PARTICLE-OBSTACLE INTERACTIONS AT LOW REYNOLDS NUMBER: IMPLICATIONS FOR MICROFLUIDIC APPLICATIONS. Sumedh R. Risbud

Transcription

1 PARTICLE-OBSTACLE INTERACTIONS AT LOW REYNOLDS NUMBER: IMPLICATIONS FOR MICROFLUIDIC APPLICATIONS by Sumedh R. Risbud A dissertation submitted to The Johns Hopkins University in conformity with the requirements for the degree of Doctor of Philosophy. Baltimore, Maryland October, 2013 c Sumedh R. Risbud 2013 All rights reserved

2 Abstract In this dissertation, we analyse the motion and interactions of a suspended (spherical) particle moving past a fixed (spherical or cylindrical) obstacle, as a means to better understand the underlying mechanisms of microfluidic separation techniques. The particle trajectory is characterised by an offset, which is defined as its distance from the line passing through the obstacle centre along the direction of motion far away from the obstacle. Our analysis reveals the existence of extremely small minimum surface-to-surface separation between the particle and the obstacle, even for offsets of the order of particle radius. We derive the mathematical relationship between the minimum surface-to-surface separation and the offset. Extremely short-range non-hydrodynamic interactions such as solid-solid contact due to surface roughness, electrostatic or steric repulsion, etc. play a crucial role in determining the fate of the particle trajectory downstream to the obstacle. We model the short-range interactions as a hard-wall potential and use the aforementioned mathematical relationship to define a critical parameter that depends on the range of the non-hydrodynamic interactions. The critical parameter, being of the same order of magnitude as the ii

3 ABSTRACT particle radius, serves as an experimentally measurable macroscopic indicator for the microscopic characteristic length-scale over which the non-hydrodynamic interactions are relevant. We use the critical parameter to understand the behaviour of particle trajectories in two contexts, viz. - a suspended particle negotiating a square array of obstacles and a suspended particle passing through a constriction. In the former case, we use the aforementioned theoretical framework to derive the condition necessary for the trajectories to be periodic in the square array. We show that the periodicity condition implies directional locking of the trajectories, wherein, a particle driven at a range of angles with respect to the principal axes of the obstacle array (forcing angles), exhibits the same periodicity (i.e., gets locked in the period). The theory also predicts sudden transitions from one periodicity to another, at sharply defined transition angles with respect to the lattice, leading to experimentally and computationally observed devil s staircase-like behaviour of the locking directions as a function of the forcing angles. This theory, therefore, highlights and corroborates the underlying mechanism of the microfluidic separation system known as deterministic lateral displacement. We also extend the application of such an obstacle array to a size-based bandpass filter, which forces particles within an arbitrary range of radii to traverse along a locking direction that is significantly different than that corresponding to the particles with radii outside the range. In the analysis of trajectory of a spherical particle passing through a constriction iii

4 ABSTRACT between a fixed obstacle and a plane wall, we investigate (computationally and experimentally) the effect of particle size and the aperture of the constriction on the critical parameter. We observe that the critical parameter increases with increasing particle radius and decreasing aperture, which is the underlying mechanism for the microfluidic separation technique called pinched flow fractionation. Furthermore, our experiments indicate that the increment in the critical parameter for a fixed particle radius, exhibits a non-linear, monotonically increasing dependence on the aperture, implying that the separation can be enhanced by decreasing the aperture of constriction. The sensitivity of the critical parameter with respect to variation in size of the aperture for a suspended particle driven by a flow is more pronounced than that for a particle driven by a constant force in a quiescent fluid. Increasing the magnitude of (particle and/or fluid) inertia leads to particle trajectories reaching closer to the obstacle, resulting in a larger critical parameter in the presence of the non-hydrodynamic interactions. (Sumedh R. Risbud) Thesis Advisor: Dr. German Drazer Primary Reader: Dr. Kathleen J. Stebe Secondary Reader: Dr. Joëlle Fréchette iv

5 Acknowledgments It takes a village to raise a child, they say. In no uncertain terms I deem it true when it comes to this work. It makes the task of acknowledging everyone involved in this project all the more difficult, the act of doing so superficial, but equally necessary nonetheless. I begin by expressing my immense gratitude to my advisor Prof. German Drazer, who has been an embodiment of the word mentor in its truest sense. Due to his patience, I have been able to sail through the five years of the doctoral programme quite smoothly. May it be an academic issue, or a personal one, it has always dissolved away within a single conversation with him. If it wasn t for my wife, Dr. Rohini Gupta, the leisurely pace of the life in the programme would not have been so. She made me forget that, the same period in the outside world had been turbulent and hectic. She has been a wonderful support system, or a back-end, as we call it. May she continue to be the pivot in my life, as she has been during these five years. I feel privileged to have had collegues like Dr. Jorge A. Bernate and Dr. Raghavenv

6 ACKNOWLEDGMENTS dra Devendra, as well as a flat-mate like my highschool-friend Dr. Purushottam D. Dixit. Their help and presence on many occasions during the five years has proven to be priceless. I should highlight the way my life has been enriched by many technical discussions I have had with Rohini, Purushottam, Jorge, Raghavendra as well as Dr. Safir Merchant, Dr. Daniel Beltran, Mr. David Broesch, Mr. Robert Clay Wright, Mr. Elad Firnberg, Mr. Christian Pick, Mr. Anjor Kanekar and Mr. Harshavardhan Agashe on statistical mechanics, fluid dynamics, microfabrication, international politics, beer brewing, swimming, nature of American academic system, advisors, world history and religion. I wish to thank Prof. (technical or otherwise). Dilip Asthagiri for all the advise, help and discussions He has been a second advisor to me during my stay at Hopkins. I also thank the friends I made during this time in Baltimore, Mr. Aditya Nayak, Mr. Ambhi Ganesan, Mr. Krishna Praneeth Kilambi, Mr. Amod Jog, Mr. Hari Menon, Dr. Jatinder Singh Randhawa, Dr. Prabha Shalini Raman, Ms. Harleen Saini, Ms. Shruthi Ramkumar, Ms. Shreya Saxena, Ms. Ramya Gopal, and Mr. Salman Mukhtar, for making Baltimore a city worth remembering, no matter where I go next. I enjoyed (literally) thousands of s exchanged over past three years, on the same thread, between a group of old friends, Dr. Onkar Dalal, Dr. Varun Kanade, Dr. Mandar Gadre, Dr. Gayatri Natu, Ms. Gireeja Ranade, Dr. Purushottam Dixit, Mr. Anjor Kanekar and Mr. Karthik Shekhar. In addition, since vi

7 ACKNOWLEDGMENTS a healthy portion of my work and socialization occurred in Carma s Cafe, One World Cafe, and Charles Village Pub, I feel indebted to these fine establishments around Hopkins. The administrative staff at the department has been very coopertive over the years. I would especially like to thank Ms. Lindsay Spivey, Ms. Susannah Porterfield and Ms. Caroline Qualls, for securing a very crucial part of the student life, especially for an international student. Finally, I would like to thank Prof. Andrea Prosperetti, Prof. Kathleen J. Stebe, Prof. Robert Leheny, Prof. Joelle Fréchette, Prof. Michael Bevan, Prof. Zachary Gagnon for devoting time to critique this doctoral work and serving as the thesis committee. vii

8 Dedication To the elders who taught me the importance of simplicity and hard work, my parents, r tn к (Mr. Ratnakar G. Risbud) and (Mrs. Sudha R. Risbud), my paternal grandfather, r (late Mr. Govind S. Risbud) and, my maternal grandparents, к (Dr. Vinayak G. Ranade) and (late Smt. Mrudula V. Ranade) viii

9 Contents Abstract Acknowledgments List of Tables List of Figures ii v xii xiii 1 Introduction A bird s-eye-view Salient findings The critical impact parameter b c Implications of b c for microfluidic applications I Particle-Obstacle Interactions: Theory 10 2 A brief survey of motion at zero Reynolds number Meaning of zero Reynolds number: fluid motion Motion of a suspended particle Trajectory and distribution of spherical particles around an obstacle Introduction Model systems and formulation of the problem Problem geometry and symmetries Mobility functions for the problem Mobility functions: near contact (ξ 1) Mobility functions: large separations (ξ 1) Steady-state distribution of particles around a fixed obstacle Asymptotic behaviour of the probability distribution near contact 40 ix

10 CONTENTS Asymptotic behaviour of the probability distribution at large distances Minimum surface-to-surface separation Eulerian approach: Conservation of the number of particles Lagrangian approach: Trajectory analysis Asymptotic behaviour of minimum separation near contact, ξ min Discussion and Summary Acknowledgements A Mobility fuctions A and B A.1 Spherical obstacle A.1.1 Constant force acting on the moving sphere A.1.2 A freely suspended sphere in a uniform ambient velocity field A.2 Cylindrical obstacle B Derivation of the equation for the minimum separation II Implications of Theory for Microfluidic Applications 68 4 Analysis of directional locking in a sparse array of obstacles Introduction System description, assumptions and abstractions Directional locking The periodicity-condition The first locking direction after [1, 0] The last locking direction before [1, 1] Design rules and separation resolution in DLD for simple staircase structures Design constraints and separation resolution in DLD Extension and application of theory Directional locking with noisy trajectories Size-based band-pass filtering with DLD Summary Irreversibility and pinching in deterministic particle separation Introduction System and particle trajectories Effect of pinching A discussion on the effect of inertia A Methodology A.1 Materials x

11 CONTENTS 5.A.2 Characterization of the particles trajectories B Measurement of b in and b out C Relationship Between b c and Particle Reynolds Number Analysis of the trajectory of a sphere through a geometric constriction I Introduction System definition and parameter space Hard-core model for non-hydrodynamic interactions Stokes regime and the critical offset Inertia effects on the critical offset Results and implications of the hard-core model Effect of pinching in the Stokes regime Effect of inertia in the absence of pinching Effect of pinching and inertia Summary A An alternate perspective: the minimum separation as a function of the initial offset A.1 Pinching gap, inertia and minimum separation A.2 Minimum separation, critical offsets and non-hydrodynamic interactions Analysis of the trajectory of a sphere through a geometric constriction II Introduction The system The minimum separation and the model for non-hydrodynamic interactions Results The fixed offset simulations The hard-wall potential model and the fixed offset simulations The fixed minimum separation simulations Size-based separation Summary Conclusion and summary 164 Bibliography 167 Vita 181 xi

12 List of Tables 3.1 Coefficients b i and γ i from (3.23) and (3.24). Constant force. Spherical obstacle Coefficients b i and γ i from (3.23) and (3.24). Uniform velocity field. Spherical obstacle Coefficients k 0 and k 1 from Uniform velocity field. Spherical obstacle xii

13 List of Figures 3.1 Schematic representation of the problem. (top) The small circle of radius a on the left represents the moving sphere, with a corresponding incoming impact parameter b in. The circle of radius b with its centre at the origin of coordinates represents the fixed obstacle (either a sphere or a cylinder). The empty circle represents the position of the suspended particle as it crosses the symmetry plane normal to the x axis. The surface-to-surface separation ξ and its minimum value ξ min are also shown. (bottom) Representation of the conservation argument used to calculate the minimum separation (see 3.4). The unit vector in the radial direction d is shown, as well as the velocity components in the plane of motion. The surfaces over which the flux is conserved are shown, both far upstream (S ) and at the plane of symmetry (S 0 ). The geometry is shown for the cases of a spherical and a cylindrical obstacle as indicated Excess probability distribution (p(ξ) 1) as a function of the dimensionless surface-to-surface separation, for radius-ratio β = 1. (top) Comparison between both flow cases for a spherical obstacle with n = 3 and a cylindrical obstacle. (bottom) Comparison between both flow cases for a spherical obstacle with n = 2 and a cylindrical obstacle Excess probability distribution (p(ξ) 1) as a function of dimensionless surface-to-surface separation, for different aspect ratios β, a constant force and a spherical obstacle (n = 3). Inset: Enlarged view for large values of ξ, depicting symmetric behaviour about β = 1 for a fixed value of ξ, i.e., filled and open symbols almost overlap Exponents of ln(ξ 1 min ) from (3.12), (3.22) and (3.24) as a function of β for both cylindrical and spherical obstacles. The exponent for the case of a uniform velocity driving a particle past a sphere ( and ) is evaluated at discrete values of β = 1/8, 1/4, 1/2, 1, 2, 4, 8, as explained in the text xiii

14 LIST OF FIGURES 3.5 Minimum separation ξ min versus incoming impact parameter b in. Comparison between particle-particle simulations (Frechette & Drazer, 2009) and the analytical result given in (3.20): different symbols depict the simulation data as described in the legend, while the solid lines represent (3.20). (left) Average radius, (a + b)/2, is used to nondimensionalize the axes, (right) radius of the obstacle is used to nondimensionalize the axes, ξ = [(1 + β)/2β]ξ, (inset) Enlarged view of the interval (0.5, 1.5) of b in Minimum separation ξ min versus incoming impact parameter b in for different geometries and flow-fields when the particle and the obstacle are of the same radius. The incoming impact parameter is made dimensionless using the average radius, (a + b)/2. As explained in the text, for a cylindrical obstacle, the minimum separation decreases for a given incoming impact parameter in case of a confined system Open and filled symbols represent approximate scaling (3.24) compared with full governing relation (3.20) when a particle is driven by a constant driving force past a spherical obstacle for radius-ratios β = 1/8, 1/4, 1/2, 1, 2, 4 and 8. (#) represents a particle being driven past a cylindrical obstacle by a constant force, and (*) represents a particle being driven by a uniform velocity field past a spherical obstacle, for β = 1. The multiplicative constant λ 1 is determined by fitting the data generated with (3.20) for small values of ξ. The fitted values are 1.91, 1.88, 1.84, 1.86, 1.99, 2.33, 2.08 for β = 1/8, 1/4, 1/2, 1, 2, 4, 8, respectively, in the case of the open and filled symbols. For a cylindrical obstacle (#) we obtained λ 1 = 1.58 and for the case of uniform flow (*) λ 1 = (a) A spherical particle of radius a negotiating a portion of a square array of obstacles of circular cross-section with radius b [adapted from (Frechette & Drazer, 2009)]: the length of a unit-cell is l, the driving field F oriented at an angle θ as shown, drives the particle through the array. The principal lattice-directions are indicated with Cartesian axes X and Y. (b) A few example particle trajectories exhibiting directional locking [adapted from (Frechette & Drazer, 2009)]: results of Stokesian dynamics simulations with a = b, l = 5a and the range of non-hydrodynamics interactions ϵ = 10 3 (see 4.2 for a discussion on non-hydrodynamic interactions). Counter-clockwise, from X-axis to Y -axis, the trajectories can be seen to be locked in directions [1, 0], [3, 1], [1, 1], [2, 3] and [1, 2] (the inset shows the migration directions). The dot-dashed lines are to guide the eye and highlight [3, 1] and [2, 3] locking directions xiv

15 LIST OF FIGURES 4.2 Qualitative assessment of the dilute assumption: a schematic comparing an exact and an approximate trajectory around two successive obstacles. See the text for description of b in, b out and ξ min. The approximate trajectory is constructed as a union of a straight incoming part maintaining a constant b in, a circular region with radius r 0 = (a + b)(1 + ξ min ), and another straight outgoing part maintaining 2 a constant b out. Thus, the essential features of the exact trajectory are preserved by construction (a)[adapted from (Balvin et al., 2009)] Three kinds of particle trajectories in the presence of short-range repulsive non-hydrodynamic interactions. (b) Depiction of abstraction of the physical particle-obstacle system to a system in which a point-particle traverses past an obstacle of radius b c. The outgoing part of the trajectories with b in < b c is tangent to the obstacle in the abstract system Schematic depicting three possibilities leading to periodic trajectories (see text). In (a) and (b), the trajectories repeat after p obstacles along X-axis and q obstacles along Y -axis. In (c), the period along X-axis is (p 1 + p 2 ) and that along Y -axis is (q 1 + q 2 ) Migration direction (tan α) versus forcing direction (tan θ) portraying devil s staircase-like structure representing directional locking. The empty circles represent individual particle-particle simulations under the dilute approximation, the line represents the solution of (4.6), [p, q], such that the integer pair [p, q] is the closest integer pair to [0, 0]. The filled circles with error bars correspond to the data from microfluidic experiments (Devendra & Drazer, 2012) (a) 1-step staircase with transition [1, 0] [1, 1] at θ 1 (b) 2-step staircase [1, 0] [2, 1] [1, 1] with transitions at θ 1 and θ Three combinations of the simplest staircase structures possible: (a) one particle exhibits 1-step staircase, the other exhibits 2-step staircase, with the former transition lying between the two transitions of the latter, (b) and (c) both particles exhibit 2-step staircase structures Migration direction (tan α) as a function of forcing direction (tan θ). The horizontal steps represent the staircase in the deterministic case. (a) Results of computing the staircase for noisy trajectories according to (4.6) and (4.17) (see text). (b) Numerical solutions of steady state Fokker Planck equation with periodic boundary conditions for a point particle, reproduced from (Herrmann et al., 2009) xv

