DYNAMICS OF SUSPENSIONS OF RODLIKE POLYMERS WITH ...

DYNAMICS OF SUSPENSIONS OF RODLIKE POLYMERS WITH HYDRODYNAMIC INTERACTIONS

By JOONTAEK PARK

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009 1

c 2009 Joontaek Park °

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To my father, Jong-Oh Park, my mother, Won-Sun Paek, and my ‘great’ brother, Jung-Taek Park. I thank them for their love and support.

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ACKNOWLEDGMENTS I would like to acknowledge all the people who have been instrumental in helping me to complete my doctoral degree. First of all, I devote a ‘BIG THANKS’ to my advisor, Dr. Jason E. Butler, the greatest teacher, boss, and mentor I have ever had. His patient guidance, generous spirit, enthusiasm, and, above all, faith in me enabled me to thrive as both a doctoral student and a researcher. In addition, his wellspring of knowledge and ideas, engaging teaching strategies and research projects, and his intelligence and wisdom have been constant sources of inspiration. I will always keep in mind his favorite proverb: “Be positive.” I am also indebted to my committee members, Dr. Anthony J. C. Ladd, Dr. Dmitry I. Kopelevich, and Dr. Renwei Mei, for not only their advice, feedback, and endeavors to improve this dissertation but also for serving as role models for the type of independent researcher and professor I wish to become. Their graduate courses are among the best courses I have ever taken throughout my college years. I wish also to give thanks to my colleagues, Dr. Jonathan M. Bricker, Dr. Osman Berk Usta, Hyun-Ok Park, Mike Roy, Xuan-Thinh Phan, Todd Mock, Dr. Jonghoon Lee, Dr. Byung-Jin Chun, Gaurav Misra, Rahul Kekre, Carlos Silvera-Batista, Dr. Philip D. Cobb and Dr. Gunjan Mohan, for all their help and precious and productive discussions. I am fortunate to have had the opportunity to work with and to receive encouragement and advice from a number of professors in Korea: Dr. Ho-Nam Chang, Dr. Sang-Yup Lee, Dr. Hyun-Gyu Park, Dr. Seung-Man Yang at the Korea Advanced Institute of Science & Technology; Dr. Cha-Yong Choi, Dr. Byung-Gee Kim at Seoul National University; Dr. Sung-Ok Baek at Yeungnam University and Dr. Won-Ji Chung at Changwon University. This work was supported by the National Science Foundation through a CAREER Award (CTS-0348205), graduate and research assistantships from the University of Florida, the Department of Chemical Engineering. I also acknowledge the usage of computational time in the Chemical Engineering Computer System at UF, Ray W. Fahien

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Graduate Teaching Awards for the best teaching assistant in 2008, travel grants from the American Physical Society’s Division of Fluid Dynamics as well as the Graduate Students Council of UF, support from KOSEF (M06-2004-000-10029-0), and invited talks at KAIST and POSTECH. Once again, I thank my family, Jong-Oh Park, Won-Sun Paek, and Jung-Taek Park, and blessings from my late grandparents, Yong-Hak Park, Kwae-Soon Kim, Jang-Soo Paek, and Yi-Bun Seo.

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TABLE OF CONTENTS page ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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INCLUSION OF HYDRODYNAMIC INTERACTIONS FOR RIGID ROD SUSPENSIONS . . . . . . . . . . . . . . . . . . . . . . . . .

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2.1 2.2 2.3 2.4 2.5

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KINETIC THEORY FOR CROSS-STREAM MIGRATION OF A RIGID ROD UNDER RECTILINEAR FLOW NEAR A WALL . . . . . . . . . . . . . . . . .

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3.1 3.2

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Governing Equation . . . . . . . . . . . . . . . . . . 4.2.2 Simulation Details . . . . . . . . . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Transverse Velocity Due to Shear Flow . . . . . . . 4.3.2 Distributions in Simple Shear Flow . . . . . . . . . 4.3.3 Steady State Distribution in Parabolic Flow . . . . 4.3.4 Steady State Distribution in Oscillatory Shear Flow

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DYNAMIC SIMULATION OF A RIGID ROD IN SHEARING FLOWS BETWEEN TWO PARALLEL WALLS . . . . . . . . . . . . . . . . . . . . . . 4.1 4.2

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38 40 41 43 48 48 49 52 54 56

3.4 3.5

Introduction . . . . . . . . . . . . . . . . . . Model . . . . . . . . . . . . . . . . . . . . . 3.2.1 Equation of Motion near a Wall . . . 3.2.2 Distribution Function . . . . . . . . . Steady State Distribution . . . . . . . . . . 3.3.1 Simple Shear Flow . . . . . . . . . . 3.3.2 Pressure-driven Flow . . . . . . . . . 3.3.3 Discussion on Migration Mechanisms Transient Distribution . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . .

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3.3

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Introduction . . . . . . . . . . . . . . . . . . . . . Hydrodynamic Interactions between Rigid Rods . Sedimenting Orbits . . . . . . . . . . . . . . . . . Short-time Diffusivities of Rigid Rod Suspensions Conclusion . . . . . . . . . . . . . . . . . . . . . .

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58 59 59 62 65 65 68 75 78

4.4 5

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Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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STRESS OF RIGID ROD SUSPENSIONS IN SHEARING FLOWS BETWEEN TWO PARALLEL WALLS . . . . . . . . . . . . . . . . . . . . . .

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5.1 5.2 5.3 5.4 5.5

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83 84 86 88 91

INHOMOGENEOUS DISTRIBUTIONS OF A RIGID ROD IN ROTATIONAL FLOWS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.1 6.2 6.3 6.4 6.5 7

Introduction . . . . . . . . . . . . . . . . . . . . . . . Calculation of Extra Particle Stress from Simulations Theoretical Prediction of Extra Particle Stress . . . . Discussion: Comparison of Stress Calculations . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . .

Introduction . . . . . . . . . . . . Migration in the Radial Direction Steady State Distributions . . . . Stress Calculation . . . . . . . . . Conclusion . . . . . . . . . . . . .

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CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

APPENDIX A

COMPONENTS OF GRAND MOBILITY TENSOR . . . . . . . . . . . . . . . 106

B

DETAILS OF CALCULATIONS FOR CHAPTER 3 . . . . . . . . . . . . . . . 108 B.1 B.2 B.3 B.4 B.5

C

Linearization of the Green’s Function for a Wall . . Average of Brownian Contributions . . . . . . . . . Numerical Calculation of Orientation Moments . . Asymptotic Solutions at Low Peclet Number Limit Crank-Nicolson Method . . . . . . . . . . . . . . .

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108 109 110 111 112

ADDITIONAL TESTS FOR CHAPTER 4 . . . . . . . . . . . . . . . . . . . . . 115 C.1 The Effect of Torque on Migration . . . . . . . . . . . . . . . . . . . . . . 115 C.2 Excluded Volume Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

D

A SEDIMENTING CLOUD OF RIGID RODS IN VISCOUS FLUID . . . . . . 120 D.1 D.2 D.3 D.4

Introduction . . . . . Simulation Methods Results . . . . . . . . Ongoing Work . . . .

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120 121 121 123

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

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LIST OF FIGURES Figure

page

1-1 Concentration regimes for suspensions of rigid rods . . . . . . . . . . . . . . . .

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1-2 Physical models for rigid rod-like particle . . . . . . . . . . . . . . . . . . . . . .

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2-1 A schematic diagram for a Stokeslet, f , acting at the origin, O, in an infinite fluid, which induces a velocity field, u∗ . . . . . . . . . . . . . . . . . . . . . . . .

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2-2 A schematic diagram for a line distribution of Stokeslets over a slender-body in an infinite fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2-3 Rigid rods α and β in an infinite fluid with imposed velocity field, u∞ . . . . . .

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˜ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4 Relation between T and F

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2-5 Sedimentation of two rigid rods which are aligned parallel to applied force in a box size 15L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2-6 Demonstration of two rods sedimenting parallel to gravity . . . . . . . . . . . .

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2-7 Sedimentation of two rigid rods which are aligned perpendicular to gravity . . .

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2-8 Motions of 2 rods on xy-plane. Gravity acts in the negative z-direction. . . . . .

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2-9 Demonstration of two rods sedimenting perpendicular to gravity . . . . . . . . .

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2-10 Simulation results of DTS /DTO for suspensions of rigid rods with A = 50 as a function of periodic box size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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S 2-11 Simulation results of DR /DTO for suspensions of rigid rods with A = 50 as a function of periodic box size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2-12 Periodic box size corrected DS /DO as a function of nd L3 . . . . . . . . . . . . . .

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3-1 A rigid polymer suspended in a shear flow near a solid boundary. . . . . . . . .

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3-2 Transverse velocity, r˙y , as a function of polymer orientation. . . . . . . . . . . .

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3-3 Ensemble averages of orientation moments of a slender-body as functions of Pe r .

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3-4 Contribution of the shear flow and Brownian torque to the migration of a rigid polymer in simple shear flow as a function of Pe r . . . . . . . . . . . . . . . . . .

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3-5 Ensemble averages of orientation moments of a slender-body as functions of Pe r .

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3-6 The steady state center-of-mass distributions in simple shear flow. . . . . . . . .

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3-7 The steady state depletion layer thickness Ld /L of a rigid fiber . . . . . . . . . .

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3-8 Center-of-mass distribution for a rigid polymer with A = 10 in pressure-driven flow. The height of the channel is 8L. . . . . . . . . . . . . . . . . . . . . . . . .

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3-9 Comparison of migration mechanisms in shear flow . . . . . . . . . . . . . . . .

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3-10 The transient center-of-mass distributions of a rigid fiber in simple shear flows. .

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4-1 Schematic diagrams for simulation: Arrays of closely packed fibers form the walls in a periodic boundary system . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4-2 Velocity profiles of simulated shear flows . . . . . . . . . . . . . . . . . . . . . .

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4-3 Transverse velocity r˙y due to shear flow as a function of angle φ with pz = 0, ry = 1.0L, and H = 6L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4-4 Transverse velocity due to shear contribution as a function of distance from a wall for a fixed angle under a constant shear rate . . . . . . . . . . . . . . . . .

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4-5 The simulation results and theoretical solutions for the transient center-of-mass distributions of a rigid in simple shear flows of Pe = 1.2 × 103 . . . . . . . . . .

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4-6 The simulation results and theoretical solutions for the transient center-of-mass distributions of a rigid fiber in simple shear flows of Pe = 1.2 × 104 . . . . . . .

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4-7 The simulation results and theoretical solutions for the transient center-of-mass distributions of a rigid fiber in simple shear flows of Pe = 4.8 × 104 . . . . . . . ­ ® 4-8 The simulation results with and without hydrodynamic interactions for p2y of a rigid fiber in simple shear flows . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9 Rotations of rigid rods in shear flow: Jeffery orbit motion and pole-vault motion.

73 74 75

4-10 The simulation results and theoretical solutions for the center-of-mass distributions of a rigid fiber in parabolic flows . . . . . . . . . . . . . . . . . . . . . . . . . . 76 ­ ® 4-11 The simulation results with hydrodynamic interactions for p2y of a rigid fiber in parabolic flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4-12 The simulation results for the center-of-mass distributions of a rigid fiber as in oscillatory shear flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ­ ® 4-13 The simulation results for p2y of a rigid fiber in oscillatory shear flows . . . . .

80 81

5-1 The extra particle shear stresses for rigid fibers under simple shear flow between two walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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P (ry ) for a rigid fiber with φ = 3◦ , 18◦ , pz = 0 and 5-2 The particle shear stresses σxy A = 10 in a gap of H = 6L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6-1 Schematic diagrams for a torsional flow . . . . . . . . . . . . . . . . . . . . . . .

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6-2 A drift velocity a force-free rigid rod in a torsional flow . . . . . . . . . . . . . .

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6-3 Average orientation moments of a rigid rod in torsional flows . . . . . . . . . . .

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6-4 Center-of-mass distributions of a rigid rod in torsional flow . . . . . . . . . . . .

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6-5 σ Pxz of a rigid rod in a torsional flow . . . . . . . . . . . . . . . . . . . . . . . . . 102 B-1 Schematic diagram of finite meshes for solving Eq. 3–27 using Crank-Nicolson scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 C-1 Transverse velocity due to torque contribution . . . . . . . . . . . . . . . . . . . 116 C-2 Trajectory of a non-Brownian rod near a wall . . . . . . . . . . . . . . . . . . . 118 D-1 Sequential images of a sedimenting cloud of rigid rods . . . . . . . . . . . . . . . 122

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DYNAMICS OF SUSPENSIONS OF RODLIKE POLYMERS WITH HYDRODYNAMIC INTERACTIONS By Joontaek Park May 2009 Chair: Jason E. Butler Major: Chemical Engineering Dynamics of suspensions of rigid rod-like polymers are studied by theoretical analysis and simulation considering hydrodynamic interactions. The motions of rigid rods in Newtonian suspending fluids are modeled using slender-body theory and and assumption of zero Reynolds number. Hydrodynamic interactions are included using the linearized force distribution along a rod axis. The orbits in trajectories of a pair of sedimenting rods and the concentration dependent trend of short-time diffusivities are simulated to validate the simulation method used in this study. A kinetic theory is developed to investigate cross-stream migration of a rigid polymer undergoing rectilinear flow in the vicinity of a wall. Hydrodynamic interactions between the polymers and the boundary result in a cross-stream migration. In simple shear flow, polymers migrate away from the wall, creating a depletion layer in the vicinity of the wall which thickens as the flow strength increases relative to the Brownian force. In pressure-driven flow, an off-center maximum in the center-of-mass distribution occurs due to a competition between hydrodynamic interactions with the wall and the anisotropic diffusivity induced by the inhomogeneous flow field. A rigid polymer within simple shear flow, parabolic flow, and oscillatory flow at high P´eclet numbers (weak Brownian motion) is simulated to enable comparisons with the steady and transient distributions predicted by the theory and simulation. The simulation and theoretical results are in good agreement at sufficiently high shear rates, validating

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approximations made in the theory. The effect of the inhomogeneous distribution on the effective stress is also investigated. A theoretical analysis on cross-stream migration of a rigid polymer in torsional flow is also investigated. Due to the curvature of radial shear flow, a rigid rod migrates towards the center axis. An analogy is made between flexible polymers migrating in torsional flows and the particle stress is also calculated for the purpose of determining the practical impact of the migration upon rheological measurement in parallel-plate geometries.

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CHAPTER 1 INTRODUCTION Rigid, rod-like polymers, macromolecules and particles (simply rigid rods or rigid fibers) can be found in many areas of common existing and new emerging technologies. Many macromolecules of biological origin have rod-like structure: a short DNA fragment less than 100 base pairs (' 50nm) (Newman et al., 1974; Tirado et al., 1984; Wang et al., 1991; Tinland et al., 1997), xanthan gum which is a helical polysaccharide chain of molecular weight ∼ 1.8 × 106 (Davidson, 1980), actin (Schmidt et al., 1996), and collagen (Oh and Park, 1992; Claire and Pecora, 1997). Even micro-organisms, such as fd bacteriophages and tobacco mosaic viruses, have rod-like structure (Caspar, 1963; Chen et al., 1980; Tracy and Pecora, 1992; Cush and Russo, 2002). Synthetic rigid polymers or nanotubes are used for improvement of material properties (Adams et al., 1989). Composites of Kevlar and carbon fiber greatly increase tensile strength and stiffness (Gustin et al., 2005). Semiconductor CdSe nanorods is used in solar cells (Huynh et al., 2002). Carbon nanotubes have enormous applications due to those unique properties (Salvetat et al., 1999; Colbert and Smalley, 1999; Vigolo et al., 2000). Compared to suspensions of spherical particles, suspensions of rigid rods exhibit a much larger range of behaviors since the orientation of a rigid rod makes its configuration more likely to be affected by flow fields and interactions with other particles at lower volume fraction than that of a spherical particle. Therefore, suspensions of rigid rods produce much stronger non-Newtonian effects, such as normal stress differences, shear thinning and thickening, than a suspension of spherical particles at a similar volume fraction (Larson, 1999). Compared to flexible polymers (Larson, 2005), rigid polymers have different physical behavior, such as formation of liquid crystal structure at high concentration. These characteristic physical properties dependent on concentration make

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Figure 1-1. Concentration regimes for rod-like polymer: A) dilute (nd L3 < 1), B) semi-dilute (1 < nd L3 < L/d), C) concentrated isopropic (L/d < nd L3 400. Fifty configurations were required for convergence to within 0.01% for N < 400. The aspect ratio of rods in these simulations was fixed to 50. The number of Gauss-Legendre quadrature points on the rods for integrating hydrodynamic interactions was set to 14 to produce a convergence within 0.02% at nd L3 = 150.

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A

3

ndL = 5 (0.9879 ±0.0001) 3

ndL =10 (0.9764 ±0.0001) 3

ndL =30 (0.9366 ±0.0002)

T

0.9

3

D T/D

O

ndL =40 (0.9224 ±0.0006) 3

S

ndL =50 (0.9071 ±0.0007)

0.85

0.8 0.2

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0.25

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0.35

0.8

0.45

0.5

0.55

S

D T/D

O

T

B

0.75 3

ndL = 70 (0.8799 ±0.0010) 3

ndL = 90 (0.8594 ±0.0006) 3

ndL =110 (0.8397 ±0.0009) 3

ndL =130 (0.8274 ±0.0011) 3

0.7 0.45

ndL =150 (0.8114 ±0.0006)

0.5

0.55

0.6

0.65

-1

(b/L)

Figure 2-10. Simulation results of DTS /DTO for suspensions of rigid rods with A = 50 dependent inversely with periodic box size, b/L. Values extrapolated at infinite box size limit (or box size corrected) for each number density and corresponding regression standard errors are shown in the legend. A) nd L3 = 5 ∼ 50 and B) nd L3 = 70 ∼ 150.

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1

A

3

ndL = 5 (0.9957 ±0.0000) 3

ndL =10 (0.9916 ±0.0000)

0.99

3

ndL =30 (0.9762 ±0.0001)

0.98

3

S

D R/D

O

R

3

ndL =40 (0.9690 ±0.0002) ndL =50 (0.9630 ±0.0003)

0.97

0.96

0.95

0.02

0.04

0.06

0.08

B

0.1

0.12

3

ndL = 70 (0.9514 ±0.0002) 3

ndL = 90 (0.9407 ±0.0003)

0.94

3

ndL =110 (0.9320 ±0.0002) 3

S

D R/D

O

R

3

ndL =130 (0.9246 ±0.0003)

0.93

ndL =150 (0.9181 ±0.0006)

0.92

0.91

0.9 0.1

0.15

0.2

0.25

0.3

-3

(b/L)

S O Figure 2-11. Simulation results of DR /DR for suspensions of rigid rods with A = 50 show a dependence on the inverse cube of periodic box size, (b/L)3 . Values extrapolated at infinite box size limit (or box size corrected) for each number density and corresponding regression standard errors are shown in the legend. A) nd L3 = 5 ∼ 50 and B) nd L3 = 70 ∼ 150.

