dynamics of flowing polymer solutions …dynamics of flowing polymer solutions under confinement by...

DYNAMICS OF FLOWING POLYMER SOLUTIONS UNDER CONFINEMENT

by

Hongbo Ma

A dissertation submitted in partial fulfillment of

the requirements for the degree of

Doctor of Philosophy

(Chemical Engineering)

at the

UNIVERSITY OF WISCONSIN–MADISON

2007

i

To my parents, Xuemin Ma and Xinyang Li,

and my sisters, Hongge Ma and Hongxia Ma,

For your love, your encouragement and your patience.

To my wife, Jue Guo,

For the love and the wonderful time together.

ii

ACKNOWLEDGMENTS

Thanks to my advisor, Michael D. Graham and

Juan J. de Pablo for their support, patience, and guidance.

Thanks to former group members Richard Jendrejack and

Yeng-Long Chen for their help and many discussions.

Thanks to current group members of the Graham Group –

Samartha Anekal, Juan Pablo Hernandez-Ortiz, Aslin Izmitli, Wei Li,

Mauricio Lopez, Pratik Pranay, Matthias Rink, Patrick Underhill, Li Xi, Yu Zhang.

iii

This work was supported through the NSF/NSEC program.

iv

TABLE OF CONTENTS

Page

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

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Molecular Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Chain Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 133.2 Brownian Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 163.3 Hydrodynamic Interactions . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 173.4 Excluded Volume Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 20

4 Migration Near Solid Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .224.2 Illustration of Migration Mechanism . . . . . . . . . . . . . . . .. . . . . . . . . 274.3 Kinetic Theory for a Dumbbell in Dilute Solution . . . . . . .. . . . . . . . . . . 294.4 Steady State Depletion Layer near a Single Wall . . . . . . . .. . . . . . . . . . . 374.5 Temporal and Spatial Evolution of the Depletion Layer ina Semi-Infinite Domain . 414.6 Plane Couette Flow and Plane Poiseuille Flow . . . . . . . . . .. . . . . . . . . . 494.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53

5 Brownian Dynamics Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 575.2 Point-Dipole Theory of Polymer Migration . . . . . . . . . . . .. . . . . . . . . 635.3 Polymer Model and Simulation Method . . . . . . . . . . . . . . . . .. . . . . . 645.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 70

5.4.1 Single Wall Migration in Simple Shear . . . . . . . . . . . . . .. . . . . 705.4.2 Slit Confinement: Shear Flow . . . . . . . . . . . . . . . . . . . . . .. . 73

v

Page

5.4.3 Highly Confined Polymer Chains . . . . . . . . . . . . . . . . . . . .. . 815.4.4 General Flux Expression for Dumbbells . . . . . . . . . . . . .. . . . . . 86

5.5 Effect of Finite Reynolds Number on Wall-induced Hydrodynamic Migration . . . 895.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90

6 Simulating Polymer Solution Using Lattice-Boltzmann Method . . . . . . . . . . . 92

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 926.2 Lattice-Boltzmann Method . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 94

6.2.1 Velocity Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.2.2 Equilibrium Velocity Distribution . . . . . . . . . . . . . . .. . . . . . . 966.2.3 Collision Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 976.2.4 External Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.2.5 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 100

6.3 Polymer Chain Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 1026.3.1 The Bead-spring chain Model . . . . . . . . . . . . . . . . . . . . . .. . 1026.3.2 Coupling of the Polymer Chain and the Solvent . . . . . . . .. . . . . . . 1046.3.3 Equation of Motion for Polymer Beads . . . . . . . . . . . . . . .. . . . 105

6.4 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 1066.5 Chain Migration in Dilute Polymer Solution Flow in a Slit. . . . . . . . . . . . . 1086.6 Complications of the Lattice-Boltzmann Method . . . . . . .. . . . . . . . . . . 113

6.6.1 Fluid Relaxation Time . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1166.6.2 Grid Size Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1186.6.3 Reynolds Number Effect . . . . . . . . . . . . . . . . . . . . . . . . . .. 122

6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .124

7 Polymer Chain Dynamics in a Grooved Channel. . . . . . . . . . . . . . . . . . . . 126

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 1267.2 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 1297.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 1307.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 132

7.4.1 Hydrodynamic Interactions . . . . . . . . . . . . . . . . . . . . . .. . . . 1327.4.2 Chain Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1347.4.3 Peclet Number Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 137

7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .140

8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

vi

LIST OF FIGURES

Figure Page

1.1 Optical Mapping method developed by David Schwartz’s group at the University ofWisconsin-Madison. YAC clone 5L5 derived from human chromosome 11, was di-gested with EagI and MluI, stained with the fluorochrome YOYO-1, and visualizedby fluorescence microscopy. Five fragments are generated from the 360-kb parentmolecule. Courtesy of the Laboratory for Molecular and Computational Genomics,University of Wisconsin-Madison. . . . . . . . . . . . . . . . . . . . . .. . . . . . 2