16 LIST OF FIGURES 4.9 Underlying mechanism of band-pass filtering. The dimensionless critical parameter b c /b as a function of the dimensionless particle radius a/b obtained from experimental data for liquid drops (Bowman et al., 2012). The inter-obstacle spacing l = 5b. Horizontal lines correspond to the values of b c yielding transition from [1, 0] as well as transition to [1, 1], as shown. (a) For forcing angle θ = 24.5 < arctan(1/2) and a > a 1, [p, q] = [1, 0] is the integer pair closest to [0, 0] that satisfies (4.6). Thus particles with radii a > a 1 satisfy l sin θ b c, and are locked in [1, 0], as shown in the bottom plot (tan α versus a/b). (b) For forcing angle θ = 28.5 > arctan(1/2), particles with radii a 1 < a < a 2 satisfy l(cos θ sin θ) b c and those with a 2 < a satisfy l sin θ b c. Consequently, the bottom plot shows a narrow band locked in [1, 1]-direction (a) Experimental setup, (b) illustration of reversible (b in = b out ), critical (b in = b c ), and irreversible (b in < b c ) trajectories for a given particle (the dashed circle represents excluded volume) past an obstacle (filled circle), and (c) measurement of b out as a function of b in. The inset illustrates the determination of b c. In (c), circular and square symbols represent delrin particles with d = 3 mm and 6.35 mm, respectively (a) Critical impact parameter, b c, for delrin particles as a function of the width of the pinching gap, D. Particle diameter d is in mm. (b) Linear relationship between b c (corrected by the particle radius) and relative pinching d/d. In both panels, the data points correspond to average and standard deviation of 5 independent measurements. Dashed lines are to guide the eye Critical impact parameter, b c, for steel particles as a function of the width of the pinching gap, D. Particle diameter d is in mm. Dashed lines are to guide the eye Critical impact parameter, b c, as a function of particle size for delrin particles in fluids with different viscosities. Open and filled symbols represent particles with Reynolds number less than and larger than 1, respectively (see Figure 5.6 in appendix 5.A) Measurement of b out as a function of b in. Circular, square, and triangular symbols represent particles with d = 3 mm, 6.35 mm, and 7.18 mm, respectively. The inset illustrates b out < b in (deviation from a slope of 1) for stainless steel particles with d = 6.35 mm and 7.18 mm b c as a function of the Reynolds number. For each type of symbol, from left to right, symbols represent measurements in fluids with kinematic viscosity 350, 28 and 15 cst xvi

17 LIST OF FIGURES 6.1 (a) System, simulation box and boundary conditions. The schematic is to scale, i.e., the box is 16 obstacle/particle radii in length. The top wall creates the pinching gap. In the absence of walls, we use the periodic boundaries for the box, and the obstacle is at the centre of the box. (b) The part of the simulation box indicated with dashed lines in (a) is enlarged here. The initial offset b in, the final offset b out and the aperture of the pinching gap D are indicated. The dashed circle depicts the position of the particle when the particle-obstacle separation is minimum (in the absence of inertia). (c) Enlarged view of the particle (dashed) and the obstacle (solid) surfaces showing the minimum separation aξ min between them Trajectories corresponding to the three possible types of particle-obstacle collisions, adapted from Balvin et al. (2009). The particle travels from left (incoming half of the trajectory) to right (outgoing half of the trajectory). The dashed circle represents the excluded volume inaccessible to the particle centres In presence of significant inertia (in this particular case, particle inertia, St = 1.11), the trajectories are asymmetric even in the absence of non-hydrodynamic interactions. Such asymmetry leads to two critical offsets, viz.- b c,in, b c,out as shown Various trajectories with b in = 2.0, exhibiting the effect of pinching in the Stokes regime (Re = and St = ). The vertical dashed lines at x ±7a represent the initial and final x-coordinates of the particle, respectively Various trajectories with b in = 2a, exhibiting the effect of inertia in the absence of pinching. The vertical dashed lines at x ±7a represent the initial and final x-coordinates of the particle, respectively The dimensionless surface-to-surface separation between the particle and the obstacle, as a function of the dimensionless x-coordinate of the particle. The plot shows the combined effect of particle and fluid inertia on the dimensionless separation along particle trajectories for b in = 2a, in the absence of pinching. The vertical dashed line depicts the symmetry plane of the system at x = Effect of reducing the aperture the of pinching gap (d/d ) on the minimum surface-to-surface separation between the particle and the obstacle, for different magnitudes of inertia and a fixed initial offset, b in = 2a The final offset b out (corresponding to b in = 2a) as a function of Stokes (top) and Reynolds (bottom) numbers, for different apertures of the pinching gap xvii

18 LIST OF FIGURES 6.9 The critical final-offset b c,out corresponding to ϵ = 0.3, as a function of Reynolds and Stokes numbers, in the presence and the absence of a pinching wall The minimum separation ξ min as a function of the initial offset b in, Re and St are negligible The minimum separation ξ min as a function of the initial offset b in, in the absence of pinching. (Left) effect of particle inertia when fluid inertia is negligible, (right) effect of fluid inertia when particle inertia is negligible The minimum separation ξ min as a function of the initial offset b in showing the effect of significant fluid as well as particle inertia in the absence of pinching The dimensionless surface-to-surface separation ξ as a function of the dimensionless x-coordinate of the particle. The x-coordinate at which the minimum separation is attained along a trajectory, varies with inertia. (Left) Effect of particle inertia, (right) effect of fluid inertia. The minimum separation is attained before the x = 0 symmetry-plane in both cases The effect of inertia on the ξ min b in relationship in the presence of a pinching wall. The aperture of the pinching gap is 3d (i.e., d/d = 0.333). (Left) The effect of particle inertia, (right) the effect of fluid inertia The effect of inertia on the ξ min b in relationship, in the presence of pinching. The aperture of the pinching gap is 1.7d (i.e., d/d = 0.588). (Left) The effect of particle inertia, (right) the effect of fluid inertia Combined effect of the pinching wall, particle and fluid inertia on the ξ min b in relationship. (Left) the aperture of the pinching gap is 3d (i.e., d/d = 0.333), (right) the aperture is 1.7d (i.e., d/d = 0.588) The outgoing offset b out as a function of the incoming offset b in. (Left) effect of particle inertia, (right) effect of fluid inertia The minimum separation as a function of the final offset, (left) effect of particle inertia, (right) effect of fluid inertia Effect of both significant fluid as well as particle inertia on the ξ min b out relationship, in the absence of pinching The individual effect of particle and fluid inertia on the ξ min b out relationship in the presence of pinching. The aperture of the pinching gap is 3.0d (i.e., d/d = 0.333). (Left) The effect of particle inertia, (right) the effect of fluid inertia The individual effect of particle and fluid inertia on the ξ min b out relationship in the presence of pinching. The aperture of the pinching gap is 1.7d (i.e., d/d = 0.588). (Left) The effect of particle inertia, (right) the effect of fluid inertia xviii

19 LIST OF FIGURES 6.22 The effect of inertia on the ξ min b out relationship, in the presence of pinching for two different apertures of the pinching gap. (Left) The aperture of the pinching gap is 3.0d (i.e., d/d = 0.333), (right) the aperture is 1.7d (i.e., d/d = 0.588) Simulation box and relevant length scales. (a) The simulation box with the coordinate system. Two parallel walls perpendicular to the Y -axis form a channel in all simulations. (b) This is an enlarged view of the relevant part of the simulation box. (left) In set I simulations, the initial position of all particle centres is the same (as depicted by the concentric circles). The initial and the final offsets (b in, b out ) are indicated, along with the particle and obstacle radii and the size of the constriction aperture D. The dashed circle shows the position of the closest approach along the particle trajectory, in absence of significant inertia. (right) In set II simulations, the particles begin their motion just before the apex of their trajectories, and attain the same minimum surface-to-surface separation (as depicted by the internally tangential circles). Radii of the particle and the obstacle are a and b, respectively. (c) The minimum separation between particle and obstacle surfaces, as highlighted by a dashed box in (b) Three types of particle trajectories in the presence of non-hydrodynamic interactions in the Stokes regime. Reproduced from Balvin et al. (2009) The dimensionless minimum separation ξ min versus the dimensionless particle radius α: (a)a constant force drives the particles past the obstacle in a quiescent fluid, (b) a constant body-force-density acts on the fluid carrying the particle past the obstacle, emulating a fluid flowing under a constant pressure-drop The dimensionless surface-to-surface separation ξ between the particle and the obstacle as a function of the dimensionless x-coordinate. As mentioned in the text, two simulations for each particle size are carried out for the purposes of linear interpolation, such that both attain minimum separation approximately close to ξ min = 0.05, one less than 0.05, and the other greater than Four different line-styles correspond to the four particle sizes used in the simulations. The two groups of trajectories (one corresponding to ξ min > 0.05 and the other corresponding to ξ min < 0.05) can be clearly seen in the figure. This particular plot depicts the case when b = 20 lattice units, D/b = 2.9 and the particles are carried around the obstacle by fluid under the action of a constant force-density acting on the fluid. Both fluid as well as particle inertia are negligible xix

20 LIST OF FIGURES 7.5 The critical offset b c as a function of the particle radius. The critical offset is evaluated at ϵ = 0.05 by linear interpolation between two simulated trajectories for each particle size, leading to the error-bars depicted above. (a) A constant force drives the particles past the obstacle in a quiescent fluid, (b) a constant body-force-density acts on the fluid carrying the particle past the obstacle xx

21 Chapter 1 Introduction 1.1 A bird s-eye-view This work originates from the tacit notion that microfluidics has advanced swiftly within the last decade. The flows involved in microfluidics are typically Stokes flows, in that, the corresponding Reynolds numbers are negligible (with the exception of the recent field of inertial microfluidics, where Re O(1) O(10) may be encountered). The interaction of a micro-particle with an obstacle in its path determines the fate of its subsequent trajectory. This is the basis for microfluidic separation platforms, since the interaction is typically sensitive to size, shape, density or any other discriminatory property possessed by the particles. A particularly attractive micro-separation platform functions without application of any external field (such as electric or magnetic). Purely hydrodynamic as well as short-range non-hydrodynamic interactions 1

22 CHAPTER 1. INTRODUCTION (examples of the latter are: solid-solid contact due to surface roughness, repulsion due to electric double layer, steric hindrance, etc.) should play a dominant role in such stand alone platforms. The theoretical tools required for the analysis of such zero/low Re systems have existed for many decades. They were initially developed for the analysis of microstructure in dilute suspensions, estimation of suspension viscosity, and characterization of suspension rheology. In this dissertation, we apply the theoretical framework to microfluidic systems. For a better understanding of the mechanisms underlying the micro-separation techniques, the first part of this dissertation is dedicated to theoretical investigation of hydrodynamic interactions between a particle and an obstacle in its path. We model the particle-obstacle pair using theoretically tractable shapes, viz.- circular cross sections. Specifically, we investigate the motion of a suspended non-brownian spherical particle driven by a constant force or a uniform flow in an unbounded fluid, around a fixed spherical or cylindrical obstacle in the limit of negligible particle as well as fluid inertia. The particle trajectory begins parallel to the direction of the constant force or the uniform flow, far away from the obstacle. There exists an initial lateral offset between the particle trajectory and the coordinate axis passing through the centre of the obstacle and parallel to the driving force or the uniform flow. We derive expressions for the particle distribution around the obstacle as well as the distance of closest approach between the particle and obstacle surfaces. We find that the particle distribution is isotropic, a result similar to the one reported in the context of sheared 2

23 CHAPTER 1. INTRODUCTION dilute suspensions. Further, we see that the minimum separation attained along a particle trajectory decreases exponentially with the initial offset in the trajectory. We use a hard-wall repulsive core potential to represent the short-range repulsive non-hydrodynamic interactions, and relate the range of these interactions with the offset in the trajectory, through the minimum attained surface-to-surface separation. We observe that the fore-aft symmetry of the particle trajectories with an offset less than a critical value is broken in the presence of the non-hydrodynamic interactions. The analysis of such trajectories gives rise to the definition of a critical parameter, a macroscopic parameter characterizing the asymmetry in the trajectory. The critical parameter is a function of the particle and obstacle sizes and the range of the nonhydrodynamic interactions between them. It provides us with an experimentally tangible link to the microscopic range of the non-hydrodynamic interactions. The second part of the dissertation portrays the practical implications of the fact that the critical parameter is a function of the particle size and range of nonhydrodynamic interactions. We begin the second part by investigating a system consisting of a periodic planar square array of obstacles of circular cross section. A spherical particle negotiates such an array while being driven in a non-collinear (and perhaps incommensurate) direction with respect to the principal directions of the array. We have observed experimentally and numerically, that the particle traces a periodic trajectory even if the driving force is incommensurate with the lattice. The trajectory exhibits a fixed rational slope on average, with respect to the lattice axes, 3

24 CHAPTER 1. INTRODUCTION for a range of forcing directions. We call this phenomenon directional locking, akin to the mode locking behaviour observed in other dynamical systems subjected to periodic forcing. The lattice array can be approximated as being dilute or sparse for a sufficiently large lattice spacing, such that the particle interacts with only one obstacle at a time. Therefore, we can directly apply the concept of the critical parameter for an interacting particle-obstacle pair and show that macroscopic as well as microfluidic experimental data agrees well with the predictions of a theory founded upon the dilute assumption. We predict, and indeed observe, that due to the presence of the critical parameter and the broken fore-aft symmetry of particle trajectories even with negligible inertia, the average migration angle as a function of the forcing angle follows a functionality similar to a devil s staircase. The presence of such a functionality is a hallmark of mode locking behaviour. Further, size-dependence of the critical parameter leads to a different staircase structure for different sizes of particles. Therefore, one may expect different sizes of particles traversing different average migration directions through the same array of obstacles, albeit the forcing direction remains fixed with respect to the lattice axes. This separation mechanism is exploited in deterministic lateral displacement (DLD), a popular microfluidic separation technique. Our second model system captures the essential features of microfluidic systems that exploit the motion of a particle through a constriction. Examples include, pinched flow fractionation in micro-separations, some particle focusing methods and micromodels employed in studying clogging of microchannels in porous media. The 4

25 CHAPTER 1. INTRODUCTION model system consists of a spherical particle traversing a trajectory that passes through a constriction (pinching gap) created by a spherical (or cylindrical) obstacle and a plane wall. We observe that the particle reaches closer to the obstacle as the aperture of the constriction is decreased. In the presence of short-range repulsive non-hydrodynamic interactions, since surface-to-surface separations smaller than the range of interactions are unattainable, we infer that the critical parameter must increase as the particle tries to reach closer to the obstacle upon decreasing the aperture of the constriction. We observe experimentally as well as computationally, that the critical parameter increases with the particle size, thus founding the basis for sizebased separations. Furthermore, not only the critical parameter but the increment in the critical parameter is also dependent on the particle size. For a given size of the aperture of constriction and a fixed range of repulsive non-hydrodynamic interactions, the larger the particle size the more it feels the effect of the constriction. Therefore, the increase in the critical offset (upon decreasing the aperture) is more pronounced for a larger particle than a smaller one, thereby enhancing the size-based separation. We further explore the effect of small (but not negligible) magnitude of inertia on the distance of closest approach, and consequently the critical parameter. The focus of this investigation is the potential of the system to achieve separation based on particle density rather than size (note that the motion of a denser particle corresponds to higher magnitude of inertia). In the case without a constriction, we observe again, that the particles reach closer to the obstacle as the magnitude of inertia increases, 5

26 CHAPTER 1. INTRODUCTION which translates to an increasing critical parameter. However, we note that the foreaft symmetry of particle trajectories is broken. Furthermore, when the trajectories are subjected to non-negligible magnitudes of inertia in the presence of a constriction, we see that the influence of inertia is attenuated. Overall, more effective density-based separation can be achieved in the absence of a constriction, rather than its presence. 1.2 Salient findings The critical impact parameter b c 1 Before delving into the detailed theoretical derivations and numerical and experimental results mentioned above, we first list the important findings of this work. It can be gauged from the discussion thus far, that the most fundamental concept upon which this work rests, is the existence of a critical impact parameter, denoted by b c throughout this dissertation. Here we elaborate on the concept in depth. In chapter 3, we derive the following relationship between the initial offset in the particle trajectory (b in ) and the minimum separation attained by the particle from the obstacle (ξ min ): b in = (a + b)(1 + ξ min 2 ) exp A(ξ) B(ξ) ξ min (2 + ξ)a(ξ) dξ 1 The material presented in this subsection has been reproduced with some additions in 3.5, 4.2, 6.3.1, and 7.3 to preserve the continuity and completeness of the journal articles corresponding to the respective chapters. 6

27 CHAPTER 1. INTRODUCTION where, A(ξ) and B(ξ) denote the hydrodynamic mobility in two orthogonal directions in the plane of motion of the particle, as a function of the dimensionless surface-tosurface separation ξ between the particle (radius a) and the obstacle (radius b). The equation reflects the exponential dependence of the offset on the minimum separation, implying that extremely small minimum separations (ξ min 10 3 ) would be attained between the particle and the obstacle surfaces for moderate offsets (b in 1). We argue that the dependence of the critical parameter on the range of the non-hydrodynamic interactions ϵ can be obtained by relating the former with the offset b in in the particle trajectory and the latter with the minimum separation ξ min attained by the particle from the obstacle. The above relationship can be combined with a simple hard-wall model for nonhydrodynamic interactions, in which the inter-particle potential is approximated by a hard-wall repulsion with dimensionless range ϵ. Under this approximation, the range of the potential limits the minimum attainable separation between the particle and the obstacle, but does not affect the hydrodynamic interaction between them. In our problem, the model implies that, independent of the impact parameter b in, dimensionless separations smaller than ϵ are unattainable due to the hard-wall potential. On the other hand, the potential does not affect the particle trajectory downstream to the obstacle. Therefore, if b in = b c is the incoming impact parameter corresponding to ξ min = ϵ, any trajectory with b in < b c reaches the same minimum separation ϵ and collapses onto the outgoing trajectory corresponding to b in = b c. Thus, b c and ϵ are 7

