this time changes depending on the ! 69!applied constant pressure). Figure 4.13 is a schematic of the experimental setup used for single-cycle transient flow experiments outlined in this section. The same setup as previously described in section 4.2 with the high-speed camera and its accompanying settings was used. Data was acquired using the defocusing-based technique described in section 4.2  to correlate the apparent size of the particles to their vertical position in the channel [167]. Images of the particles were taken in bright-field after it was found that the resolution and exposure time settings made it difficult to contrast the fluorescing beads with the background. Also, taking images in bright-field allowed for verification of the 3D defocusing method without the need for fluorescence microscopy. The Fluigent pressure control system was used to regulate applied pressure into the channel and through cross-sectional area. Three Reynolds numbers (Re = 16.52, 22.95, 32.12) were chosen to create an overall representation of transient effects with changing flow rate.      ! 70! Figure 4.13. Experimental schematic. The microfluidic chip is placed on the stage, where a bright field source (for imaging PS beads) are able to capture information. The majority of the setup is similar to the calibration study and steady-state studies previously conducted. Data was collected and analyzed using Phantom CV 2.2 software (Vision Research, USA). Schematic is not to scale [110].  4.3.1.1 Single Cycle Results The velocity of the flow during the transient period was obtained by tracking small 2 µm red fluorescent PS beads using the developed 3D defocusing method. A solution of the beads was fed into the microfluidic channel, and a series of data sets were acquired for the chosen Reynolds numbers (Re = 16.52, 22.95, 32.12). This experimental procedure was repeated at the same pressures (Re) with the larger 15.5 µm PS beads to obtain the particle velocities. To highlight the effects of the transient flow in manipulating particle positions, the relative velocity between the particle and the fluid was measured. Data was restricted to particles in the channel center to avoid complexity related to the double-parabolic velocity profile in the ! 71!channel. The velocity of the fluid and the particles was measured during the transient flow time period. Figure 4.14 provides an example of the observed trend for Re = 32.12.    Figure 4.14. Relative velocity of fluid (as tracked using 2 µm beads) and of 15.5 µm beads vs. time following the time at which the outlet pressure was matched via the Fluigent to the inlet pressure. In this example, the maximum difference between fluid velocity and bead velocity occurs at a time approximately 18 ms after the Fluigent outlet and inlet pressure are set to be equalized. Similar plots were derived for all Re tested [110].  The relative velocity data results indicate that by changing the flow rate, a relative velocity is created which, according to the inertial focusing force theories presented, can be used to manipulate the focusing positions of the particles. When the pressure gradient is abruptly set to zero, the steady parabolic flow is expected to come to rest after trest ~ 2h2/ν = 5 ms where h is half the channel height, and ν is the kinematic viscosity [92]. The fluid velocity at the center of channel, and the particle velocity passing through the channel center is shown in Figure 4.14. The difference between the observed time-scale in Figure 4.14 and trest can be attributed to the physical delay of the pressure control system. The particle response time is calculated as ! 72!τ = (ρpdp2)/(18µf). The Stokes number (St = τ/ trest) is a good criterion for evaluating the particle slip with respect to its streamline (in undisturbed flow) passing through the center of each particle. For the range of liquids and particles used in this study, Stokes number is found to be smaller than 0.01, and therefore it is expected that the particles have the same velocity as the streamlines passing through their centers [92]. In the case of Re = 32.12, the time around 18 ms is the period that results in the highest deviation from the force balances typically seen in steady-state flow experiments. Therefore, data from the transient flow studies was collected at this time period (and a slightly different time period for other Reynolds number cases). To investigate the effects of the induced relative velocity and as a source of comparison, the focused positions of the 15.5 µm particles were measured for both steady-state and transient flows. Using the 3D defocusing method, captured images for the transient experiment were translated into a 2D cross-sectional plot of the channel that illustrates the difference between the equilibrium focusing positions of steady-state and transient flow rates. The results were also normalized to the same number of data points to ensure statistical equivalence. The focusing positions for steady state and transient flow rates are shown in Figure 4.15 for three Reynolds numbers (Re = 16.72, 22.95 and 32.12).      ! 73! Figure 4.15. Transient study results. Comparison of steady-state flow rate data (blue) with transient data (orange) in the channel cross-section for Reynolds numbers chosen. Plots are of raw data (not probability density plots) and are normalized for data point size to maintain statistically significant and comparable results of both steady-state and transient results [110].    The difference in the observed focused positions can be discussed from several perspectives. In the transient measurements, the velocity of flow during the capturing period is lower than the velocity during the steady state flow.  