1.2 Schematic of the Direct Linear Analysis (DLA) method developed by US Genomics[26]. Shown in the figure is a cross-section of the microfluidic DNA stretching mi-crochip. Fluorescent tagged DNA molecules are stretched atthe entrance of a ta-pered channel due to the collision with the posts and the elongational flow. When thestretched DNA molecules travel through the narrow channel,the positions of the tagsites are read out by laser detectors. This method has a claimed resolution of±0.8kbresolution and throughput of 30-60 million bp/min. . . . . . . .. . . . . . . . . . . 3

2.1 Schematic of different regimes of confinement: (a) single wall confinement, (b) weakconfinement:2h ≫ Rg, (c) strong confinement:2h ∼ Rg, and (d) extreme confine-ment:2h ∼ Lp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Various coarse-graining levels of a polymer chain. (a) Atomistic model. At this level,all the atoms in a polymer molecule as well as the solvent molecules explicitly presentin the model. (b) Bead-rod model. Discretizing the polymer chain into segments andlumping up a fairly large amount of atoms within each segmentinto a bead whichis connected to each other by rigid rods lead to the so-called“bead-rod” model. (c)Bead-spring model. Following the same logic further, representing a group of beadsand rods by one larger bead and connecting those larger beadsby elastic springs givethe bead-spring model. This is the coarse-graining level wewill work on. . . . . . . . 8

3.1 Bead-spring chain model of a polymer moleclue. The springs account for the resis-tance to the stretch due to the entropic effect, and the beadsrepresent the interactionsites along the chain contour. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 14

vii

Figure Page

4.1 Illustration of the position vectors used for a point force above a plane wall. . . . . . . 28

4.2 Velocity field due to a point force in thex direction located at(x, y) = (−5a, 5a),wherea is the bead radius. The plane wall is aty = 0. The lines correspond tostreamlines, while the light and dark area indicate regionswhere the wall-normal ve-locity is positive (away from the wall) and negative (towards the wall), respectively.Also shown is a “bead” of radiusa located at(x, y) = (5a, 5a) - this can be thoughtof as the other end of a relaxing dumbbell oriented parallel to the wall. . . . . . . . . . 30

4.3 Steady state concentration profiles scaled by the bulk value in uniform shear flowabove a single wall at different Weissenberg numbers. The concentration profilesare calculated using a FENE-P dumbbell model with finite extensibility parameterb = 600 and hydrodynamic interaction parameterh∗ = 0.25. . . . . . . . . . . . . . . 40

4.4 Depletion layer thickness vs. Weissenberg number in a uniform shear flow above asingle wall for FENE-P dumbbell with finite extensibility parameterb = 600 andhydrodynamic interaction parameterh∗ = 0.25. The straight line is the high Weis-senberg number asymptote,Ld/Rg ∼ Wi2/3. . . . . . . . . . . . . . . . . . . . . . . . 42

4.5 Temporal development of the concentration profile in uniform shear flow above asingle wall atWi = 10. A FENE-P dumbbell model with finite extensibility parameterb = 600 and hydrodynamic interaction parameterh∗ = 0.25 is used. . . . . . . . . . . 44

4.6 Similarity solution for time evolution of the concentration profile in uniform shearflow above a single wall. The full numerical solutions including diffusion forWi =100 at two different times,t = 10λH andt = 1000λH, are also plotted for comparison.A FENE-P dumbbell with finite extensibilityb = 600 and hydrodynamic interactionparameterh∗ = 0.25 is used when solving for the numerical solutions. . . . . . . . . .46

4.7 Similarity solution for spatial development of the concentration profile in uniformshear flow above a single wall. The full numerical solutions including the diffusionfor Wi = 100 at two different downstream positions,x = 10(kB/H)1/2 andx =10000(kB/H)1/2, are also shown for comparison. A FENE-P dumbbell with finiteextensibility b = 600 and hydrodynamic interaction parameterh∗ = 0.25 is usedwhen solving for the numerical solutions. . . . . . . . . . . . . . . .. . . . . . . . . 48

4.8 Steady state concentration profiles atWi = 2, 10 and100 in plane Couette flow in aslit with width 2h = 30

√kBT/H. Length is scaled by

√kBT/H and concentration

by its value at the centerline of the slit,nc. Migration effects due to the two walls ofthe slit are superimposed by taking the “single-reflection”approximation. . . . . . . . 51

viii

Figure Page

4.9 Steady state concentration profiles atWi = 2, 10, 100 in lane Poiseuille flow in a slitwith width 2h = 30


√kBT/H and concentration by

its value at the centerline of the slit,nc. Migration effects due to the two walls of theslit are superimposed by taking the “single-reflection” approximation. . . . . . . . . . 52

4.10 Steady state concentration field for plane Poiseuille flow in the entrance region ofa slit with width 2h = 300

√kBT/H at Wi = 20. Only half of the slit is shown.