28 CHAPTER 1. INTRODUCTION related through the same relationship as that between b in and ξ min : b c = (a + b)(1 + ϵ 2 ) exp ϵ A(ξ) B(ξ) (2 + ξ)a(ξ) dξ, The above equation can be used to compute the critical impact parameter, knowing the hydrodynamic mobility functions and the range of non-hydrodynamic interactions a priori. On the other hand, it can also be understood as the definition of the length scale ϵ quantifying the effective surface roughness, incorporating all shortrange repulsive non-hydrodynamic interactions (see chapter 3) Implications of b c for microfluidic applications In chapter 4, we apply the concept of the critical parameter to directional locking of particle trajectories in a square lattice of obstacles. If the lattice spacing is l, and a particle is driven at an angle θ with respect to one of the principal directions of the lattice, then we derive the following inequality that the system variables should satisfy as the necessary condition for a periodic particle trajectory: q cos θ p sin θ b c l, where, the integers p and q denote periodicities of the particle trajectory in the two principal directions of the lattice. The above inequality directly leads to the devil s staircase-like dependence of the average migration angle of the particle trajectory on 8

29 CHAPTER 1. INTRODUCTION the forcing angle θ. Since the critical parameter b c is a function of particle radius, particles of different sizes migrate in trajectories of different periodicities, leading to size-based separation. Chapters 5, 6 and 7 are dedicated to the investigation of a particle trajectory passing through a constriction created by a plane wall and an obstacle of circular crosssection. We have mentioned the key finding of this work that the critical parameter is observed (computationally as well as experimentally) to increase as the size of the particle increases, the aperture of the constriction decreases and/or the inertia associated with the particle motion increases. Therefore, separations based on size as well as density are possible. Furthermore, when inertia is negligible, the change in the critical parameter due to change in the size of the aperture itself is a function of the radius of the particle: increase in the critical parameter for a larger particle is greater than that for a smaller particle. Therefore, size-based particle separation is possible, and is observed to get enhanced by the presence the constriction. 9

30 Part I Particle-Obstacle Interactions: Theory 10

31 Chapter 2 A brief survey of motion at zero Reynolds number 2.1 Meaning of zero Reynolds number: fluid motion We know that the governing equation for fluid motion, in general, is the momentum balance per unit volume of the fluid (in other words, Newton s second law per unit volume of fluid), also otherwise known as the Navier-Stokes equation: u ρ t + u u = p + τ + f. Here, ρ is the fluid density, u is the fluid velocity, p is the pressure, τ is the shear stress tensor, and f are body-forces acting on the fluid per unit volume. For a Newtonian 11

32 CHAPTER 2. MOTION AT ZERO RE fluid above equation becomes, u ρ t + u u = p + µ 2 u, where, µ represents the viscosity of the fluid (a constant, since the fluid is Newtonian), and the body-forces arising from the gradient of a potential (such as gravity) have been incorporated in the first gradient term to yield a modified pressure. Further, we know that the equation can be rendered dimensionless as, where, Re = ρu cl c µ u Re t + u u = p + 2 u, is the Reynolds number for the problem at hand, U c and l c being the characteristic velocity and length, respectively. In the context of microfluidics, or even microhydrodynamics of suspensions, the characteristic length is, l c 10 5 m. Velocities typically do not exceed U c 10 3 m/s. Assuming the working fluid to be water, one gets Re In this case, the dimensionless equation can be simplified by neglecting the term with Reynolds number altogether (i.e., negligible fluid inertia), as a zeroth-order approximation, p + 2 u = 0 µ 2 u = p. This is the Stokes equation, the governing equation of fluid motion at zero Reynolds number. Coupled with the continuity equation for an incompressible flow, u = 0 and suitable boundary conditions, one specifies the problem completely. An important feature of the Stokes equation is the absence of an explicit dependence of the velocity or pressure fields on time. In absence of a pressure gradient (i.e., flows 12

33 CHAPTER 2. MOTION AT ZERO RE driven by the boundary condition, or a movement of the boundary), the equation is invariant under time-reversal, and the velocity fields satisfying this equation are called kinematically reversible. In the presence of a pressure gradient, one reverses the direction of the gradient along with time to observe the kinematic reversibility of the flow. 2.2 Motion of a suspended particle Consider a particle suspended in an unbounded quiscent fluid. Also suppose that fluid inertia can be neglected (Re 0), yielding a velocity field satisfying the Stokes equation. If the particle velocity is U, then the velocity field should satisfy the following boundary conditions, u = U... on the particle surface, u = 0... at infinity. Since the Stokes equations are linear in fluid velocity, one can propose (Happel & Brenner (1965), Ch. 5), u = V U and p = µp U for a tensor function V and a vector function P. Thus, the equation of motion (Stokes equation), continuity and 13

34 CHAPTER 2. MOTION AT ZERO RE the boundary conditions become: 2 V = P V = 0 V = I... on the particle surface, V = 0... at infinity. Here, I the identity dyadic. Note that, by construction, the functions V and P are independent of any fluid properties (density, viscosity, etc.) as well as the particle velocity U. Further, using the definition of these functions, the stress tensor can be written as, Π = pi + µ u + ( u) T = µ I P + V + T ( V) U. Here we define, P = I P + V + T ( V) as a triadic tensor for the sake of brevity. Therefore, in terms of P, the drag force acting on the particle can be written as, F drag = µ ds P U. Particle surface resistance tensor As mentioned in the equation above, the integral can be separated from the fluid property µ and the particle velocity U, to be defined as the resistance tensor R = ds P, making R a function of particle shape and size only. Using this notation, the drag on the particle can be succinctly written as F drag = µr U. 14

35 CHAPTER 2. MOTION AT ZERO RE The properties of the resistance tensor are well documented in the literature (Happel & Brenner, 1965; Kim & Karrila, 1991). The idea has been extended from translational motion of the particle, to rotational and coupled motions. With each mode of the particle motion, the resistance tensor grows in size. It can also be extended to a suspension of particles, as well as particles in a bounded domain. In each case, the resistance tensor remains a function only of the geometry, shapes and sizes of the particles and their relative configuration. For two particles under translation, rotation and shear, the grand resistance tensor R relates the drag forces, torques and stresslets acting on the particles to the particle velocities and the shear-field as (Kim & Karrila, 1991), F 1 F 2 T 1 T 2 S 1 S 2 v U 1 v U 2 Ω ω 1 = µr. Ω ω 2 E E Here, F j, T j, and S j are the force, torque and stresslet on particle j, whereas U j, ω j are its translational and angular velocities. The particles are placed in the ambient flow-field given by v = U + Ω x + E x. The previous expression can be rearranged to suit a particular problem at hand. For example, in chapter 3, for the case of a suspended particle moving around a 15

36 CHAPTER 2. MOTION AT ZERO RE fixed obstacle under the action of a constant force in a quiescent fluid, we have used U 2 = ω 2 = T 1 = S 1 = S 2 = v = 0, and F 1 = constant. In absence of a shear field, only translational and rotational parts of the grand resistance tensor remain. This part, relating the forces and torques to translational and angular velocities of the particles, is invertible. The inverse is called the mobility matrix. Specifically, for the previously discussed case of a single particle moving with velocity U in an infinitely unbounded fluid, we have, F drag = µr U U = 1 µ M F drag, where, M = R 1 is the mobility tensor for the particle. Equipped with the expression for the drag force, the equation of motion for a particle driven by a constant force F can be written as, m du dt = F + F drag = F µr U The equation above can be rendered dimensionless as, St du dt = F R U, where St is the Stokes number associated with the particle motion. For a spherical particle of radius a and mass m, moving through a fluid of viscosity µ with velocity U, the Stokes number is given by the ratio of the characteristic time for particle motion to that for fluid motion: St = m/6πµa a/u. Thus neglecting particle inertia (St 0), we get F = R U. Note that, in a mobility problem, F is the externally applied known force, and particle velocity is to be determined. Thus, we invert the resistance tensor to get, U = M F. We 16

37 CHAPTER 2. MOTION AT ZERO RE employ this formalism in It is interesting to observe that Stokes number being zero makes the particle motion kinematically reversible and without an explicit time dependence, similar to the motion of the fluid when Reynolds number is zero. We will invoke the argument of kinematic reversibility frequently in this work as follows: in practical microfluidic applications, the particle trajectories seldom exhibit kinematic reversibility, a tell-tale sign that interactions other than pure hydrodynamics in Stokes flow are at play (e.g., non-negligible inertia effects, short-range repulsive interactions like solid-solid contact due to surface roughness, electrostatic double layer, etc.). However, this insight is repeatedly lost in the explanations of particle behaviour in the literature, that are based on behaviour of fluid streamlines in the absence of particles. In summary, when we assume negligible particle as well as fluid inertia, we neglect the terms accompanying the Stokes and the Reynolds numbers in the equations of motion of a particle and the fluid, respectively. This leads to a linear system in the particle velocity, or the externally applied force on the particle. The particle trajectories can be then computed without resorting to solve for the fluid motion, if the resistance or mobility tensors for the problem are known. From the perspective of microfluidics, most of the relevant species can be modelled as being spherical in shape (e.g., blood cells, macromolecules, silica particles, polymer particles in a suspension). Therefore, in this work, we exclusively work with spherical particles. Fortunately, the resistance/mobility functions for spherical particles around spherical (and in some 17

38 CHAPTER 2. MOTION AT ZERO RE cases cylindrical) obstacles are well documented in the literature (Jeffrey & Onishi, 1984; Claeys & Brady, 1989; Kim & Karrila, 1991; Nitsche, 1996). 18

39 Chapter 3 Trajectory and distribution of spherical particles around an obstacle Introduction As remarked earlier, the interaction between suspended particles and obstacles encountered in their flow is essential for the understanding of the transport of particulate suspensions in natural porous systems as well as in engineered porous media used in different applications. In porous media, the behaviour of individual par- 1 Fair-use statement: This chapter is adapted from an article in Journal of Fluid Mechanics of the Cambridge University Press, [Risbud, S. R. & Drazer, G Trajectory and distribution of suspended non-brownian particles moving past a fixed spherical or cylindrical obstacle. J. Fluid Mech. 714, ]. Please refer to the end of the dissertation for fair-use license agreement. 19

40 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES ticles at the pore-scale determines the migration, dispersion and ultimate fate of suspended particles (Lee & Koplik, 1999). Lagrangian methods in filtration theory, for example, rely on calculating the trajectory of individual particles in simplified representations of the pore space (Spielman, 1977; Adamczyk, 1989a; Ryan & Elimelech, 1996; Jegatheesan & Vigneswaran, 2005). Particle capture by single cylindrical and spherical collectors is investigated to understand filtration in fibrous media and packed beds, respectively (Spielman, 1977). In many filtration methods, such as deep bed filtration, the particles are typically smaller than the characteristic scale of the pores, and get collected or deposited onto the obstacles in their path. In order to gain insight into these systems, several studies have focused on the trajectories followed by small particles near large spherical or cylindrical collectors (Adamczyk & van de Ven, 1981; Adamczyk, 1989a,b; Goren & O Neill, 1971; Gu & Li, 2002; Li & Marshall, 2007). Trajectory studies have also shown that not only particle capture but transit times may also be dominated by single particle-obstacle interactions (Lee & Koplik, 1999). Other studies modelling particle transport in porous media have focused on the similar problem of the trajectory followed by individual particles as they move through narrow channels, such as in the constricted-tube model (Burganos et al., 1992, 2001; Chang et al., 2003). Studies of particle migration, dispersion and capture using Eulerian methods also rely on calculating the detailed particle-fibre or particle-grain interactions in model porous media (Koch et al., 1989; Phillips et al., 1989, 1990; Shapiro et al., 1991; Nitsche, 1996). In this work, we investigate the prop- 20

41 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES erties of particle trajectories as the particles move past a fixed spherical or cylindrical obstacle driven by either a constant force or a uniform velocity field in the presence of particle-obstacle hydrodynamic interactions and for arbitrary particle-to-obstacle size ratios. Recently, with the advent of microfluidic technology, there is a renewed interest in the understanding and characterization of the motion of individual particles past solid obstacles. The possibility of designing the stationary media with nearly arbitrary geometry and chemistry, with features of micron and sub-micron size, has led to microfluidic separation techniques that are primarily based on the species-specific particle-obstacle interactions. In deterministic lateral displacement, for example, a mixture of suspended particles driven through a two-dimensional (2D) periodic array of cylindrical obstacles spontaneously fractionates as different species migrate in different directions (Huang et al., 2004). The migration angle depends on the particleobstacle interactions, and can be accurately described based on the trajectory followed by individual particles as they move past a fixed obstacle (Frechette & Drazer, 2009; Balvin et al., 2009; Herrmann et al., 2009; Bowman et al., 2012). Particle-obstacle interactions are also essential in separation methods based on ratchet effects induced by asymmetric arrays of obstacles (Li & Drazer, 2007). Other methods, such as pinched flow fractionation (Yamada et al., 2004) and some particle focusing systems (Hewitt & Marshall, 2010; Xuan et al., 2010) can also be investigated from the perspective of particle-obstacle interactions and the effect that both hydrodynamic and 21

42 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES non-hydrodynamic forces have on the trajectories followed by different species (Luo et al., 2011). In all cases, the individual particle trajectories or the asymptotic distribution of suspended particles as they are driven past spherical or cylindrical obstacles contains the information necessary for calculation of the relevant average transport properties, such as migration speed and angle. However, an analytical treatment describing these two aspects of particle motion has not been developed. In this work, we provide such an analysis, within the purview of low Reynolds number hydrodynamics, wherein Stokes equations are assumed to describe the motion of the fluid, and particle inertia is neglected. Further, we consider the non-brownian regime, since in many applications separation devices in particular high throughput is advantageous, leading to a routine occurrence of high Péclet numbers and the motion of the particles can be approximated as deterministic. In addition, the results derived here are of relevance to suspension rheology and flows, specially in active micro-rheology, where a probe particle is driven through a quiescent suspension of particles. In turn, we shall show that it is possible to extend analytical results established in suspension rheology to the motion of a spherical particle driven past a fixed obstacle. In suspension flows, each individual particle is moving through a random distribution of other spheres. Furthermore, in the dilute approximation, the analysis of the relative motion (and distribution) between any two spheres of the suspension suffices to characterize its properties. Such a relative motion is analogous to the problem of a sphere moving with respect to a fixed ob- 22

43 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES stacle in say, for example a microfluidic device. (Note that in the latter case the dilute approximation assumes not only a dilute suspension but also that the distance between obstacles is sufficiently large.) Therefore, before proceeding, it is convenient to briefly review some of the most relevant results concerning two-particle interactions in the limit of zero Reynolds and infinite Péclet numbers. Batchelor & Green (1972) first obtained the pair distribution function of spheres in a sheared suspension. Batchelor later extended his work by deriving the pair-distribution function in the case of a sedimenting polydispersed suspension of spheres (Batchelor, 1982; Batchelor & Wen, 1982; Batchelor, 1983). Davis & Hill (1992) used the resulting expression for the pair distribution function to analyze the hydrodynamic diffusion of a sphere sedimenting through a dilute suspension of neutrally buoyant particles. Almog & Brenner (1997) further employed the same pair distribution function to study the apparent viscosity experienced by a sphere moving through a quiescent suspension. They considered both the micro-rheological setting of a falling ball viscometer as well as a non-rotating sphere moving with a prescribed velocity through the suspension. Khair & Brady (2006) studied the motion of a single Brownian particle through a suspension, and extended the analysis of Batchelor (1982) to obtain the pair-probability distribution function for finite Péclet numbers. In this work, we consider the trajectory followed by a suspended particle driven by a constant external force (e.g. buoyancy force) or a uniform velocity field as it moves past a fixed sphere or cylinder in the limit of zero Reynolds and infinite Péclet 23

44 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES numbers. We derive an expression for the minimum particle-obstacle separation attained during the motion, as a function of the incoming impact parameter between the sphere trajectory and the centre of the obstacle (offset b in in figure 3.1). The minimum distance between the particle and obstacle surfaces, that would result from a trajectory determined solely based on hydrodynamic interactions, dictates the relevance of short-range non-hydrodynamic interactions such as van der Waals forces, surface roughness (solid-solid contact), etc. In fact, the resulting scaling relation derived for small b in shows that extremely small surface-to-surface separations would be common during the motion of a particle past a distribution of periodic or random obstacles. This highlights the impact that short-range non-hydrodynamic interactions could have in the effective motion of suspended particles. In this context, we further demonstrate that the particle attains smaller surface-to-surface separations from the obstacle when confined in a channel, with walls parallel to the direction of motion of the particle. We also calculate the distribution of particles around a fixed obstacle in the dilute limit, by extending the existing analytical treatment in the case of sheared suspensions. We show that the steady state distribution is isotropic, which is a somewhat surprising result given the anisotropy induced by the driving field, but it is analogous to the distribution obtained in the cases of sheared and sedimenting suspensions investigated by Batchelor (Batchelor & Green, 1972; Batchelor, 1982). The steady state distribution of particles provides the necessary information to obtain macroscopic transport properties, such as the average velocity of the suspended 24