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S Simulation results are shown in Figure 2-10 for DTS and Figure 2-11 for DR . Though

diffusivities were calculated at the same number density, the results varied with b, the size of the cubic periodic box. This box size effect must be corrected to attain the values for an unbounded solution. The translational diffusivities show linear dependence on the S inverse of box size, while DR depends inversly on the cube of the box size, approximately.

These trends were used to approximately correct the box size effect by extrapolation to an infinite box size. The linear trends of current simulation data have correlation coefficients of mostly over 99.9%. Phillips et al. (1988) extrapolated to an infinite number of spherical particles using the scaling ∼ N −1/3 for translational diffusivity and ∼ N −1 for rotational diffusivity. Since the number density is nd L3 = N (L/b)3 , scalings used in extrapolations at the infinite b and the infinite N under the same number density are identical. Mackaplow (1995) also reported similar trends for the mobility of prolate spheroids and explained that the contribution to translational mobility is mainly through the Stokeslet which is a function of 1/r, where r is the distance between centers of mass of particles. The rotational mobility depends on dipole moments which are a function of 1/r3 . Other formula for correcting the box size effect of suspensions of spherical particles used ratios between solvent viscosity and effective viscosity (Ladd, 1990). Mackaplow (1995) tried to find a method for suspensions of rigid rods, however a formula only for translational mobility is derived and the results corrected by this formula lack consistency. Therefore, the method to correct the box size effect for suspensions of rigid rods is not developed yet. The short-time diffusivities considering hydrodynamic interactions with correction S of the box effect are plotted in Figure 2-12. The ratio, L2 DR /DTS , which is expected to

be an important parameter for the long-time diffusivity scaling (Cobb and Butler, 2005) is also plotted. As the number density increases, both diffusivities decreased because hydrodynamic interactions hinder the motion of rods. The translational diffusivity decays S faster with increase of concentration than DR . This can also be explained by the mobility

dependence on 1/r and the random distribution of rods. The value of DTS was derived

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S from the translational mobility which has a 1/r dependence while the value of DR was

captured from rotational mobility, which has a slower 1/r3 dependence. The different S dependencies on concentration make the ratio of L2 DR /DTS increase from 9 at infinite

dilution to about 10 at 150 nd L3 . According to the findings of Cobb and Butler (2005) that the scaling exponent of the long-time diffusivity decreased at higher ratios, inclusion of hydrodynamic interactions is expected to decrease the exponent. Converting the number density dependent simulation data for DST into volume fraction, φv , the volume fraction dependent data gives a scaling of 1 − 2.66φv at the dilute limit. Compared to the theoretical scaling 1 − 1.83φv for spherical particles by Batchelor (1972), the DTS of rigid rods with A = 50 has a stronger dependence on concentration, as other physical properties of rigid rods are all more concentration sensitive than those of spherical particles. Other simulation or experimental data for DS for rod-like particles are not available yet. As concentration increases to isotropic concentrate regime, sensitivities to concentration are reduced. This trend suggests that hydrodynamic screening arises at higher concentrations. 2.5

Conclusion

The equation of motion for rigid rods in a suspension is derived by incorporating hydrodynamic interactions in the mobility matirx. The examples of simulations using Eq. 2–22 are shown for dynamics of a pair of sedimenitng rods and calcualtion of short-time diffusivities in dilute to concentrate suspensions. The results confirm the validation of the model equation derived in Chapter 2 and the code which is developed using the equation.

36

1.1

2

S

S T

L D R/D S DR S DT

1

S

D /D

O

1.05

0.95

0.9

0.85

0.8

0

10

20

30

40

50

60

70

80

ndL

90

100 110 120 130 140 150

3

Figure 2-12. Periodic box size corrected DS /DO considering hydrodynamic interactions for suspensions of rigid rods with A = 50 as a function of nd L3 .

37

CHAPTER 3 KINETIC THEORY FOR CROSS-STREAM MIGRATION OF A RIGID ROD UNDER RECTILINEAR FLOW NEAR A WALL 3.1

Introduction

In Chapter 3, a kinetic theory is developed for cross-stream migration of rigid polymers in dilute solutions undergoing rectilinear flow near a solid boundary. Flexible polymers in dilute solution migrate across streamlines in simple shear and pressure-driven flows. Though the origin and direction of the migration were controversial (Agarwal et al., 1994), recent work (Ma and Graham, 2005; Khare et al., 2006; Usta et al., 2006, 2007) has clarified that flexible polymers primarily migrate away from bounding walls due to a hydrodynamic lift force. The local shear flow extends the polymer, generating tension in the chain and an additional flow field around the polymer. The flow field becomes asymmetric near a no-slip boundary and results in a net drift away from the wall for both simple shear and pressure-driven flow. In inhomogeneous flows, the variation in the local shear rate alters the position dependent conformation and consequent diffusivity transverse to the flow. This additional mechanism results in a weak displacement of the polymers away from the centerline in pressure-driven flow, though the net migration still occurs away from the wall unless hydrodynamic interactions with the wall are screened, as occurs for highly confined polymers. This net migration towards the centerline, which has been observed qualitatively in experiments by Fang et al. (2005), has implications for the design and operation of microfluidic devices (Whitesides and Stroock, 2001; Stone et al., 2004). As one effect, the lift force hinders adsorption of polyelectrolytes (Hoda and Kumar, 2007b,a, 2008). The mechanism and direction of migration remains unclear for rigid polymers, or Brownian rigid rods, in dilute solution. Measurements on semi-rigid xanthan molecules in pressure-driven flow indicate migration away from the wall, resulting in a depletion layer (Ausserre et al., 1991). Mechanisms not associated with hydrodynamic interactions with the bounding wall also alter the depletion layer. Simulations and models which consider

38

only steric effects predict that sufficiently strong shear enhances the depletion layer beyond that of the equilibrium distribution (de Pablo et al., 1992; Schiek and Shaqfeh, 1995). Also, gradients in the diffusivity, caused by the coupling of the anisotropic mobility of the center of mass of a rod and differences in the shear rate across a channel with pressure-driven flow, shift the distribution closer to the walls (Nitsche and Hinch, 1997; Schiek and Shaqfeh, 1997a), similar to the mechanism for flexible polymers. Most recently, simulations of rigid polymers in pressure-driven flow predicted migration away from the wall (Saintillan et al., 2006a), though the depletion layer is larger than predicted by steric interactions alone. The authors proposed that a subtle combination of orientation effects and hydrodynamic interactions with the walls produces the overall migration. To confirm the results of Saintillan et al. (2006a) and clarify the origins of the observed migration, a theoretical model equation predicting the distribution of rigid polymers near solid boundaries is developed in Section 3.2. The theory contains approximations similar to those made for flexible polymers (Ma and Graham, 2005): the polymer distribution function is factorized into a product of a center-of-mass and orientation distribution and a far-field approximation for the hydrodynamic interaction with the bounding wall is made. Using the model equation, distributions of rigid polymers in simple shear flow and pressure-driven flow between parallel walls are presented in Section 3.3. Results show that rigid polymers migrate away from the wall due to hydrodynamic interactions with the wall in the both shear flows. For pressure-driven flow, the off-center maximum in the center-of-mass distribution due to a competition between hydrodynamic interactions with the wall and the anisotropic diffusivity induced by the inhomogeneous flow field. The theory is also developed to give the time evolution of the distribution in simple shear flow in Section 3.4.

39

Figure 3-1. A rigid polymer suspended in a shear flow of strength γ. ˙ Notations describing geometry of a rod are same as defined in Chapter 2 except that sα and sβ indicate an evaluation point and a source point on a rod, respectively. A no-slip boundary is located at y = 0. 3.2

Model

The evolution of a rigid polymer in solution (Figure 3-1) is governed by a continuity equation for the distribution function, Ψ (r, p, t): ∂Ψ ∂ ∂ ˙ . = − · (˙rΨ) − (I − pp) : (pΨ) ∂t ∂r ∂p

(3–1)

For the rigid fiber, Ψ is factorized into a product of a center-of-mass, n, and orientation, ψ, distribution, Ψ (r, p, t) = n (r, t) ψ (r, p, t) , where n =

R

(3–2)

Ψdp. Integrating Eq. 3–1 over the orientation distribution and solving for the

steady state gives 0=−

∂ · (n h˙ri) , ∂r

40

(3–3)

where the angle brackets h· · · i indicate an ensemble average over orientation,

R

· · · ψdp.

Determining n from Eq. 3–3 necessitates development of an explicit expression for the transverse velocity, r˙y . 3.2.1

Equation of Motion near a Wall

In the absence of inertia and for a Newtonian suspending flow, Eq. 2–7 gives an expression for the center-of-mass velocity, r˙ , approximated by slender-body theory. In Eq. 2–7, a local fluid velocity on a rod u (s) under shear flow near a solid boundary includes contributions from the imposed shear flow, γ, ˙ and the disturbance velocity generated by the force on the polymer which is reflected by the wall, Z ∞

∗

L/2

u (sα ) = u (sα ) + u (sα ) = γ(s ˙ α )ˆ x+

G(sα , sβ ) · f (sβ )dsβ ,

(3–4)

−L/2

ˆ is a unit vector in the x-direction. For the Green’s function in Eq. 3–4, the where x solution of Blake (1971) for a planar wall with the Oseen-Burger tensor removed is used. The coordinate sβ indicates the location of a point force on the polymer, whereas sα is the point of evaluation of the reflected disturbance in velocity. Combining Eq. 3–4 and the linearized force density, Eq. 2–16, followed by a substitution into Eq. 2–7 gives a complete expression for r˙ , approximated at the stresslet level: " Z # Z L/2 1 L/2 ∞ 6ξ (1b) r˙ = u (sα )dsα − G · pp · u∞ (sα )sα dsα + L −L/2 cs L5 αβ −L/2 · ¸ 1 (0) 6ξ (1b) (1a) −1 ξ (I + pp) + 2 G αβ − G · pp · G αβ · F + L cs L5 αβ · ¸ 12 (1b) 72ξ (1b) (2) G − G · pp · G αβ · (T × p) , L4 αβ cs L8 αβ

(3–5)

where cs = 1 +

6ξ p · G (2) · p. L4

(3–6)

Recall that double integral of Green’s functions are indicated as G, as defined in Eq. 2–20. Under the assumption that a distance of a polymer from a wall is comparable to its own length (ry & L), the Green’s function is linearized about the center of the polymer.

41

Replacing the Green’s functions in Eq. 3–5 with the linearized expressions, shown in Appendix B, retaining leading orders in each contributions and extracting the transverse component of the velocity give, ¡ ¢ r˙y = γλ ˙ (ry ) px py 1 − 3p2y + ξ −1 (ˆ y + py p) · F +

24 −1 ˆ − p) · (T × p) , ξ λ (ry ) (3py y L2

(3–7)

ˆ is a unit vector where ry and py indicate y-components of vectors r and p, respectively, y in y-direction and

" # L3 1 1 λ (ry ) = − , 128 ln(2A) ry 2 (ry − H)2

(3–8)

which linearly superposes the effect of the other wall that is at a distance of H. There exists a Green’s function for two walls (Liron and Mochon, 1976), however its analytical manipulation is too complicated and superposition approximation qualitatively correct for H > 5L (Usta et al., 2007). The first term on the right hand side of Eq. 3–7 corresponds to the shear flow contribution to the transverse motion. The Brownian force on the center-of-mass results in a fluctuation in the y-position, but less obvious is the prediction of a transverse velocity due to the Brownian torque which appears as the third contribution in Eq. 3–5. The stresslet coefficient within Eq. 2–16 couples the shear flow to a transverse motion. In this sense, the stresslet for the rigid polymer is analogous to the spring force in the theory for a flexible polymer; the inability of the rigid polymer to stretch or compress along its major axis generates an additional flow field within the fluid. This flow field is reflected by the wall and creates a transverse motion of the polymer. This contribution to the polymer motion appears in the first term of Eq. 3–7: ¡ ¢ 2 r˙y = γλ(r ˙ y )px py 1 − 3py ,

(3–9)

which matches previous results given without derivation Saintillan et al. (2006a). Figure 3-2 shows r˙y /γλ ˙ as a function of orientation with respect to the wall for a polymer in the

42

0.6 θ =90

0.4

o

o

θ ≅144.74

+

θ ≅35.26

-

-

0.2

ry/(γλ)

o

+ o

θ =180

θ=0

o

0

-0.2 -0.4 0

20

40

60

80

100

120

140

160

180

θ (deg) Figure 3-2. Transverse£ velocity, p 2 r˙y , 2as¤ a function of polymer orientation, −1 θ = cos px /( px + py ) , with pz = 0. The inset shows the directional dependence on the polymer angle. plane of shear (pz = 0). For a force and torque-free polymer rotating in shear flow, Eq. 3–9 suggests the center-of-mass will oscillate perpendicular to the wall (Yang and Leal, 1984; Hsu and Ganatos, 1994; Olla, 1999). Symmetry of the orientation distribution must be broken to produce a net velocity either away or towards the wall (Olla, 1999). In the case of a rigid polymer, even very weak Brownian fluctuations break the symmetry. 3.2.2

Distribution Function

The Brownian force and torque in Eq. 3–7 are expressed in terms of the distribution function, F = −kB T

∂ ln Ψ ∂r

and T × p = −kB T (I − pp) ·

∂ ln Ψ . ∂p

(3–10)

After performing an average of r˙y over the orientation distribution function, the average transverse velocity of a rigid fiber, hr˙y i, is multiplied by n to give the particle flux as a

43

function of position, " ­ 2® # ­ ®¤ ¡ ­ ®¢ ∂ py k T ∂n B n hr˙y i = nγλ ˙ (ry ) hpx py i − 3 px py 3 − 1 + p2y +n ξ ∂ry ∂y ¡ ­ ® ¢ kB T −72n 2 λ (ry ) 3 p2y − 1 , (3–11) ξL £

of which detailed derivations are in Appendix B. Integrating Eq. 3–3, using Eq. 3–11 and the condition of no particle flux through the bounding walls (n hr˙y i = 0), gives an equation for the steady state distribution, ¡­ ® ­ ®¢ λ (ry ) K S + K B + hΓi ∂ ln n = . ∂ry hD∗ i

(3–12)

Here, hKi contains the contributions to migration due to hydrodynamic interaction with the bounding walls, ­ S® ­ B® K + K , · ¸ £ ­ ®¤ ­ 2 ®¢ 72 ¡ 3 = hpx py i − 3 px py γ˙ + 1 − 3 py γ, ˙ Pe

hKi =

(3–13)

where the P´eclet number, Pe, is defined as Pe =

γL ˙ 2 γL ˙ 2ξ = kB T DTO

(3–14)

and DTO is the translational diffusivity of a slender-body in an infinite fluid (Doi and ­ ® Edwards, 1986). The quantity K S is the contribution to transverse motion from the ­ ® imposed shear flow due to including the stresslet coefficient and K B is the contribution arising from the Brownian torque. At Pe = 0 or Pe = ∞, where the orientation distribution is symmetric, hKi becomes 0. However, we will show that the asymmetric orientation distribution at high values of Pe results in a positive value for hKi and a migration away from the bounding walls. The term hD∗ i represents the diffusion component acting perpendicular to the wall, 1 + hpy 2 i 2 hD i = L γ, ˙ Pe ∗

44

(3–15)

and hΓi corresponds to the contribution arising from the anisotropic diffusivity of a rigid polymer,

· ¸ ∂ hpy 2 i 2 hΓi = − L γ, ˙ ∂y Pe

(3–16)

that was previously identified by Nitsche and Hinch (1997) and Schiek and Shaqfeh (1997a). The ensemble averages of orientation moments within Eq. 3–12 can be calculated by numerically solving the governing equation for the orientation distribution. The equation is derived by applying Eq. 3–2 to Eq. 3–1 at steady state, n (I − pp) :

∂ ∂ ˙ (pψ) = − · (n˙rψ) . ∂p ∂r

(3–17)

Approximating r˙ ψ by h˙ri in Eq. 3–17 and applying Eq. 3–3 gives (I − pp) :

∂ ˙ (pψ) = 0. ∂p

(3–18)

Equation 3–18 assumes that the orientation distribution equilibrates much faster than particles migrate or diffuse across streamlines. Further assumptions that rotation is not influenced by hydrodynamic or steric interactions with the bounding walls gives a position independent expression for the rotational velocity, p˙ = γp ˙ y (ˆ x − px p) −

12 ∂ ln ψ (I − pp) · . 2 ξL ∂p

(3–19)

Replacing p˙ in Eq. 3–18 with Eq. 3–19 and simplifying gives, (I − pp) :

∂ 2ψ ∂ = Pe r (I − pp) : · [py (ˆ x − px p) ψ] , ∂p∂p ∂p

(3–20)

where the rotary P´eclet number, Pe r , is defined as Pe r =

γξL ˙ 2 γ˙ Pe = = , O 12kB T 12 DR

(3–21)

O and DR is the rotational diffusivity of a slender-body in an unbounded fluid (Doi and

Edwards, 1986). 45

2



0.3

3

Per/30 2

(1/3)-(Per /504) Per/70

0.2

0.1

0 10

-2

-1

10

10

0

1

10

10

2

3

10

10

4

Per Figure 3-3. Ensemble averages of orientation moments of a slender-body as functions of Pe r . Symbols denote the numerical solutions and lines are the asymptotic solutions at low Pe r limit. Numerical solution of Eq. 3–20 as a function of Pe r gives the orientation moments shown in Figure 3-3. Our numerical solution was obtained from a series of Brownian dynamics simulations for a single fiber in an infinite fluid using the algorithm of Cobb and Butler (2005), which is described in Appendix B. The asymptotic solutions at low Pe r are also calculated and plotted on Figure 3-3 to confirm the numerical results. Using the numerical results in Figure 3-3, the contributions of hK S i and hK B i to the center-of-mass distribution function are plotted in Figure 3-4. The value of hK B i/ hD∗ i approaches a limiting value of 72, whereas hK S i/ hD∗ i increases indefinitely. Since the orientation moments are independent of position for simple shear flow, hKi and hD∗ i are constants depending only on Pe and

∂ ∂y

hpy py i is zero.