Therefore, based on the results of the previous steady-state flow studies, it was expected that the transient measurement show more scattered behavior in z-direction (channel height) [5, 108]. Interestingly, it appears that the constant flow rate has a larger z-axis scatter than the transient counterparts. The lack of scattering in the transient results is an interesting observation which highlights the potential of using transient flow in achieving higher throughput microfluidic particle focusing. Furthermore, despite the trends observed in previous studies [167], the transient results show some additional horizontal spreading in the y-axis. The use of transient flows results in a short-term increase in the relative velocity of the particles, thereby increasing the lift force as defined in equation 7. The observed difference is hypothesized to be due to the created slip velocity and the resulting slip-shear effects. Moreover, the difference could be caused by introduction of transient forces in the lateral direction and changing the balance of main lateral forces including the shear gradient and wall forces. We ! 74!hypothesize that the combination of more spreading in the y-axis and less spreading in the z-axis in the transient data can be attributed to the larger lift force due to the increased relative velocity term. This increased lift force affects the overall balance of the forces as described in equation 6, which in turn explains the new inertial focusing positions.  4.3.1.2 Effects of Density Difference with Transient Inertial Focusing As described by equation 11 from Section 2.1.5, transient inertial focusing theory involves the relative velocity terms found within the shear-slip lift force, which is related to the density of both fluid and particles in the system. As a proof of concept and means of verification, this study was used to determine the effect of altering the density difference while using transient flows on particle focusing positions in straight channels. For these experiments, a (PDMS) microfluidic chip with a straight rectangular (100 µm wide x 85 µm high) channel design was used as in the previous experiments [145]. To explore density effects on transient flows, three test solutions were made: (1) DI water and PS beads, (2) DI water with PLGA beads, and (3) 80% ethanol, 20% DI water and PS beads (see Table 3.2 for density differences). All beads used were approximately 15 µm in diameter. To evaluate the density difference effects across a variety of flow conditions, each solution was tested at three Reynolds numbers (Re = 16.52, 22.95 and 32.12). These are the same Reynolds numbers chosen for the steady-state study and single-cycle transient studies. Transient conditions were created in the same way as in the single cycle study. Particles maintained a relative velocity to the flow for a short period of time (see Figure 4.14). Data sets of several thousand beads were collected for each solution/particle choice and each Re using a high-speed camera. The vertical position of ! 75!each bead was determined using defocusing-based particle tracking as in previous work [110, 152].  Post-processing of the image sets into 2D histograms (Figure 4.16) and consequently bar graphs (Figures 4.17 and 4.18) revealed trends in particle density peak positions and peak widths for each solution/particle (density difference) choice and Reynolds number. The particle positions are measured at a particle entry length LE = 2 cm away from the inlet, or LE/DH ~ 220, where DH is the hydraulic diameter. Based on the results from previous studies, and also our own calculations (LE = 4 cm) this entry length is not sufficient for achieving stable focusing positions [100]. Therefore, the observed focusing positions are not the final equilibrium positions, and the particles are still continuing their lateral motion.   Figure 4.16. Particle Distribution Histogram Illustration. Histograms were developed from the particle distribution plots as shown in Figure 4.15, for both channel width and height and across all solution/particle choices and Re [168].  ! 76! Figure 4.17. Particle Distribution Peak Width and Position Plots, comparing effects of density mismatch due to variation in particle density. 0% ethanol PS (Δρ = 0.05 g/mL) and 0% ethanol PLGA (Δρ = 0.3 g/mL) [168].   Figure 4.18. Particle Distribution Peak Width and Position Plots, comparing effects of density mismatch due to variation in fluid density. 0% ethanol PS (Δρ = 0.05 g/mL) and 80% ethanol PS (Δρ = 0.21 g/mL) [168].  We believe that the observed differences in Figures 4.17 and 4.18 are caused by different phenomena. While the difference in Figure 4.17 may be due to the change in the response time of the particle, the difference in Figure 4.18 is mainly due to the changes in the magnitude of the slip-shear lift force acting on the particles (FSS!∝ ρf) [92] and the resulting changes in the dynamics of the lateral motion of the particles. The presented analysis of transient inertial flows in microfluidic channels highlights the effect of density difference on the particle positions in the channel. Experimental results have verified the changes in particle positioning due to increasing density difference in transient regimes.  ! 77!4.3.2 Multi Cycle Transient Study As a next step to the single-cycle transient study conducted, a multi-cycle study was designed. This study was developed to determine if including multiple cycles of transient flow would further exaggerate changes in particle focusing positions compared to steady-state cases. We hypothesize that since the inertial forces are related to non-linear Navier-Stokes equations, any resulting displacement of focusing positions from steady-state results found during a single cycle would remain intact during further constant flow (due to the non-reversible nature of Poiseuille flow as described by Segre and Silberberg [94]). Therefore, we hypothesize that a continuous cyclic transient flow should move the particles into focusing positions differing from both steady-state and single-cycle transient results.  