The concentration is scaled by its bulk valuen0 before entering the slit. Migrationcontributions due to two walls of the slit are superimposed by taking the “single-reflection” approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 54

5.1 Schematic of different regimes of confinement: (a) Single wall confinement, (b) weakconfinement:2h ≫ Rg, (c) strong confinement:2h ∼ Rg, and (d) extreme confine-ment:2h ∼ Lp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2 Time evolution of axially averaged fluorescence intensity of fluorescent labeled T2-DNA solution as a function of cross-sectional position. Thechannel walls are aty =±20µm. The solution is undergoing oscillatory pressure-drivenflow at a maximumstrain rate of 75s−1 and a frequency of 0.25Hz in a 40µm× 40µm microchannel [31].The bright band at the center indicates higher concentration of T2-DNA molecule andthe dark region represents the depletion layer near the channel walls. . . . . . . . . . . 61

5.3 Steady-state chain center-of-mass concentration profiles predicted by theory, using theStokeslet-doublet (far-field) approximation, and the BD simulation atWi = 0, 5, 10and 20 in simple shear flow. The concentration is normalized using its value aty/(kBT/H)1/2 = 90. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.4 Migration velocity scaled with the point-dipole value for different dumbbell (force-dipole) sizes, as a function of distance from the wall. . . . . .. . . . . . . . . . . . . 74

5.5 Near-field center-of-mass steady-state concentrationprofiles predicted by theory, us-ing the Stokeslet-doublet (far-field) approximation and finite-size dumbbells, and theBD simulation atWi = 5 in simple shear flow. . . . . . . . . . . . . . . . . . . . . . 75

5.6 Steady-state chain center-of-mass concentration profiles predicted by theory, usingthe Stokeslet-doublet (far-field) approximation, and the BD simulation of 10 springschains, atWi = 5 and10 in simple shear flow. . . . . . . . . . . . . . . . . . . . . . 76

ix

Figure Page

5.7 Steady-state chain center-of-mass concentration profiles predicted by theory, usingfar-field and single-reflection approximations, and the BD simulation atWi = 0, 5 and20 in shear flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.8 Steady-state chain center-of-mass concentration profiles predicted by the BD simula-tion at Wi = 20 in shear flow, for different polymer discretizations:Ns = 1, 5 and10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.9 Schematic of two different discretization levels of a same molecule (a) dumbbell: theeffect of the molecule on the solvent is approximated as two point forces with largeseparation; (b) chain: the effect of the molecule on the solvent is approximated asseveral point forces with smaller separation. . . . . . . . . . . .. . . . . . . . . . . . 80

5.10 Steady-state chain center-of-mass concentration profiles predicted by the theory, usingfar-field and single-reflection approximations, and the BD simulation atWi = 20 inshear flow. The steady-state chain center-of-mass concentration profile at equilibrium(Wi = 0) and the bead-distribution from the simulation atWi = 20 are also shown. . . 82

5.11 Steady-state chain center-of-mass concentration profiles predicted by the BD simula-tion of chains (Ns = 10) for a highly confined polymer solution,2h = 2.9Rg. . . . . . 83

5.12 Steady-state bead-concentration profiles predicted by the BD simulation of chains(Ns = 10) for a highly confined polymer solution,2h = 2.9Rg. . . . . . . . . . . . . 84

5.13 Polymer stretch as a function of the wall-normal direction, y, for Wi = 0 (no flow);2h = 2.9Rg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.14 Polymer stretch in the flow direction,x, as a function of the wall normal direction,y;2h = 2.9Rg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.15 Polymer stretch in the confined direction,y, as a function of the wall normal direction,y; 2h = 2.9Rg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.16 Schematic of the hydrodynamic migration mechanism (a)Rey ≪ 1: wall-inducedmigration – momentum diffusion to the wall and back to the particle is fast; (b)Rey ≫1: No wall-induced migration – the shear flow distorts the velocity perturbation dueto the particle so that the particle is not affected by the presence of the wall. . . . . . 91

x

Figure Page

6.1 The set of discrete velocities in a D3Q19 model shown in a lattice cube. The solidparallelogram represents thexy plane, the dashed rectangle theyz plane, and thedotted parallelogram thexz plane. The D3Q19 model consists of a zero velocityrepresented by the cube center, six velocities with magnitude unity represented by thearrows pointing to the centers of the cube faces, and12 velocities with magnitude

√2

represented by the arrows pointing to the cube-edge centers. . . . . . . . . . . . . . . 95