The scalings with Pe r of the orientation moments at high Pe r shown in Figure 3-5 agree closely with those derived for slender bodies in shear at high Pe r (Leal and Hinch,

46

Moment Contribution

10

< K >/ S / B /

3

2

10

10

1

0

10

10

10

-1

-2 -1

10

10

0

1

10

10

2

10

3

10

4

Per

­ ® Figure 3-4. Contribution of the total hKi, the shear flow K S , and the Brownian torque ­ B® K to the migration of a rigid polymer in simple shear flow plotted as a function of Pe r . 1971; Hinch and Leal, 1972; Brenner, 1974). Following the approach of Chen and Koch (1996), the coefficients of the scalings are determined from a best fit to the numerical data over the range of 102 ≤ Pe r ≤ 4 × 104 . Although we fit a wider range than of Pe r than that fitted by Chen and Koch (1996) (102 ≤ Pe r ≤ 103 ), the reported coefficients for ­ ® the orientation moments hpx py i and p2x p2y match. Agreement was also found between the numerical data produced by the Brownian dynamics algorithm and calculations for Pe r ≥ 103 by Chen and Jiang (1999) and for 0 < Pe r ≤ 103 by Asokan et al. (2002). Asymptotic solutions at low Pe r limit are also calculated as described in Appendix B. They matched the numerical results at low Pe r , as shown in Figure 3-3. The relationships determined from the data in Figure 3-3 are used within Eq. 3–12, which is then integrated

47

for the case of simple shear with hΓi = 0 to give ·

¸ HL3 hKi n (ry ) = C exp , 128 ln(2A)ry (ry − H) hD∗ i where the normalization constant C is determined so that

R

(3–22)

ndry = 1 (0.1L ≤ ry ≤

(H − 0.1) L and n (0.1) = n (H − 0.1) = 0). The values of hD∗ i and hKi are given by

and

h i hD∗ i = Pe −1 + 0.94Pe −4/3 L2 γ˙

(3–23)

h i hKi = 0.82Pe −1/3 + 42Pe −1 − 200Pe −4/3 γ, ˙

(3–24)

where the correlations for the orientation moments have been used. Note that all Pe r have been converted to Pe using Eq. 3–21. Equation 3–22 is a fully analytic expression for simple shear between two bounding walls which corresponds to a numerical result for a single bounding wall with simple shear. 3.3 3.3.1

Steady State Distribution

Simple Shear Flow

Setting A = 10, the steady state center-of-mass distributions between two parallel walls under simple shear flows for Pe ≥ 1.2 × 103 are calculated using Eq. 3–22; the results are displayed in Figure 3-6 for H = 6L and 10L. The result shows that a net migration away from the wall exists at large Pe. The large depletion layer can exceed that predicted for rigid polymers at high Pe which interact with the bounding wall only through excluded volume (de Pablo et al., 1992). As mentioned after Eq. 3–13, the symmetry of the orientation distribution is broken and hKi is positive. As Pe increases, the ratio between the migration and the diffusion, hKi / hD∗ i, increases, therefore the rigid fibers distribute more strongly towards the center of the channel. Comparing the distributions in the wider and narrower channel demonstrates that reducing the level of confinement increases the depletion layer. The depletion layer Ld , defined as the point ry at which n returns to the bulk value, is calculated from Eq. 3–22 and plotted in Figure 3-7

48

10

-1 -2

-3

10

-4

10

2

,

3



10

10

-5

10

2

3

10

10

4

10

Per

-2

10

2

10

3

10

4

10

5

Per Figure 3-5. Ensemble ­averages of orientation moments of a slender-body as functions of ® Pe r : (2) p2y ≈ 0.41Pe r−1/3 , (◦) hpx py i ≈ 0.36Pe −1/3 and (¦) r −1 3 hpx py i ≈ 0.83Pe r . Symbols denote the numerical solutions and lines are the scalings with coefficients determined by the best fit to the numerical data. for A = 10 and different values of Pe and H. The increased hydrodynamic lift associated with increasing the shear rate, to give a higher value of Pe, generates a larger depletion layer. Furthermore, increasing the channel width while holding Pe constant extends the depletion layer further into the channel. Though not plotted, the depletion layer also depends upon the aspect ratio since the hydrodynamic lift varies as 1/ ln(2A) as seen in Eq. 3–7. Consequently, increasing the aspect ratio reduces the depletion since the hydrodynamic interaction between the wall and fiber become screened. 3.3.2

Pressure-driven Flow

To obtain distributions in pressure-driven flow, Eq. 3–12 is integrated numerically, with the orientation moments calculated from Eq. 3–20 at each position according to the local shear rate, kB T Pe γ(r ˙ y) = ξL2

49

µ

2ry 1− H

¶ ,

(3–25)

0.25

B

A

0.4

0.2 0.3

n(ry)

n(ry)

0.15 0.2

0.1

Pe = 1200 Pe = 12000 Pe = 48000 Pe = 120000

0.1

0

0

0.5

1

1.5

2

2.5

Pe = 1200 Pe = 12000 Pe = 48000 Pe = 120000

0.05

0

3

ry/L

0

1

2

3

4

5

ry/L

Figure 3-6. The steady state center-of-mass distributions n (ry ) of a rigid fiber predicted by Eq. 3–22 as a function of distance from a wall, ry /L, with A = 10 and A) H = 6L and B) H = 10L in simple shear flow.

Ld / L

10

1 H=50L H=10L H=8L H=6L 10

3

10

4

5

10

Pe Figure 3-7. The steady state depletion layer thickness Ld /L of a rigid fiber of aspect ratio A = 10 predicted by Eq. 3–22 as a function of Pe and H in simple shear flow.

50

where Pe is the Peclet number corresponding to the mean shear rate. Figure 3-8 shows the results of the calculations for A = 10. Near the center of the channel, where the shear rate is small, the contribution due to the anisotropic diffusivity results in a weak migration of polymers toward the wall. Specifically, hpy py i decreases from a value of 1/3 at the center of the channel and approaches zero near the wall. As a result, the migration is balanced by the hydrodynamic interactions of polymers with the boundaries to give an off-center maximum for n(ry ). The effect of the anisotropic diffusivity becomes increasingly important relative to the wall interactions for polymers with larger aspect ratios. Consequently, the maximum value of n(ry ) will move towards the wall as A increases. When neglecting both steric and hydrodynamic interactions with the wall however, a migration of polymers toward the wall is predicted for arbitrary values of Pe (Nitsche and Hinch, 1997). Similarly, under conditions of highly confined polymers coupled with weak pressure-driven flow, models that include steric wall effects (Schiek and Shaqfeh, 1997a) predict migration towards the wall. In these cases, the migration occurs solely because of the anisotropic diffusivity of the polymer as a result of the inhomogeneous flow field. Such a mechanism predicts no migration in simple shear flow where the orientation distribution is spatially uniform, contrary to our results. Furthermore, the qualitative difference in predicted migration in pressure-driven flow highlights the importance of including long-range hydrodynamic interactions. The theoretical results for pressure-driven flow are only qualitatively similar to those from simulations which include hydrodynamic interactions with the wall (Saintillan et al., 2006a). While Saintillan et al. (2006a) obtained Ld > L at Pe = 300, the result predicted here gives a much smaller value of Ld ∼ 0.5L at the same Pe. Since the theoretical model does not include the excluded volume effect, such as a repulsive wall force, direct comparison by simulation will be performed in the next chapter.

51

2

Pe =3.0x10 4 Pe =1.2x10 4 Pe =4.8x10 5 Pe =1.2x10 5 Pe =2.4x10

0.25

n(ry)

0.2

0.15

0.1

0.05

0

0

1

2

3

4

5

6

7

8

r y /L Figure 3-8. Center-of-mass distribution for a rigid polymer with A = 10 in pressure-driven flow. The height of the channel is 8L. 3.3.3

Discussion on Migration Mechanisms

A previous theoretical investigation (Nitsche and Roy, 1996) found that hydrodynamic interactions acting upon a Brownian fiber sheared near a wall have little effect on the depletion layer. That theory considered the limit of weak shear flows only, however. At high rates of shear, the kinetic theory in Chapter 3 predicts migration arising from three mechanisms as shown in Eq. 3–12, two of them related to hydrodynamic interactions. The third corresponds to the gradient mechanism identified by Nitsche and Hinch (1997) and Schiek and Shaqfeh (1997a) that does not depend upon far-field hydrodynamic interactions and moves the rigid fibers closer to the bounding walls on average. Mechanisms which screen hydrodynamic interactions between the fiber and wall would render this term dominant. Screening can occur for fibers of very high aspect ratio and for a separation between bounding walls which is of the length of the fiber or smaller.

52

Figure 3-9. Comparison of migration mechanisms in shear flow: A) Stresslet contribution, B) Brownian torque contribution of rigid rod and C) restoring spring force contribution of flexible dumbbell. The thin arrows indicate actions on polymers by the shear flow and the thick arrows represent corresponding resistances of polymers. As compared in Figure 3-9, the stronger (Figure 3-9A) of the two hydrodynamic mechanisms identified in this chapter is similar to the mechanism described for flexible polymers (Figure 3-9C). The inability of the rigid fiber to deform with the fluid creates an additional flow field which is reflected by a bounding wall to create a transverse motion. This transverse motion can occur towards or away from the bounding wall depending upon the instantaneous orientation of the rod. The center-of-mass of a force and torque free rod tumbling in a Jeffery orbit (Jeffery, 1922) due to an imposed shear flow will oscillate transverse to the wall since the symmetry of the orientation distribution is preserved. This coupling of a shearing flow and a transverse motion has been noted and investigated in past theories and simulations for rod-like particles (Yang and Leal, 1984), prolate spheroids (Hsu and Ganatos, 1976; Olla, 1999), and even for oblate spheroids (Mody and King, 2005). Producing a net migration requires breaking the symmetry of the orientation distribution for the particle tumbling in the shear flow. Kinetic theory predicts that weak Brownian motion alters the rotational dynamics of a rigid fiber in shear flow (Leal

53

and Hinch, 1971; Stover et al., 1992; Hijazi and Zoaeter, 2002), changing the orientation distribution in a manner that results in a net migration of the fiber away from the wall. The weaker (Figure 3-9B) of the two hydrodynamic mechanisms of migration newly predicted by the kinetic theory does not have an analog within the theory for the flexible dumbbell. This secondary migration arises from a hydrodynamic coupling of the Brownian rotation of the rigid fiber and bounding wall. A particle with an isotropic orientation distribution fluctuates transversely to the wall in response to this interaction, but the shear flow creates a preferential orientation and a consequent lift away from the wall. 3.4

Transient Distribution

The evolution of the center-of-mass distribution under simple shear flow is obtained from the time dependent version of Eq. 3–3, ∂n ∂ = − (n hr˙y i) . ∂t ∂y

(3–26)

Using the formula for average fiber flux derived in Eq. 3–11 with the orientation moments obtained in Figure 3-3 under the same assumptions made in the previous Section, the governing equation becomes ∂n ∂ 2n ∂ = hD∗ i 2 − hKi [nλ (ry )] . ∂t ∂y ∂y

(3–27)

This equation is similar to the evolution equation for the center-of-mass distribution of a flexible polymer as derived by Ma and Graham (2005). For the flexible polymers, hKi is related to the coupling of the restoring spring force and shear flow which induces a flow disturbance and migration away from the wall while hKi in Eq. 3–27 relates the coupling of the stresslet coefficient, or inability of the fiber to stretch, and Brownian torque with the shear flow. Equation 3–27 is solved numerically using the Crank-Nicolson scheme with an initially uniform distribution and the boundary condition of no flux of the center-of-mass through a wall. The detailed numerical formulas are described in Appendix B. The time

54

A

n(ry)

0.2

t=100 t=500 t=1500 ~Pe t=2500 ~2Pe t=5000 ~steady state

0.1

0

0

0.5

1

1.5

2

2.5

3

ry/L B

n(ry)

0.3

0.2

t=1000 t=5000 t=20000 t=48000 ~Pe t=96000 ~2Pe ~steady state

0.1

0

0

0.5

1

1.5

2

2.5

3

ry/L Figure 3-10. The transient center-of-mass distributions of a rigid fiber predicted by Eq. 3–27 as a function of distance from a wall, ry /L, with A = 10 and H = 6L in simple shear flows of A) Pe = 1.2 × 103 and B) Pe = 4.8 × 104 . The units of time t are γ˙ −1 .

55

step, ∆t = 0.01γ˙ −1 , and position step, ∆y = 0.001L give convergent numerical results that are shown in Figure 3-10. At short times, a peak appears near each wall. As time proceeds, the peak gradually moves towards the center and becomes more blunt. At a time comparable to Pe γ˙ −1 , the peaks from the two walls merge and the distribution is fully developed at a time of about 2Pe γ˙ −1 . Since the migration velocity due to hydrodynamic interactions is much larger near a wall, the rigid fibers accumulate near the wall faster than diffusion can smooth the profile. At higher Pe, migration is much stronger than diffusion; therefore, the peak is much sharper near the wall for Pe = 4.8 × 104 than Pe = 1.2 × 103 . As time passes, the migration and diffusion of fibers balances and the distribution reaches steady state. An upper bound on the time tS to achieve steady state can be estimated as the time for a rigid fiber to diffuse from the wall to the center of the channel, H 2 /4 tS = ∼O 2 hD∗ i

µ

PeH 2 8L2 γ˙

¶ ,

(3–28)

where hD∗ i is approximated using the leading order of Pe −1 in expression, Eq. 3–23. This estimate for tS at H = 6L is about twice as large as indicated by the results in Figure 3-10. Ma and Graham (2005) argued that a flexible polymer needs to diffuse only a distance equivalent to the depletion width, whereas we find that the distributions at t = 2Pe γ˙ −1 are virtually identical to the steady distributions. 3.5

Conclusion

A kinetic theory has been developed for the migration of rigid polymers in rectilinear flows. Results from the theory are similar to results from theories describing flexible polymer systems which consider hydrodynamic interactions with the wall, despite differences in flexibility. For simple shear flow near a single wall, results indicate that hydrodynamic interactions with the wall cause a net cross-stream migration of polymers away from the wall. Results for pressure-driven flow, which include contributions from the anisotropic diffusivity of polymers, show an off-center maximum for the center-of-mass

56

distribution. For transient distributions, initially uniform distribution shows a peak near a wall at short times. As time proceeds, the peak gradually moves towards the center and finally vanishes to reach steady state distribution. Though the theory presents general trends, improvements to the quantitative prediction can be made. For example, steric effects (Schiek and Shaqfeh, 1997a) as well as improvements to the wall interactions (Liron and Mochon, 1976) can be included. Validation of multiple approximations made in the theory will be confirmed by comparison with simulation in Chapter 4.

57

CHAPTER 4 DYNAMIC SIMULATION OF A RIGID ROD IN SHEARING FLOWS BETWEEN TWO PARALLEL WALLS 4.1

Introduction

While the kinetic theory in Chapter 3 qualitatively predicts the migration observed in the simulations of Saintillan et al. (2006a), there are quantitative disagreements. Therefore multiple approximations made in the theory should be investigated to validate their use. These include solving for the orientation distribution separately from the center-of-mass and an evaluation of the hydrodynamic interaction with the wall that lacks rigor in favor of simplicity. Excluded volume interactions of the rigid fiber with the wall were also neglected in the theory. Just as Brownian dynamics simulations have been performed to test kinetic theories for the migration of flexible polymers (Hern´andez-Ortiz et al., 2006; Hoda and Kumar, 2007a), we test the theory by performing simulations with and without hydrodynamic interactions of a rigid fiber with weak Brownian motion suspended in rectilinear flows between two walls. Comparisons between simulations demonstrate that the depletion layer is affected by hydrodynamics only for very large rates of shear; at moderate rates of shear where the hydrodynamic mechanism for migration is relatively weak, the thickness of the depletion layer is controlled by excluded volume interactions of the rod with the bounding walls. The distributions obtained by the simulation with hydrodynamic interactions are in good agreement with those predicted by the theory for sufficiently high rates of shear rather than by the simulation of Saintillan et al. (2006a). A rigid fiber confined between two bounding walls is simulated as shown in Figure 4-1. To include hydrodynamic interactions of the bulk particle with bounding walls, we represent each wall as a periodic array of slender bodies and calculate interactions between all particles using methods described in Chapter 2. This approach differs from calculating the Green’s function for two planar walls of infinite extent (Liron and Mochon, 1976; Staben et al., 2003) as done by Saintillan et al. (2006a). However, representing

58

Figure 4-1. Arrays of closely packed fibers form the walls in a periodic boundary system of dimension bx , by and bz . A) A view from the vorticity-gradient (zy)-plane, B) view of simple shear flow, and C) parabolic flow from the flow-gradient (xy)-plane. the walls by closely packed arrays of particles has been used with success in Stokesian dynamics (Nott and Brady, 1994; Singh and Nott, 2000; Bricker and Butler, 2007) and the method can be extended to perform simulations of non-dilute systems which is a focus of ongoing work. In Section 4.2, the equations governing the motion of the rigid fibers are modified from Eq. 2–22 and additional details of the simulation method are given. In Section 4.3, simulated distributions in various shearing flows are presented. 4.2 4.2.1

Simulation

Governing Equation

As demonstrated in Figure 4-1, particles in the bulk, as well as walls are simulated as rigid rods using governing equation, Eq. 2–22, which is for rigid rods in periodic suspensions is also used for this simulation. In this simulation, the Green’s function given by Beenakker (1986), BP , which is the Ewald-summed expression of Green’s function by Rotne and Prager (1969) and Yamakawa (1970) in a periodic system, is used. Therefore,

59

Eq. 2–26 should be changed accordingly,    BP (sα , sβ ) if α 6= β    G (sα , sβ ) = BP (sα , sβ ) − B (sα , sβ ) if α = β and sα = 6 sβ ,      B0P if α = β and sα = sβ where B0 is the limiting form given by Beenakker (1986) and B is  ³ ´  I xx  H (x, 0) + a2c − 3 if |x| > 2ac 12πµ |x|3 |x|5 B (x, 0) = , h³ ´ i  9|x| 3xx  1 1 − I + if |x| < 2a c 6πµac 32ac 32ac |x|

(4–1)