45 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES particles, from the pore-scale point-wise velocity (Brenner & Edwards, 1993). The organisation of the chapter is as follows: in the next section we formulate the problem, describe the relevant geometries and briefly summarise the available results for the mobility functions corresponding to these geometries. In 3.3 we derive the particle-obstacle pair-distribution function in terms of the mobility functions, and consider the limiting cases of nearly touching particle-obstacle pairs ( 3.3.1) as well as widely separated pairs ( 3.3.2). In 3.4 we derive an expression for the minimum surface-to-surface separation attained during the course of motion of a particle around an obstacle. The derivation follows two distinct paths outlined in and 3.4.2; the former follows an Eulerian approach by deriving the expression using the probability distribution obtained in the preceding section, while the latter arrives at the same expression using a Lagrangian approach by calculating the particle trajectory. Then in 3.4.3, we discuss the scaling of the minimum surface-to-surface separation in the limiting case of particles nearly touching the obstacle due to small incoming impact parameters. Finally, in 3.5, we present a possible application of the relationship between the minimum separation and the incoming impact parameter to determine an effective hydrodynamic surface roughness. 25

46 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES 3.2 Model systems and formulation of the problem As discussed before, we intend to characterise the deterministic transport of a dilute suspension of spherical particles through an array of obstacles. Here, we refer to a dilute suspension in the sense that the particle-particle hydrodynamic interactions within the suspension can be neglected. In the case of an unbounded suspension, this approximation is accurate for particle volume fraction below 2% (Batchelor & Green, 1972). In the case of a suspension under geometric confinement, such as in a microfluidic device, the confinement screens the particle-particle hydrodynamic interactions, and higher volume fractions might still be accurately described by the dilute limit approximation. In addition, we assume that the obstacles are sufficiently separated that particles interacts with only one obstacle at any instant. That is, we consider the situation in which a single particle negotiates an isolated fixed obstacle. In order to obtain the steady-state distribution of particles we shall assume that the incoming particles in suspension are spatially uncorrelated, i.e., they follow a uniform distribution far away from the obstacle. In addition, we neglect the effect of both fluid and particle inertia (zero Reynolds and Stokes numbers), as well as the Brownian motion of the suspended particles (infinite Péclet number). This leads to the consideration of pair hydrodynamic interactions between the suspended particle and a solid obstacle in the Stokes regime. We investigate the effect of the driving field (either a constant 26

47 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES force or a uniform ambient velocity field), the obstacle type (either cylindrical or spherical), and the particle-obstacle aspect ratio. From a Lagrangian perspective we are interested in the trajectory followed by individual particles and, in particular, the distance of closest approach, r min, as a function of the incoming impact parameter, b in (see figure 3.1 for a schematic representation). From an Eulerian perspective we are interested in the distribution of particles around the fixed obstacles. Note that in the problem considered here, the single-particle probability density around a fixed obstacle is analogous to the two-particle probability or pair distribution function in the case of suspension flows Problem geometry and symmetries A schematic view of the problem under investigation is depicted in figure 3.1. A suspended sphere of radius a moves towards the obstacle parallel to one of the Cartesian coordinate axes, say the x-axis, from x. An obstacle of radius b (spherical or cylindrical) is held fixed with its centre at the origin of coordinates. The centre-to-centre separation is given by the radial coordinate r, and the corresponding surface-to-surface separation, nondimensionalised by the mean of the two radii is ξ = 2 (r a b) / (a + b). The minimum (dimensionless) separation reached between the surfaces, ξ min, occurs when the particle crosses the plane of symmetry perpendicular to the x-axis. We refer to the initial perpendicular distance between the line of motion and the x-axis as the incoming impact parameter and denote it by b in (it corresponds 27

48 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES to the far upstream y-coordinate of the particle, as x ). The motion is assumed to be caused by a uniform vector field F acting on the sphere, which can be a constant body force (F F, say, gravity) or a uniform incoming velocity field far from the obstacle (F 6πµav ). In either case, the sphere is torque free. We note that in the case of a spherical obstacle, the symmetry of the problem results in the planar motion of the suspended particle. The centre of the particle moves in the plane formed by F and the radial position vector at any time, and the problem is axisymmetric (the unit vector in the radial direction is indicated by d as shown in figure 3.1; the plane of motion is the xy- plane in the figure). In the case of a cylindrical obstacle, the motion of the particle parallel to the axis of the cylinder is decoupled from that perpendicular to the axis, due to translational symmetry. Here, we only consider the planar motion that occurs in the absence of a velocity component along the axis of the cylinder. Further, since the driving field F points along the positive x-axis, all trajectories in the problem are open, extending to infinity in both directions Mobility functions for the problem In Stokes flows, the velocity of the suspended particle U is linear in the driving field F, such that U = M F for an appropriate mobility tensor M that depends on the geometry of the particle-obstacle system. Moreover, since the particle motion is contained in the xy-plane, the velocity can be decomposed into two mutually 28

49 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES Y b in a ( a+b 2 r (position vector) ( ( a+b 2 b ( min X _ U U d (unit vector) Spherical Obstacle a+b b Cylindrical Obstacle a+b b S S 0 S 0 Figure 3.1: Schematic representation of the problem. (top) The small circle of radius a on the left represents the moving sphere, with a corresponding incoming impact parameter b in. The circle of radius b with its centre at the origin of coordinates represents the fixed obstacle (either a sphere or a cylinder). The empty circle represents the position of the suspended particle as it crosses the symmetry plane normal to the x axis. The surface-to-surface separation ξ and its minimum value ξ min are also shown. (bottom) Representation of the conservation argument used to calculate the minimum separation (see 3.4). The unit vector in the radial direction d is shown, as well as the velocity components in the plane of motion. The surfaces over which the flux is conserved are shown, both far upstream (S ) and at the plane of symmetry (S 0 ). The geometry is shown for the cases of a spherical and a cylindrical obstacle as indicated. 29

50 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES orthogonal components in that plane; a radial component along d and a tangential component perpendicular to it (Batchelor, 1982; Davis & Hill, 1992). Then, given the symmetries of the system, the particle velocity can be represented as, U = (A(r)dd + B(r)(δ dd)) F, (3.1) where δ is the identity dyadic, and A and B are scalar functions of the separation r (or equivalently, ξ), and also depend on the aspect ratio β = b/a. It is well known that no analytical expressions are available for the radial (A) and tangential (B) mobility functions that are valid throughout the entire range of separations (r (a + b, ) or ξ (0, )). Instead, it is a common practice to derive expressions in two limiting cases, the near-field (ξ 1) and the far-field (ξ 1) limits (Jeffrey & Onishi, 1984; Kim & Karrila, 1991), and use some matching or interpolation procedure for intermediate separations. Lubrication theory and multipole expansions (equivalently, Lamb s general solution, method of reflections) are typically employed to derive mobility functions in the near-field and far-field, respectively. Available results in these limiting cases are briefly discussed below, as it will be useful for subsequent derivations. A more detailed discussion of the mobility functions used in this work is presented in appendix 3.A. 30

51 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES Mobility functions: near contact (ξ 1) For sufficiently small separations, lubrication theory yields the following functional forms for the leading order of the radial and tangential mobility functions: A(ξ) a 0 ξ, (3.2) B(ξ) b 1 ln(1/ξ). (3.3) Expressions are available for the coefficients a 0 and b 1 above, for any generic pair of convex surfaces that can be approximated by a quadratic form (Cox, 1974; Claeys & Brady, 1989). In particular, Kim & Karrila (1991) provide the following expressions for a 0 and b 1 for a pair of spheres: a 0 = k 0 6πµa b 1 = k 1 6πµa β , β β where, the constants k i = 1 for the case of a constant force acting on the particle, but are functions of β in the case of a uniform velocity driving the particle. They can be written in terms of the O(1)-terms of the respective resistance functions, using the notation due to Kim & Karrila (1991):, k 0 = A X (1 + β) AX 12, k 1 = A Y (1 + β) AY β β + 1 B11 Y (1 + β)2 B12 Y. The numerical values of k i for various values of β are tabulated in appendix 3.A. 31

52 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES Nitsche (1996) applies expressions derived by Claeys & Brady (1989) to the case of a sphere moving relative to a cylinder of the same radius, in the case of a constant force driving the particle. We apply the same expressions to arbitrary ratios of radii to obtain: a 0 = β 6πµa 4 b 1 = 1 6πµa β /2, β 1/ Mobility functions: large separations (ξ 1) In the case of a constant force driving the particle past a spherical obstacle, we use the far-field expansions of the mobility functions provided by Kim & Karrila (1991) to obtain the following expressions for the functions A(r) and B(r) in powers of (a/r): A(r) = a 6πµa 4 β r B(r) = 1 6πµa 1 9 a 2 16 β 3 r 8 2 3β a + 2 9β3 4 r β + 3β 3 a r 4 + O 1 r 6, 4 + O 1 r 6. Analogously, in the case of a uniform velocity field and a spherical obstacle we obtain, A(r) = 1 1 3β 6πµa 2 B(r) = 1 6πµa 1 3β 4 a + 1 β + β 3 a r 2 r a 1 β + β 3 a r 4 r 3 + O 1 r 5, 3 + O 1 r 5. Note that, the leading order terms in the limit a b describe the streamlines of a uniform flow past a fixed spherical obstacle. 32

53 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES Finally, in the case of a constant force acting on a sphere moving past a cylindrical obstacle we use the following empirical far-field forms for radial and tangential mobility functions proposed by Nitsche (1996): A(r) = π 6πµa 128 B(r) = 1 6πµa 1 48π 128 a/r ln(r/aβ) a/r ln(r/aβ) , (3.4). (3.5) Here, we do not consider the case of a uniform velocity field driving a particle past an unbounded cylinder, since in this case, the description of the far-field motion in terms of mobility functions is not possible due to Stokes paradox (Happel & Brenner, 1965). In practice, however, the motion of the particle far from the obstacle would probably be dictated by the physical boundaries of the system (e.g. the walls of the microfluidic device). A similar screening argument will become relevant when we discuss the farfield asymptotic probability distribution in In order to understand the role of confinement, we shall consider the motion of a particle past a cylindrical obstacle between two parallel walls forming a channel of width 2l 0. We shall assume that the particle moves in the mid-plane, driven by a constant force and thus the far-field mobility is dictated by the hydrodynamic interaction with the channel walls (Happel & Brenner, 1965): M wall (r) = a a a a πµa l 0 l 0 l 0 l 0 As explained in appendix 3.A, at intermediate separations we interpolate between this far-field mobility and the mobility of a sphere in the vicinity of an infinite cylinder. 33

54 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES The latter is obtained by an interpolation between (3.4) (or (3.5)) and the lubrication regime discussed in (Nitsche, 1996). 3.3 Steady-state distribution of particles around a fixed obstacle From an Eulerian point-of-view, we are interested in the steady-state distribution of particles around a fixed obstacle. The basic assumption in this investigation is that the particles are uniformly distributed far away from the obstacle, leading to a uniform flux of incoming particles at infinity. The probability density of finding a particle at a given position r from the obstacle centre (i.e., the origin) is then analogous to the pair distribution function studied in dilute suspensions (Batchelor & Green, 1972; Batchelor, 1982; Davis & Hill, 1992; Almog & Brenner, 1997). Thus, we talk of the pair distribution function p(r, t), referring to particle-obstacle pairs, with the fixed obstacle located at the origin and the particle at position r. Further, p(r, t) is the normalised distribution function, such that the actual probability density is given by n p p(r, t), where n p is the uniform number density of particles far from the obstacle. Following Batchelor & Green (1972), we start with the conservation equation for the number of particles, which in terms of the pair distribution function can be written 34

55 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES p(ξ) Cyl, constant F Sph, constant F, n = 3 Sph, uniform v, n = 3 Almog & Brenner (1997) p(ξ) Cyl, constant F Sph, constant F, n = 2 Sph, uniform v, n = ξ Figure 3.2: Excess probability distribution (p(ξ) 1) as a function of the dimensionless surface-to-surface separation, for radius-ratio β = 1. (top) Comparison between both flow cases for a spherical obstacle with n = 3 and a cylindrical obstacle. (bottom) Comparison between both flow cases for a spherical obstacle with n = 2 and a cylindrical obstacle. 35

56 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES as follows: dp dt = p + U p = p U. (3.6) t Then, we write the divergence of the particle velocity in terms of its radial component using (3.1), (n 1)(A B) U = + 1 da U r, (3.7) ra A dr where n equals 2 or 3 depending on the dimensionality of the problem (see the discussion below). Substituting this expression into (3.6) we obtain, dp (n 1)(A B) dt = pu r + 1 ra A da dr 1 dq = pu r q dr = pdr dt 1 dq q dr = p dq q dt, (3.8) where, following Batchelor & Green (1972) we introduced a function q(r) defined such that p(r, t)/q(r) is a constant of motion (its material derivative is zero). This result is valid for both types of obstacles and both driving fields, with q(r) determined by the mobility functions specific to each case. Further, p(r, t)/q(r) is constant under both steady as well as unsteady conditions and, as a result, the distribution of particles at a position r and time t can be determined from a previous position of the same material point at a prior time. In addition, in steady state p(r)/q(r) is constant along trajectories, which implies a stationary, radially symmetric distribution of particles (i.e., p(r) p(r)), a surprising result, but one that is analogous to that obtained by Batchelor & Green (1972). An alternative way to arrive at the same result is to propose a function q(r) such that the field qu becomes solenoidal and therefore p/q is constant along particle trajectories. We can obtain a solenoidal field by mul- 36

57 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES tiplying the velocity with a function q of the magnitude r only, which indicates an isotropic distribution of particles in steady state. Finally, solving for q(r) and imposing the asymptotic limit p(r ) 1, we obtain a general expression for the pair distribution function in terms of the mobility functions A and B, p(r) = A A(r) exp r (n 1)(A B) dr, (3.9) ra where A = 1/(6πµa) is the asymptotic value of the radial mobility function for r. The precise effect of geometry and driving field is captured by the hydrodynamic mobility functions A and B, as well as the effective dimensionality n of the problem. An infinitely long cylindrical obstacle naturally imposes a two dimensional geometry (n = 2). However, there are two independent possibilities in the case of a spherical obstacle. First, the fully 3-dimensional (3D) problem of a dilute suspension moving past a spherical obstacle corresponds to n = 3. The second problem corresponds to the case in which the suspended particles are restricted to move in the xy-plane, i.e., the plane of the paper in figure 3.1. This simplification corresponds to n = 2, and is sometimes used to approximate the motion past a cylindrical fibre (Phillips et al., 1989, 1990). In figure 3.2, we show the probability density function for an aspect ratio β = 1 and various possibilities in terms of driving field, geometry of the obstacle, and the associated dimensionality. The top plot shows that the distribution of particles around a cylindrical obstacle is almost five-fold larger than that in the case of a spherical obstacle with n = 3 for a given separation from the obstacle. The distribution is 37

58 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES observed to be less sensitive to the effect of the driving field (spherical obstacle, n = 3), with that for the particles driven by a constant force being higher than when they are driven by a uniform velocity field. The same plot also shows a good agreement between our calculations for a spherical obstacle (with n = 3) and those of Almog & Brenner (1997), for the case of a particle driven by a uniform velocity field. Note that, Almog & Brenner (1997) have analysed the equivalent case of a particle moving with a prescribed constant velocity in a suspension of neutrally buoyant spheres. In the bottom plot we compare the probability distribution around a cylindrical obstacle with that around a spherical obstacle when the incoming particles are restricted to move in the xy-plane (n = 2). The distributions exhibit similar behaviour at small separations. However, we will show that the asymptotic scaling of the probability distributions in the limit of small separations is different for each geometry of the obstacle. We also note that all the distributions are in fact divergent at contact (ξ 0). We shall discuss the asymptotic behaviour and this divergence of the probability distributions in more detail in In figure 3.3, we compare the probability distribution function for different aspect ratios between the suspended particle and a spherical obstacle at the centre (n = 3) when the particle is driven by a constant force. The distribution of particles increases monotonically with the aspect ratio β for relatively small separations, ξ 1. However, for large separations the probability distribution function exhibits a symmetric scaling behaviour with respect to the aspect ratio, in that the same leading order expressions 38

59 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES p(ξ) β = 1/8 β = 1/4 β = 1/2 β = 1 β = 2 β = 4 β = ξ Figure 3.3: Excess probability distribution (p(ξ) 1) as a function of dimensionless surface-to-surface separation, for different aspect ratios β, a constant force and a spherical obstacle (n = 3). Inset: Enlarged view for large values of ξ, depicting symmetric behaviour about β = 1 for a fixed value of ξ, i.e., filled and open symbols almost overlap. 39

60 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES are obtained for β and β 1, as discussed in below. As mentioned before, the mobility functions used to calculate the probability distribution plotted in figures 3.2 and 3.3 in the case of a spherical obstacle were obtained from the literature for the limiting cases of small and large separations. For intermediate separations, a transition is made between the two limits using interpolation. This interpolation region is depicted by dotted lines in figures 3.2 and 3.3. For the cylindrical obstacle, we have used interpolated functions empirically computed by Nitsche (1996) over the entire range of separations (see appendix 3.A) Asymptotic behaviour of the probability distribution near contact In order to investigate the asymptotic behaviour of the probability distribution near contact, we first rewrite (3.9) in terms of the dimensionless surface-to-surface separation ξ, where, p(ξ) = A exp {(n 1)H(ξ)}, (3.10) A(ξ) H(ξ) = ξ (A B) d ξ. (3.11) ξ A 2 For small separations, we can split the domain of integration for H(ξ) at some arbitrary value ξ 0 1 such that the expressions for the mobility functions A and B can be approximated using lubrication theory in the near-field part of the integral, 40