The experimental setup for the multi-cycle study differs significantly from the single cycle study, mainly due to the necessity for a quick, reproducible actuation of pressure control within the microfluidic device. The Fluigent system used in the single-cycle study has a relatively slow response time, especially once the flow is ‘stopped’ using an opposing flow (this response time varies between 40-100 ms depending on initial pressure conditions). Also, once ‘stopped’, the pressure in the channel may not reach completely zero when the next wave of pressure is pulsed through. Therefore, a different system was developed to ensure a rapid cycle time with complete on/off pressure regulation. Previously, a group within Dr. Cheung’s lab built a pressure control setup with multiple valves in order to create multi-channel flow profiles through sequential valve pressure cycling for use with optical resonators [169]. The setup used for this work was re-purposed to create a single-valve multi-cycle setup used in this study. The original project used a printed circuit board (PCB) with seven octal solenoid drivers (ST Microelectronics, Geneva, Switzerland, part number ! 78!L9822E) which were connected to an Arduino Uno microcontroller that communicated with three of the drivers at any one time. Using serial peripheral interface (SPI), each driver transmitted data to several pneumatic solenoid valves (Pneumadyne, MN, USA, model 3/2 N.C., part number N375.3). This hardware was controlled via a graphical user interface (GUI) through MATLAB as well as Arduino. In order to use only one valve, the MATLAB GUI was eliminated from the overall design, and control was provided through the Arduino Serial Monitor only. Several edits were made to the Arduino code to eliminate redundancies of operation of multiple valves. The code runs in a simple binary conversion system. The code assigns a driver several decimal numbers, which are then converted into binary. Each bit of the binary number corresponds to a single valve. If the bit is 0, then the valve is set as off, and if the bit is 1, then the valve is set as on. For example, if the driver were assigned the decimal number 3, which has a binary equivalent of 00011, the first two valves would be set as on, and the next three valves would be set as off. We manipulated the values assigned to the driver in order to only turn a single valve on and off at a user-defined time interval. Appendix D below shows a portion of the Arduino code used. The hardware setup was simplified to the PCB, Arduino Uno, and a single valve. Pressure was regulated through the same pressure source as the other studies (found in the lab), with the addition of a pressure gauge/controller (Parker Watts, OH, USA, R25-02AK). Finally, the high-speed camera was used to acquire images of the beads as in the single cycle study. Figure 4.19 below shows a flow diagram of the setup.   ! 79! Figure 4.19. Diagram of multi-cycle study setup. Red lines represent data/computer interactions, and blue lines represent physical/pressure interactions.  4.3.2.1 Multi Cycle Results The multi-cycle transient study included three Reynolds numbers for rectangular channels: 16.52, 87.02, and 112.02. These Reynolds numbers were chosen for two reasons: 16.52 was chosen previously and provides comparison to other studies, and the other Re are higher values than previously used, in order to increase the overall perspective of the studies in this work and determine the effects of transient conditions for high-Re flows. The cycle time for the valve was set to 20 ms (10 ms at Re = Remax and 10 ms at Re = 0). Figure 4.20 below is an example of the Reynolds number (velocity) profile as expected from the setup, and as measured experimentally using the 2 µm beads as in Section 4.3.1.1.   ! 80! Figure 4.20. Reynolds profile for Remax = 87.02. The profile is a square waveform with a period of 20 ms. Experimental results using the solenoid valve fit well, however there is slight delays during onset and offset of the pressure. Data was captured and collected across a time interval of approximately 1.3 seconds, however final results (Figures 4.21 and 4.22) were restricted to regions where the Re = Remax.  Results were collected in a similar way to the single cycle study, and images were post-processed in the same manner. Images were collected after approximately 1-5 seconds (50-250 cycles) from when the valve cycling began, for approximately 1.3 seconds (65 cycles). Data was extracted for sections where Re = Remax. Data was further separated by picking several cycles along the time scale. The results therefore compare steady-state focusing positions to positions found after a few (10-20) cycles, hereon known as ‘early cycle’ flows, and also to positions found after many (55-65) cycles, hereon known as ‘late cycle’ flows. Cross-sectional plots were created, and from these histogram plots of channel width and height as in the density study (Section 4.3.2.1). Particle density peaks in the histogram plots for early and late cycles were ! 81!compared when peak positions were within 10% of the total channel width of one another. Peak widths and positions were determined using full-width half-maximum (FWHM) of Gaussian curves overlaid on peaks in the histograms. The histogram results were then converted to bar graphs comparing early cycle and late cycle results as in Figures 4.17 and 4.18 and are shown in Figure 4.21 below for comparison.   Figure 4.21. Rectangular channel transient multi-cycle results. Early cycle data (in red) and late cycle data (in yellow) when compared to steady-state results. Includes peak positions and widths from histograms of both channel width and height (100 x 85 µm).  