(4–2)

where ac is an effective radius (Rotne and Prager, 1969; Yamakawa, 1970). For dynamic simulations of suspensions of hydrodynamically interacting rigid fibers, the solution of Hasimoto (1959) for the periodic Oseen-Burger tensor has been widely used (Harlen et al., 1999; Butler and Shaqfeh, 2002; Saintillan et al., 2006b). However, this results in a mobility tensor which sometimes lacks the property of being positive definite when a pair of fibers is in close proximity. The solution of Beenakker (1986) ensures a positive definite matrix for any configuration of the interacting particles regardless of separation distance, so it has been applied to these simulations due to the presence of closely packed arrays of fibers which form the walls. Tornberg and Gustavsson (2006) also used Beenakker’s solution in the simulation of sedimenting rigid fibers. Comparing the velocity disturbance generated by a force on a fiber by integrating Oseen-Burger tensor and Beenakker’s solution along the length of the rod demonstrates that the error is negligible at a distance of three times the fiber diameter. For the specific case of a force acting perpendicularly to a rod centered at the origin, the velocity disturbances in the perpendicular direction agree within 5% at a distance of two diameters for a point located on the plane lying perpendicular to the rod and centered on the rod. We identify two types of particles in this simulation: wall fibers (W) and a bulk fiber (B). Closely packed arrays of the wall fibers form the two planar boundaries as shown in Figure 4-1. The bulk fiber is contained in the space between the upper wall and the

60

bottom wall in a periodic box, but not in the space between the wall and its adjacent image wall. The subscripts used in Eq. 2–22 are replaced with the types of particles. The shear flow is not given as the imposed flow field u∞ , but is generated by the flow disturbance u∗ , induced by moving wall fibers. Hence, solving Eq. 2–22 requires the following manipulation of a resistance problem:  FW    T W × pW    SW   SB



−1

 TF MW W

TT MW W

MTWSW

MTWSB

       MRF MRT MRS MRS   WW WW WB   WW   ·  =   MEF MET MES MES    WW  WW WB  WW    ES ES EF ET MBW MBW MBW MBB      TF TT ˙  rW   MW B MW B          p˙ W   MRF MRT   F B WB      WB   − ·  ,  0   MEF MET  T B × pB   W   W B  WB       EF ET 0B MBB MBB

(4–3)

where stresslet coefficients on wall fibers are designated by the vector S W . Since the fluid motion is generated by moving the wall particles, the rates of strain corresponding to an imposed flow have been set to zero, 0, in Eq. 4–3; this does not mean that the fibers experience no straining flow, but that the straining flow is generated by the relative motion of the particles only. Using the results from Eq. 4–3, the motion of the bulk fiber confined between the two parallel walls is given by   ˙  rW         p˙ W  FB      r˙ B  = P · +Q·  ,     T B × pB p˙ B  0W    0B

61

(4–4)

where P represents the response of the motion of the bulk fiber to the shear flow, −1

 TF MW W

 TF TT TS TS  MBW MBW MBW MBB P = RT RS RS MRF BW MBW MBW MBB

and Q is the mobility of the bulk rod,  TF  MW B   MRF  WB Q = −P ·   MEF  WB  MEF BB

MTWTW

MTWSW

MTWSB

   RT RS RS  RF   MW W MW W MW W MW B ·  EF ES ES  MW W MET W W MW W MW B  MEF MET MES MES BW BW BW BB

       

,

(4–5)

 MTWTB MRT WB MET WB

    TT TF    MBB MBB  + .  RT M MRF  BB BB 

(4–6)

MET BB

The Brownian forces and torques acting on the bulk fiber can be obtained from the inverse of the mobility matrix Q by satisfying the fluctuation-dissipation theorem,   r FB 2kB T   B · W,  = ∆t T B × pB

(4–7)

where B · BT = Q−1 ,

(4–8)

the thermal energy is kB T , and the vector W contains random numbers generated from a uniform distribution of zero mean and a variance of one. Cholesky decomposition is used to evaluate B as explained by Butler and Shaqfeh (2005) and Saintillan et al. (2006c). 4.2.2

Simulation Details

This simulation method can be used to study fibers flowing at any value of the P´eclet number, but simulations are performed at relatively high values where the phenomena of interest is predicted to strongly impact the distribution of the fibers. We use the given algorithm to simulate the dynamics of a rigid fiber having an aspect ratio of A = 10 for all cases. Likewise, the simulations calculate the specific case of a wall separation of H = 6L

62

and one (bulk) fiber is simulated at a time, with averages performed over multiple runs and blocks of time to generate distributions as stated in Section 4.3.2. The bounding walls are constructed of 2bx bz /Ld fibers of aspect ratio 10. The mobility matrices, consisting of dual integrals over the particle lengths, are generated using a five point and six point Gaussian quadrature for the cases of simple shear and parabolic flow, respectively. The number of summation for periodic Green’s function is set to 3 with ac = 1.66d for simple shear flow and ac = 2.05d for parabolic flow. To generate simple shear flow, the wall particles are assigned a velocity in the x-direction, r˙x , but with opposite signs for particles in the ‘top’ and ‘bottom’ wall; all other velocities are set to zero. For a simple shear flow, the velocity, r˙x = ±H γ/2, ˙ generates a shear flow of γ. ˙ Both walls translate in the positive x-direction at the same velocity of H γ/3 ˙ to produce a parabolic flow with a local shear rate of µ γ˙ (y) = 2γ˙

¶ 2y −1 , H

(4–9)

where γ˙ is the mean shear rate. In both cases the result corresponds closely to the desired, theoretical profiles and there is no net flow through the yz-plane since the periodic sums were constructed to ensure zero net flow through any plane. The flow profiles, shown in Figure 4-2, were generated using Eq. 2–15 with the linearized force distribution in Eq. 2–16 and in the absence of a bulk particle. In addition, we confirmed that fluid does not flow perpendicular to the walls. Lubrication interactions dominate when the bulk particle approaches within one fiber diameter of the wall. This short range interaction is added directly to the resistance matrix using methods identical to those of Butler and Shaqfeh (2002). The interaction itself depends upon the relative orientation of the rod with respect to the wall and is evaluated using the method of Claeys and Brady (1989). The lubrication interaction is

63

supplemented by a short-ranged repulsive force, fR = ±0.01Lξ γ˙

exp (−10h/L) ˆ, y 1 − exp (−10h/L)

(4–10)

where h is the closest distance between the fiber and the bounding wall, the force acts ˆ is the unit vector in the y-direction. When to push the fiber away from the wall, and y evaluating the separation between the bulk fiber and the wall, the end tip of the fiber is regarded as a hard sphere having the same diameter as the fiber. This repulsive force is included with the forces and torques acting on the bulk fiber and the wall fibers. Simulation results confirm that the particle does not cross the wall. We also tested other algorithms for maintaining the excluded volume (de Pablo et al., 1992; Hijazi and Khater, 2001) in Appendix C.2, but the differences in the distributions were small. To distinguish the effects of the hydrodynamic and excluded volume interactions on the depletion layer, simulations are also performed in the absence of hydrodynamic interactions with the walls. These simulations are performed using Eq. 2–7 and Eq. 2–9 with the velocity uB (sB ) replaced by the theoretical expression for simple shear flow or parabolic flow as appropriate. The Brownian forces and torques are calculated using the fluctuation dissipation theory, but with the mobility of a single rod suspended in an infinite fluid. The same repulsive force given in Eq. 4–10 prevents the fiber from crossing the wall. Equation 4–4 is integrated using the modified midpoint method; this method integrates the trajectories of the stochastic motion accurately at first order without requiring an explicit evaluation of the divergence of the mobility (Fixman, 1978; Grassia et al., 1995; Morse, 2004). The time step for integrating positions is set to 0.001γ˙

−1

.

Note that many of the submatrices in Eq. 4–5 need to be calculated only once. The self-mobilities and any lubrication interactions for the bulk fiber are updated at each time step, but the long-range hydrodynamic interactions between the bulk particle and wall particles are updated less frequently at intervals of 0.05γ˙

64

−1

. Tests on this

3

B

A 2

lo

h

es pl

1

rf ea

2

w

Simple shear flow 1

m Si

γ (y)

Flow velocity in the x-direction

3

0

.

-1

0

low cf

li bo

ra Pa

-1

Parabolic flow -2

-2 wall fibers (upper)

wall fibers (bottom)

-3 0

1

2

3

4

5

6

-3 0

y/L

1

2

3

4

5

6

y/L

Figure 4-2. Simulation results (broken lines) of A) the velocity profile in the x-direction (units of Lγ) ˙ and B) the local shear rate γ(y) ˙ (units of γ) ˙ of the shear flow generated by moving walls as functions of distance from the bottom wall, y/L. Fibers have an aspect ratio of A = 10 and the walls are separated by a gap of H = 6L in a periodic box of dimensions bx : by : bz = 3L : 8L : 3L for simple shear flow and bx : by : bz = 3L : 12L : 3L for parabolic flow. Theoretical values are plotted as solid lines. integration scheme demonstrated convergence of the results while greatly decreasing the computational expense of the calculations. 4.3

Results

Comparisons between the results of the simulation and the theory are presented in Section 4.3. We demonstrate that the calculations are in good agreement for the prediction of the configuration dependent lift velocity (Section 4.3.1), the steady and unsteady distribution in simple shear flow (Section 4.3.2), and the steady distribution in parabolic flow (Section 4.3.3). 4.3.1

Transverse Velocity Due to Shear Flow

The primary mechanism for migration of a rigid fiber in a strong shear flow is the transverse velocity, coupled with a preferential orientation, resulting from hydrodynamic interactions with the wall. An explicit expression for the transverse velocity due to shear

65

10

-3

Simulation Theory

ry/(Lγ)

.

0

^ y

.

)φ -10

-3

0

20

40

60

80

100

φ (degree)

120

^ x

140

160

180

Figure 4-3. Transverse velocity r˙y due to shear flow as a function of angle φ with pz = 0, ry = 1.0L, and H = 6L. Simulation conditions are the same as those in Figure 4-2B with simple shear flow. The inset illustrates a rigid fiber with an angle φ and pz = 0. as a function of configuration is embodied in the first term of Eq. 3–7, ¡ ¢ r˙y = γλ ˙ (ry ) px py 1 − 3p2y .

(4–11)

The response of a fiber to the shear flow can be determined from the simulation using Eq. 4–4 by setting FB to 0 and extracting the y-component of the velocity,   r˙ W      0W    ˆ·P · r˙y = y .   0  W    0B

(4–12)

Comparisons between the theoretical analysis Eq. 4–11 and computational result Eq. 4–12 appear in Figures 4-3 and 4-4.

66

-4

5×10 -4

Theory (bx:by:bz)=(4:8:4)

4×10

(bx:by:bz)=(3:8:3)

-4

0

1×10

-4

2×10

-4

.

(bx:by:bz)=(2:8:2)

3×10

ry/(Lγ)

.

0.5

1

2

1.5

2.5

3

ry/L Figure 4-4. Transverse velocity r˙y due to shear contribution as a function of distance from a wall, ry /L, for a fixed angle (φ = 3◦ and pz = 0) under a constant shear rate γ. ˙ Results are shown for the theory and three simulations with different periodic dimensions. Figure 4-3 compares the dependence of the transverse velocity on angle φ (angle between the x-axis and projection of pB onto the xy-plane as illustrated) for a fiber fixed at a distance 1L from the wall. The results match qualitatively, with the largest quantitative discrepancies occurring at orientations corresponding to the secondary maximum of r˙y . The quantitative differences between the calculations of transverse velocity also depend upon the position of the fiber and size of the periodic box as shown in Figure 4-4. Causes of the quantitative discrepancies clearly include the approximations made for the evaluation of the Green’s function used in the analytic expression. The theory overestimates the transverse velocity near the center due to superimposing the effect of the two bounding walls. The theory was derived by linearizing the Green’s function, assuming that the fiber is far from the wall in comparison to its length; the increasing discrepancy

67

with the simulated results as the fiber approaches the wall can be attributed to the breakdown of this assumption. Though the solution given by the numerical calculation avoids key problems associated with the theoretical evaluation, the ‘bumpy’ wall formed by the array of fibers and the use of the periodic Green’s function represent sources of error. With regard to the periodicity, changing the box height by or ratio between H and by (distance between one wall and neighboring image wall) makes little difference, but changing the area of the xz-plane alters the results as shown in Figure 4-4. Due to periodicity, the transverse velocity predicted by the simulation is lower than expected for a single rod suspended between two plates of infinite extent. We also performed tests on the components of r˙y arising from forces and torques (Appendix C.1) given in Eq. 3–7. Comparisons of these terms with the responses from simulations also demonstrate a qualitative match and quantitative differences similar to those observed for the shear contribution. 4.3.2

Distributions in Simple Shear Flow

Simulations of the rigid fiber in simple shear flow were performed with a total of 32 different runs for each value of Pe. Initial orientations and positions were chosen randomly and positions were sampled during the simulations to produce the center-of-mass distribution. The results are compared with the transient and steady state distributions predicted in Chapter 3. Figure 4-5 shows the time evolution of center-of-mass distribution in simple shear flow for Pe = 1.2 × 103 . The distributions exhibit a maximum near the wall at short times (Figure 4-5A), with a larger maximum produced by the simulation than the theory. While the theory predicts attainment of a steady state distribution around t = 2Pe γ˙ −1 , the distribution averaged over times of t = 2500 ∼ 5000γ˙ −1 (Figure 4-5B) still shows significant variations around the predicted distribution which is at steady state. The simulated distribution conforms closely to the theoretical distribution at steady state (Figure 4-5C) when averaged over long times.

68

The good agreement shown in Figure 4-5 between the results of the theory and simulation with hydrodynamic interaction is fortuitous. Simulations with and without hydrodynamic interactions with the walls at Pe = 1.2 × 103 produce a nearly identical distribution as shown in Figure 4-5C. The similarity establishes that excluded volume, which the theory does not consider, is controlling the distribution; the flow strength at Pe = 1.2 × 103 is too low to generate clear evidence of the hydrodynamic lift. The results in Figure 4-5C are also consistent with previous investigations (de Pablo et al., 1992; Schiek and Shaqfeh, 1995) of the changes in the depletion layer due to shear and excluded volume. We note that the distribution becomes uniform at a value of about 0.6L ∼ 0.7L, rather than 0.5L, since the end of the rod is a hemisphere that extends an additional distance of 0.1L beyond 0.5L and the surface of the wall is corrugated. Simulated distributions with hydrodynamic interactions at Pe = 1.2×104 are shown in Figure 4-6. The transient maximum near the wall appears early in the simulation results (Figure 4-6A) and agrees with the theoretical prediction within the averaging errors. The intermediate distribution (Figure 4-6B) fluctuates around the corresponding theoretical distribution and the distribution for t > 17000γ˙ −1 (Figure 4-6C) closely corresponds to the theoretical steady state. The steady distribution from simulations without hydrodynamic interaction at Pe = 1.2 × 104 is plotted in Figure 4-6C. As compared to the corresponding result at Pe = 1.2 × 103 in Figure 4-5C, the depletion layer increases because of the higher rate of shear, but still does not extend beyond ry ≈ 0.65L, where the distribution becomes uniform. Comparing the distributions from simulations at Pe = 1.2 × 104 with and without hydrodynamic interaction demonstrates that the hydrodynamic interactions strongly influence the distribution in a manner consistent with the theoretical prediction. Figure 4-7 shows distributions from simulation with hydrodynamic interaction at Pe = 4.8 × 104 . Although the averaging errors are large at short times, the transient, off-center maximum is detectable in the mean values. The simulation was terminated at

69

A

0.25

n(ry)

0.2

0.15

0.1 t=100 (theory) t=1000 (theory) t=0~2500 (simulation)

0.05

0

0

0.5

1

1.5

2

2.5

3

ry/L B 0.25

n(ry)

0.2

0.15

0.1 t=3000 (theory) t=2500~5000 (simulation)

0.05

0

0

0.5

1

1.5

2

2.5

3

ry/L C 0.25

n(ry)

0.2

0.15

0.1 Steady state (theory) t=5000~30000 (simulation) no hydrodynamic interaction

0.05

0

0

0.5

1

1.5

2

2.5

3

ry/L

Figure 4-5. The simulation results (symbols) and theoretical solutions (lines) for the transient center-of-mass distributions of a rigid fiber as a function of distance from a wall, ry /L, with A = 10 and H = 6L in simple shear flows of Pe = 1.2 × 103 . Simulation results are averaged over the time ranges (units of γ˙ −1 ) as indicated in each legend.

70

A 0.25

n(ry)

0.2

0.15

0.1 t=3000 (theory) t=0~6000 (simulation)

0.05

0

0

0.5

1

1.5

2

2.5

3

ry/L B

n(ry)

0.2

0.1 t=15000 (theory) t=6000~15000 (simulation) 0

0

0.5

1

1.5

2

2.5

3

ry/L C 0.25

n(ry)

0.2

0.15

0.1 Steady state (theory) t=17000~25000 (simulation) no hydrodynamic interaction

0.05

0

0

0.5

1

1.5

2

2.5

3

ry/L

Figure 4-6. The simulation results (symbols) and theoretical solutions (lines) for the transient center-of-mass distributions of a rigid fiber as a function of distance from a wall, ry /L, with A = 10 and H = 6L in simple shear flows of Pe = 1.2 × 104 . Simulation results are averaged over the time ranges (units of γ˙ −1 ) as indicated in each legend.

71

t ∼ Pe γ˙ −1 owing to the computational burden, though the distribution is continuing to develop. The distribution from simulation without hydrodynamic interaction in Figure 4-7C demonstrates a clear difference as compared to the one with hydrodynamic interaction, indicating again the role played by the hydrodynamic lift force. ­ ® Figure 4-8 shows the orientation moment p2y calculated from the simulations as a function of position and Pe; the orientation moments for unbounded flow, which ­ ® are used to make the theoretical prediction, are also shown. The values of p2y from simulations with and without hydrodynamic interaction have no discernible differences regardless of distance from the wall, confirming the assumption made within the theory that hydrodynamic interaction with the walls does not affect the orientation distribution. The wall affects the orientation through excluded volume interactions for ry < 0.7L, resulting in substantial differences with the orientation distribution assumed within the theoretical calculations. ­ ® The overshoot of p2y around 0.65L for finite shear rates is an interesting feature observable in Figure 4-8. The action of the short range repulsive force with the wall on the end of a rod rotating in the shear flow creates a ‘pole-vault’ type of motion that displaces the rods, oriented nearly perpendicular to the wall, outwards to give the enhanced ­ ® probability of p2y in the range of ry = 0.6L ∼ 0.7L. Though the effect is likely caused by the repulsive force, attempts to eliminate the overshoot by altering the strength and range of the repulsive force were unsuccessful. We note that ‘pole-vault’ motions have been observed by others in simulations in the absence of Brownian motion (Stover and Cohen, 1990; Mody and King, 2005) and even in experiments (Holm and S¨oderberg, 2007). In the vicinity of a wall (ry . 0.7L), the steric effects control the orientation distribution and ignoring the excluded volume within the theory is clearly in error. However, the neglect of excluded volume has a limited effect on the theoretical prediction of the center-of-mass distribution for values of Pe above 1.2 × 103 . The primary reason is

72

A

n(ry)

0.2

0.1 t=12000 (theory) t=0~12000 (simulation) 0

0

0.5

1

1.5

2

2.5

3

ry/L 0.4

B

n(ry)

0.3

0.2

0.1 t=24000 (theory) t=12000~36000 (simulation) 0

0

0.5

1

1.5

2

2.5

3

ry/L 0.4

C

n(ry)

0.3

0.2

0.1

0

0

Steady state (theory) t=42000~60000 (simulation) no hydrodynamic interaction 0.5

1

1.5

2

2.5

3

ry/L

Figure 4-7. The simulation results (symbols) and theoretical solutions (lines) for the transient center-of-mass distributions of a rigid fiber as a function of distance from a wall, ry /L, with A = 10 and H = 6L in simple shear flows of Pe = 4.8 × 104 . Simulation results are averaged over the time ranges (units of γ˙ −1 ) as indicated in each legend.