61 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES -0.1 Cylidrical obstacle, constant force Spherical obstacle, constant force, n = 2 Spherical obstacle, constant force, n = 3 Spherical obstacle, uniform velocity, n = 2 Spherical obstacle, uniform velocity, n = 3 Exponent of ln(ξ min -1 ) Radius ratio (β = b/a) Figure 3.4: Exponents of ln(ξ 1 min ) from (3.12), (3.22) and (3.24) as a function of β for both cylindrical and spherical obstacles. The exponent for the case of a uniform velocity driving a particle past a sphere ( and ) is evaluated at discrete values of β = 1/8, 1/4, 1/2, 1, 2, 4, 8, as explained in the text. 41

62 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES where ξ 1. From this lubrication region we obtain the leading order term for the probability distribution of particles at small separations, p(ξ) p 0 ξ 1 (ln(ξ 1 )) (n 1)α... α = b 1 2a 0, (3.12) where p 0 is a constant that depends on the geometry and driving field, and from 3.2.3, α = 2k 1 /k 0 β for the case of a spherical obstacle, while α = 6β(1 + 2β) 1 (1 + β) 1 for the case of a constant force driving the particle past a cylindrical obstacle. The divergence of the probability distribution at small separations is dominated by the ξ 1 term, which explains the apparent linear behaviour observed in figure 3.2 and the weak dependence on the aspect ratio observed in figure 3.3. The asymptotic expression (3.12) is similar to the results obtained by Batchelor (1982) and Khair & Brady (2006). They obtain an expression of the form p(ξ) ξ l (ln(ξ 1 )) m, wherein l( 1) arises due to the presence of O(ξ 0 ) terms in their tangential mobility functions. These terms are exactly zero when the obstacle is fixed, leading to l = 1, as obtained in (3.12). Figure 3.4 shows the exponent (n 1)α in the equation above, as a function of β for both types of obstacles. As mentioned in 3.2.3, k 1 = k 0 = 1 for the case of a constant force driving the particle past a spherical obstacle. This leads to α = 2/β, corresponding to the two straight lines in the log-log plot presented in the figure. Further, for the case of a uniform velocity driving the particle, k 0 and k 1 can be computed using the numerical values of the O(1) terms in the corresponding resistance functions, which are available at discrete values of the aspect ratio (Jeffrey & Onishi, 1984; Kim & Karrila, 1991). This leads to a numerical evaluation of the 42

63 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES exponent at these values, as shown in the figure (see appendix 3.A for a table of values of k 0 and k 1 ). We note that for very small values of the aspect ratio β i.e., when the obstacle is very small compared to the particle we get k 0 k 1 1, and the exponent for the case of a spherical obstacle becomes equal for both types of driving fields. This is consistent with a constant drag acting on the suspended particle outside the lubrication region. Finally, we note the presence of a diffusive boundary layer around the obstacle. A local Péclet number for the particle, P e L, can be written as the ratio between the radially inward convective flux (j C = n P p(ξ)u r ) and the radially outward diffusive flux (j D = n P Ddp/dr). At small separations we can use the asymptotic behaviour of p(ξ) to obtain, P e L = j C = n P pu r j D n P Ddp/dr = p(ξ) dp/dξ (a + b)f 2kT ξp e 0, where, P e 0 = (a+b)f/2kt is the system Péclet number. This indicates the existence of a boundary layer around the obstacle for ξ 1/P e 0, within which the diffusive mode of transport dominates over the convective one. Thus, the expressions derived above for the probability distribution, under the deterministic assumption, are not valid within this boundary layer. Batchelor (1982) highlights this issue for sedimenting suspensions. Nitsche (1996) discusses the presence of such a boundary layer in the context of the diffusion of a spherical particle close to a cylindrical fibre of comparable size and Khair & Brady (2006) have quantified this region in the context of micro-rheology. 43

64 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES Asymptotic behaviour of the probability distribution at large distances In order to obtain the leading order behaviour of the probability distribution far from the obstacle, we use the expressions for the far-field mobility functions discussed in Substituting these far-field expressions for the mobility functions A and B in (3.9), we obtain the asymptotic probability distribution for ξ 1, p F Sph(r) a 2 a 4 32 β + O, (3.13) r r p v Sph(r) β a r + 27 a 2 a 3 32 β2 + O, (3.14) r r p F Cyl(r) (a/r) a + O r ln r aβ 2, (3.15) corresponding to a spherical obstacle (n = 2) and constant driving force, a spherical obstacle (n = 2) and a uniform velocity field driving the particle, and a cylindrical obstacle and a constant driving force, respectively. Further, for a spherical obstacle with n = 3, the following expressions for the asymptotic distribution are obtained: p F Sph(r) β a r 2 + O a r 4, (3.16) p v Sph(r) a 5 a 6 8 β2 + O, (3.17) r r 44

65 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES Note that both (3.13) and (3.16) are symmetric about β = 1 for a given (albeit large) separation ξ. This can be observed in figure 3.3 for ξ 10 (see inset). When the particle and the obstacle can be treated as point forces separated far apart, the expressions for the mobility functions (given in 3.2.4) become symmetric in their radii a and b, which leads to the symmetry shown above. In contrast, the mobility functions and the corresponding probability distribution remain asymmetric in the case of a uniform velocity field driving the particles. Analogous expansions in the case of suspension flows are discussed by Batchelor (1982) and by Almog & Brenner (1997) in the context of micro-rheology. However, there is an important difference with these cases. The expressions for n = 2, (3.13), (3.14) and (3.15) (for n = 3, (3.16)) indicate that the integral R [p(r) 1]2πrdr (a+b) (correspondingly, R (a+b) [p(r) 1]4πr2 dr) diverges for R. Physically, this integral indicates a divergence in the total excess number of particles in the suspension at steady state, induced by the long-range nature of the hydrodynamic interactions with the fixed obstacle. As we mentioned earlier in 3.2.4, in the case of microfluidics either the channel walls and/or the presence of other suspended particles would effectively screen such long-range interactions, thus leading to convergent integrals. On the other hand, the asymptotic particle distribution around a spherical obstacle for the case of a uniform velocity field and n = 3 given by (3.17), yields a finite excess number of particles. We attribute this to the rapidly decaying divergence of the particle velocity 45

66 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES in this case, given by (3.7, n = 3), U ln[p(r)] 27 a 5 8 β2... r 1. r 3.4 Minimum surface-to-surface separation We now turn to the dependence of the dimensionless minimum surface-to-surface separation, ξ min (equivalently, the corresponding centre-to-centre separation r min ), on the incoming impact parameter b in, following both Eulerian and Lagrangian approaches. The Eulerian approach entails invoking the conservation of the number of particles entering and exiting the region defined by the revolution of the trajectory of a particle around the x-axis (the region created by the translation of a particle trajectory along the axis of the cylinder in the case of a cylindrical obstacle). The Lagrangian approach involves the explicit calculation of the trajectory of an individual particle, followed by the evaluation of the position of the particle as it crosses the plane x = Eulerian approach: Conservation of the number of particles We consider the situation depicted in the bottom half of figure 3.1. On the left, it shows the volume of revolution obtained by revolving around the axis of motion (x- 46

67 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES ξ min β = 1/8 β = 1/4 β = 1/2 β = 1 β = 2 β = 4 β = 8 ~ ξmin b in /[(a+b)/2] b in /b Figure 3.5: Minimum separation ξ min versus incoming impact parameter b in. Comparison between particle-particle simulations (Frechette & Drazer, 2009) and the analytical result given in (3.20): different symbols depict the simulation data as described in the legend, while the solid lines represent (3.20). (left) Average radius, (a + b)/2, is used to nondimensionalize the axes, (right) radius of the obstacle is used to nondimensionalize the axes, ξ = [(1 + β)/2β]ξ, (inset) Enlarged view of the interval (0.5, 1.5) of b in 47

68 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES ξ min Cylindrical obstacle, constant F Spherical obstacle, constant F Spherical obstacle, uniform v Cylindrical obstacle confinement 10 radii apart, constant F Cylindrical obstacle confinement 6 radii apart, constant F b in /[(a+b)/2] Figure 3.6: Minimum separation ξ min versus incoming impact parameter b in for different geometries and flow-fields when the particle and the obstacle are of the same radius. The incoming impact parameter is made dimensionless using the average radius, (a + b)/2. As explained in the text, for a cylindrical obstacle, the minimum separation decreases for a given incoming impact parameter in case of a confined system. 48

69 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES axis) the trajectory followed by a spherical particle moving past a spherical obstacle. The conservation of particles inside this volume of revolution takes a simple form, due to the fact that the flux of particles across the surface of revolution is zero; since trajectories do not cross in Stokes flow. (The construction of such a volume of revolution corresponds to the fully 3D case with n = 3. The 2D case with n = 2 can be treated in a manner completely analogous to that of a cylindrical obstacle.) Then, we calculate the flux at two cross sections that are perpendicular to the x-axis, and over which the velocity is exactly normal to the surface. These two surfaces are depicted in figure 3.1: one is the cross section far upstream, indicated by S, and the otheris the annular region corresponding to the cross section at x = 0, indicated by S 0 in the figure. (Note the excluded volume of radius (a + b)). On the bottom right corner of the figure, we show the analogous case of a cylindrical obstacle, in which the flux is considered per unit length in the z-direction. In both cases the local flux of particles is given by p(r)u. Therefore, the conservation of particles in terms of surface integrals takes the form, pu x ds = S p U ds = S p 0 U 0 ds = S 0 pu x=0 ds, S 0 (3.18) where U and U 0 are the magnitude of the particle velocity far upstream and at x = 0, respectively. The integration over S is simple, the probability distribution tends to unity and the velocity is uniform. Specifically, the velocity is asymptotically radial and equal to A F, that is F/6πµa or v, depending on the driving field. The flux is, therefore, S A F, with S = π (n 2) b (n 1) in. At the plane of symmetry, the 49

70 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES velocity is tangential to the obstacle, U 0 = B(r)F. Then, substituting in (3.18) and performing suitable algebraic simplifications we obtain, b n 1 rmin in n 1 = B(r) a+b A(r) rn 2 exp r (n 1)(A B) d r dr, (3.19) ra where the limits of integration of the annular region are the excluded volume radius (a + b) and the radial position of the particle at x = 0, which corresponds to the minimum radial distance reached during the course of the motion, r min. We can simplify the equation further, as shown in appendix 3.B to yield, b in = r min exp {H (ξ min )}, (3.20) where H is the function defined in (3.11) Lagrangian approach: Trajectory analysis The differential equation for the particle trajectory in terms of y and r coordinates can be integrated in a straightforward manner by separation of variables, dy dr = U y U r = U r sin θ + U θ cos θ U r = sin θ B(r)F sin θ 1 A(r)F cos θ cos θ = y r B(r). (3.21) r A(r) Thus, upon integration between (y = b in, r ) and (y, r) we get, ln bin y = r A(r) B(r) dr = H(ξ) b in = y exp{h(ξ)}. ra(r) Evaluating at x = 0 (where y = r min and ξ = ξ min ), we retrieve (3.20). The expression (3.20) explicitly relates the minimum separation between particle and obstacle surfaces with the incoming impact parameter, and is a general result 50

71 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES applicable to systems in which the velocity can be decomposed as in (3.1). It is interesting to note that it has the same form for the relation between b in and r min (or, ξ min ) for two and three dimensional geometries, which can be explained by the fact that the motion is planar in both cases. Again, the entire information about the driving field and the geometry of the problem is implicit in the mobility functions A(r) and B(r). In figure 3.5, we compare the minimum separation obtained using (3.20) with that obtained from the numerical integration of the trajectory followed by a spherical particle moving around a fixed spherical obstacle (Frechette & Drazer, 2009). Excellent agreement is observed for all the aspect ratios considered, β = 0.125, 0.25, 0.5, 1, 2, 4, 8. Note that the same mobility functions were used in both cases, hence the agreement is independent of the accuracy of the mobility functions themselves. In order to illustrate (3.20) in a practical manner, we also present the minimum separation for particles of different size when the size of the obstacle is fixed (see right hand panel in figure 3.5). We observe that for a constant incoming impact parameter, b in /b O(1), the minimum attained separation decreases with increasing particle radius. However, an asymptotic analysis at very small separations reveals that for a given (sufficiently small) incoming impact parameter, the minimum separation decreases monotonically with decreasing particle radius. This can be corroborated from the behaviour of the exponent involved in the asymptotic scaling of the incoming impact parameter (3.22) shown in figure 3.4 (see below for a detailed discussion). The inset in the right 51

72 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES hand panel of figure 3.5, in fact, portrays the imminent reversal of this trend in the form of the criss-crossing curves. Figure 3.6 shows ξ min as a function of b in for β = 1 for five different cases. We can see that for a constant driving force and a given b in the particle gets closer to a spherical obstacle, compared to a cylindrical one. It also shows that a constant force drives the particle closer to a spherical obstacle than a uniform velocity field. Further, we show the case of an obstacle-particle pair confined between two parallel walls, when the obstacle is a cylinder and the walls are perpendicular to the axis of the cylinder. This case is the most relevant to model the motion of a particle past a cylindrical post in a microfluidic channel. As explained in and appendix 3.A, at intermediate separations we compute the mobility of the particle in this system by interpolating between the mobility of a sphere moving along the mid-plane between the two walls (Happel & Brenner, 1965) and that for a particle moving close to an infinite cylinder (Nitsche, 1996). We observe that the presence of such confinement, as well as increasing the extent of the confinement from 10 particle radii to 6 particle radii, decreases the attained minimum separation for a given incoming impact parameter. The underlying reason is that, in the far-field, the mobility is isotropic in the plane of motion and, therefore, the motion of the particle is not affected by the obstacle. The interaction with the obstacle becomes significant for particle-obstacle separations of the order of the channel width. Thus, for a narrower channel, the particle reaches closer to the obstacle before its mobility is affected by the presence of the obstacle. The minimum separation attained by the 52

73 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES particle along its trajectory determines the relevance of short-ranged repulsive forces during its motion. Thus, in terms of the effect of confinement, the above observation implies that the particle will experience an earlier onset of non-hydrodynamic effects upon confinement between two parallel walls Asymptotic behaviour of minimum separation near contact, ξ min 0 Here, we calculate the asymptotic behaviour of the relation (3.20) in the limit of very small separations, in a manner similar to that discussed in for the probability distribution. Specifically, using the lubrication approximation for the mobility functions in 3.2.3, and substituting in (3.20) we get: b in (a + b)/2 λ 0 (ln(1/ξ min )) α... α = b 1 2a 0, (3.22) where λ 0 is a constant that incorporates the contributions from the far-field. The behaviour of the exponent ( α) is depicted in figure 3.4. As discussed earlier, we can see from the behaviour of the exponent that for incoming impact parameters small enough to follow (3.22), the minimum separation decreases monotonically with decreasing particle radius. However, the asymptotic form of the tangential resistance function 1/B(r), derived using (3.3), is only logarithmically divergent, and valid only when extremely small separations are attained. Thus, for reasonably small values of ξ min, albeit ξ min 1, 53

74 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES the expression (3.22) is not an accurate approximation. Therefore, in addition to the leading order of the scalar resistance functions from which B(ξ) is obtained by matrix inversion, we also retain their O(1) terms. Consequently, the tangential mobility function takes the form: B(ξ) 1 s(ξ)(b 1 + b 2 s(ξ))... s(ξ) = 6πµa 1 + b 3 s(ξ) + b 4 s(ξ) 2 1 ln(1/ξ). (3.23) We continue to use (3.2) as an approximation for the radial mobility function, so that A(ξ) a 0 ξ. This leads to the following approximate expression for the scaling: b in (a + b)/2 λ 1 (s(ξ min )) α (γ 1 + s(ξ min )) α γ 0 (γ 2 + s(ξ min )) γ 0, (3.24) where, as before, the exponent α is as defined in (3.12) and (3.22), and the constant λ 1 captures the far-field behaviour. The constants γ i are functions of the coefficients b i in the tangential mobility function (see appendix 3.A for their tabulated values). In figure 3.7, we compare the limiting behaviour given by (3.24), with λ 1 as the only fitting parameter, with the numerical evaluation of the governing relation (3.20), and observe that there is excellent agreement over a considerable range of separations. 3.5 Discussion and Summary Using (3.20) it is straightforward to extend the concept of hydrodynamic surface roughness (henceforth, HR) of spheres introduced by (Smart & Leighton, Jr., 1989), to the motion of a suspended particle past a fixed obstacle. In their original work, 54

75 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES ξ min # Approximate Scaling (3.24) Exact expression (3.20) β = 1, uniform v β = 1, cylinder β = 1/8 # β = 1/4 β = 1/2 β = 1 β = 2 β = 4 β = b in /[(a+b)/2] Figure 3.7: Open and filled symbols represent approximate scaling (3.24) compared with full governing relation (3.20) when a particle is driven by a constant driving force past a spherical obstacle for radius-ratios β = 1/8, 1/4, 1/2, 1, 2, 4 and 8. (#) represents a particle being driven past a cylindrical obstacle by a constant force, and (*) represents a particle being driven by a uniform velocity field past a spherical obstacle, for β = 1. The multiplicative constant λ 1 is determined by fitting the data generated with (3.20) for small values of ξ. The fitted values are 1.91, 1.88, 1.84, 1.86, 1.99, 2.33, 2.08 for β = 1/8, 1/4, 1/2, 1, 2, 4, 8, respectively, in the case of the open and filled symbols. For a cylindrical obstacle (#) we obtained λ 1 = 1.58 and for the case of uniform flow (*) λ 1 =