The results of this study show significant evidence of the effects of transient flows in a multi-cycle format. This is especially true for higher-Re flows. In the Re = 16.52 case, both early cycle and late cycle transient results show particle distribution peak positions and widths to be similar to steady-state conditions. However, as Re increases, these similarities become less apparent. For Re = 87.02, the differences between transient and steady-state conditions are more apparent, as is the difference between early and late cycle transient flows. Even more exaggerated results are ! 82!found for Re = 112.02. From the histogram plots of Re = 87.02 and 112.02 (Figure 4.22), it appears that the middle of the channel becomes an unavailable focusing position for particles in transient flow conditions, especially for higher-Re flows, with preference towards a 4-region equilibrium along the channel width. This is significant since this allows for manipulation of particles to only annulus positions even at higher-Re, excluding central focusing.     Figure 4.22. Rectangular channel histogram for multi-cycle transient particle focusing. Steady-state (blue), early cycle (red), late cycle (yellow) data was collected and converted into a 2D histogram to determine particle density peak locations and widths. Bin size of 1 x 0.85 µm (100 bins). Results emphasize the difference in peak positions, especially for steady-state compared to transient cases at Re = 112.02. The y-axis of each graph refers to the count of particles within each bin, normalized to the maximum number. !  ! 83!Chapter 5: Conclusions & Future Work    5.1 Results Summary Simplification of current techniques is a hallmark of engineering and design. Particle analysis in microfluidic devices has reached exciting possibilities with three-dimensional tracking, however most systems available are complex in either hardware or theory. In this work, we have chosen to address this with a simplified approach to 3D particle tracking via a defocusing-based method. The method was developed using previous theoretical work [138] and calibration curves to define the relationship between particle vertical positions and apparent particle diameter found in images taken at a fixed focal plane. A single-camera approach was used to acquire images, and post-processing using known methods [157] was conducted to determine particle positions within large data sets. Error related to the calibration curve acquisition was discussed, including a section on peak-locking in image post-processing. It was determined that peak-locking is not present using our post-processing techniques. Several studies allowed for verification of this method. As a proof-of-concept example for the defocusing method, a gravitational force study in rectangular channels to determine the effects of gravity on non-neutrally buoyant particles in microfluidic flows was conducted. Results show that the defocusing-based method is accurate to within 1 µm of the predicted particle locations.  Following the gravitational force study, several inertial focusing studies were conducted using the defocusing-based method, for further verification as well as to investigate new regimes of inertial particle focusing. Sheathless flow focusing has garnered a great deal of attention in recent years [71, 92] due to its simplistic manipulation of particle positions in space within microfluidic devices. Firstly, a steady-state inertial focusing study was conducted, related to ! 84!previous work [5, 109] by determining the 3D particle positions within a rectangular or cylindrical channel geometry at a range of Reynolds numbers (i.e. flow rates). The results show similar trends to previous theoretical and experimental work, further verifying the defocusing-based method.  While proceeding with the steady-state studies, investigation of the theories related to inertial focusing revealed the effect of relative velocity on the inertial lift forces involved in the final equilibrium positions of particles due to inertial flows. This led to a new study of transient flow profiles, which increase the relative velocity of the particles relative to the flow and therefore, in theory, affect their final focusing positions. An initial study using a single pressure gradient cycle was conducted as a proof-of-concept, revealing the potential for new focusing regimes caused by the presence of a cyclic velocity profile in the flow. Along with the theoretical investigation of relative velocity, it was found that this velocity is related to both the particle and fluid densities. Therefore, a further single-cycle study was conducted using several particle/solution variations that further exaggerated the density difference between particle and fluid, thereby causing additional changes to the overall particle focusing positions in transient flows. Evidence of these changes is shown, confirming the relationship between density, relative velocity, and inertial lift forces within inertial focusing phenomena. Finally, a multi-cycle transient study was conducted with a new experimental design involving a cyclic flow profile pattern to further exaggerate the relative velocity term over time. The study was done using rectangular channels at a wide range of flow rates. This was hypothesized to affect the particle focusing positions when compared to steady-state and single-cycle transient conditions. Results show this to be the case, especially in a rectangular channel geometry and at high Reynolds numbers.  ! 