6.2 In the single-time-relaxation model, the velocity distribution at each site relaxes to-ward the equilibrium one at each time step. Without the external force, the equilib-rium velocity distribution consists simply equal amount offluid particles for each ofthe discretized velocities. The figure shows the two processes that occur during eachtime step: the streaming and the relaxation. First, the incoming velocity distributionassembles at a lattice site as the particles in the neighboring sites stream along theirdirections of motion to that site. Second, the incoming distribution relaxes due tothe particle collisions, according to the single-time-relaxation rule, towards the equi-librium distribution. (a) Whenτs = 1, the incoming velocity distribution relaxes tothe equilibrium distribution in one time step. (b) Whenτs = 2, the post-relaxationdistribution is halfway between the incoming and the equilibrium distributions. . . . . 99

6.3 Bounce-back rule for a solid-fluid interface. The arrowsshows the velocity directionand their lengths are proportional to the magnitude of the velocity distribution in thatdirection. (a) Bounce-back rule for a stationary solid boundary. (b) Bounce-back rulefor a moving solid boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 103

6.4 Relaxation of a stretched polymer molecule in bulk solution. The mean square stretchof the chain< X2 > is plotted against time for a chain ofNs = 10 at two dif-ferent temperatureskBT = 0.001, and 0.0002. An exponential decay fitting of< X(t)2 >=< X(∞)2 > +X0 exp(t/λ) gives the chain relaxation time asλ = 426for kBT = 0.001 andλ = 2037 for kBT = 0.0002, in lattice units.X0 andλ are thefitting parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 109

6.5 Mean square displacement of the center-of-mass of a polymer chain withNs = 10 asa function of time in bulk solution. The simulation parameters areµ = 0.2, ζ = 0.6,andkBT = 0.001. A linear fitting to the diffusion equation< [(r(t)− r(0)]2 >= 6Dtgives the chain diffusion coefficient asD = 1.73 × 10−4, in lattice units. . . . . . . . 110

6.6 Steady state chain center-of-mass distribution of a dilute polymer solution undergoingsimple shear flow confined in a slit at Weissenberg number of 0,10, 100, and 200.The center-of-mass distributions are normalized such thatthe area under the curvesare all unity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 112

xi

Figure Page

6.7 Steady state chain center-of-mass distribution of a dilute polymer solution undergo-ing shear flow confined in a slit. The solid line is the equilibrium chain center-of-massdistribution, the dotted line is the chain center-of-mass distribution obtained from sim-ulations with free draining model (FD) atWi = 50, and the dashed line is the chaincenter-of-mass distribution obtained from simulations with hydrodynamic interactions(HI) at Wi = 50. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.8 Steady state chain center-of-mass distribution of a dilute polymer solution undergoingshear flow confined confined in a slit atWi = 50. The solid line is the chain center-of-mass distribution obtained from Lattice-Boltzmann Method, and the dashed line fromBrownian Dynamics simulation with hydrodynamic interactions. . . . . . . . . . . . . 115

6.9 Viscous flow of a fluid near a wall suddenly sheared. At timet = 0, the bottom solidsurface is set in motion in the positivex direction with velocityv0 . . . . . . . . . . . 117

6.10 Velocity profile in dimensionless form for flow near a wall suddenly sheared. (a)Results from Lattice-Boltzmann Method withτs= 1.1. (b) Results from Lattice-Boltzmann Method withτs= 10.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.11 Contour plot of the wall normal component of the steady state flow field due to astretched dumbbell (white beads connected by dotted line) confined in a slit. (a) Finiteelement solution. (b) Result from Lattice-Boltzmann Method with τs = 1.1. (c) Resultfrom Lattice-Boltzmann Method withτ = 3.5. . . . . . . . . . . . . . . . . . . . . . 120

6.12 Comparison of the wall normal component of the steady state flow field due to astretched dumbbell confined in a slit. (a) Slice of the flow field along wall-normaldirection atx = 20. (b) Slice of the flow field along the wall-tangential direction aty = 5. The dotted lines in (b) indicates the positions of the two beads of the stretcheddumbbell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.13 Steady state chain center-of-mass distribution of a dilute polymer solution undergoingshear flow confined in a slit atWi=10. Line styles correspond to grid resolution of∆x = 1.0µm (dashed),∆x = 0.50µm (solid), and∆x = 0.25µm (dotted). . . . . . . 123

6.14 Steady state chain center-of-mass distribution of a dilute polymer solution undergoingshear flow confined in a slit atWi = 10. Line styles correspond to Reynolds numbersof Re= 10 (dotted),Re= 2 (dashed),Re= 0.4 (solid), andRe= 0.04 (dash-dotted). . . 125

7.1 Schematic of a grooved channel. Shown in the figure is thexy plane cross-section.The simulation domain is periodic inx andz directions. . . . . . . . . . . . . . . . . 128

xii

Figure Page

7.2 Stream lines corresponding to the flow field generated by shearing the upper wall ofthe grooved channel in positivex direction. The contour variable is the velocity inxdirection. Note that the magnitude of the velocity inside the groove is much smallerthan outside. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .130