73

0.3

Pe = 0 Pe =1200 Pe =12000 Pe =48000

2



0.2

0.1

0

0

0.5

1

1.5

2

2.5

3

ry/L Figure 4-8. The simulation results with (filled symbols) and without symbols) ­ (open ® hydrodynamic interactions for the orientation moment p2y of a rigid fiber as a function of distance­ from ® a wall, ry /L, with A = 10 and H = 6L. The 2 expected values of py for unbounded flow are plotted as lines. that few fibers remain within a distance of 0.7L of the bounding walls for the higher shear rates due to the hydrodynamic migration away from the wall. The qualitative features of the center-of-mass distributions predicted by the theory appear in the simulations for Pe = 1.2 × 104 and 4.8 × 104 : the transient maximum near the walls forms at short times, then moves towards the center while broadening and disappears at t ∼ Pe γ˙ −1 . Quantitative comparisons of transient profiles are hindered by the larger errors in some of the results, but the overall good agreement verifies that the errors associated with assumptions made in the theory do not have severe, adverse affects on the prediction so long as the depletion layer extends beyond the range at which the rotation of the rod is hindered by the wall. This mechanism which induces the depletion layer solely by excluded volume effect of the wall is demonstrated in Figure 4-9. A rigid rod in shear flow is found to be rotate

74

Figure 4-9. Rotations of rigid rods in shear flow of strength γ: ˙ A) Jeffery orbit motion far from a wall and B) pole-vault motion ignoring hydrodynamic interactions with a wall. The dotted arrow indicates a trajectory of center-of-mass of a rod under each rotation. following Jeffery orbit (Figure 4-9A). However, if the distance of a rod from a wall is close so that its rotation is restricted by the wall, the rotation pushes the rod giving a slight increase of center-of-mass in a direction normal to the wall. This type of motion is called ‘pole-vault’ motion (Figure 4-9) (Stover and Cohen, 1990; Mody and King, 2005). 4.3.3

Steady State Distribution in Parabolic Flow

Simulations were performed under parabolic flow to produce a long-time distribution for three values of Pe = γL ˙ 2 /DT as shown in Figure 4-10. The number of runs, initial configuration, and sampling method matches those of simulations for simple shear flow presented in Section 4.3.2. The distributions are compared with the steady distributions obtained by numerically integrating Eq. 3–12. The orientation moments used within Eq. 3–12 depend upon position according to the local shear rate as given by solution to Eq. 3–20; alterations in the orientation due to hydrodynamic interactions or excluded volume are ignored as was done for the calculations in simple shear. At Pe = 1.2 × 103 , the theory and simulation with hydrodynamic interaction generate a similar concentration profile, although the results from simulations are noisy due to averaging errors. However, the simulation in the absence of hydrodynamic interaction gives nearly the same result. As in the case of simple shear at a similar Pe, the

75

A 0.25

n(ry)

0.2

0.15

0.1 Steady state (theory) t=5000~20000 (simulation) no hydrodynamic interaction

0.05

0

0

0.5

1

1.5

2

2.5

3

ry/L B 0.25

n(ry)

0.2

0.15

0.1 Steady state (theory) t=26000~36000 (simulation) no hydrodynamic interaction

0.05

0

0

0.5

1

1.5

2

2.5

3

ry/L 0.4

C

n(ry)

0.3

0.2

0.1 Steady state (theory) t=56000~76000 (simulation) no hydrodynamic interaction 0

0

0.5

1

1.5

2

2.5

3

ry/L

Figure 4-10. The simulation results (symbols) and theoretical solutions (lines) for the center-of-mass distributions of a rigid fiber as a function of distance from a wall, ry /L, with A = 10 and H = 6L in parabolic flows of A) Pe = 1.2 × 103 , −1 B) 1.2 × 104 and C) 4.8 × 104 . The time (units of γ˙ ) ranges at which the simulation data were taken are indicated in each legend.

76

Pe =1200 Pe =12000 Pe =48000

0.2

2



0.3

0.1

0

0

0.5

1

1.5

2

2.5

3

ry/L ­ ® Figure 4-11. The simulation results with hydrodynamic interactions (symbols) for p2y of a rigid fiber as a function of distance from a wall, ry /L, with A = 10 and H = 6L in parabolic flows of Pe = 1.2 × 103 (◦), Pe = 1.2 × 104 (2) and Pe = 4.8 × 104 (¦). The corresponding values for unbounded flow that account for the position-dependent rate of shear are plotted as dashed lines. hydrodynamic lift has little effect upon the near-wall distribution for this and lower values of Pe. Clear differences are observed between simulations with and without hydrodynamic interaction for Pe = 1.2 × 104 and 4.8 × 104 , indicating that the excluded volume interactions are not causing the strong depletion near the wall. The good agreement between the results from the simulation with hydrodynamic interaction and the theory, which does not consider excluded volume, provides further evidence that the balance between Brownian diffusion and the hydrodynamic lift primarily controls the distribution for these higher values of Pe. The simulations of a Brownian, rigid fiber in parabolic flow by Saintillan et al. (2006a) were performed at Pe = 50 and 300. The authors’ suggestion that the results were due to a hydrodynamic lift, similar to that identified for flexible polymers, motivated the kinetic theory of Chapter 3 as well as Park et al. (2007). The current results imply

77

that the depletion observed by Saintillan et al. (2006a) was mostly due to steric effects. However, the simulations of Saintillan et al. (2006a) were also performed in a wider channel of H = 8L and without periodic boundaries, where the hydrodynamic lift would be expected to be stronger. The results of the simulations without hydrodynamic interactions and theory exhibit an off-center maximum due to the gradient in diffusivity induced by the variation in shear rate across the channel (Nitsche and Hinch, 1997; Schiek and Shaqfeh, 1997a). Evidence of the off-center maximum is difficult to detect in the simulations with interactions due to large averaging errors, though Figure 4-10B seems to demonstrate a reduction in the distribution near the centerline of the channel. The lack of a clear, off-center maximum may be due in part to the orientation distribution. Figure 4-11 shows that the gradient ­ ® in p2y , which causes the migration towards the wall through the term hΓi in Eq. 3–12, is smaller than assumed in the theoretical description. Note that Figure 4-11 also shows an ­ ® overshoot in p2y at ry = 0.6L ∼ 0.7L, similar to the results for the case of simple shear flow. 4.3.4

Steady State Distribution in Oscillatory Shear Flow

Results in Section 4.3.2 and 4.3.3 provide that Eq. 3–7, which models the motion of a rod near a wall, is valid at the bulk region in a channel (ry & 0.5L) while excluded volume effect dominates near a wall. It suggests that numerically integrating Eq. 3–7 considering excluded volume effect represented by Eq. 4–10 enables simulation of a rigid rod near a wall at reduced computational burden. Here, we this concept to study rods in oscillatory shear. The motion of an upper wall oscillating with frequency ω in the x-direction is given by Ux = γ max Hω cos(ωt),

78

(4–13)

where γ max is strain amplitude. The P´eclet number for oscillatory shear flow is defined as Pe O =

ωL2 . DTO

(4–14)

Distributions in oscillatory shear flows are obtained by a modified simulation. Theoretical analyses on rigid rod suspensions in oscillatory shear have been limited to low Pe O and small amplitude limits (Kirkwood and Auer, 1951; Ullman, 1969). Park and Fuller (1985) and Schiek and Shaqfeh (1997b) added the effect of confinement by the wall but not hydrodynamic interactions. Here, simulations are performed at high Pe O with considering hydrodynamic interactions with the wall. To compare the effect of oscillatory shear on distributions with the previous ones of other shear flows, flows of Pe O = 1200 and Pe O = 48000 are chosen. Results at various γ max are shown in Figure 4-12. At Pe O = 1200, distributions are very similar with ones shown in Figure 4-5C for simple shear flow. Therefore migration due to hydrodynamic interaction with a wall is still too weak to give any effects. It is observed that increase of γ max slightly thickens depletion layer. Even at higher Pe O of 48000, where migration due to hydrodynamic interaction is clear, distributions are very similar with one without hydrodynamic interactions shown in Figure 4-7C. This result demonstrates that oscillatory flow removes hydrodynamic lift effect. A limited increase of depletion layer with increasing amplitude is also observed at Pe O = 48000. The lack of migration in oscillatory shear can be explained by changes in the orientation distribution. In Chapter 3, it was stated that broken symmetry of the orientation distribution due to Brownian rotation provides the configuration which favors migration away from a wall. The average orientation moment, hpx py i, which controls the migration in steady shear, is found to be ∼ O (10−5 ) for Pe O = 1200 and ∼ O (10−3 ) for Pe O = 48000, which are much smaller than values for simple shear flows > 10−2 found in Figure 3-5. Oscillatory shear seems to recover the symmetry of the orientation distribution again and nullify the migration due to hydrodynamic interactions.

79

0.25

A 0.2

n(ry)

0.15

0.1 max

γ = 0.1 max γ = 1.0 max γ =10.0

0.05

0

0

0.5

1

1.5

2

2.5

3

ry/L 0.25

B 0.2

n(ry)

0.15

0.1 max

γ = 0.1 max γ = 1.0 max γ =10.0

0.05

0

0

0.5

1

1.5

2

2.5

3

ry/L Figure 4-12. The simulation results for the center-of-mass distributions of a rigid fiber as a function of distance from a wall, ry /L, with A = 10, H = 6L, and varying γ max in oscillatory shear flows of A) Pe O = 1.2 × 103 , and B) 4.8 × 104 .

80

0.3

0.1 1.0 10 0.1 1.0 10

max

0.25

PeO = 1200, γ

0.2

PeO = 48000, γ

= =

2



max

0.15

0.1

0.05

0

0

0.5

1

1.5

2

2.5

3

ry/L ­ ® Figure 4-13. The simulation results for p2y of a rigid fiber as a function of distance from a wall, ry /L, with A = 10 and H = 6L in oscillatory shear flows of Pe O = 1.2 × 103 (open symbols) and Pe O = 4.8 × 104 (closed symbol). Each value of γ max is indicated in legend. ­ ® Figure 4-13 shows the distribution of p2y with varying amplitude. As the amplitude ­ ® decreases, p2y approaches 1/3, which is for no flow. Small amplitude or small frequency implies that a rod does not have enough time to be aligned by the flow. Therefore the depletion layer is similar to that of simple shear flow without hydrodynamic interactions (de Pablo et al., 1992). 4.4

Conclusion

Detailed comparisons with simulations demonstrate that an extended version of the kinetic theory in Chapter 3 can adequately predict the center-of-mass distribution for a rigid fiber suspended between two plane walls in either simple shear or parabolic flow of sufficiently high strength, or Pe. For these large Pe, the good agreement validates that the multiple approximations made in the theory do not have a severe, adverse affect upon the predictions for the conditions studied. The success of the theory at high rates of shear is 81

due to the hydrodynamic interaction lifting the particles a significant distance from the wall where the lack of quantitative differences in the instantaneous prediction of the lift velocity are small (Section 4.3) and the particle orientation distribution conforms to that of an unbounded flow (Figures 4-8 and 4-11). The theory can not be applied with confidence if Pe . 1.2 × 103 for the case studied here of A = 10 and H = 6L. The steric interactions between the rod and bounding walls, which are ignored in the current version of the kinetic theory, control the thickness and shape of the depletion layer at these lower rates of shear where the hydrodynamic lift force is too weak to move the particle out of range of the wall. A theory capable of making accurate predictions over a wider range Pe should include the steric interactions as done in previous investigations (de Pablo et al., 1992; Hijazi and Khater, 2001). Simulations presented in the current work were limited to the conditions of A = 10 and H = 6L. The theory, which relies on a superposition of the Green’s function for plane walls, will become continually less accurate as the gap width H becomes smaller with regard to the particle length L. Simulations of the migration of flexible polymers (Usta et al., 2007) indicate that theoretical predictions for the rigid rod distribution would be greatly in error for H . 5L. Conversely, the theory is expected to become more accurate as the ratio H/L increases and comparisons with simulations of Saintillan et al. (2006a) on parabolic flow suggest that the hydrodynamic lift force remains of importance at lower Pe within larger channels, again conforming to expectations.

82

CHAPTER 5 STRESS OF RIGID ROD SUSPENSIONS IN SHEARING FLOWS BETWEEN TWO PARALLEL WALLS 5.1

Introduction

It has been shown in Chapters 3 and 4 that the dynamic behavior of rigid rods suspended in a confined domain is affected by the wall through hydrodynamic interactions and excluded volume to give an inhomogeneous distribution in the channel. As a consequence, the rheological properties of the confined suspension can differ from those of unbounded suspensions due to the interactions between the solid boundaries and the bulk particles. A non-local theory for calculation of particle stress of rigid rods in a confined system has been developed by Schiek and Shaqfeh (1995). However hydrodynamic interactions with the boundaries were not considered and the calculation is limited to low Pe, or weak flow. Since the viscosity changes with particle concentration due to hydrodynamic interaction (Shaqfeh and Fredrickson, 1990), the effect of hydrodynamic interaction with the boundaries on particle stress will be investigated in Chapter 5. An effect of the inhomogeneous distributions obtained at high Pe on the stress is also expected. The effective stress of a Newtonian suspension of viscosity µ, sheared at a rate γ, ˙ is σ = −P I + 2µE + σ P , where P is the fluid pressure, E is the rate of strain tensor which satisfies γ˙ =

(5–1) 1 2

(E : E),

and σ P is the average contribution due to the presence of the particle. The stresslet for the rigid slender body suspended between the bounding walls, SB , is given by (Batchelor, 1971)

Z

L/2

SB = − −L/2

·

¸ 1 fB (sB ) pB − I{fB (sB ) · pB } sB dsB 3

83

(5–2)

for an instantaneous configuration. Using the linearized expression for the line force density, fB (sB ), on the bulk fiber Eq. 2–16 and integrating along the axis gives µ SB = − (T B × pB ) pB − SB

¶ 1 pB pB − I . 3

(5–3)

Note that we ignore the direct contribution of repulsive forces between the wall and bulk particle when calculating the stress. Consequently, the torque T B in Eq. 5–3 contains contributions only from Brownian motion. The relation between the stresslet coefficient SB and the particle stress is also made clear in Eq. 5–3. Calculating the effective stress contribution of the rigid fiber requires averaging Eq. 5–3 over the orientation and center-of-mass distributions. This average is calculated from the simulations by sampling the distributions at steady state (Section 5.2) and then from theory at the same level of approximation as used in Chapter 3 for predicting the center-of-mass distribution (Section 5.3). Results are presented comparing the calculated value of the total contribution, as well as individual contributions (Section 5.4). 5.2

Calculation of Extra Particle Stress from Simulations

Once the distribution has approached steady state (t > Pe γ˙ −1 ), the instantaneous value of the particle stresslet from Eq. 5–2 is sampled NR times from the simulation results and averaged, NR nd X σ = [SB ]l , NR l=1 P

(5–4)

to give the mean expected value for the extra particle stress. Here nd is the number density N/V . The number of rods in each periodic cell is N = 1 and the volume V represents the space between the bounding walls where the bulk particle resides, rather than the entire volume of the periodic cell. Calculating this volume while accounting for the bumpy walls gives V = bx bz H −

84

πL3 NW , 4A2 2

(5–5)

where NW is the total number of wall particles. The number density is consequently N/V = 1.876 × 10−2 L−3 in all of these simulations. Note that the shear stress for the confined flow can be calculated from the simulation results in two ways: as above from the particle stresslets or by normalizing the total force acting on the walls in the x-direction by the area. The forces on the wall due to the fluid in the absence of a particle must be subtracted from the total values in order to isolate the contribution from the particles. The expected value of the fluid stress matches within 1.38% of the result from simulations in the absence of particles and the mean value of the particle contribution to the shear stress as calculated from the stresslets and the wall forces agrees within at least 0.04% for all cases. Figure 5-1A shows the total contribution of the particle stress for values of Pe between 1.2 × 103 and 1.2 × 105 ; the values of σ Pxy have been normalized by nd L3 µγ. ˙ Even at the lowest Pe of 1.2 × 103 , where the particle stress is largest, the normalized σ Pxy is only 8.58 × 10−3 , demonstrating the relatively small impact of the dilute suspension of rods on the overall rheology. The extra particle shear stress, σ Pxy , can be separated into components, σ Pxy = σ PxyB + σ PxyS ,

(5–6)

arising from the Brownian torque σ PxyB and the shear flow σ PxyS . Each is generated independently from Eq. 5–3, σ PxyB

NR nd X ˆ ]l = [−py (T B × pB ) · x NR l=1

(5–7)

NR nd X = [−SB px py ]l , NR l=1

(5–8)

and σ PxyS

then averaged as described in Eq. 5–4. The result for each contribution is shown in Figures 5-1B and 5-1C. Not surprisingly since Pe is large, the contribution from shear exceeds that from the Brownian rotation by orders of magnitude.

85

10

-2

B

A 10

-3

σxy /(ndL µγ)

.

3 PB

P

10

σxy /(ndL µγ)

3

.

-4

10

10

-3

10

3

10

4

10

10

5

-5

-6

10

3

10

4

Pe 10

10

5

Pe

-2

10

C

10

D

-3

-4

10

10 -3

10

3

10

4

10

5

10

PW

-6

-7

σxy

PS

10

10

10

.

-5

3

3

10

/(ndL µγ)

σxy /(ndL µγ)

.