85!5.2 Future Work This thesis has outlined a 3D defocusing-based particle tracking method for use in a wide variety of applications. It has also provided several verifications of this method, including studies of inertial focusing of particles and gravitational force effects. Finally, this thesis outlines new regimes of inertial focusing using transient flow profiles. With the introduction of these efforts, there is a significant opportunity for new research into each of the areas discussed. More specifically, improvements on the tracking method, as well as further investigation of the transient flow effects on equilibrium focusing positions of particles represent major avenues of future work. Finally, applications of both the tracking method and the transient inertial effects should be considered.  5.2.1 Refinement of Tracking Method Several improvements are recommended upon thorough examination of the defocusing-based tracking method described in this thesis. In this work, both fluorescent and bright field microscopy were used to examine particle flows. Specifically, bright field was chosen for all transient experiments as it was found that the intensity profile of particles flowing at high flow rates or rapidly stopping was very difficult to discern from the background intensity of the images. However, bright field microscopy offers its own challenges, namely in post-processing the images to remove light scattering artifacts due to the entire image area being illuminated. One remedy for these issues is the use of a high intensity light source during fluorescence microscopy experiments, such as a laser source with a low pulse length, resolution time and high repetition rate. ! 86!The image post-processing algorithm used in this work based on mean-square displacement (MSD) includes many filter and particle recognition steps and has proven to be computationally robust. However, in some cases, especially when particles overlap or remain in only a few image frames (as in high-Re flows), tracking becomes more difficult. New methods for particle tracking are continuously developed for image processing, including more sophisticated trajectory determination algorithms beyond mean-square displacement analysis. New filters, such as the Kalman filter [170], allow for progressive modeling of particle trajectories for specific flow profiles, allowing for a framework to be masked onto images to help analyze their potential motion patterns [171]. Time-averaging or phase-averaging techniques can also be employed to register unique signal-to-noise correlations and determine signal peaks within ‘fuzzy’ or incoherent image data [25]. Another consideration for future work with our tracking method is the quantifying of noise and abnormalities. In general, we did optimize the thresholding of our current system for noise removal, however, we did not intentionally add noise to our images to determine how effectively our algorithms can adhere to the filter used. Similarly, images with particles overlapping were discarded. A type of force optimization to determine how our codes handle these situations could be used to even further optimize our method.  Mean-square displacement methods alone are sometimes insufficient for analyzing active systems such as living cells. This work did not include cellular studies, however this application would be an obvious next step. MSD methods cannot reveal underlying mechanical responses of cells or the mechanisms that cause these active systems to move in unique ways. Moreover, MSD typically conceals segments of a total trajectory [172]. Therefore, development of supplementary tools for particle tracking have been developed [173]. The use of any of these ! 87!secondary algorithms along with or supplanting the MSD method used could lead to a more thorough investigation of unique particle focusing positions.  5.2.2 Evaluation of Transient Inertial Flows The hypotheses that structure the transient inertial flow studies in this work have been developed with a thorough investigation of fluid mechanics and the principles behind inertial flows already developed. However, this is an entirely new area of research that has yet to be subjected to rigorous self-evaluation or critical peer-reviewed analysis. The studies shown in this work reflect only preliminary explorations of the effects created when transient flows are introduced into microfluidic systems. In fact, there is very little work in general on microfluidics that include externally-applied changing flow profiles to create unique focusing positions. Therefore, a full review of the hypotheses expressed here should be conducted, in order to fully substantiate the experimental results and perhaps provide new avenues to explore in the future. Furthermore, the use of transient flows could be expanded or included in a variety of other microfluidic devices. For example, varying channel geometry (i.e. cylindrical channels, spiral or curved channels, or channels with other aspect ratios) or particle choice could result in new advantages. Exploring higher Re flows seems to exaggerate the transient effects, and therefore it is recommended that more research be conducted in this flow regime.  Finally, within the multi-cycle work, it is of interest to explore the transient region between Re = 0 and Re = Remax. This would be similar to the approach from the single-cycle study, where the maximum velocity difference between particles and fluid was determined and focusing positions found during these time regimes were investigated. This study was not conducted in this work as the amount of data collected at these time regimes was too small. As ! 88!notable from Figure 4.20, this time regime is on the order of several hundred microseconds, while the high-speed camera used captures frames approximately every 140 µs. Therefore, either a higher frame rate is necessary, or many more data sets at the same frame rate are required to reach reasonable statistical significance.      