7.3 Steady state chain center-of-mass distribution in a flowing polymer solution confinedin a grooved channel at effective Weissenberg number of (a)Wi = 0, (b) Wi = 5,and (c)Wi = 10. Note the strong depletion downstream of upstream horizontal wall,which is clearly related to the steric depletion layer near the walls. . . . . . . . . . . 131

7.4 Slice of the two dimensional steady state chain center-of-mass distribution in flowingpolymer solution confined in a grooved channel. The slice is taken alongy directionat x = 20, which is the center of the channel inx direction. The vertical dotted lineindicates the position of the groove top edge. . . . . . . . . . . . .. . . . . . . . . . 133

7.5 Steady state chain center-of-mass distribution in a dilute polymer solution confined ina grooved channel. The dash-dotted line is the distributionobtained from free draining(FD) simulation, and the solid line is the result from simulation with hydrodynamicinteractions (HI). Both simulations are performed withWi = 10. . . . . . . . . . . . . 135

7.6 Steady state center-of-mass distribution of isolated beads in shear flow in a groovedchannel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.7 Snapshots of polymer chains in flowing solution confined in a grooved channel attime t = 711∆t, 740∆t, 756∆t, and766∆t, chronologically from top to bottom. Thearrows point to the polymer chain that approaches the corner. . . . . . . . . . . . . . . 138

7.8 Schematic of a chain crossing the boundary layer near theseparatrix at the top edgeof the groove. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .139

xiii

ABSTRACT

This thesis focuses on the dynamics and transport of flowing polymer solutions near surfaces or

confined to small geometries. Combining the theoretical analysis and simulation approaches, we

explore the dynamics of dilute polymer solutions under three types of confinements: single-wall

confinement, slit and grooved channel.

Starting from the single-wall confinement, we develop a kinetic theory based on a dumbbell

model of the dissolved polymer chains. It is shown that hydrodynamic interactions between the

chains and the wall lead to migration away from the wall in shear flow. The depletion layer thick-

ness is determined by the normal stresses that develop in flowand can be much larger than the

size of the polymer molecule. Numerical and similarity solutions show that the developing con-

centration profile generally displays a maximum at an intermediate distance from the wall. Using

single-reflection approximation, the kinetic theory for single-wall confinement is extended to slit

geometry.

Our Brownian Dynamics (BD) simulations results confirm thatthe kinetic theory captures the

correct far-field (relative to the walls) behavior. Once a finite-size dipole is used, the theory im-

proves its near-wall predictions. In the regime2h ∼ L > Rg, the results are significantly affected

by the level of discretization of the polymer chain, becausethe spatial distribution of the forces

exerted by the chain on the fluid acts on the scale of the channel geometry.

Finally, We consider the chain center-of-mass distribution in a dilute linear polymer solution

during flow in a channel with grooves running perpendicular to the flow direction. A simulation

method which couples a bead-spring chain model of the polymer molecule to a Lattice-Boltzmann

xiv

fluid is implemented. We observe that in flow, polymer chains leave the groove, leading to lower

concentration there than in the bulk. Furthermore, a band ofincreased concentration formes near

the wall containing the grooves. The degree of depletion of chains from the groove increases

significantly with increasing Weissenberg number. Our results show that the chain connectivity

and the complex flow field are the primary reasons for these observations.

1

Chapter 1

Introduction

The dynamics and transport of polymer solutions near surfaces or confined to small geometries

is a long-standing research topic with many applications. For example, in polymer enhanced-oil-

recovery, polymer solutions flood through rock layers to improve volumetric sweep efficiency and

reduce channeling, leading to more oil produced in less time. In various chemical and biological

analysis, like gel-permeation and gel-electrophoresis, polymer mixtures flow through porous media

(some of which are made of polymers themselves), and the mixture is separated based on the

mobility difference of the species.

Aside from those traditional applications, the recent emergence of microfluidic devices [6, 36,

90, 116, 129, 67, 146, 34] in micron and nanometer scale for single molecule manipulation and

analysis of DNA have fueled considerable interest in the structure and dynamics of confined DNA

solutions [35, 11, 75, 72, 74, 133, 152, 153, 139, 144, 105]. Two particularly relevant examples are

Optical Mapping and Direct Linear Reading. Shown in Figure 1.1 is the Optical Mapping device

developed in our collaborator David Schwartz’s lab (University of Wisconsin-Madison, Genetics

Department) [116]. Combining the confinement, shear flow andelectric field, they are able to

stretch the DNA molecules and deposit them onto a solid surface. The immobilized DNA are

digested using restriction enzyme, and by visualizing the cleavage location along the DNA chain,

the DNA map is constructed. In another DNA mapping method, the Direct Linear Reading shown

in Figure 1.2 developed by US Genomics [26], fluorescent dyedDNA molecules are stretched due

to the collision with the posts and the elongational flow at the entrance region of the channel, and

then pass the detector, where the gene position is read out.