-8

-9

-10

10

3

Pe

10

4

10

5

Pe

Figure 5-1. The extra particle shear stresses for rigid fibers with A = 10 under simple shear flow between two walls with a gap of H = 6L calculated from simulation (°), simulated orientation moments (H), and theory (¥). Results shown include A) the total particle stress σ Pxy , B) the Brownian torque contribution σ PxyB , C) the shear flow contribution σ PxyS , and D) the additional contributions due to presence of the wall σ PxyW . 5.3

Theoretical Prediction of Extra Particle Stress

To compare the stress calculation from the simulation as described in Section 5.2 with the theoretical calculation, the approach of the previous theory in Chapter 3 which uses the approximate, linearized Green’s function is applied directly to calculate the stress using Eq. 5–3. For the Brownian torque, the expression as given in Eq. 3–10 is used. The explicit result for the stresslet coefficient is given by £ ¡ ¢ ¤ πµL3 γ˙ ˆ · F B, SB = − px py − λ (ry ) 2py pB + p2y − 1 y 6 ln(2A)

86

(5–9)

where the first term is the direct contribution of the shear flow. Upon substitution of Eq. 3–10 for the Brownian force, SB = −

£ ¡ ¢ ¤ ∂ ln Ψ πµL3 γ˙ ˆ · px py + kB T λ (ry ) 2py pB + p2y − 1 y , 6 ln(2A) ∂rB

(5–10)

it becomes clear that the second term is related to the gradient in the center-of-mass distribution which is created by the hydrodynamic interaction of the rod and bounding walls. Replacing T B and SB in Eq. 5–3 with Eq. 3–10 and Eq. 5–10 gives an expression for the stresslet, SB

µ ¶ πµL3 γ˙ 1 ∂ ln Ψ + = kB T pB (I − pB pB ) · px py pB pB − I ∂pB 6 ln(2A) 3 ¶ µ ¡ ¢ ¤ ∂ ln Ψ 1 £ ˆ · , −kB T λ (ry ) pB pB − I 2py pB + p2y − 1 y 3 ∂rB

(5–11)

which must be averaged to give the mean stress. The average is performed over the expected configuration in the system of volume V , ZZ P

σ = nd

ΨSB dpB drB .

(5–12)

By factorizing the distribution function Ψ = nψ under the same assumption that orientation distribution is not influenced by the wall and equilibrates faster than n, this equation is completed in terms of the ensemble average over the orientation distribution, ψ, µ ¶ σP 4π π 1 = (3 hpB pB i − I) + hpx py pB pB i − I hpx py i (5–13) nd L3 µγ˙ Pe ln(2A) 6 ln(2A) 3 · µ ¶ ¸ Z ­ 2® 1 ® ­ 2 4π ∂ ln n + nλ (ry ) dry hpB pB i + I py − − 3 py pB pB . Pe ln(2A) ∂ry 3 The first two contributions correspond to those of a dilute system of Brownian fibers in an unbounded flow as derived by Hinch and Leal (1976). The first (details of its derivation is in Appendix B.4) represents the affect of the Brownian torque on the effective stress,

87

which vanishes at high Pe in comparison to the second term representing the effect of the shear flow. The third term in Eq. 5–13 is a contribution to the effective stress from the non-uniform distribution of the center-of-mass of the rods caused by the migration. The shear stress component (xy) of σ P can be evaluated as a function of Pe from Eq. 5–13 using the analytical expression obtained in Chapter 3 for n (ry ) and numerically integrating λ (ry ) ∂n/∂ry over the channel height. The results are plotted in Figure 5-1A. The additional contribution from the inhomogeneous distribution is labeled σ PxyW (the third term in Eq. 5–13) and is small as seen in Figure 5-1D. 5.4

Discussion: Comparison of Stress Calculations

Figure 5-1A shows that the effective particle stress as calculated from the simulation results and the theoretical description closely agree. The close agreement, given the small contribution from the inhomogeneous distribution in the theory, implies that the effective particle stress is close to that of a dilute suspension of Brownian fibers in the absence of bounding walls. To investigate the effects of the bounding walls, as well as periodic images, on the results calculated from the simulations, we calculate the dilute contribution evaluated directly from the simulations. The contribution from the Brownian torque, given by σ PxyB 12π = hpx py i , 3 nd L µγ˙ Pe ln(2A)

(5–14)

is calculated using the moment of px py as sampled from the simulations and averaged over the channel height. The result, plotted in Figure 5-1B, shows that the dilute contribution is slightly lower than the value calculated directly from the simulation. Likewise, Figure 5-1C shows the contribution from the shear where the value ­ 2 2® σ PxyS π = p p 3 nd L µγ˙ 6 ln(2A) x y

88

(5–15)

was calculated from the orientation moments extracted from the simulation. The result is only slightly lower than the prediction made directly from the simulation results as given by Eq. 5–8. The difference in the total particle stress from the simulations as given by Eq. 5–6 and the dilute value as calculated from the orientation moments produced by the simulations by combining Eq. 5–14 and Eq. 5–15 is given in Figure 5-1D. This quantity represents the additional stress due to the walls, inhomogeneous distribution for the center of mass, and interactions with the periodic images. Figure 5-1D shows that the value predicted from the theory for the contribution beyond that of a dilute suspension in the absence of walls is much smaller than that predicted by the simulations. The lack of agreement shown in Figure 5-1D clearly arises from the approximations made within the theory for the Green’s functions. The theory predicts an additional contribution to the stress from the presence of the inhomogeneous distribution, but does not fully account for the reduction in the stresslet coefficient due to direct interactions with the bounding walls and other mechanisms. Improved approximations for the evaluation of the Green’s function could improve the agreement with the simulation results. That the presence of walls can affect the rheology of suspension systems is a well known result (Happel and Brenner, 1965). For the specific case of Brownian rods, Schiek and Shaqfeh (1995) calculated the stress for a bound suspension at low values of Pe using a nonlocal theory and accounting for the excluded volume, but not hydrodynamic interactions which alter the probability distribution. The increasing confinement of the rods reduced the effective viscosity further as compared to an unbounded suspension. The analysis presented in Eq. 5–1-5–3 is based upon a local approximation, but does account for the dependence of the stress upon the position of the rod. For the conditions studied here, the difference between the bounded and unbounded results is about 2% at best. A direct investigation of the dependence of the stress upon particle position

89

1.05 1.04 1.03 1.02 1

1.01

P

P

[σxy (ry)] / [σxy ](unbounded)

ο

φ=18 ο φ=3

0.5

1

1.5

2

2.5

3

ry/L P Figure 5-2. The particle shear stresses σxy (ry ) for a rigid fiber with φ = 3◦ , 18◦ , pz = 0 and A = 10 in a gap of H = 6L, normalized by the theoretical values with the same configurations in unbounded flow, as a function of a distance from a wall, ry /L. Calculations are from the simulation with simple shear flow with a force and torque free fiber.

is given in Figure 5-2, which shows the instantaneous value of the effective particle stress as calculated from the simulations normalized by the dilute value for two different orientations. The particle stress is only about 1% higher than the unbounded value at ry = 0.5L and vanishes to less than 0.3% of the value at the center. Only a small amount of additional stress is found to exist even for particles very near a wall, consequently the fact that the average particle stress closely matches the dilute values is not surprising at high Pe where the particle migrates away from the bounding walls. The limited effect of the walls on the rheology of dilute suspensions of rods has been noted by others (Attansio et al., 1972; Ganani and Powell, 1985; Petrie, 1999; Moses et al., 2001; Zurita-Gotor et al., 2007).

90

5.5

Conclusion

The bounding walls alter the particle contribution to the total stress little as compared to the dilute values in an unbounded system owing to the strong migration away from the bounding walls at high Pe. Differences between evaluations of the particle stress from the simulations and a theoretical calculation at the same level of approximation used to predict the distribution are attributable to the approximate evaluation of the Green’s function and can likely be improved upon. Investigation of the effect of multiple particles on the distribution and the stress is ongoing now.

91

CHAPTER 6 INHOMOGENEOUS DISTRIBUTIONS OF A RIGID ROD IN ROTATIONAL FLOWS 6.1

Introduction

Studies on migration of a flexible polymer in inhomogeneous velocity fields, especially rotational flows in Stokes regime (Aubert and Tirrell, 1980; Aubert et al., 1980; Brunn, 1984; Brunn and Chi, 1984), have shown that flexible polymers in rotational flows migrate toward the axis of rotation, resulting in inhomogeneous distributions. For rigid polymers, Aubert and Tirrell (1980) and Aubert et al. (1980) speculated that a similar migration would happen for ellipsoids in inhomogeneous velocity fields, citing a formula predicting motions of rigid ellipsoids (Happel and Brenner, 1965; Brenner, 1966). However, quantitative and qualitative analyses of the dynamics of migration, center-of-mass distribution, and particle stress contribution of rigid polymers in inhomogeneous velocity fields, such as rotational flows, have not been made. A rigid polymer in one type of rotational flow, a torsional flow, is studied in Chapter 6. The equation of motion which considers the inhomogeneous flow field is derived in Section 6.2. Prediction of the center-of-mass distributions in the radial direction are made in Section 6.3 and the particle stress contribution is evaluated from the resulting distribution in Section 6.4. 6.2

Migration in the Radial Direction

An inhomogeneous velocity field at a certain point x is expressed in a series expansion, 1 u (x) = u (O) + x · ∇u + xx : ∇∇u + · · · . 2

(6–1)

The velocity of a rigid rod is approximated by applying Eq. 6–1 at its center-of-mass. Integrating the fluid velocity along the rod using Eq. 2–7 gives the center-of-mass velocity

92

Figure 6-1. Schematic diagrams for a torsional flow in A) Cartesian coordinates and cylindrical coordinates, and in views from B) the xy-plane, C) and the ˆ ˆ = θˆ and y ˆ = R. xz-plane. A rigid rod is located at r = (0, R, 0.5H) so that x in the inhomogeneous velocity field, ¸ s2 r˙ = ξ u (0) + sp · ∇u + pp : ∇∇u + · · · . ds 2 −L/2 ¯ 2 ¯ L (6–2) = ξ −1 (I + pp) · F + u (0) + pp : ∇∇u¯¯ + · · · , 24 −1

1 (I + pp) · F + L

Z

L/2

·

s=0

which will be used to predict the velocity of a rod in a torsional flow. Figure 6-1 illustrates a torsional flow, where a liquid between two parallel disks of equal radii of Rmax and separated by a gap of H is sheared by rotating the top disk with angular velocity Ω. In cylindrical coordinates, the axis of rotation is in the z-direction, the ˆ and θˆ indicate the radial and angular directions, respectively. The origin unit vectors R O is located at the center of bottom disk which is stationary. Cartesian coordinates are also shown in Figure 6-1. For convenience, the rod is positioned so that the y-direction corresponds to the radial direction and the x-direction to the flow direction. There are no flows in the radial and z-directions. The flow at a position x is only in the angular direction, uθ (x) =

Ω Rz = γ˙ (R) z, H

93

(6–3)

where γ˙ (R) is a local shear rate at a position R =

p

x2 + y 2 in the radial direction. The

flow rate can be also expressed in Cartesian coordinates: u (x) =

Ω z (−yˆ x + xˆ y + 0ˆz) , H

(6–4)

which is then substituted into Eq. 6–2 to get the quadratic term which contributes to migration, Ω (p · ∇)2 z (−yˆ x + xˆ y) H Ω ˆ + (px z + pz x) y ˆ] = (p · ∇) [− (py z + pz y) x H 2Ω ˆ + px y ˆ) . = pz (−py x H

pp : ∇∇u =

(6–5)

If the center-of-mass of a rigid rod is fixed on the y-axis, such that r = (0, R, 0.5H) as in Figure 6-1, the radial velocity at a given H, Rmax , and γ˙ max = γ˙ (Rmax ) = ΩRmax /H becomes r˙y = −

L2 γ˙ max px pz + ξ −1 (ˆ y + py p) · F . max 12R

(6–6)

An identical result to Eq. 6–6 can be derived from direct integration of local velocities, u (s), along a rod. For a force and torque free rod at r = (0, R, 0.5H), the velocity in the radial direction becomes 1 L

Z

L/2

Ω uy (s) ds = − LH −L/2

Z

L/2

(z + spz ) spx ds = − −L/2

L2 γ˙ max px pz , 12Rmax

(6–7)

which shows that Eq. 6–6 is valid as long as the slender-body approximation holds. Using Eq. 6–7, the dependence of migration velocity on polymer orientation φ, the angle between the x-direction and projection of p onto the xz-plane, is demonstrated in Figure 6-2. A rigid polymer moves towards the axis or the edge depending upon the instantaneous orientation. Since the orientation distribution is symmetric, a non-Brownian rod does not experience any net migration. However, Brownian rotation breaks the symmetry to

94

max 2

L )]

0.04

0

ry[R

max

/(γ

.

.

-0.04 0

20

40

60

80

100

φ (degree)

120

140

160

180

Figure 6-2. A drift velocity r˙y , units of Rmax /γ˙ max L2 , of a force-free rigid rod in a torsional flow as a function of φ. Negative velocity indicates a migration towards the axis. produce an average angle of small positive value, which also corresponds to migration toward the axis, as in rectilinear shear flows in Chapter 3. As long as the position of the center-of-mass is not too close to either the axis or the edge (0 < R ¿ Rmax ) or to the bottom or the top disk (0 ¿ z ¿ H), such that orientation is not affected by the boundaries, the migration velocity depends on the instantaneous configuration: only orientation, not center-of-mass position. In a torsional flow, curvature (∼ 1/R) is canceled by the local shear rate (∼ R) to make the migration independent of R explicitly. At a given γ˙ max , migration velocity decreases at larger disks. 6.3

Steady State Distributions

The distribution of the center-of-mass is derived from a continuity equation, Eq. 3–1. Similar approximations made in Chapter 3 are applied. The rotation of a rod is assumed to not be influenced by steric and hydrodynamic interactions with the boundaries. The

95

distribution function is also factorized into center-of-mass and orientation distributions as in Eq. 3–2. With no net migration in the angular and z-direction, the center-of-mass distribution in the radial direction satisfies 0=

D E´ −1 ∂ ³ nR R˙ . R ∂R

(6–8)

Applying a boundary condition of no flux at the edge of the plates in the radial direction gives

D E 0 = n R˙ ,

(6–9)

which can be obtained from Eq. 6–6. Replacing F in Eq. 6–6 with Eq. 3–10, followed by integrating over the orientation distribution gives " ­ ®# D E 2 max ¡ ­ 2 ®¢ ∂n ∂ p2y L γ ˙ k T B n R˙ = −n hpx pz i − 1 + py +n . 12Rmax ξ ∂R ∂R

(6–10)

Further manipulation with substitution of Pe max = r gives a differential equation,

ξ γ˙ max L2 12kB T

(6–11)

­ I® K + hΓi ∂ ln n ­ ® , = ∂R 1 + p2y

(6–12)

­ I® Pe max r K = − max hpx pz i , R

(6–13)

where

which relates the migration due to the inhomogeneous velocity of a torsional flow, and ­ ® ∂ p2y hΓi = − , (6–14) ∂R which is the anisotropic diffusivity term which causes migration from low to high shear regions (Nitsche and Hinch, 1997; Schiek and Shaqfeh, 1997a). Integration of Eq. 6–12 over R gives a center-of-mass distribution in the radial direction, requiring calculation of average orientation moments. The orientation

96

distribution function also satisfies Eq. 3–20 due to similar approximations made for factorization of distribution functions in Eq. 3–2. To solve Eq. 3–20, the equation of motion for rotation of a rigid slender-body in torsional flow is required, which can be obtained by substitution of Eq. 2–9 with Eq.6–4 for a rod with position r = (0, R, 0.5H). The resulting rotation is p˙ =

¤ £¡ ¢ 12 γ˙ max γ˙ max 2 ˆ ˆ ˆ ˆ ˆ z + y ] + y − p p x − p x − p p r [p Rp 1 − p T × p. x z x x y z y z x Rmax Rmax ξL2

(6–15)

The first term on the right hand side represents translation of orientation: The orientation relative to the center-of-mass, flowing in the angular direction, does not change. The second term corresponds to the change of orientation due to shear flows of rate γ˙ (R) in the x-direction with the gradient in the z-direction. These two terms imply that orientation distribution depends only dependent on R and is same with that of simple shear flow. As in Chapters 3 and 4, average orientation moments of a rigid rod of r = (0, R, 0.5H) in torsional flows of various Pe max are also obtained by performing the same numerical r method. Results are shown as a function of R/Rmax in Figure 6-3. Three orientation ­ ® moments are chosen: p2y and hpx pz i are used for integration of Eq. 6–12 and hp2x p2z i will be used in the calculation of stress. As Pe max changes, the average orientation moments r shift according to the local Peclet number, Pe r (R) = RPe max /Rmax . r Figure 6-4 shows the center-of-mass distributions in the radial direction in torsional flows of various Pe max , which is obtained from integration of Eq. 6–12 using orientation r R moments from Figure 6-3. The distribution function n is normalized such that ndR = 1 and scaled with Rmax . As Pe max increases, n (r) distributes towards the axis. As in simple r shear flow, Brownian rotation breaks the symmetry of orientation distribution to give positive values of hpx pz i, which induce migration towards the axis as predicted in Eq. 6–6. Compared to flexible polymers (Brunn, 1984), rigid polymers require larger Pe max for r effective migration. Since the curvature affect orientation moments little as shown in

97

max

Per

= 200

Per

= 400

Per

=1000

max max



0.1

0.05

0 0.001

0.01

max

0.1

1

0.1

1

0.1

1

R/R

0.3



0.25

0.2

0.15

max

Per

= 200

max Per = 400 max Per =1000

0.1 0.001

0.01

max

R/R

0.08



0.06

0.04

max

Per

= 200

Per

= 400

Per

=1000

max max

0.02 0.001

0.01

max

R/R

Figure 6-3. Average orientation moments of a rigid rod in torsional flows at three values of Pe max as a function of R/Rmax . r

98

30

n(R)R

max

1.5

max

20

1

n(R)R

0.5 0

0.2 0.4 0.6 0.8

R/R

max

1 max

Per

10

0

=200

max Per =400 max Per =1000

0

0.1

0.2

0.3

0.4

0.6 0.5 max

0.7

0.8

0.9

1

R/R

Figure 6-4. Center-of-mass distributions of a rigid rod in torsional flow of various Pe max as r max a function of R/R . Inset shows the distribution without inhomogeneous velocity field migration. ­ ® Figure 6-3 and K I does not depend on R explicitly, all steady state distributions at various Rmax are matched if scaled with Rmax . The anisotropic diffusivity term hΓi in Eq. 6–12, which induces an opposing migration in pressure-driven flow (Nitsche and Hinch, 1997; Schiek and Shaqfeh, 1997a; Park et al., 2007), is expected to generate migration away from the axis since the shear rate increases in the radial direction for torsional flows. However, its effect is too weak to give any apparent effect. The inset of 6-4 shows the distribution obtained by integrating ­ ® Eq. 6–12 without the inhomogeneous velocity field term K I . In pressure-driven flow, the off-center maximum due to competition between two opposite migrations only appears near the centerline where hydrodynamic lift decays with ∼ ry−2 (Saintillan et al., 2006a; Park et al., 2007). However, as shown in Eq. 6–7 the migration effect due to the inhomogeneous velocity field does not decay and overcomes the competition.