5.2.3 Applications Many applications are available for both the tracking method and transient flow profiles described in this work. Perhaps the most prominent example is biological applications. Investigating cellular focusing profiles using transient flows should be a top priority in future work. This includes the use of the defocusing-based tracking method with biological specimens. New parameters must be developed to account for the deformability and varying morphology of cells in using both the tracking method and in the investigation of any transient flow profiles. A foundation for this effort in establishing the effects of deformability has been made in steady-state inertial focusing [174, 175]. Applications for inertial focusing are already abundant. For example, a “CTC-iChip” has been developed through the Harvard Medical School and Massachusetts General Hospital which is presumably en route to commercialization by Johnson & Johnson. This device uses inertial focusing to sort rare CTCs from whole blood at 107 cells per second [176]. Another example is a platform developed at UCLA and commercialized by CytoVale which employs inertial focusing with deformability cytometry to offer new classes of biomarkers [177]. Finally, Clearbridge BioMedics, a spin-off from the National University of Singapore (NUS), has developed the ClearCell FX system, using inertial focusing in spiral channels to separate larger cancer cells from other blood components [178]. ! 89!The applications of both the tracking method and transient flows could be significant in fields such as flow cytometry, particle mixing, particle filtration and separation, chromatography, particle reactors, and PIV and PTV analyses. Furthermore, devices implementing the transient flows could vary from electrochemical mixers to cellular assays. In essence, the particle tracking method could be used in any circumstance where particles of a known size range must be located and tracked within space and time in a flow profile. 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Rep., vol. 3, 2013.    ! 104!Appendix A: DOF Calculation  !"# = !2Nc (1+M)M! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(22) ! = ! fNA !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(23) f = 20 mm, NA = 0.3 mm, c = 0.02 mm (standard), M = 10. Therefore, DOF = 293 µm.  Appendix B: 3D-printed Cylindrical Channel Study   An extensive study was conducted using the Objet 3D Printer (Stratasys, MN, USA). Several Solidworks designs were created for cylindrical channels, in order to determine the resolution of the printer, as well as if issues such as material spreading occurred during the printing routine. Results showed that for channel designs within the micron scale (<300-400 µm in diameter), the printer failed to produce simple straight cylindrical channel designs in a robust and reproducible fashion. Several designs, including ‘tall’ ellipses were used to try to take into account spreading issues, but reproducibility was difficult. Issues were also found when using different 3D printers (named ‘Agassi’ and ‘Sampras’ below). Some of the results are included here. Channel dimensions were measured with optical profilometry (Veeco WYKO NT9100 and DEKTAK 150, NY, USA). Experiments were designed by myself and executed by Anna Lee. All figures and graphs shown were created by Anna Lee. More details from the study are available upon request.   ! 105! Figure B-1. 400x400um channel. Left: Sampras 3D printer, right: Agassi 3D printer. Credit: Anna Lee.    Figure B-2. Height/Radius measurements for several cylindrical channel samples. ‘Row’ refers to the channel # (as 8 channels were made on each SolidWorks file/3D mold). Credit: Anna Lee. ! 106!  Figure B-3. Optical profilometry screen-capture for 250 µm diameter channel. Variability in channel width along channel length is clear. Credit: Anna Lee.    Appendix C: MATLAB Code for 3D Defocusing Particle Tracking Method  This is an example of the script created for use with the 3D tracking method. Two variables, the bpass variable and disk variable, were optimized in each study to threshold particles as effectively as possible from the background, avoid misshapen/overlapping particles, and generally obtain an approximated diameter as close to the observed diameter as possible from the original images.  %Script for particle tracking brightfield images   clear all clear memory close all ! 107!clc   %Applied pressure pr = 157;   %Constant to Transient switch image # imsw = 334;   %image # imnum = 700;   %bpass variable bp = 8;   %disk variable dv = 4;   %Import matrix of TIF files tifim = ls('*.tif');   %Import bkgd bg = imcomplement(imread(tifim(1,:))); bg = bg(:,:,1);   %Test image for particle picks img = imcomplement(imread(tifim(imnum,:))); img = img(:,:,1); ! 108!a = img-bg; figure,imshow(a); a1 = imopen(a,strel('disk',1)); figure,imshow(a1);   %Run filter and dilation/erosion on image b = bpass(double(a1),3,bp); figure,colormap('gray'),image(b);   c = imopen(b,strel('disk',dv)); figure,colormap('gray'),image(c);   %Find blob and check to see how fit is on original image d = im2bw(c); e = bwperim(d,8); a2 = a; a2(e) = max(max(a2))/5; figure,imshow(a2,[0 max(max(a2))/5]);   %Calculate centroid and diameter of blob f = regionprops(d,'Centroid','EquivDiameter'); cnt = cat(1, f.Centroid); hold on plot(cnt(:,1),cnt(:,2),'y*'); diam = cat(1, f.EquivDiameter); diam = 1.5*diam hold off   ! 109!%%   %Start matrix to hold cnt files after each loop A1 = []; A2 = [];   %Constant Data Set for i = 2:imsw i      %Import image img = imcomplement(imread(tifim(i,:))); img = img(:,:,1);   %subtract bkgd a = img-bg; a1 = imopen(a,strel('disk',1));   %Run filter and dilation/erosion on image b = bpass(double(a1),3,bp); c = imopen(b,strel('disk',dv)); d = im2bw(c);   %Calculate centroid and diameter of blob f = regionprops(d,'Centroid','EquivDiameter'); cnt = cat(1, f.Centroid); diam = cat(1, f.EquivDiameter); ! 