76 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES Smart & Leighton, Jr. (1989) related the time taken by a sphere to fall away from a flat surface under the action of gravity to an effective surface roughness. In our case, we can establish a relationship between the net lateral displacement experienced by a suspended sphere moving past an obstacle and the length scale at which surface roughness effects become dominant over hydrodynamic forces. This relationship is based on the simple but useful excluded-annulus model in which the inter-particle potential is approximated by a hard sphere repulsion with range (1 + ϵ)a (Brady & Morris, 1997). In this approximation, the effective roughness determines the minimum separation between particles but has no effect on the hydrodynamic interaction between them. Such hard sphere potentials (in some cases including tangential friction) are widely used to model roughness effects on suspension properties, including micro-structure (Rampall et al., 1997; Brady & Morris, 1997; Drazer et al., 2004; Blanc et al., 2011), shear-induced dispersion (da Cunha & Hinch, 1996; Drazer et al., 2002; Ingber et al., 2008), sedimentation (Davis, 1992; Davis et al., 2003), rheology (Wilson & Davis, 2000; Bergenholtz et al., 2002), and transport in periodic systems (Frechette & Drazer, 2009). In our problem, the excluded annulus model implies that, independent of the impact parameter b in, separations smaller than aϵ are unattainable due to the hard sphere potential. On the other hand, the repulsive potential has no effect as the particle separates from the obstacle. Therefore, if b in = b ϵ is the incoming impact parameter corresponding to ξ min = ϵ, any trajectory with b in < b ϵ reaches the same minimum separation ϵ and collapses onto the outgoing trajectory corresponding 56

77 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES to b in = b ϵ, thus inducing a net lateral displacement of magnitude b in b ϵ. In the case of a dilute suspension flowing past a fixed obstacle, this corresponds to the presence of a wake of width 2b ϵ behind the obstacle, analogous to that observed by (Khair & Brady, 2006). The excluded-annulus model implies that the wake behind a fixed obstacle is related to the HR by (3.20), b ϵ = (a + b) 1 + ϵ exp {H (ϵ)}. (3.25) 2 Alternatively, we can view the equation above as the definition of the HR, which serves as an effective hard-wall potential resulting from all non-hydrodynamic short-range repulsive interactions. The experimental measurement of b ϵ would yield such length scale ϵa for deterministic systems with negligible particle and fluid inertia. A similar surface characterization method has been employed in the case of colloidal particles by numerically solving the particle trajectories (Dabroś & van de Ven, 1992; van de Ven et al., 1994; Wu & van de Ven, 1996; van de Ven & Wu, 1999; Whittle et al., 2000). We note, however, that due to the presence of a diffusive boundary layer (see 3.3.1), (3.25) is valid for sufficiently large Péclet numbers, such that the thickness of the boundary layer is smaller than the repulsive core, i.e., ξ BL 1/P e < ϵ. In summary, we have investigated the problem of a spherical, non-brownian particle negotiating a spherical or cylindrical obstacle in the absence of particle and fluid inertia. The particle is driven by a uniform ambient velocity field or a constant force acting on it, and its motion is entirely contained in the plane formed by the driving force and the radial vector joining the centres of the obstacle and the 57

78 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES particle. Given this planar nature of the motion and the symmetry of the problem, the particle velocity renders itself for decomposition into a radial component (along the centre-to-centre line) and a tangential component (perpendicular to the centreto-centre line), with the corresponding mobility functions being dependent on the centre-to-centre distance only. Based on these properties, and extending an approach introduced by (Batchelor & Green, 1972) in the context of sheared suspensions, we have derived the steady-state probability distribution of particles around the obstacle, assuming a uniform distribution of incoming particles at infinity. We showed the distribution to be radially symmetric for both obstacle types and both driving fields, analogous to the isotropic pair distribution functions obtained in sheared suspensions, sedimentation and microrheology. The asymptotic form of the distribution funtions diverges at contact, suggesting the presence of a boundary layer around the obstacle in which Brownian transport is not negligible; existence of such a boundary layer has also been reported in the context of sedimenting suspensions and microrheology (Batchelor, 1982; Nitsche, 1996; Khair & Brady, 2006). In addition, the asymptotic distribution of particles close to the obstacle is similar for both driving fields, and depends only weakly on the dimensionality and geometry of the problem. Further, our numerical results indicate that this asymptotic behaviour becomes dominant at small separations, highlighting the relevance of other (non-hydrodynamic) interactions. The other asymptotic limit, that of large separations, yields a divergent excess number of particles around the obstacle, suggesting that screening by other parti- 58

79 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES cles or container walls would eventually become important for the description of the distribution of particles far from the obstacle. We have also derived an expression for the minimum particle-obstacle separation attained in the course of motion, as a function of the incoming impact parameter, using both Eulerian and Lagrangian approaches. We have shown that a smaller minimum separation is attained by particles moving in a confined channel, and the separation decreases with the extent of the confinement. The asymptotic behaviour in the limit of small impact parameters (particles nearly touching the obstacle) shows that the minimum surface-to-surface separation decays exponentially (with a negative power of the impact parameter). The exponent governing this asymptotic relationship varies monotonically with particle radius, and indicates that for a given obstacle size and sufficiently small incoming impact parameter, a smaller particle reaches closer to the obstacle than a larger one. Further, the exponential nature of the relationship shows that extremely small surface-to-surface separations can be frequently encountered in the motion of particles through an array of obstacles, which, could easily lead to a dominant role of non-hydrodynamic interactions in microfluidic systems. Interestingly, the exponential decay of the minimum separation as a function of the impact parameter is independent of the dimensionality of the problem and depends only weakly on the geometry of the obstacle and aspect ratio. 59

80 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES 3.6 Acknowledgements We would like to thank Profs. A. Acrivos, A. Prosperetti, A. Sangani and J. F. Morris for useful discussions. Partially funded by NSF grant no. CBET

81 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES 3.A Mobility fuctions A and B In this appendix, we describe the radial, A(r), and tangential, B(r), mobility functions used in this work. We also provide the tabulated values of the different coefficients used in the expressions provided for the mobility functions at small surface-to-surface separations. 3.A.1 Spherical obstacle In the case of a spherical obstacle we follow the notation used by Jeffrey & Onishi (1984) and write A and B in terms of the scalar functions x a αβ, xb αβ, xc αβ, ya αβ, yb αβ and yαβ c. The subscript αβ denotes a function relating the motion of sphere α to the force or torque acting on sphere β. Consequently the values of α and β can be 1 or 2. In our case, we use 1 for the moving particle and 2 for the obstacle. The expressions for A and B are given below for both driving fields. The near-field and far-field approximations presented in and 3.2.4, respectively, were obtained from the equations below using the expressions tabulated by Kim & Karrila (1991). 3.A.1.1 Constant force acting on the moving sphere A = 1 µ x a 12 x a 21 + x a 11x a 22 x a 22 (3.26) 61

82 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES β b 1 b 2 b 3 b 4 γ 0 γ 1 γ 2 1/ / / Table 3.1: Coefficients b i and γ i from (3.23) and (3.24). Constant force. Spherical obstacle. B = 1 y11 a y a y21y a 22 c y21y b 22 b 12 µ y22y a 22 c + (y22) + ya 21y21 b b 2 y b 22 + ya 22y b 21 y b 22 y21y a 22 c y21y b 22 b y22y a 22 c + (y22) b 2 (3.27) In table 3.1, we provide the numerical values of the coefficients b i and γ i used in for the case of a constant force driving the particle: 62

83 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES β b 1 b 2 b 3 b 4 γ 0 γ 1 γ 2 1/ / / Table 3.2: Coefficients b i and γ i from (3.23) and (3.24). Uniform velocity field. Spherical obstacle. 3.A.1.2 A freely suspended sphere in a uniform ambient velocity field A = 1 1 xa 12 6πµa x a 22 (3.28) B = 1 1 yb 21 + ya 22y21 b y12y a 22 b y22 c 6πµa y22 b y22 b y22y a 22 c (y22) b 2 (3.29) In table 3.2, we enlist the values of the coefficients b i and γ i used in when a freely suspended particle is driven by a uniform velocity field: In table 3.3, we tabulate the values of the coefficients k 0 and k 1 defined in 3.2.3, in the expressions for the mobility functions near contact for the case of a uniform ambient velocity past 63

84 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES β 1/10 1/8 1/5 1/4 1/2 1 k k β k k Table 3.3: Coefficients k 0 and k 1 from Uniform velocity field. Spherical obstacle. a spherical obstacle. 3.A.2 Cylindrical obstacle Unlike the hydrodynamic interactions between two spheres, those between a cylinder and a sphere of comparable radius (external to the cylinder) are scantily documented in the literature. (Nitsche, 1996) considers the motion of a sphere near a cylindrical fibre, for the case of a constant force acting on the sphere suspended in a quiescent fluid. First, representing the sphere by a point particle and the cylinder by a line of singularities, Nitsche obtains the far-field expressions for A and B in (3.4) and (3.5). For small separations between the sphere and the obstacle, the framework established by (Cox, 1974) and generalized by (Claeys & Brady, 1989) yields the near-field 64

85 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES expressions, as documented for the case of a sphere and a cylinder of the same radius (i.e., β = 1) by (Nitsche, 1996). From these expressions, the coefficients b i and γ i used in are given as follows: b 1 = , b 2 = 0, b 3 = , b 4 = 0 and γ 0 = 0, γ 1 = and γ 2 is immaterial. Further, Nitsche (1996) provides an approximate functional form for the entire range of separations (for β = 1) using a hyperbolic tangent as a weighting function between the far-field and near-field regions, A(r) = 1 1 r/a tanh 2 ln 6πµa π 1 tanh 0.09 cosh 2 ln 2 ln 1 a 117π r ln 2r a r/a 2 12π (r/a 2) 151π 60 2 ln 1 r/a 2, 1.57 (3.30) 3 (r/a 2) B(r) = 1 1 r/a tanh 2 ln 1 a 6πµa r + 1 r/a π tanh 2 ln π 2 ln r/a cosh 2 ln π 8 1 ln 2r 3 (r/a 2) (3.31) a Finally, we compute the mobility of a sphere in a particle-obstacle system confined between two parallel walls that are perpendicular to the axis of the cylinder. To this end, we use an interpolation technique similar to the above expressions, i.e., we interpolate between the mobility of a sphere along the mid-plane between two 65

86 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES parallel walls (Happel & Brenner, 1965) and the mobility of a sphere in the vicinity of a cylinder: where, effective mobility 1 in confinement = tanh 2 ln 1 tanh 2 ln r/a 2 2l 0 /a r/a 2 2l 0 /a [M wall (l 0 )] [M cyl (r)], M wall (r) = a a a a , 6πµa l 0 l 0 l 0 l 0 and M cyl (r) A(r) or B(r) from (3.30) or (3.31). Note that 2l 0 involved in the hyperbolic tangent functions is the separation between the parallel walls. This represents the length-scale at which the interpolation switches from the mobility of a sphere between parallel walls to that in the vicinity of a cylinder. 3.B Derivation of the equation for the minimum separation Here we simplify the following equation: b n 1 rmin in n 1 = B(r) a+b A(r) rn 2 exp r (n 1)(A B) d r dr (3.32) ra 66

87 CHAPTER 3. TRAJECTORY AND DISTRIBUTION OF PARTICLES First, we multiply both sides by (n 1) and then write the RHS of the equation as: rmin b n 1 in = r n 1 (n 1)(A B) exp a+b ra r rmin + (n 1)r n 2 exp a+b Then, integrating the first term by parts we obtain, b n 1 in = (r min ) n 1 (n 1)(A B) exp d r r min ra r d r (n 1)(A B) ra (n 1)(A B) d r ra (a+b) n 1 exp The last term above can be written formally as the following limit, a+b dr (3.33) dr. (n 1)(A B) d r ra (3.34) lim (a + b) 1 + ξ n 1 exp {(n 1)H (ξ)} = 0 (3.35) ξ 0 2 Where H(ξ) was defined in (3.11) of and the limiting behavior can be obtained using the near-field lubrication expressions for A and B. Therefore, we get b n 1 in = r n 1 min exp (n 1)(A B) r min ra d r (3.36) Taking (n 1) th root on both sides, one obtains the required relation (3.20) in terms of r. Simplification of this equation in terms of dimensionless separation is obtained by substituting ξ = 2 (r a b) /(a + b). 67

88 Part II Implications of Theory for Microfluidic Applications 68

89 Chapter 4 Analysis of directional locking in a sparse array of obstacles Introduction One of the essential unit operations in micro-total-analysis-systems (µtas) is separation of species about to be analysed downstream. Early microfluidic separation strategies involved miniaturization of the macroscopic separation methods, e.g., size exclusion and hydrodynamic chromatography. However, at the micro-scale, when the particle and the pore sizes are comparable, it is possible to exploit the specificity of the interactions between them at the pore-level. Moreover, current micro-fabrication 1 Fair-use statement: At the time of this writing, the present chapter is being prepared for submission as a journal article, [Risbud, S. R. & Drazer, G. Analysis of directional locking in a sparse square array of obstacles]. 69

90 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING techniques enable design and fabrication of highly regular micro-structures to act as separation-matrices as opposed to random micro-structure of the conventional separation media. For example, in entropic trapping (Han & Craighead, 2000) larger species spend more time in entropic wells designed as an array of trenches in a micro-device. In pinched flow fractionation (Yamada et al., 2004), species entering a constriction and exiting into a sudden expansion experience a lateral displacement from their trajectories that is a function of their size. Deterministic lateral displacement (Huang et al., 2004) (DLD) employs a periodic array of solid obstacles, through which species of different sizes migrate in different spatial directions in the presence of the same driving force. This effect can also be achieved with a periodic array of optical traps (soft potentials instead of solid obstacles, (MacDonald et al., 2003)). In this work, we focus on the mechanism underlying separations in such DLD systems. Although DLD systems have been studied extensively (Huang et al., 2004; Inglis et al., 2008; Korda et al., 2002; Lacasta et al., 2005; Gopinathan & Grier, 2004), the understanding of the underlying mechanism is heuristic at best, and lacks a theoretical basis incorporating the hydrodynamic interactions between the particles and the obstacles. Further, the effect of short-range non-hydrodynamic repulsive interactions (such as solid-solid contact due to surface roughness, electrostatic repulsion, steric repulsion, etc.) is seldom taken into account. Note that some early explanations of the separation mechanism in DLD systems are based on the flow streamlines and flow-lanes (Huang et al., 2004), and fail to account for the observed irreversible 70

91 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING trajectories of particles around the obstacles even at negligible Reynolds numbers (particle trajectories should be fore-aft symmetric and kinematically reversible for Re 1). Despite of our extensive computational and experimental study of DLD systems (Balvin et al., 2009; Frechette & Drazer, 2009; Herrmann et al., 2009; Koplik & Drazer, 2010; Bowman et al., 2012; Devendra & Drazer, 2012) where the experiments include microfluidic as well as macroscopic platforms at low Reynolds number a sound theoretical analysis (howsoever approximate) incorporating finite particle size as well as the non-hydrodynamic interactions has not been attempted until now. A theoretical framework describing the DLD systems can not only lead to rational design of the DLD devices, but also gives a way to interesting modifications of the technique, such as a size-based spatial band-pass filter for micro-particles. Therefore, here we present our theoretical findings related to the deterministic lateral displacement systems. We consider a square array of equal-sized obstacles, and a spherical particle of arbitrary radius driven by a field through the array (either a constant force, or a flow field) at an arbitrary angle (henceforth, forcing angle) with respect to the principal lattice directions of the square array. We assume negligible particle as well as fluid inertia, and infinite Péclet number (non-brownian particles, deterministic trajectories). We also assume a dilute limit for the obstacle array, such that the inter-obstacle spacing is sufficiently large and the particle negotiating the array interacts with only one obstacle at a time (henceforth, the dilute assumption ). Under the dilute as- 71

92 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING sumption, the particle-obstacle pair of finite size can be replaced by a kinematically equivalent pair in which the particle has zero radius (point-particle) and the obstacle radius is equal to a critical parameter, which incorporates the hydrodynamic as well as short-range repulsive non-hydrodynamic interactions between the pair (we have studied the critical parameter both computationally (Frechette & Drazer, 2009) as well as theoretically (Risbud & Drazer, 2013)). Using the kinematically equivalent representation of the system under the dilute assumption, the complex dynamical problem reduces to simple geometric manipulations. Using geometric arguments, we derive bounds on the ratio of the critical parameter to the size of the lattice unit cell. These bounds directly lead to the explanation for experimentally and computationally observed devil s-staircase-like dependence of the average migration angle on the forcing angle. Further, the same framework can be used to uncover size-based spatial band-pass filtering of particles through the array, such that particles of a certain intermediate size migrate with an average migration angle larger than that corresponding to particle sizes smaller as well as larger than them. The chapter is organized as follows: in 4.2 we introduce the system under consideration, the system variables, and the dilute approximation. We also explain the model for short-range repulsive non-hydrodynamic interactions leading to the definition of the critical parameter b c, and establish the kinematically equivalent abstract model for the particle-obstacle pair. In 4.3, we use the abstract model to derive a periodicity condition for particle trajectories. We apply the same periodicity condition 72