2

Figure 1.1 Optical Mapping method developed by David Schwartz’s group at the University ofWisconsin-Madison. YAC clone 5L5 derived from human chromosome 11, was digested with EagIand MluI, stained with the fluorochrome YOYO-1, and visualized by fluorescence microscopy.Five fragments are generated from the 360-kb parent molecule. Courtesy of the Laboratory forMolecular and Computational Genomics, University of Wisconsin-Madison.

3

Figure 1.2 Schematic of the Direct Linear Analysis (DLA) method developed by US Genomics[26]. Shown in the figure is a cross-section of the microfluidic DNA stretching microchip. Fluores-cent tagged DNA molecules are stretched at the entrance of a tapered channel due to the collisionwith the posts and the elongational flow. When the stretched DNA molecules travel through thenarrow channel, the positions of the tag sites are read out bylaser detectors. This method has aclaimed resolution of±0.8kb resolution and throughput of 30-60 million bp/min.

4

The basic physics behind all the above processes or devices is the interactions between poly-

mer and confinement. However, despite the scientific importance and various applications, our

understanding of such interactions is still very limited. Predictive methods capable of describing

the conformation and motion of polymer chains in confined geometry are desired for the concep-

tion and design of novel processes and devices. In this thesis, we present some contributions we

have made in developing kinetic theory and simulation techniques for confined flowing polymer

solutions. We put efforts on the coupling between polymer, solvent, and the confinement with an

emphasis on the hydrodynamics interactions.

This dissertation is organized as follows. In Chapter 2, Theproblem in which we are interested

is formulated by the conservation equations which govern the flow of a dilute solution of linear

polymer. Connectivity, solvent effects, and hydrodynamicinteractions are introduced in Chapter

3. Following that, a kinetic theory model is developed to explain the shear-induced migration in

flowing polymer solution near a solid wall in Chapter 4. The migration mechanism is elaborated

based on the hydrodynamic interactions. This model is also generalized to flowing polymer solu-

tions in a slit. In Chapter 5, we introduce Brownian Dynamicssimulation method with fluctuating

hydrodynamic interactions for simulating the confined polymer solution flow. The results are com-

pared with the theoretical predictions in Chapter 4. The assumptions in the theoretical model are

evaluated. In Chapter 6, a Lattice-Boltzmann based method capable of simulating the polymer

solution flow in complex geometries and/or with high concentration is implemented. The Lattice-

Boltzmann Method is utilized to investigate the chain center-of-mass distribution in polymer so-

lutions flowing through a smooth slit, where the strength andcomplications of Lattice-Boltzmann

Method are discussed. Finally, in Chapter 7, dynamics and transport of polymer solutions in a

grooved channel is investigated using Lattice-Boltzmann Method. The effects of chain connec-

tivity, hydrodynamics interactions and Peclet number on the chain dynamics and center-of-mass

distribution are discussed.

Chapter 4 through Chapter 7 correspond to different publications. Those sections are therefore

self-contained, and some repetition should be expected.

5

Chapter 2

Problem Statement

The system we address is confined flow of a complex fluid consisting of dilute solutions of

monodisperse linear polymer in an incompressible Newtonian solvent. Depending on the ratio

of the characteristic length of the confinement and the characteristic length scale of the polymer

molecule, there are several primary regimes of confinement.Consider a flexible polymer chain

at equilibrium in solution, confined between two infinite walls separated by a distance2h. When

the slit width is much larger than the equilibrium polymer radius of gyrationRg, the chain adopt

its unperturbed isotropic coil conformation at equilibrium. We call this the weak confinement

regime; it is illustrated in Figs. 2.1(a) and (b). We note that during flow another length scale, the

contour lengthL of the molecule, can become comparable to the degree of confinement. When the

slit width is reduced to about the unperturbed chain dimension of Rg, the free arrangement of the

polymer chain is restricted by the walls and deviations fromthe bulk equilibrium coil conformation

are expected. This regime is called strong or high confinement, and is shown in Fig. 2.1(c). If the

slit width is reduced further to the order of the chain persistence lengthLp, then the chain dynamics

is extremely restricted [145, 123], as shown in Figure 2.1(d). In this thesis, we focus on the weakly

and highly confined regimes.

The challenge of developing modeling tools for polymer solution, or complex fluid in general,

lies in the presence of a wide range of length and time scales in the system. For example, a simple

process involving a dilute polymer solution contains time and length scales of the solvent, polymer,

fluid deformation, and of course, the process. Even the polymer molecules themselves contain a

huge number of degrees of freedom.