110!diam = 1.5*diam;   [row,col] = size(diam); q = ones(row,1); p = i*q;   A1 = [A1;[cnt diam p]]; end   %Transient Data Set for i = imsw+1:length(tifim(:,1)) i      %Import image img = imcomplement(imread(tifim(i,:))); img = img(:,:,1);   %subtract bkgd a = img-bg; a1 = imopen(a,strel('disk',1));   %Run filter and dilation/erosion on image b = bpass(double(a1),3,bp); c = imopen(b,strel('disk',dv)); d = im2bw(c);   %Calculate centroid and diameter of blob ! 111!f = regionprops(d,'Centroid','EquivDiameter'); cnt = cat(1, f.Centroid); diam = cat(1, f.EquivDiameter); diam = 1.5*diam;   [row,col] = size(diam); q = ones(row,1); p = i*q;   A2 = [A2;[cnt diam p]]; end   %% %Data processing B1 = [A1 ((sqrt(2.527*(((A1(:,3)*10.^-6).^2)-(2.4427896*10.^-10))))*10.^6)+30]; for k = 1:length(B1)     if real(B1(k,5)) > 30         B1(k,5) = real(B1(k,5));     else         B1(k,5) = NaN;     end end B2 = [A2 ((sqrt(2.527*(((A2(:,3)*10.^-6).^2)-(2.4427896*10.^-10))))*10.^6)+30]; for k = 1:length(B2)     if real(B2(k,5)) > 30         B2(k,5) = real(B2(k,5)); ! 112!    else         B2(k,5) = NaN;     end end   figure hold on plot(B2(:,2),B2(:,5),'gx') plot(B1(:,2),B1(:,5),'rx') hold off   %%   %Write Average values to XLS xlswrite([num2str(pr),'.xls'],B1,1); xlswrite([num2str(pr),'.xls'],B2,2);  Code developed by Grier and Crocker function res = bpass(image_array,lnoise,lobject,threshold) %  % NAME: %               bpass % PURPOSE: %               Implements a real-space bandpass filter that suppresses  %               pixel noise and long-wavelength image variations while  %               retaining information of a characteristic size. ! 113!%  % CATEGORY: %               Image Processing % CALLING SEQUENCE: %               res = bpass( image_array, lnoise, lobject ) % INPUTS: %               image:  The two-dimensional array to be filtered. %               lnoise: Characteristic lengthscale of noise in pixels. %                       Additive noise averaged over this length should %                       vanish. May assume any positive floating value. %                       May be set to 0 or false, in which case only the %                       highpass "background subtraction" operation is  %                       performed. %               lobject: (optional) Integer length in pixels somewhat  %                       larger than a typical object. Can also be set to  %                       0 or false, in which case only the lowpass  %                       "blurring" operation defined by lnoise is done, %                       without the background subtraction defined by %                       lobject.  Defaults to false. %               threshold: (optional) By default, after the convolution, %                       any negative pixels are reset to 0.  Threshold %                       changes the threshhold for setting pixels to %                       0.  Positive values may be useful for removing ! 114!%                       stray noise or small particles.  Alternatively, can %                       be set to -Inf so that no threshholding is %                       performed at all. % % OUTPUTS: %               res:    filtered image. % PROCEDURE: %               simple convolution yields spatial bandpass filtering. % NOTES: % Performs a bandpass by convolving with an appropriate kernel.  You can % think of this as a two part process.  First, a lowpassed image is % produced by convolving the original with a gaussian.  Next, a second % lowpassed image is produced by convolving the original with a boxcar % function. By subtracting the boxcar version from the gaussian version, we % are using the boxcar version to perform a highpass. %  % original - lowpassed version of original => highpassed version of the % original %  % Performing a lowpass and a highpass results in a bandpassed image. %  % Converts input to double.  Be advised that commands like 'image' display  % double precision arrays differently from UINT8 arrays. ! 115!  % MODIFICATION HISTORY: %               Written by David G. Grier, The University of Chicago, 2/93. % %               Greatly revised version DGG 5/95. % %               Added /field keyword JCC 12/95. %  %               Memory optimizations and fixed normalization, DGG 8/99. %               Converted to Matlab by D.Blair 4/2004-ish % %               Fixed some bugs with conv2 to make sure the edges are %               removed D.B. 6/05 % %               Removed inadvertent image shift ERD 6/05 %  %               Added threshold to output.  Now sets all pixels with %               negative values equal to zero.  Gets rid of ringing which %               was destroying sub-pixel accuracy, unless window size in %               cntrd was picked perfectly.  Now centrd gets sub-pixel %               accuracy much more robustly ERD 8/24/05 % %               Refactored for clarity and converted all convolutions to ! 116!%               use column vector kernels for speed.  Running on my  %               macbook, the old version took ~1.3 seconds to do %               bpass(image_array,1,19) on a 1024 x 1024 image; this %               version takes roughly half that. JWM 6/07 % %       This code 'bpass.pro' is copyright 1997, John C. Crocker and  %       David G. Grier.  It should be considered 'freeware'- and may be %       distributed freely in its original form when properly attributed.     if nargin < 3, lobject = false; end if nargin < 4, threshold = 0; end   normalize = @(x) x/sum(x);   image_array = double(image_array);   if lnoise == 0   gaussian_kernel = 1; else         gaussian_kernel = normalize(...     exp(-((-ceil(5*lnoise):ceil(5*lnoise))/(2*lnoise)).^2)); end   ! 117!if lobject     boxcar_kernel = normalize(...       ones(1,length(-round(lobject):round(lobject)))); end    % JWM: Do a 2D convolution with the kernels in two steps each.  