93 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING b a F θ Y ` X ` (a) (b) Figure 4.1: (a) A spherical particle of radius a negotiating a portion of a square array of obstacles of circular cross-section with radius b [adapted from (Frechette & Drazer, 2009)]: the length of a unit-cell is l, the driving field F oriented at an angle θ as shown, drives the particle through the array. The principal lattice-directions are indicated with Cartesian axes X and Y. (b) A few example particle trajectories exhibiting directional locking [adapted from (Frechette & Drazer, 2009)]: results of Stokesian dynamics simulations with a = b, l = 5a and the range of non-hydrodynamics interactions ϵ = 10 3 (see 4.2 for a discussion on non-hydrodynamic interactions). Counter-clockwise, from X-axis to Y -axis, the trajectories can be seen to be locked in directions [1, 0], [3, 1], [1, 1], [2, 3] and [1, 2] (the inset shows the migration directions). The dot-dashed lines are to guide the eye and highlight [3, 1] and [2, 3] locking directions. to derive expressions for the simplest locking directions in In 4.4, we apply the above periodicity condition corresponding to particles exhibiting the simplest directional locking behaviour, and comment on the resolution of separation between such particles. In 4.5, we show the possibility of extension of our theoretical framework to noisy particle trajectories ( 4.5.1), and size-based band-pass filtering of particles as a possible application of DLD systems ( 4.5.2). 73

94 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING 4.2 System description, assumptions and abstractions Figure 4.1 depicts the system under investigation. We consider a suspended spherical particle of radius a negotiating a square array of obstacles with circular cross section of radius b, under the action of a driving field F (either a constant force, or a uniform flow away from the lattice). The field is oriented at an angle θ with respect to one of the principal axes of the array (say, the X-axis as shown in the figure). The lattice spacing is l. The domain of the forcing angle is restricted to θ [0, π ], since 4 the system possesses a reflection symmetry in the X = Y line. We work in the Stokes regime, i.e., we neglect fluid inertia (negligibly small Reynolds number) and particle inertia (negligibly small Stokes number). We consider the deterministic limit (large Péclet number, non-brownian limit). Further, we assume that the obstacles are sufficiently widely separated, and the interactions between a single obstacle and the particle need be taken into account for trajectory calculation (figure 4.2). Figure 4.2 depicts the variables of the problem; the incoming and outgoing impact parameters are denoted by b in and b out, respectively. The dimensionless minimum surface-to-surface separation attained by the particle from the obstacle is denoted by ξ min in the figure. The functional relationship between b in and ξ min explicitly incorporates the hydrodynamic mobility of the particle around the fixed obstacle, thereby taking into consideration the hydrodynamic interactions 74

95 ( ( CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING Exact and approximate trajectories b in b out ( a+b 2 ξ min r 0 = (a+b)( 1 + ξ min 2 Figure 4.2: Qualitative assessment of the dilute assumption: a schematic comparing an exact and an approximate trajectory around two successive obstacles. See the text for description of b in, b out and ξ min. The approximate trajectory is constructed as a union of a straight incoming part maintaining a constant b in, a circular region with radius r 0 = (a + b)(1 + ξ min ), and another straight outgoing part maintaining 2 a constant b out. Thus, the essential features of the exact trajectory are preserved by construction. (Risbud & Drazer, 2013). Apart from the hydrodynamic interactions between the particle and the obstacle that arise from their finite size, we also take into account the short-range repulsive non-hydrodynamic interactions such as solid-solid contact due to surface roughness, electrostatic repulsion, steric repulsion, etc. We have seen in that these nonhydrodynamic interactions can be modelled as a hard-wall potential with a dimensionless range ϵ around the obstacle, such that the particle surface cannot approach the obstacle surface closer than the range (Davis, 1992; da Cunha & Hinch, 1996; Rampall et al., 1997; Brady & Morris, 1997; Wilson & Davis, 2000; Bergenholtz et al., 2002; Drazer et al., 2002; Davis et al., 2003; Drazer et al., 2004; Ingber et al., 2008; Frechette & Drazer, 2009; Blanc et al., 2011). In 1.2.1, we have also argued 2 The material presented in this section is similar to that in 1.2.1, 3.5, 6.3.1, and 7.3. However, it has been repeated here to preserve the continuity and completeness of the chapter. 75

96 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING that the presence of such non-hydrodynamic repulsion leads to the occurence of the critical parameter b c. This can be further elaborated with the aid of figure 4.3(a). As shown, the particle trajectories can be categorized as follows (from top to bottom): (a) the trajectories that attain surface-to-surface separations ξ min > ϵ are unaffected by the presence of the non-hydrodynamic interactions, (b) the trajectory that corresponds to the minimum attained surface-to-surface separation ξ min = ϵ grazes the obstacle and serves as the critical trajectory, and (c) the trajectories that would have approached a minimum surface-to-surface separation ξ min < ϵ, however, are forced to circumnavigate the obstacle by maintaining a constant separation equal to ϵ on the approaching side due to the hard-core potential. The final type of trajectories collapse onto the critical trajectory downstream of the obstacle, breaking their fore-aft symmetry. Thus, the critical trajectory (of type (b) described above) defines the critical impact parameter as b in = b c, with the minimum separation corresponding to this impact parameter as the range of the non-hydrodynamic interactions ξ min = ξ c = ϵ. Therefore, in the presence of short-range repulsive non-hydrodynamic interactions, the relationship between b in and ξ min also relates the critical impact parameter b c to the range of the interactions ϵ (Balvin et al., 2009; Herrmann et al., 2009; Frechette & Drazer, 2009; Luo et al., 2011; Risbud & Drazer, 2013; Risbud et al., 2013). Using the hard-wall model for the non-hydrodynamic interactions combined with the dilute assumption, we can thus replace the physical particle-obstacle system with its kinematic equivalent shown in figure 4.3. The obstacle radius b can be replaced 76

97 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING Particle F. Particle (a) bc (b) Figure 4.3: (a)[adapted from (Balvin et al., 2009)] Three kinds of particle trajectories in the presence of short-range repulsive non-hydrodynamic interactions. (b) Depiction of abstraction of the physical particle-obstacle system to a system in which a pointparticle traverses past an obstacle of radius b c. The outgoing part of the trajectories with b in < b c is tangent to the obstacle in the abstract system. 77

98 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING by b c and the particle can be reduced to a point particle. As shown, since the particle trajectories with incoming impact parameter b in > b c remain fore-aft symmetric, one can replace them with straight lines uninfluenced by the obstacle. The trajectories with b in < b c (that would intersect the new, abstract obstacle), get laterally displaced by (b c b in ), and continue as tangents to the obstacle, parallel to the forcing direction. It is interesting to note that the hydrodynamic as well as non-hydrodynamic interactions are incorporated in the single parameter b c. 4.3 Directional locking The defining feature of deterministic lateral displacement is directional locking of particle trajectories (see figure 4.1(b)). In a square array of obstacles (e.g., DLD devices), the particle follows a periodic trajectory with a periodicity of (say) p obstacles in X-direction and q obstacles in Y -direction for a range of values of θ, and some integers p & q. In such a case, the trajectory is said to be locked in the [p, q] direction for that range of θ. The migration angle α is defined by, tan α = q p. Equipped with the abstraction of the particle-obstacle pair described in the previous section, we now consider a square array of obstacles with radius b c, separated by the lattice spacing l. Figure 4.4 shows the geometric set-up essential for the derivations that follow. The figure shows two coordinate systems, XY -system with 78

99 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING its axes parallel to the principal axes of the lattice as well as x y -system with x -axis parallel to the direction of the driving field F. Since we have a point-particle traversing in a straight line parallel to the direction of the driving field, it is evident that particle-obstacle interaction (a collision ) is possible only if the particle trajectory intersects the obstacle, i.e., only if the distance d of the obstacle centre from the trajectory is less than the obstacle radius. Note that, as shown in figure 4.4, d is the same as the incoming impact parameter b in for the corresponding obstacle. It is evident that, with respect to the sign of y -coordinate of the point of collision, the two kinds of collisions are top (y > 0) and bottom (y < 0). Therefore, a given periodic trajectory, can exhibit periodicity in exactly three distinct modes: (a) all successive collisions satisfy y > 0 (top-top collisions, figure 4.4(a)), (b) all successive collisions satisfy y < 0 (bottom-bottom collisions, figure 4.4(b)), or (c) collisions alternately satisfy y > 0 and y < 0 (top-bottom-top collisions, figure 4.4(c); equivalently, the fourth kind, bottom-top-bottom collisions can also lead to periodicity, however, they are mathematically equivalent to top-bottom-top collisions). As shown in figure 4.4, we choose an arbitrary obstacle, which has undergone a collision, as the origin of the XY -system. In the case of top-top and bottom-bottom collisions (figure 4.4(a) and (b)), we assume that the period is p in X-direction, and q in Y -direction, for some integers p & q. Hence the coordinates of the centre of the next obstacle are (pl, ql) in figures 4.4(a) and (b). In the case of periodic trajectories arising from top-bottom-top (equivalently, bottom-top-bottom) collisions 79

100 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING (figure 4.4(c)), we assume that p 1 and p 2 are the alternate periods in X-direction, while q 1 and q 2 are in Y -direction, again, for some integers p 1, p 2, q 1 & q The periodicity-condition For a top-top collision, the equation of the trajectory in XY -system is, Y = X tan θ + b c sec θ. Since the centre of the next obstacle, (pl, ql), lies in the lower half-plane of the trajectory, it satisfies ql < pl tan θ + b c sec θ. Therefore, the normal distance between the obstacle centre and the trajectory in the case of top-top collisions is, d T T = pl tan θ ql + b c sec θ 1 + tan 2 θ = pl sin θ ql cos θ + b c (4.1) For a top-top collision, in x y -system on the second obstacle, the incoming impact parameter must satisfy 0 b in = d T T < b c. Therefore, (4.1) yields, ql cos θ pl sin θ b c (4.2) A similar procedure for bottom-bottom collisions dictates that the trajectory is described by Y = X tan θ b c sec θ. The obstacle centre (pl, ql), lies in the upper half-plane of the trajectory satisfying ql > pl tan θ b c sec θ. Therefore, d BB = ql pl tan θ + b c sec θ 1 + tan 2 θ = ql cos θ pl sin θ + b c (4.3) 80

101 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING Y y' x' Y } b in = d b c y' b c x' q` b in = d } q` b c θ θ p` ^_a X Y b c θ θ p` ^_b X y' } b in = d 2 x' Y b in = d 1 y' } b c x' p 2` b c q 2` X q 1` b c θ θ p 1` X ^_c Figure 4.4: Schematic depicting three possibilities leading to periodic trajectories (see text). In (a) and (b), the trajectories repeat after p obstacles along X-axis and q obstacles along Y -axis. In (c), the period along X-axis is (p 1 + p 2 ) and that along Y -axis is (q 1 + q 2 ). 81

102 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING The bounds on b in in x y -system dictate, b c < b in = d BB 0. Imposing these bounds on d BB in (4.3), we obtain, b c ql cos θ pl sin θ (4.4) For top-bottom-top collisions leading to periodicity, we can similarly arrive at b c < p 1 l sin θ q 1 l cos θ + b c 0 and 0 p 2 l sin θ q 2 l cos θ b c < b c for the first (top-bottom) and the second (bottom-top) collisions, respectively. Since these always occur successively in a periodic trajectory, we can add the two inequalities to yield, b c (q 1 + q 2 )l cos θ (p 1 + p 2 )l sin θ b c (4.5) Thus, (4.2), (4.4) and (4.5) can be combined into a single inequality: if a trajectory is p-q-periodic for some arbitrary integers p and q, i.e., it is p-periodic in X-direction and q-periodic in Y -direction, then p & q satisfy, ql cos θ pl sin θ b c q cos θ p sin θ (b c /l) (4.6) Note that (4.6) is the necessary condition for periodicity of a trajectory in a strict mathematical sense, but it is not the sufficient condition. Which means, if a trajectory is known to exhibit [p, q]-locking, then the pair [p, q] must satisfy (4.6). Conversely, there may exist many pairs [p, q] which satisfy (4.6), for a given forcing angle θ and parameters b c and l. However, physically, the trajectory would become periodic after a collision with the obstacle closest to the one at the origin, i.e., only if the pair [p, q] is the closest possible pair to the origin [0, 0] satisfying (4.6). 82

103 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING In figure 4.5, we compare the migration angles (tan α = q/p) obtained by solving (4.6) for [p, q] (using Mathematica R ) for different forcing angles, with those obtained from trajectory calculations using particle-particle simulations under the dilute assumption (Frechette & Drazer, 2009). The critical parameter b c, used in both cases corresponds to particle size the same as that of the obstacle (a = b), the range of non-hydrodynamic interactions ϵ = 10 3 a, and the inter-obstacle spacing l = 5a. We see an excellent agreement between the theoretical solution and the simulations. The same figure also shows the agreement between data from microfluidic experiments (Devendra & Drazer, 2012) and theory. Note that the ratio b c /l corresponding to the experimental data shown in the figure is approximately equal to that used in theoretical calculations. Equality in (4.6) holds at the ends of the steps of the staircase. Therefore, both transition-forcing-angles from one locked migration direction to the next as well as the migration direction itself can be computed assuming equality in (4.6). For example, in the simplest possible cases, such calculations yield: (i) The first transition angle from [1, 0]: sin θ = sin θ = b c l (4.7) (ii) The last transition angle to [1, 1]: cos θ sin θ = b c l 2 sin( π 4 θ) = b c l (4.8) 83

104 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING tan α Microfluidic experiments Partilcle-particle simulations Theory, inequality (4.6) tan θ Figure 4.5: Migration direction (tan α) versus forcing direction (tan θ) portraying devil s staircase-like structure representing directional locking. The empty circles represent individual particle-particle simulations under the dilute approximation, the line represents the solution of (4.6), [p, q], such that the integer pair [p, q] is the closest integer pair to [0, 0]. The filled circles with error bars correspond to the data from microfluidic experiments (Devendra & Drazer, 2012). 84

105 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING The first locking direction after [1, 0] If the locking direction after the first transition is [p 1, q 1 ], then q 1 cos θ p 1 sin θ b c l. However, from (4.7), sin θ = b c /l. Also, increasing θ counter-clockwise from X- to Y -axis, the first transition should be from a locking direction along the zeroth row of obstacles along X-axis (i.e., [1, 0]) to the first row of obstacles along X-axis (i.e., [p, 1] for some integer p). Therefore, q 1 should satisfy q 1 = 1 (see figure 4.1(b)). Thus, cot θ p 1 1. Since p 1 is an integer, we get, p 1 = cot θ.... floor function (4.9) Thus, the locked direction after first transition is tan α = q 1 /p 1 = 1/ cot θ, where θ is given by (4.7) above The last locking direction before [1, 1] If the locking direction before the final transition is [p f, q f ], then q f cos θ p f sin θ b c l. From (4.8), cos θ sin θ = b c /l. Further, increasing θ counter-clockwise from X- to Y -axis, the last transition from [p f, q f ] to [q f, q f ] (i.e., [1, 1]) should satisfy p f = q f +1. Thus, the above inequality becomes, sin θ q f = = cos θ sin θ sin θ cos θ sin θ qf 1. Since qf is an integer, 85 tan θ 1 tan θ. (4.10)

106 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING [1,1] 1 1 (a) Tan α Tan α [1,0] θ Tan θ [1,1] (b) [2,1] [1,0] θ 1 θ 2 Tan θ Figure 4.6: (a) 1-step staircase with transition [1, 0] [1, 1] at θ 1 (b) 2-step staircase [1, 0] [2, 1] [1, 1] with transitions at θ 1 and θ 2. Therefore, the locked direction before the final transition to [1, 1] is given as where, θ is the solution of (4.8). tan α = q f /p f = tan θ 1 tan θ tan θ 1 tan θ + 1, 4.4 Design rules and separation resolution in DLD for simple staircase structures We first derive the constraints on the ratio b c /l for a particle to exhibit exactly one transition (figure 4.6(a)) and exactly two transitions (figure 4.6(b)), based on (4.6) and We have shown in that the first transition angle is given by sin θ = b c /l corresponding to the transition from [1, 0] locking direction, and the last transition angle to [1, 1] locking direction is given by 2 sin (π/4 θ) = b c /l. We equate these two conditions to get the only transition resulting in the 1-step staircase 86

107 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING (figure 4.6(a)): sin θ 1 = π 2 sin 4 θ 1 θ 1 = arctan , (4.11) and, b c l = 1 5. (4.12) Thus, a square lattice with unit cell length l can be designed for a particle of a given radius a, such that the particle exhibits a 1-step staircase structure: by using obstacles of radius b and range of non-hydrodynamic interactions ϵ that correspond to the critical parameter b c a function of a, b and ϵ satisfying b c = l/ 5. In the case of a 2-step staircase (figure 4.6(b)), the locking direction after the first transition from [1, 0] is [2, 1], which is obviously the same as the locking direction before the final transition to [1, 1]. Using 4.3.1, , , tan α = 1 2 = 1 cot θ 1 = tan θ 2 1 tan θ 2 tan θ 2 1 tan θ The first set of equations above yields, cot θ 1 = 2 and & sin θ 1 = b c /l tan θ 2 1 tan θ 2 = 1. Thus, using sin θ 1 = b c /l, we obtain, 1 3 < tan θ tan θ 2 < 2 3 (4.13) 1 < b c 10 l 1. 5 (4.14) Thus, if a square lattice satisfies (4.14) for a particle of radius a, a critical parameter b c (a function of a, b and ϵ) and unit cell l, then the particle exhibits locking with a 2-step staircase structure with the corresponding two transition angles satisfying (4.13). 87