6

(b) 2h >> Rg

Rg

2h

(c) 2h ~ Rg

Rg

2h

Lp

(d) 2h ~ Lp

2h

Rg

(a) Single wall oo

Figure 2.1 Schematic of different regimes of confinement: (a) single wall confinement, (b) weakconfinement:2h ≫ Rg, (c) strong confinement:2h ∼ Rg, and (d) extreme confinement:2h ∼ Lp.

7

Insights into the modeling of complex fluids are given by the fact that although there are many

processes happening in very different scales simultaneously, usually we are only interested in a

simple set of such processes. For example, in the Optical Mapping, we are concerned with the

dynamics of the DNA chain, not the individual covalent bond.Therefore, coarse-grained models

are widely used to investigate the dynamics of complex fluids. The coarse-graining enables us

to reduce the complexity of the problem, and work on a level which has sufficient details and at

the same time, tractable. In the kinetic theory of macromolecules [15], the solvent is treated as

continuum medium and the polymer molecules are coarse-grained into simple mechanical models,

beads connected by rods or springs. The continuum medium affects the dynamics of the chain

through thermal fluctuations which causes the Brownian motion of the chain, and the chain in turn

acts on the solvent through the microscopic contribution tothe stress tensor.

There are different levels of coarse-graining of the polymer chain. At atomistic level, all the

atoms on the polymer molecules as well as the solvent molecules explicitly present in the model.

While straightforward and representing the system faithfully, this level of modeling is computa-

tional demanding and accessible only for very small time andlength scale, typically picosecond

and nanometer. Discretizing the polymer chain into segments and lumping up a fairly large amount

of atoms within each segment into a bead which is connected toeach other by rigid rods lead to

the so-called “bead-rod” model. Although losing some molecular details, the bead-rod model can

simulate the chain dynamics on a much larger time and length scale. An obvious drawback of

the bead-rod model is that the rigid constraints are computationally challenging in many circum-

stances. To reach time and length scale of seconds andµm, bead-rod model is still too expensive.

Following the same logic, representing a group of beads and rods by one larger bead and connect-

ing those larger beads by elastic springs give the bead-spring chain model. Carefully calibrated

bead-spring chain model greatly enlarges the accessible time and length scales of computer simu-

lation of polymer solutions. This is the coarse-graining level we will work on. Figure 2.2 illustrates

the various levels of coarse-graining of a polymer chain.

In our work, the polymer chain is discretized as a sequence ofNb beads connected byNs =

Nb − 1 springs as shown in Figure 2.2(c). When polymer chain is stretched, the entropy will

8

Figure 2.2 Various coarse-graining levels of a polymer chain. (a) Atomistic model. At this level,all the atoms in a polymer molecule as well as the solvent molecules explicitly present in the model.(b) Bead-rod model. Discretizing the polymer chain into segments and lumping up a fairly largeamount of atoms within each segment into a bead which is connected to each other by rigid rodslead to the so-called “bead-rod” model. (c) Bead-spring model. Following the same logic further,representing a group of beads and rods by one larger bead and connecting those larger beads byelastic springs give the bead-spring model. This is the coarse-graining level we will work on.

9

be reduced; the springs model the resistance to the stretch due to the entropic effect. The beads

represent the interaction sites along the chain where, for example, the drag force, spring force and

excluded volume force are exerted on. The contour length of the molecule is given byL = Nkbk,

wherebk is the Kuhn length characterizing the stiffness of the chainandNk is the number of Kuhn

segments in the molecule. A vectorr with length of3Nb contains the Cartesian coordinates of the

Nb beads, withri denoting the position vector of theith bead.

The core of the kinetic theory is the configurational distribution functionΨ(t, r), which gives

a description of the probability of the polymer chain takinga given configurationr at a given time

t. The vectorr contains the3Nb Cartesian coordinates ofNb beads. From this distribution, one

can obtain all the structural information about the polymerchain and also the interplay between

the polymer chain and the solvent. The configurational distribution function is governed by a

“diffusion equation,”

∂Ψ

∂t= −∇r · rΨ, (2.1)

r = v +1

kBTD · F −D · ∇r ln Ψ, (2.2)

wherekB is Boltzmann’s constant,T is the absolute temperature, and∇r ≡ ∂/∂r is the gradient in

configuration space. Thev(r, t) denotes the unperturbed external imposed velocity field at each of

theNb beads, which is the solution to the incompressible Navier-Stokes equations in the absence

of the polymer molecules. The flow perturbation caused by themotion of the beads,v′(r, t), enters

the diffusion equation through the product of the3Nb × 3Nb diffusion tensorD. The vectorF

contains the3Nb components of the total non-hydrodynamic, non-Brownian forces acting on the

beads which include the spring force, the excluded volume force and the chain-wall interactions.