99

In spite of the same steady distributions for larger disks at the same Pe max , the r time to achieve steady state would be longer for larger Rmax /L. For example, a single carbon nanotube of length 2.5 × 10−7 m sheared between disks of Rmax = 2.5cm has Rmax /L = 105 . Transient distribution and prediction of the time to achieve steady state are also being studied as an ongoing work now. However, if the geometry is in microscale, such as Rmax ∼ 100L, this migration surely becomes more effective. Hence, it can provide a new concept to the recent efforts on development of ‘rheometer-on-a-chip’ (Poole et al., 2007; Pipe et al., 2008) and may have relavance to studies on dynamics in microchannels with curvature, such as microbends (Davison and Sharp, 2008; Gulati et al., 2008). In Chapter 6, hydrodynamic interactions with the wall are not included. However, the broken symmetry of orientation due to Brownian motion should be coupled to cause migration. Therefore, hydrodynamic interactions will induce only migration in the z-direction and not affect the migration in radial direction in dilute concentration. 6.4

Stress Calculation

Particle stress contributions in a torsional flow can be calculated considering the inhomogeneous distribution obtained in Section 6.3. Equation 5–3, which represents a stresslet S of a rigid slender-body, is rewritten using Eq. 3–10 for a shear flow in the x-direction generating stress in the z-direction; ¶ µ ∂ ln Ψ πL3 µγ˙ 1 S = kB T p (I − pp) · + px pz pp − I . ∂p 6 ln(2A) 3

(6–16)

Averaging S over the expected configuration in the system of volume V gives the mean particle stress,

ZZ P

σ = nd

ΨSdpdV.

(6–17)

Factorizing the distribution function Ψ = nψ and taking the average over the orientation distribution gives

Z P

σ = nd

hSi ndV,

100

(6–18)

where πµL3 γ˙ max R hSi = kB T (3 hppi − I) + 6 ln(2A) Rmax

µ ¶ 1 hpx pz ppi − I hpx pz i . 3

(6–19)

Extracting the shear stress component xz from Eq. 6–18 gives particle stress contributions σ Pxz σ PxzS σ PxzB = + nd L3 µγ˙ max nd L3 µγ˙ max nd L3 µγ˙ max Z Z ­ 2 2® π π = R px pz ndV + hpx pz i ndV 6Rmax ln (2A) Pe max ln (2A) r π π R hp2x p2z i + hpx pz i, (6–20) = max max 6R ln (2A) Pe r ln (2A) which is factorized into shear stress contribution σ PxzS and Brownian contribution σ PxzB . Interpolation of the orientation moments from Figure 6-3 and n (R) obtained from Eq. 6–12 are used for the integration in Eq. 6–20. As in Figure 6-4, using the entire terms of Eq. 6–12 gives an inhomogeneous distribution due to the inhomogeneous velocity fields, ­ ® while neglecting K I gives an almost uniform distribution. The stress contributions are also calculated using the both distributions for comparison. Figure 6-5 shows the resulting particle stress contributions σ Pxz for both the inhomogeneous and uniform distributions. As particles migrate towards the axis, where the shear rate is lower, with increasing Pe max , the Brownian contributions, σ PxzB , r considering inhomogeneous distributions are larger than those of uniform distributions due to simple integration of hpx pz i at lower Pe r (R). However, the shear contributions, σ PxzS , show the opposite behavior due to integration of an additional factor of R in 6–20. As σ PxzS becomes more dominant over σ PxzB , the total shear thinning effect considering the inhomogeneous distribution becomes more obvious than that considering the uniform distribution. This enhanced shear thinning behavior due to migration may explain shear thinning behavior observed in experiments (Bricker et al., 2008). 6.5

Conclusion

Dynamics of a rigid polymer in a torsional flow are investigated in Chapter 6. The relation between migration direction and polymer orientation is clarified, showing that net 101

-2

max

)

10

10

-3

P

3

σxz /(ndL µγ

.

10

-4

10

-5

100

σxz

P

σxz

PS

σxz

PB

σxz

P

σxz

PS

σxz

PB

} }

Uniform distribution

Inhomogeneous distribution 1000 max Per

Figure 6-5. Particle stress contributions σ Pxz of a rigid rod in a torsional flow with Rmax = 100L as a function of Pe max in the inihomogeneous distributions (open r symbols) and the uniform distribution (closed symbols). migration requires breaking of the symmetry of the orientation distribution in the shear direction. The center-of-mass distribution in the radial direction is also predicted from the equation of motion. Rigid polymers are distributed toward the axis and more strongly at higher Pe max . This distribution affects the particle stress contribution and results in an r enhanced shear thinning. This theory is being improved for the transient distribution and stress now.

102

CHAPTER 7 CONCLUSIONS Rigid rod-like polymers suspended in Newtonian fluids were studied theoretically as well as numerically. Applying a method of simulation adapted from previous work (Harlen et al., 1999; Butler and Shaqfeh, 2002, 2005), it was found that a pair of sedimenting rods, which have a certain initial configuration, have orbits in their trajectories due to hydrodynamic interactions. The short-time diffusivity of rigid rods in dilute and concentrated suspensions (nd L3 = 0 ∼ 150) were also calculated by the simulation method to demonstrate the concentration-dependent reduction due to hydrodynamic interactions. A kinetic theory was developed for the migration of a rigid rod in rectilinear flows near solid boundaries. The coupling of a broken symmetry of the orientation distribution due to Brownian motion and the transverse motion induced by hydrodynamic interactions with a wall causes net migration normal to the wall to give an inhomogeneous distribution. Three mechanisms were identified for the migration of a rigid rod. A mechanism due to shear flow is a major contribution to migration away from a wall and has an analogy with the migration mechanism for flexible polymers. The second mechanism due to Brownian torque is newly found, is unique to rigid rods, and is a weaker contribution. The anisotropic diffusivity mechanism, which has been also identified by Nitsche and Hinch (1997) and Schiek and Shaqfeh (1997a), gives the opposite migration. These mechanism are combined to affect the distributions in the channel. For simple shear flows between two walls, the kinetic theory predicted that the center-of-mass would distribute towards the centerline of the channel and the depletion layer is extended into the bulk region at higher shear rates. For pressure-driven flows, the prediction showed an off-center maximum for the center-of-mass distribution. The theory also predicted that for transient behavior, an initially uniform distribution would show a peak near a wall at short times. As time proceeds, the peak would gradually move towards the center and finally vanish before reaching the steady state distribution.

103

The simulation was used to confirm the kinetic theory. The good agreement between the center-of-mass distribution obtained by the simulation and the theory demonstrates that the approximations made in the kinetic theory are not stringent. The orientation moments obtained from the simulation matched the unbounded value in the bulk region. The discrepancies both in the center-of-mass and orientation distributions near a wall show that the excluded volume effect is dominant in that region. However, at sufficiently higher Pe (Pe > 1.2 × 103 for the case studied here of A = 10 and H = 6L), it moves the rods out of the region where the approximations are not valid and the distributions become distinguishable depending upon whether hydrodynamic interactions are included in the calculation. Simulation results in oscillatory shear flows showed that migration due to hydrodynamic interactions with the wall did not occur due to the symmetric orientation distribution. The particle stress of a rigid rod in shear flows between two parallel walls was calculated from the simulation as well as the theoretical prediction at the same level of approximation used in the kinetic theory. The inhomogeneous distribution induced by the hydrodynamic interactions with the wall made the values of the particle stress very close to those of an unbounded system since the additional wall contribution to the particle stress is negligible far from a wall. The values of the additional wall contribution evaluated from the simulation were larger than those from the theoretical prediction owing to the approximate evaluation of the Green’s function and can likely be improved upon. A theoretical analysis for dynamics of a rigid rod in a torsional flow was developed. The combined effects of the average orientation due to Brownian fluctuation and the inhomogeneous velocity field generate a net migration toward the axis. The center-of-mass distribution in the radial direction was also predicted to show that rigid rods are distributed more strongly toward the axis at higher Pe max . The particle stress evaluated r from the inhomogeneous distribution resulted in an enhanced shear thinning as compared to a uniform distribution.

104

The simulation method will be extended to study the rheology and structure with semi-dilute and concentrated systems of rods at high Pe. Previous simulations, such as those of Sundararajakumar and Koch (1997) and Pryamitsyn and Ganesan (2008), examined the rheology of unbounded shear flow. The net migration caused by hydrodynamic interaction with the walls in a non-dilute system would be expected to alter the microstructure, which is directly related to the rheological properties (Shaqfeh and Fredrickson, 1990; Dhont and Briels, 2003). The results may be useful in explaining the shear thinning behavior observed for semi-dilute suspensions of fibers at high Pe (Ganani and Powell, 1985; Chaouche and Koch, 2001; Bricker et al., 2008). The simulation method is also being applied to the studies on flows of DNA fragments in nanochannels under electrophoresis and the separation of single wall carbon nanotubes in a microchannel using dielectrophoresis.

105

APPENDIX A COMPONENTS OF GRAND MOBILITY TENSOR f in Eq. 2–25 are shown in Mobility components of the grand mobility matrix M Appendix A. They are evaluated for infinite suspensions using the solutions for periodic Stokeslets. For isolated particles using a Green’s function for free space, the integrated Green’s functions in self-terms (αα or ββ), which relates interactions between self images, are 0. A mobility component, MTααF = ξ −1 (I + pα pα ) +

1 (0) G , L2 αα

(A–1)

relates translational motion of a rigid rod α, r˙ α , to the total force acting on itself and its self-periodic images, F α . It is a matrix of dimension 3 × 3. MTαβF =

1 (0) G L2 αβ

(A–2)

is a 3 × 3 matrix and relates r˙ α to F β . TT Mαα =

12 (1b) G , L4 αα

and MTαβT =

12 (1b) G L4 αβ

(A–3)

are both 3 × 3 tensors and relate r˙ α to T α and T β , respectively. MTααS =

12 (1b) G · pα , L4 αα

and MTαβS =

12 (1b) G · pβ L4 αβ

(A–4)

are 3 × 1 vectors which relate r˙ α to S. The rotation of a rigid rod α is determined using 3 × 3 tensors, MRF αα =

12 (I − pα pα ) · G (1a) αα , 4 L

MRT αα =

MRF αβ =

12 (1a) (I − pα pα ) · G αβ , 4 L

12 −1 144 ξ (I − pα pα ) + 6 (I − pα pα ) · G (2) αα , 2 L L

106

(A–5)

(A–6)

and MRT αβ =

144 (2) (I − pα pα ) · G αβ , 6 L

(A–7)

and 3 × 1 vectors, MRS αα =

144 (I − pα pα ) · G (1b) αα · pα , L6

and MRS αβ =

144 (1b) (I − pα pα ) · G αβ · pβ . 6 L

(A–8)

f to The projection tensor (I − pα pα ) is incorporated after decomposition or inversion of M avoid its singularity (Butler and Shaqfeh, 2005; Saintillan et al., 2006c). The imposed rates of strain are related to forces by 1 × 3 vectors, MEF αα = MET αα =

ξ pα · G (1a) αα , 2L2

6ξ pα · G (2) αα , L4

MEF αβ =

ξ (1a) pα · G αβ , 2 2L

and MET αβ =

6ξ (2) pα · G αβ , L4

(A–9) (A–10)

and the scalar quantities, MES αα = 1 +

6ξ pα · G (2) αα · pα , L4

and MES αβ =

107

6ξ (2) pα · G αβ · pβ . 4 L

(A–11)

APPENDIX B DETAILS OF CALCULATIONS FOR CHAPTER 3 B.1

Linearization of the Green’s Function for a Wall

Blakes’ solution (Blake, 1971) is, H (xα , xβ ) + Z (xα , xβ ) ,

(B–1)

where Z relates the point force and reflected disturbance. In more detail, ˆ ) , (B–2) ˆ ) − 2xy SD (xα , xβ − 2xy y ˆ ) + 2x2y PD (xα , xβ − 2xy y Z (xα , xβ ) = −H (xα , xβ − 2xy y where the first term cancels normal flows on the wall, and the second term, PD (potential dipole), and the third term, Sd (Stokes dipole), cancel tangential flows on the wall. Linearization around the center-of-mass gives, Z (sα , sβ ) = Z (r + sα p, r + sβ p) ≈ Z (r, r) + sα p =

∂ ∂ [Z (xα , r)]xα =r + sβ p [Z (r, xβ )]xβ =r ∂xα ∂xβ

−3 ˆy ˆ) (I + y 32πµry 3sα ˆy ˆ ) py + y ˆ p − pˆ [(I + y y] + 64πµry2 3sβ ˆy ˆ ) py − y ˆ p + pˆ + [(I + y y] . 64πµry2

Integrating the resulted linearized form twice over the length of a rod and including the effect from the two bounding walls with a gap of H using superposition under an

108

(B–3)

assumption of weakly confined system H > 5L gives, (0) G αβ (1a)

G αβ

(1b)

G αβ

· ¸ −3 1 1 ˆy ˆ) = − (I + y 32πµ ry (ry − H) # " 4 L 1 1 ˆy ˆ ) py + y ˆ p − pˆ = − [(I + y y] 2 256πµ ry (ry − H)2 " # 4 L 1 1 ˆy ˆ ) py − y ˆ p + pˆ = − [(I + y y] 2 256πµ ry (ry − H)2

(2)

G αβ = 0. B.2

(B–4) Average of Brownian Contributions

The average of the Brownian force contribution in Eq. 3–7 over orientation is ¶À ¿ µ Z −nkB T 1 ∂Ψ ∂ ln Ψ −1 = ψ (I + pp) · dp n ξ (I + pp) · −kB T ∂r ξ Ψ ∂r Z −nkB T 1 ∂ (nψ) = ψ (I + pp) · dp ξ nψ ∂r µ ¶ Z −nkB T 1 ∂n ∂ψ = (I + pp) · ψ+ n ψdp ξ nψ ∂r ∂r Z Z nkB T ∂ψ −kB T ∂n (I + pp) ψdp − (I + pp) · dp = ξ ∂r ξ ∂r · ¸ Z Z Z −kB T ∂n nkB T ∂n ∂ = (I + pp) ψdp − · (I + pp) ψdp − · (I + pp) ψdp ξ ∂r ξ ∂r ∂r −kB T ∂n nkB T ∂ hppi = (I + hppi) · − , (B–5) ξ ∂r ξ ∂r of which the y-component is used in Eq. 3–11. The average of the Brownian torque contribution in Eq. 3–7 over orientation becomes ¿

À Z 12 (1b) −12nkB T (1b) −nkB T G · ∇p ln Ψ = ψG αβ · ∇p ln ψdp L4 αβ L4 Z −12nkB T (1b) G αβ · ∇p ψdp = L4 Z Z ³ ´ ³ ´ −12nkB T 12nkB T (1b) (1b) = ∇ · G ψ dp + ∇ · G ψdp, p p αβ αβ L4 L4

(B–6)

where ∇p = (I − pp) ·

109

∂ . ∂p

(B–7)

The first term becomes Z −24nkB T (1b) ∇p · dp = p · G αβ ψdp 4 L " # £ ¡­ ® ¢ ¤ −3nkB T 1 1 ˆ , = − 2 hppy i + p2y − 1 y 2 2 32πµ ry (ry − H)

−12nkB T L4

Z

³

(1b) G αβ ψ

´

(B–8)

where Stokes’s special theorem on a spherical surface is used. The second term becomes " # Z ³ ´ £ ¡­ ® ¢ ¤ 12nkB T 3nk T 1 1 B (1b) ˆ . ∇p · G αβ ψdp = − −2 hppy i − p2y − 1 y 2 4 2 L 64πµ ry (ry − H) (B–9) Adding Eq. B–8 and Eq. B–9 completes Eq. B–6: " # ¿ À £ ¡­ 2 ® ¢ ¤ 12 (1b) −9nkB T 1 1 ˆ −nkB T G · ∇ ln Ψ = − 2 hpp i + p − 1 y p y y αβ 2 L4 64πµ ry2 (ry − H) £ ¡­ ® ¢ ¤ −72nkB T ˆ , = λ (ry ) 2 hppy i + p2y − 1 y (B–10) 2 ξL of which the y-component is also used in Eq. 3–11. Likewise, the Brownian torque contribution to the particle stress in (5–13) is derived using Stokes’s special theorem on a spherical surface: σ

PB

ZZ Z −kB T = − nψ [p∇p (ln ψ)] dpdr = nd kB T p∇p ψdp V ¸ ·Z Z ∇p · (pψ) dp − ψ∇p · pdp = nd kB T · Z ¸ Z = nd kB T 2 ψppdp − ψ (I − pp) = nd kB T (3 hppi − I) .

B.3

(B–11)

Numerical Calculation of Orientation Moments

For the calculation of the ensemble average of orientation moments, the orientation distribution function ψ is evaluated from Eq. 3–20 through Brownian dynamics simulation. Since orientation of a rod is assumed to be independent of position, rotation of a single

110

Brownian rod under simple shear flow is determined by, p˙ = γp ˙ y (ˆ x − px p) +

12ξ −1 T × p, L2

(B–12)

which is derived from Eq. 2–9. Brownian torque is generated by a random vector of unit variance W which satisfies the fluctuation-dissipation theorem in the discrete time step, µ T ×p=

kB T ξL2 6∆t

¶1/2 (I − pp) · W.