108 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING Design constraints and separation resolution in DLD In pairwise size-based separation, two particles of different sizes, say a and a (and perhaps different length-scales corresponding to the range of non-hydrodynamic interactions, say ϵ and ϵ ) exhibit two distinct critical parameters, viz.- b c and b c. Separation is possible at forcing angles such that the migration directions tan α = q/p and tan α = q /p are distinct. Further, a larger difference between the migration directions is synonymous with a higher resolution. Thus, an optimal design strategy is to maximize α α. A good approximation for α α can be obtained by rearranging (4.6) as, q 2 + p 2 sin (α θ) q 2 + p 2 (α θ) b c l, (4.15) where, the last inequality results from the small angle approximation sin(α θ) (α θ) since 0 < α, θ π/4. As an immediate consequence of (4.15), we note that the largest difference between the migration direction and the forcing direction (i.e., tan α tan θ α θ ) occurs before the first transition from α = 0 (locking [1, 0] and when p 2 + q 2 = 1), and it is always equal to α θ = θ b c /l. This observation supports our earlier experimental inference that, it is the most beneficial strategy to set forcing angle between the first transitions (θ 1 and θ 1) of the two species undergoing separation (Balvin et al., 2009; Bowman et al., 2012; Devendra & Drazer, 2012). 88

109 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING Combining (4.15) for both particles a and a, we get, α α b c b c + p2 + q 2 l p 2 + q 2 l (4.16) Inequality (4.16) not only gives an upper bound on the resolution, but also gives design constraints on the obstacle radius b and the lattice spacing l through the ratio b c /l for known locking directions ([p, q] and [p, q ]), a fixed forcing angle θ and known radii of particles (a and a ). In the following, we illustrate this result for particles with simple staircase structures, viz.- only one transition from [1, 0] to [1, 1] and two transitions [1, 0] [2, 1] [1, 1]. As shown in 4.4, any particle exhibiting 1-step staircase must have the same b c, and hence a mixture of such particles cannot be separated. Further, in case of a mixture of a particle exhibiting a 1-step staircase (radius a ) and another particle exhibiting a 2-step staircase (radius a), (4.11) and (4.13) dictate that 1 3 < tan θ 1 (tan θ 1 = 1 2 ) tan θ 2 < (figure 4.7(a)). For a mixture of particles exhibiting a 2-step staircase, using (4.7), (4.8) and (4.13), the two cases b c > b c or b c < b c lead to 1 3 < tan θ 1 < tan θ < tan θ 2 < tan θ 2 < (figure 4.7(b)) or 1 3 < tan θ 1 < tan θ < tan θ 2 < tan θ 2 < (figure 4.7(c)), respectively. 89

110 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING Therefore, figure 4.7 shows the only cases possible when a mixture of a particle exhibiting 1-step staircase and a particle exhibiting 2-step staircase are subjected to DLD (figure 4.7(a)), as well as when a mixture of two particles exhibiting 2-step staircases undergo DLD (figures 4.7(b) and (c)). It is evident from the figure that separation between primed and non-primed species is possible only if the forcing angle satisfies θ [θ 1, θ 1] [θ 1, θ 2 ] for figure 4.7(a), θ [θ 1, θ 1 ] [θ 2, θ 2] for figure 4.7(b) and θ [θ 1, θ 1] [θ 2, θ 2 ] for figure 4.7(c). Further, the figure also indicates that (π/4 arctan (1/2)) and arctan (1/2) are the only two separation resolutions ( α α ) corresponding to these cases. Thus, the maximum resolution between separated species in DLD corresponding to figure 4.7 is arctan(1/2) 26.56, and it occurs if θ [θ 1, θ 1] for figure 4.7(a), θ [θ 1, θ 1 ] for figure 4.7(b) and θ [θ 1, θ 1] for figure 4.7(c). As highlighted in the context of (4.15), this conclusion is consistent with our experimental observation, that the forcing angle between the first transition angles of the species to be separated, achieves the best resolution (Balvin et al., 2009; Bowman et al., 2012; Devendra & Drazer, 2012). 90

111 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING [1,1] [1,1] (a) (b) (c) [1,1] Tan α [2,1] [1,0] θ 1 θ' 1 Tan θ θ 2 Tan α [2,1] θ' 1 ' [1,0] θ 1 θ 2 Tan θ θ 2 Tan α [1,0] [2,1] θ 1 θ1 ' θ' 2 Tan θ θ 2 Figure 4.7: Three combinations of the simplest staircase structures possible: (a) one particle exhibits 1-step staircase, the other exhibits 2-step staircase, with the former transition lying between the two transitions of the latter, (b) and (c) both particles exhibit 2-step staircase structures. 4.5 Extension and application of theory Directional locking with noisy trajectories In the context of microfluidics, deterministic trajectories of particles (discussed thus far) are subjected to random noise through various mechanisms, such as, departure of obstacle or particle shape from perfect circular cross-section, defects in lattice, and finite Péclet numbers associated with non-negligible Brownian motion of the particles. Our abstract model ( 4.3) can be adapted to incorporate the noise by treating the ratio b c /l as a random variable sampled from a suitable probability density function f(b c /l). Modelling the exact functionality of the distribution depends upon the nature of the noise. Here, we assume f(b c /l) to be a normal distribution with mean given by the deterministic value of b c /l. We use the following approximate 91

112 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING scaling argument to estimate the variance for the same: σ 2 b c 2Dt 2 D Ul l2... t l U σ 2 b c 2l2 Pe If we assume l to be constant, and lump noise from all mechanisms in b c, we get, σ 2 b c/l = σ 2 b c /l 2 Pe (4.17) From the steps (4.1) to (4.5) it is evident that (4.6) is applicable at each consecutive pair of particle-obstacle collision. Further, due to the dilute assumption, each collision can be assumed to be an independent event in the sense of a stochastic process. Therefore, a realization of a noisy trajectory can be obtained by assigning a random (b c /l) j sampled from f(b c /l) NormalDistribution(b c /l, σ bc/l), to each collision. The integer-pair [p j, q j ] satisfying (4.6) defines the local migration direction tan α j = q j /p j corresponding to that particular collision. Averaging over multiple successive collisions gives the mean migration direction corresponding to the particular realization. In figure 4.8, we compare the results of such computation (figure 4.8(a)) with those obtained by solving the steady state Fokker Planck equation with periodic boundary conditions for a point particle in an array of obstacles, for various Péclet numbers (figure 4.8(b), (Herrmann et al., 2009)). The obstacle radius in the latter case is b = b c and the lattice spacing is the same in both cases. We observe a good agreement between the two approaches, especially considering the approximate 92

113 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING nature of scaling (4.17). The agreement validates the notion that the noise in the system can be lumped into a single variable, in this particular case b c Size-based band-pass filtering with DLD The square lattice in a DLD device can be used as a size-based bandpass filter, such that if a mixture of particles of different radii passes through the lattice, the particles with radii between two pre-specified cut-offs (say, a 1 < a < a 2 ) migrate at an angle greater than those with radii outside the cut-offs (i.e., a < a 1 or a > a 2 ). Here, we demonstrate this phenomenon using specific migration directions [1, 0] and [1, 1]: the band desired to be extracted (a 1 < a < a 2 ) migrates at [1, 1], while the particles with a > a 2 migrate at [1, 0] and the particles with a < a 1 migrate at angles less than π/4, but greater than zero. The dependence of b c on particle radius a at a constant forcing angle can be employed to understand the underlying mechanism resulting in band-pass filtering (figure 4.9). If a 1 is the smallest particle radius and [p, q] is an integer pair closest to [0, 0], such that b c1 = b c (a 1 ) satisfies b c1 = ql cos θ pl sin θ, then a monotonic functionality between b c and a implies all radii a a 1 satisfy, b c ql cos θ pl sin θ, i.e., the inequality (4.6). Thus, as shown in figure 4.9(a), particles with a a 1 satisfy b c l sin θ, and are locked in [1, 0]-direction. Note that in the same figure, 93

114 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING 1 (a) σ 2 b c /l ~ 2 / Pe (b) tan α σ 2 = 2/100 σ 2 = 2/200 σ 2 = 2/500 σ 2 = 2/ tan θ Pe = 1 Pe = 100 Pe = 200 Pe = 500 Pe = tan θ Figure 4.8: Migration direction (tan α) as a function of forcing direction (tan θ). The horizontal steps represent the staircase in the deterministic case. (a) Results of computing the staircase for noisy trajectories according to (4.6) and (4.17) (see text). (b) Numerical solutions of steady state Fokker Planck equation with periodic boundary conditions for a point particle, reproduced from (Herrmann et al., 2009). 94

115 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING 8 6 (a) θ = 24.5 deg (b) θ = 28.5 deg b c /b [1,0] b c /b = 1.25(a/b) b c /b = 1.25(a/b) [1,0] [1,1] [1,1] Tan α a 1 a 1 a 2 [2,1] [2,1] [1,0] a/b [1,0] a/b Figure 4.9: Underlying mechanism of band-pass filtering. The dimensionless critical parameter b c /b as a function of the dimensionless particle radius a/b obtained from experimental data for liquid drops (Bowman et al., 2012). The inter-obstacle spacing l = 5b. Horizontal lines correspond to the values of b c yielding transition from [1, 0] as well as transition to [1, 1], as shown. (a) For forcing angle θ = 24.5 < arctan(1/2) and a > a 1, [p, q] = [1, 0] is the integer pair closest to [0, 0] that satisfies (4.6). Thus particles with radii a > a 1 satisfy l sin θ b c, and are locked in [1, 0], as shown in the bottom plot (tan α versus a/b). (b) For forcing angle θ = 28.5 > arctan(1/2), particles with radii a 1 < a < a 2 satisfy l(cos θ sin θ) b c and those with a 2 < a satisfy l sin θ b c. Consequently, the bottom plot shows a narrow band locked in [1, 1]-direction. 95

116 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING [p, q] = [1, 1] also satisfies the inequality, above the dashed line. However, since the forcing angle θ = 25.5 is smaller than arctan(1/2), we have sin θ < cos θ sin θ, which implies that the locking condition for [1, 0]-direction is satisfied before that for [1, 1]-direction (in the figure, the solid line is below the dashed line). Thus, in the region where both [1, 0] as well as [1, 1] satisfy the corresponding necessary periodicity conditions (4.6), the particles are locked in [1, 0]-direction, since it is closer to [0, 0], thereby satisfying the sufficient condition for periodicity. For a forcing angle θ > arctan(1/2), since sin θ > (cos θ sin θ), the condition for locking in [1, 1]-direction is satisfied before the condition for [1, 0]-direction, as shown in figure 4.9(b), which leads to the band-pass filtering of the particles locked in [1, 1]-direction. Finally, the width of the band can be increased by increasing the forcing angle. 4.6 Summary In summary, we have presented a theoretical analysis of the directional locking phenomenon exhibited by particles navigating through a square array of obstacles, in the limit of negligible particle and fluid inertia. In the dilute limit for the array (i.e., a sparse array), interactions between a single obstacle and a particle are sufficient for trajectory analysis. Coupled with the dilute assumption, we have used a critical parameter (incorporating both hydrodynamic as well as short-range repulsive non- 96

117 CHAPTER 4. ANALYSIS OF DIRECTIONAL LOCKING hydrodynamic particle-obstacle interactions) to replace the physical particle-obstacle system with its kinematically equivalent abstraction. Within the abstract model, the particle is replaced by a point-particle, while the obstacle radius is scaled to be equal to the critical parameter. Due to the model, a simple geometric analysis suffices to derive the periodicity condition, both necessary and sufficient for the particle trajectory. The periodicity condition directly leads to the devil s-staircase-like behaviour of the migration direction as a function of the forcing direction. Further, using the periodicity condition, we have computed the design constraints on the ratio of the critical parameter to the lattice spacing of the square array, and commented on the resolution of deterministic separations, when the particle exhibits simple staircase structures. We have also shown the possibility to extend our analysis to the case of noisy particle trajectories (as opposed to deterministic trajectories). Finally, we have demonstrated a that our theoretical approach can be employed to design size-based band-pass filters using DLD microdevices. 97

118 Chapter 5 Irreversibility and pinching in deterministic particle separation 1 We take a brief experimental interlude in the guise of this chapter, in an otherwise predominantly theoretical and computational work comprising this dissertation. This chapter concerns our experimental observations of the trajectory of a spherical particle moving through a constriction created by a plane wall and a cylindrical obstacle. The experiments were macroscopic, however the magnitude of inertia (Reynolds number) was maintained as small as possible. The purpose of the experiments was to mimic microfluidic systems involving the motion of a particle through a constriction. The reasons for the choice of a macroscopic system, in the microscopic system s stead, 1 Fair-use statement: This chapter appeared as a letter in Applied Physics Letters of the American Institute of Physics [Luo, M., Sweeney, F., Risbud, S. R., Drazer, G. & Frechette, J Irreversibility and pinching in deterministic particle separation. Appl. Phys. Lett. 99 (6), ]. Please refer to the end of the dissertation for fair-use license agreement. 98

119 CHAPTER 5. IRREVERSIBILITY AND PINCHING IN PINCHING are elaborated in the chapter. In the subsequent chapters, we shall make an effort to qualitatively compare the experimental results with computational/numerical ones, the latter forming the content-matter of those subsequent chapters. 5.1 Introduction Microfluidic devices that provide rapid, robust, and continuous separation methods are needed to expand the range of chemical and biological analysis amenable to lab-on-chip technologies (Janasek et al., 2006; Arora et al., 2010; West et al., 2008; Li & Drazer, 2007). Among current separation strategies for particulate systems in microfluidic devices, deterministic lateral displacement (DLD) and pinched flow fractionation (PFF) are promising, as they can operate continuously and at high flow rates. In DLD, the components being fractionated become locked into periodic trajectories that travel at different orientations through an array of obstacles (Huang et al., 2004). In PFF, separation is achieved when the components in the mixture are displaced laterally as they emerge from a constricted channel into a sudden expansion (Yamada et al., 2004; Takagi et al., 2005; Sai et al., 2006; Maenaka et al., 2008; Jain & Posner, 2008; Vig & Kristensen, 2008). Although both methods have been shown to work, the underlying mechanisms leading to separation are not clear and a better understanding is essential to develop design criteria that would optimize their performance. 99

120 CHAPTER 5. IRREVERSIBILITY AND PINCHING IN PINCHING Initial explanations for the separation mechanism in DLD and PFF were based on streamlines in Stokes flow and ignored hydrodynamic interactions between the finite-size particles and the solid boundaries (cylindrical posts, channel walls) (Huang et al., 2004; Yamada et al., 2004). More recently, however, short-range particleobstacle interactions have been proposed as the underlying mechanism leading to irreversible (asymmetric) trajectories and separation of particles in DLD (Frechette & Drazer, 2009; Loutherback et al., 2009; Balvin et al., 2009; Herrmann et al., 2009; Koplik & Drazer, 2010). Similarly, asymmetric trajectories have also been observed in symmetric PFF devices (Faivre et al., 2006). Here, we demonstrate the ubiquitous nature of these short-range particle-obstacle interactions when a suspended particle moves around a fixed obstacle. In fact, we show that they lead to a separative lateral displacement that is amplified in the presence of a constriction and as such are likely to be at the core of PFF. We also highlight the deterministic nature of this technique by performing experiments in a macroscopic model (Figure 5.1(a)). Macroscopic models are well suited to study this separation method, as they allow for the investigation of a wide range of parameters such as the Reynolds number, the particle type and size, and the extent of pinching. In contrast to microfluidic devices in which feature sizes are close to the fabrication resolution, the inherent separation between these length scales in macroscopic models isolates the effects of surface roughness from geometrical irregularity and, at the same time, enhances device fidelity. 100

121 CHAPTER 5. IRREVERSIBILITY AND PINCHING IN PINCHING Figure 5.1: (a) Experimental setup, (b) illustration of reversible (b in = b out ), critical (b in = b c ), and irreversible (b in < b c ) trajectories for a given particle (the dashed circle represents excluded volume) past an obstacle (filled circle), and (c) measurement of b out as a function of b in. The inset illustrates the determination of b c. In (c), circular and square symbols represent delrin particles with d = 3 mm and 6.35 mm, respectively. 5.2 System and particle trajectories Spherical delrin particles (ρ = 1.41g cm 3 ) with diameters, d, between 2 mm and 9.5 mm were introduced at the top of an acrylic tank filled with glycerol (see Figure 5.1(a)). The entry position of the particles was such that they were forced to move around a large cylindrical obstacle (radius R = 51 mm) as they settled. For each particle trajectory, we measured the incoming (b in ) and outgoing (b out ) impact parameters far from the obstacle to ensure asymptotic values (Figure 5.1(b)). An adjustable pinching gap was created by positioning a rigid vertical wall at a variable distance D from the cylinder (see Figure 5.1(a)). In this set up, the pinching ratio (D + R)/D was varied from 2.7 (least pinched) to 21.4 (most pinched). We first investigated the symmetry of the trajectories followed by the particles in 101