It is % possible to do the convolution in only one step per kernel with  %   % gconv = conv2(gaussian_kernel',gaussian_kernel,image_array,'same');   % bconv = conv2(boxcar_kernel', boxcar_kernel,image_array,'same'); %  % but for some reason, this is slow.  The whole operation could be reduced % to a single step using the associative and distributive properties of % convolution: %   % filtered = conv2(image_array,...   %   gaussian_kernel'*gaussian_kernel - boxcar_kernel'*boxcar_kernel,...   %   'same'); % % But this is also comparatively slow (though inexplicably faster than the % above).  It turns out that convolving with a column vector is faster than % convolving with a row vector, so instead of transposing the kernel, the % image is transposed twice. ! 118!  gconv = conv2(image_array',gaussian_kernel','same'); gconv = conv2(gconv',gaussian_kernel','same');   if lobject   bconv = conv2(image_array',boxcar_kernel','same');   bconv = conv2(bconv',boxcar_kernel','same');     filtered = gconv - bconv; else   filtered = gconv; end   % Zero out the values on the edges to signal that they're not useful.      lzero = max(lobject,ceil(5*lnoise));   filtered(1:(round(lzero)),:) = 0; filtered((end - lzero + 1):end,:) = 0; filtered(:,1:(round(lzero))) = 0; filtered(:,(end - lzero + 1):end) = 0;   % JWM: I question the value of zeroing out negative pixels.  It's a % nonlinear operation which could potentially mess up our expectations ! 119!% about statistics.  Is there data on 'Now centroid gets subpixel accuracy % much more robustly'?  To choose which approach to take, uncomment one of % the following two lines. % ERD: The negative values shift the peak if the center of the cntrd mask % is not centered on the particle.   % res = filtered; filtered(filtered < threshold) = 0; res = filtered;  Appendix D: Arduino Code for Multi-Cycle Study  This portion of code initiated the pumping, using the Arduino library “TimedAction” to regulate intervals of pumping. void Start_Pumping(int A_on,int B1_on,int B2_on,int B3_on,int A_speed,int B1_speed,int B2_speed,int B3_speed){   digitalWrite(CE,LOW);   shiftOut(D4pin,SCLK,MSBFIRST,D4);     shiftOut(D5pin,SCLK,MSBFIRST,D5_B1);     shiftOut(D6pin,SCLK,MSBFIRST,D6);     digitalWrite(CE,HIGH);   //shifto(D4pin,D5pin,D6pin,SCLK,MSBFIRST,D4,D5,D6);     delay(50);      ! 120!    TimedAction PumpitA = TimedAction(5,pumpitA);     TimedAction PumpitB1 = TimedAction(125/B1_speed,pumpitB1);     TimedAction PumpitB2 = TimedAction(125/B2_speed,pumpitB2);     TimedAction PumpitB3 = TimedAction(125/B3_speed,pumpitB3);     PumpitA.enable() ;     PumpitB1.enable() ;     PumpitB2.enable() ;     PumpitB3.enable() ;    while(Serial.read() != 125){ // removed the 1 == 1 in the while brackets      if(A_on == 1){ //       noInterrupts() ;        PumpitA.check(); //       interrupts();      }      if(B1_on ==1){ //       noInterrupts() ;        PumpitB1.check(); //       interrupts();  }      if(B2_on ==1){ //       noInterrupts() ;        PumpitB2.check(); ! 121!//       interrupts();  }      if(B3_on ==1){ //       noInterrupts() ;        PumpitB3.check(); //       interrupts();    }   }  }  This is the main section of the original code that was edited. void  (){        switch (kA){     case 1:     D4 = (D4 & 224) | 3;     kA = 2;     break;     case 2:     D4 =  (D4 & 224) | 17;     kA = 3;     break;     case 3: ! 122!    D4 = (D4 & 224) | 24;     kA = 4;     break;       case 4:     D4 = (D4 & 224) | 12;     kA = 5;     break;      case 5:     D4 = (D4 & 224) | 6;     kA = 1;     break;   }  This is the code used for this experiment. void  (){    switch (kA){     case 1:     D4 = (D4 & 224) | 1;     kA = 2;     break;     case 2:     D4 =  (D4 & 224) | 0;     kA = 3; ! 123!    break;     case 3:     D4 = (D4 & 224) | 3;     kA = 4;     break;       case 4:     D4 = (D4 & 224) | 0;     kA = 5;     break;      case 5:     D4 = (D4 & 224) | 7;     kA = 6;     break;      case 6:     D4 = (D4 & 224) | 0;     kA = 1;     break;   }  Appendix E: List of Variables  In the order as presented in the thesis body:  Re – Reynolds Number ρ – Density ! 124!Q – Flow Rate DH – Hydraulic Diameter µ - Viscosity H – Channel Height W – Channel Width Rec = Channel Reynolds Number a – Particle Diameter (sometimes noted as dp) v – Flow Velocity FD – Viscous Drag Force Fw – Wall-induced Lift Force Fs – Shear-induced Lift Force Cw – Coefficient of wall lift force Cs – Coefficient of shear lift force Uf – Relative Velocity between the particle and the fluid traveling through the centre of the particle Umax – Maximum Fluid Velocity U – Particle Velocity (steady-state cases, sometimes noted as up or Up) FΩ – Rotation-induced Lift Force m – Particle Mass (sometimes noted as mp) !! – Particle Velocity (transient cases) Fss – Slip-shear Lift Force Fam – Added Mass Force Fsg – Stress Gradient Force ! 125!ρf – Fluid Density CL – Coefficient of lift force !!"# – Relative Velocity between the particle and fluid ρp – Particle Density de – Apparent diameter of particle M – Magnification of lens λ – Wavelength of particle emission no – Refractive index (sometimes noted as n) NA – Numerical aperture of lens z – Vertical position of particle zw – ‘Working’ vertical position of particle nw – Refractive index of working medium ni – Refractive index of immersion medium zo – Initial vertical position ρ(r,t) – Location of particle over space (r) and time (t) WD – Working distance of lens CV – Coefficient of variation µf – Fluid Viscosity  Fg – Force of Gravity Fb – Buoyancy Force V – Particle Volume g – Acceleration due to gravity uz – Sedimentation velocity in the z-direction ! 126!u∞ – Sedimentation velocity yo – Initial position of particle in y-direction Δz – Change in z-position of particle relative to initial position Rep – Particle Reynolds Number R – Radius of cylindrical channel h – Half the channel height ν – Kinematic Viscosity LE – Entry Length  

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