(B–13)

Equation (B–12) is then made dimensionless using the characteristic time scale t = tγ˙ to give,

µ p˙ = py (ˆ x − px p) +

2 Pe r ∆t

¶1/2

µ (I − pp) · W −

2 Pe r

¶ p,

(B–14)

where the third term on the right hand side is a correction term for numerical integration by a modified Euler method. The orientations at time t = t∗ are calculated and sampled for Ns times, where for example P hpx py i =

px (t∗ ) py (t∗ ) . Ns

(B–15)

Specifically, the averaging is performed over 1000 simulation time increments of t = 5 until a final time step of 5000. The time step was set to 5 × 10−7 , and the number of samples taken within each increment of time was Ns = 104 . This level of averaging was repeated five times using different random seed numbers to arrive at the final averaged value of the moments. B.4

Asymptotic Solutions at Low Peclet Number Limit

Equation 3–20 is solved at the low Pe r by assuming, ψ = ψ0 + ψ1 + ψ2 + · · · .

(B–16)

Rearrangement of the substituted expression gives equations at the 0th order, (I − pp) :

∂ 2 ψ0 = 0, ∂p∂p

111

(B–17)

the first order, (I − pp) :

∂ 2 ψ1 ∂ = Pe r (I − pp) : · [py (ˆ x − px p) ψ0 ] , ∂p∂p ∂p

(B–18)

(I − pp) :

∂ 2 ψ2 ∂ = Pe r (I − pp) : · [py (ˆ x − px p) ψ1 ] , ∂p∂p ∂p

(B–19)

the second order,

of Pe r equations, and so on. Solutions of each equation determine the coefficients of the series to give asymptotic expression for orientation distribution function at low Pe r limit, ψ=

¤ 1 Pe r Pe 2r £ 2 + [px py ] + px + 3p2x p2y − p2y + · · · . 4π 8π 96π

(B–20)

Averaging the orientations over ψ using integration over a spherical surface (Bird et al., 1987) gives the ensemble average orientation moments at the limit of low Pe r , such as, Z hpx py i =

­ 2 ® 1 Pe 2r py = − , 3 504 B.5

Pe r 90

(B–21)

® Pe r px p3y = . 70

(B–22)

px py ψdp = ­

Crank-Nicolson Method

Equation 3–27 is solved numerically using the Crank-Nicolson scheme, a finite difference method. Figure B-1 illustrates meshes used in this scheme. A channel of height H is divided into a total number of Y + 2 points and size of ∆y. A solution for center-of-mass at the m-th time step and the i-th position step is denoted as nm i , which is equivalent to n (i∆y) at t = m∆t. Boundary values are indicated as i = 0 and i = Y + 1. The Crank-Nicolson scheme is an implicit method, which is stable at all ratios of ∆t/∆y. It is also a mid-point method in time step, therefore its error orders are ∼ O (∆t2 ) and ∼ O (∆y 2 ). Manipulation of Eq. 3–27 gives expressions relating finite meshes. Mid-point, Rim , marked as × in figure B-1, is solved explicitly using values at m-th time step, Rim

=

−ui,1 nm i−1

µ ¶ 2 m − ui,2 + nm i − ui,3 ni+1 , ∆t 112

for i = 1, · · · , Y,

(B–23)

Figure B-1. Schematic diagram of finite meshes for solving Eq. 3–27 using Crank-Nicolson scheme. where hD∗ i hKi + λ (ry ) 2 2∆y 4∆y hD∗ i hKi 1 = − + λ (ry )0 − 2 ∆y 2 ∆t ∗ hKi hD i − λ (ry ) , = 2 2∆y 4∆y

ui,1 = ui,2 ui,3 where

" # −L3 1 1 λ (ry ) = − . 64 ln(2A) ry 3 (ry − H)3 0

(B–24)

(B–25)

Mid-points adjacent to boundaries are solved using no-flux through a wall boundary condition. A discretized expression for a boundary condition at y = 0 becomes µ

R1m

2 = − u1,2 + ∆t

¶ m nm 1 − u1,3 n2 ,

(B–26)

where µ u1,2 = −2 and u1,3 =

hKi hD∗ i + λ (ry ) 2 2∆y 4∆y

¶

hKi hD∗ i hKi 1 ∆yλ (r ) − + λ (ry )0 − , y ∗ 2 hD i ∆y 2 ∆t

hD∗ i . ∆y 2

(B–27)

113

Likewise, a boundary condition on the other wall, y = H, in discretized form is µ

RYm

=

−uY,1 nm Y −1

2 − uY,2 + ∆t

¶ nm Y,

(B–28)

where hD∗ i , and ∆y 2 µ ∗ ¶ hD i hKi hKi hD∗ i hKi 1 0 = 2 − + λ (r ) ∆yλ (r ) − λ (r ) − . (B–29) y y y 2∆y 2 4∆y hD∗ i ∆y 2 2 ∆t

uY,1 = uY,2

Finally, center-of-mass distribution, evolved at new time step, m + 1, are obtained implicitly by solving the matrix equation,  0 0 0 0  u1,2 u1,3 0   u2,1 u2,2 u2,3 0 0 0 0   .. .. ..  . . . 0 0 0  0    0 0 ui,1 ui,2 ui,3 0 0   ... ... ...  0 0 0 0    0 0 0 0 uY −1,1 uY −1,2 uY −1,3   0 0 0 0 0 uY,1 uY,2

 

  m+1 n   1     m+1     n     2     .     .     .         m+1    ·  ni =       .     ..             nm+1     Y −1       m+1 nY

 R1m   R2m   ..   .    m , Ri  ..  .    RYm−1    m RY

(B–30)

which is solved by Thomas algorithm, simplified form of Gaussian elimination for a R tridiagonal matrix. The solutions at each time step are also normalized with ndry as in steady state distributions.

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APPENDIX C ADDITIONAL TESTS FOR CHAPTER 4 C.1

The Effect of Torque on Migration

Comparison of the orientation dependent transverse migrations of a rigid rod near a wall predicted by the theory and calculated by the numerical simulation are made in Section 4.3.1. While only the orientation dependent migration due to shear flow without force and torque is demonstrated in Figure 4-3, the effect of torque under conditions of no flow and force is investigated here in Section C.1. A theoretical prediction of the Brownian torque contribution to the transverse motion can be made using an expression extracted from Eq. 3–7: r˙y =

24 ˆ − pB ) · (T B × pB ) . λ (ry ) (3py y ξL2

(C–1)

The corresponding motion is extracted from the simulation (Eq. 4–4) with γ˙ and F set to zero,









0B  r˙ B     =Q· . p˙ B T B × pB

(C–2)

The transverse velocities r˙y are calculated both theoretically using Eq. C–1 and numerically using Eq. C–2 as a function of orientation with a constant torque of   0     , T B × pB = ±  12k T B     0

(C–3)

where a torque of the positive value rotates the orientation of the rod towards parallel to the y-direction, while the negative value rotates the rod perpendicularly to the y-direction. The directions of the applied torques are illustrated in Figure C-1C. The results are plotted as functions of φ as well as ry in Figure C-1. Similarly to the comparison of theoretical and numerical calculations of shear contributions in Section 4.3.1, simulation results and theoretical predictions are in

115

A

B

0.2

0.2 Theory: -12kBT Simulation: -12kBT Simulation: +12kBT Theory: +12kbT

0.1

.

ry

ry

0.1

0

.

0

-0.1 -0.1 -0.2 0

20

40

60

80

100 120 140 160 180

φ(degree)

-0.2 0.5

1

1.5

2

2.5

3

ry/L

BT Figure C-1. Transverse velocities r˙y with units of kµL of a rigid rod with pz = 0 and A = 10 in a gap of H = 6L due to the torque contribution A) as a function of angle φ with ry = 1.0L and B) as a function of distance from a wall ry /L with φ = 3◦ . Simulation conditions are the same as those in Figure 4-2 except γ˙ = 0 and F = 0. C) Directions of applied torques and resulting migrations are illustrated.

qualitative agreement for both the angle and position dependencies and have quantitative differences in that simulation underestimates the transverse motion (Figure C-1). The transverse motions resulting from the torque applied with the positive value are always toward the wall, while those from the torque with the negative value are away from the wall. The angle dependence of migration in Figure C-1 also shows that there is no net migration for symmetric orientation distributions. As with the shear contribution in Section 4.3.1, the Brownian motion also contributes to the net migration if the distribution is assymetric. The Brownian torque is proportional to the negative gradient of the

116

orientation distribution as in Eq. 3–10. Since the orientation of rigid slender-body tends to be aligned to the shear flow direction, Brownian torque under shear flow has an average negative value so that its direction becomes dominant in opposition to the flow direction. As discussed in Section 3.3.3, due to the resistance of a rigid rod to being stretched by shear flow, shear contribution induces migration away from the wall. Likewise, the resistance to being aligned by shear flow, the Brownian contribution induces migration away from the wall. C.2

Excluded Volume Effect

The algorithms to simulate the excluded volume effect of a solid boundary on the motion of a rigid rod are tested by comparing trajectories. A non-Brownian rigid rod, for which the center-of-mass is within a half rod length from a wall, under shear flow γ˙ is simulated with three algorithms preventing the rod from crossing the boundary: •

Repulsive force: A repulsive force and torque are applied on the tips of a rod using Eq. 4–10.

•

Offset: If a rod overlaps with a wall, its center-of-mass is translated normal to the wall and its orientation is also rotated parallel to the wall such that the overlap distance becomes 0.

•

Contact force: If a rod overlaps with a wall, its configuration is recalculated with reduced ∆t to find a contact point. A contact force is evaluated at that point so that the motion of the contacting tip in the direction of penetrating the wall should be zero. Note that hydrodynamic interactions with the wall are not included in this comparison.

Starting from the same initial configuration of ry = 0.3L, φ = −0.6◦ , and pz = 0, three trajectories from each algorithm are compared in Figure C-2. The results have no significant differences: all trajectories show sudden jumps of ry to 0.6L (considering bumpy wall and spherical tip of a rod with d = 0.1L) after contact with a wall. The similar trajectories show that the repulsive force used in the simulations of Chapter 4 ensures the pole-vault motion of a rod, no penetration through a wall, and no unrealistically

117

0

-0.2

0.5

-0.4

Repulsive force Offset Contact force Repulsive force + HI

py

ry/L

0.6

0.4

-0.6

0.3

-0.8

0.2

0

50

100 .

tγ

150

200

-1 0

50

100 .

tγ

150

200

Figure C-2. Time evolutions of A) center-of-mass distance from a wall ry /L and B) orientation py of a non-Brownian rigid rod of A = 10 under shear of γ˙ near a wall. Onset illustrates the pole-vault motion of the rod. large displacement from a wall. Simulations were also performed at smaller ∆t than 0.001γ˙ −1 to confirm the consistency of ∆t used for the simulations in Chapter 4. The effect of hydrodynamic interactions on the pole-vault motion is also investigated by adding hydrodynamic interactions to the simulation of a non-Brownian rod with repulsive forces. At the start of rotation, ry is gradually reduced due to the coupling of shear and hydrodynamic interactions with a wall, which can be explained by the relation between the angle and transverse motion demonstrated in Eq. 3–9 and Figure 4-3. However, the jump of ry during the pole-vault rotation is delayed as compared to cases without hydrodynamic interactions. The time evolution of orientation in Figure C-2B shows that rotation before the wall contact is retarded by hydrodynamic interactions. After the jump, ry is elevated just a little bit due to the coupling. These simulation results shows that rotation of a non-Brownian rigid rod under shear flow near a wall is influenced by hydrodynamic interactions. However, the results from simulations of a Brownian rod in Chapter 4 do not show any significant difference from the theoretical prediction which ignored the influence of hydrodynamic interactions on rotation. In Chapter 3 and Chapter 4, the time scale for

118

changes in center-of-mass distributions was found to be ∼ O (Pe γ˙ −1 ) which is much larger at high Pe > 1000 than the delay (∼ 10γ˙ −1 ) of the pole-vault motion by hydrodynamic interactions found in Figure C-2. Therefore, approximation that rotation is not influenced by hydrodynamic interactions with a wall, made for the theory in Chapter 3 does not severely impact predictions of center-of-mass distribution at high Pe.

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APPENDIX D A SEDIMENTING CLOUD OF RIGID RODS IN VISCOUS FLUID D.1

Introduction

Simulations have been performed to investigate the deformation and breakup of a cloud of rigid rodlike particles falling under gravity through a viscous fluid in the absence of inertia and interfacial tension. The simulation results have found that an initially spherical cloud of rigid rods, like drops of spherical particles or liquids, also slowly evolves into a torus, which shatters into secondary droplets. These, in turn, transition into tori and again break apart in a cascade of events that repeats many times. However, the fluctuations in the velocities and positions of the rods accelerates the breakup process as compared to drops of liquids or spherical particles. The dispersion of swarms of particles occurs in natural processes of sedimentation of silt in rivers and on the continental shelf. The process has import for the delivery of injected particles, such as catalysts, drugs and inoculated cells, into a bulk medium. Extensive reviews of the studies regarding the deformation of drops have been conducted by Stone (1994) and Machu et al. (2001). The deformation of swarms of particles, which is analogous to that of liquid drops, has been focused on since Adachi et al. (1978) reported on this topic for the first time. The studies on particle drops, which are assumed to be spherical, have suggested that the breakup is due to inertia or an instability induced by the fluctuation of the particle velocities and positions (Nitsche and Batchelor, 1997; Schaflinger and Machu, 1999; Machu et al., 2001; Bosse et al., 2005; Metzger et al., 2007). At low Reynolds number, the fluctuation alone causes the breakup. In Appendix D, the deformation of a swarm of nonspherical particles, such as rodlike particles, is demonstrated by simulation for the first time. The simulation method is described in Section D.2. The time evolution of the deformation is illustrated in Section D.3. Ongoing work is summarized in Section D.4.

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D.2

Simulation Methods

Hydrodynamic interactions between rigid rods are included using the method adapted from one developed by Harlen et al. (1999) and Butler and Shaqfeh (2002). For far-field interactions of rigid rods in a single drop, a solution by Rotne and Prager (1969) and Yamakawa (1970) is used instead of one for periodic system (Hasimoto, 1959; Beenakker, 1986). The number of Gauss-Legendre quadrature for integrating the solution along the rod is set to 5. For short-range interactions, lubrication force and repulsive force have been considered as done by Butler and Shaqfeh (2002), with the adjusted parameters. One of the parameters used in the formulas for the lubrication force, A, is replaced with A2 /2 to ensure the positive definite mobility matrix for locally dense configurations during deformation. The parameters for repulsive force, related to its magnitude and range, are set to

ln 2A gLVp ρp 2π

and 400L−1 respectively. The lubrication force is applied to a pair of

rods within the closest distance between edges of 0.6d and the repulsive force is added within 0.2d to prevent particles from overlapping. An initial configuration is made by placing the center-of-mass of rigid rods in a spherical region with a radius of RC by setting the number of the rods NC and A. The center-of-mass and orientation of rods are randomized. The equation of motion of sedimenting rods is equivalent to that of Butler and Shaqfeh (2002), of which a more general expression is Eq. 2–28. It is solved at each instant in time and integrated over time in the same way as Butler and Shaqfeh (2002). Configurations at each time are sampled and some of them, which represent distinguishing features of the deformation, are illustrated in Figure D-1. D.3

Results

A drop of RC = 5.007L, which consists of NC = 750 rigid rods of A = 10 is simulated. Sequential images of the deformation and breakup of the drop appear in Figure D-1 with time increasing from A to E. The characteristic time is tc = RC /vC , where vC = theoretical velocity of a sedimenting cloud of spherical particles with gravity Fg .

121

NC Fg 5πµRC

is a

Figure D-1. Sequential images of a sedimenting cloud of rigid rods with RC = 5.007L, NC = 750 and A = 10 at A) tc = 23.6, B) tc = 119.5, C) tc = 191.4, D) tc = 263.3, and E) tc = 335.2. Images in the top rows are views normal to the direction of gravity; the row of images on the bottom are views from in the direction of gravity. The drop, with an initially spherical symmetry, emanates particles in a vertical tail due to the surrounding flows, which also generate a torsional vortex inside of the drop. The hydrodynamic stagnation flow at the head and hydrodynamic dispersion flatten the drop (Figure D-1A). The particle loss from the tail and the deformation depopulate the center of the drop so that it transforms into a toroidal shape (Figure D-1B). The particle leakage and the dispersion make the inner radius of the torus larger so that the hydrodynamic stagnation flow penetrates the center of torus. This torus is unstable (Figure D-1C) so that it shatters into secondary droplets (Figure D-1D), which themselves

122

transition into a torus and again break apart in a cascade of events that repeats many times (Figure D-1E). D.4

Ongoing Work

As ongoing work, simulations are being performed to investigate the effect of RC , NC , and A on the time to break up, tb , when an unstable torus disintegrates into smaller droplets. Tentative results show that

2RC ln 2A NC L

controls tb . Another investigation on the

average radial velocity of rods suggests that the fluctuations in the velocities and positions of the rods, in the absence of interfacial tension, accelerates the breakup process as compared to drops of liquids or spherical particles. This result also suggests that injecting rodlike particles reduces dispersion time as compared to spherical particles. Experiments being performed by collaborating research group have qualitatively observed the deformation procedure as shown by the simulation. Quantitative comparison of simulation and the experimental results will be performed soon.

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BIOGRAPHICAL SKETCH Joontaek Park was born in Seoul, Republic of Korea, 1973 to Jong-Oh Park and Won-Sun Paek. With the highest score on the entrance exam, he was admitted to Daeil Foreign Language High School from which he graduated at the top of his class in 1992. Joontaek attended Seoul National University in 1992. He received a Bachelor of Science in chemical technology in 1996. He continued his education in the Korea Advanced Institute of Science & Technology in 1996. He undertook research with Dr. Ho-Nam Chang in the area of biochemical engineering. He performed research on the production and separation of bacteriorhodopsin, a membrane protein applied to a light sensitive biochip, from Halobacteria. He received a Master of Science in chemical engineering in 1998. Joontaek joined SK Engineering & Construction Ltd. as a research engineer to gain industrial experiences as well as to fulfill military duty. He participated in process simulation, start-up, trouble shooting in ethanol distillation plants and the overall planning of refineries. In 2000, SKEC awarded him the “Super-Excellent” award for outstanding research. During that time, he also experimented with electric guitar playing and wrote music reviews for an on-line CD store. One of his compositions is titled “Phase Transition”. After completing his military duty, Joontaek returned to academia to pursue a PhD. He accepted an offer from University of Florida as a research assistant in 2004. He joined Dr. Jason E. Butler’s research group, which focuses on complex fluids. He performed research on dynamics of rigid rod suspensions with hydrodynamic interactions which can be applied to microfluidic devices. He won Ray W. Fahien Graduate Teaching Awards for the best teaching assistant in 2008. He received a PhD in chemical engineering in 2009. Joontaek is ready to step forward into a new world where he utilizes all his learning for his own research.

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