Note that the diffusion equation describes how the system point diffuses in the multidimensional

configuration space. It doesn’t refer to the diffusion motion of the polymer chain in the physical

space.

10

Rather than usingr, one may represent the position and configuration of the polymer molecule

by the 3 coordinates of the center-of-mass,rc, and the3Ns = 3(Nb − 1) coordinates of the con-

nector vectors,q, given as

rc =1

Nb

Nb∑

i=1

ri, (2.3)

qi = ri+1 − ri. (2.4)

Expressed in terms ofrc andq, the configurational probability distribution function,Ψ(t, rc,q), is

a function of time, position of the center of mass of the molecule, and the internal configuration of

the molecules. Then the diffusion equation becomes

∂Ψ

∂t= −∇rc · rcΨ − ∇q · qΨ. (2.5)

The number concentration of the polymer molecule is defined by the integration of the probability

distribution function over the internal degrees of freedomof the moleculesq

n(rc, t) =

∫Ψ(t, rc,q)dq, (2.6)

Ψ(t, rc,q) = n(rc, t) ˆΨ(t, rc,q). (2.7)

Accordingly, the governing equation ofn(r, t) is obtained by integrating Equation 2.5 overq

∂n

∂t= −∇rc · 〈rc〉n = −∇rc · jc, (2.8)

wherejc = 〈rc〉n is the center-of-mass flux, and the angle brackets designatean ensemble average

over the configuration variableq,

〈A〉 =

∫AΨdq. (2.9)

In the kinetic theory, the effect of the motion of a polymer bead on the solvent fluid is treated

on the point force level. Polymer beads and the fluid are coupled together through a friction

coefficient. This is justified by the fact that the length scale of the polymer molecule is much

smaller than other relevant length scales in the process. The overall fluid stress,τ (r, t), is the

summation of polymer contribution,τ p(r, t), and the solvent contribution,τ s(r, t):

τ (r, t) = τp(r, t) + τ

s(r, t). (2.10)

11

The polymer contribution to the stress tensor is given as following [15]:

τp(r, t) = n

Nb∑

i=1

∫

r

Ψ(r, t)[(ri − rc)ri]dr + n(Nb − 1)kBT I. (2.11)

A significant portion of the kinetic theory is devoted to solving the diffusion equation in various

conditions. In general, there is no exact analytical solution for the distribution function. For the

confined polymer solution system we are interested in, this is particularly true. Furthermore, the

confined solvent hydrodynamics can have significant influence on the dynamics of the polymer

chain. The solvent flow field is governed by the Navier-Stokesequation and the continuity equation

with proper no-slip boundary conditions:

ρ∂v

∂t+ v · ∇v = −∇ · [pI + τ

p] + η∇2v, (2.12)

∇ · v = 0, (2.13)

whereη is the solvent viscosity, andρ is the density of the fluid.

Simultaneously solving the equations 2.1, 2.11, 2.12 and 2.13 with proper boundary conditions

corresponding to the confinement yields the complete evolution of the fluid flow and dynamics of

the polymer chain in cases where the length scale of the polymer is much smaller than the smallest

relevant length scale in the process.

In this work, we will consider the solution of the diffusion equation for three general cases:

• Weak confinement where the characteristic length scale of the confinement is much larger

than the polymer molecule sizeRg. Polymer molecules lose the feeling of all other walls

except the nearby one. This regime can also be characterizedas single-wall confinement, and

is discussed in Chapter 4 through analytical analysis and inChapter 5 by Brownian dynamics

simulations.

• Strong confinement where the characteristic length scale ofthe confinement is comparable to

the polymer molecule sizeRg. This regime is treated in Chapter 5 using Brownian dynamics

simulations.

12

• Grooved channel where one of the slit walls is patterned to control the chain distribution.

This topic is addressed in Chapter 7.

In each of these cases we explore the transport and dynamics of dilute polymer solutions,

through the ensemble averages ofΨ. One is generally concerned with ensemble averaged prop-

erties as a function of the chain position in physical space.Particularly, we will examine the

center-of-mass distribution as a function of the position in the confinement and the mechanism for

the concentration gradient in the system.

13

Chapter 3

Molecular Interactions

The dynamics of polymer solutions in a process is determinedby intramolecular and inter-

molecular interactions, interactions with solvent and confinement, and possibly other external

fields. In this chapter, we discuss molecular connectivity (spring forces), Brownian force, ex-

cluded volume force, and the form of the hydrodynamic interaction tensor in unbounded domain

for a bead-spring model of the polymer chain as shown in Figure 3.1. Hydrodynamic interactions

in confined geometry will be discussed in Chapter 4.

3.1 Chain Connectivity

Consider a linear flexible polymer molecule with contour length L = Nkbk in a theta solvent,

whereNk is the number of Kuhn segments andbk is the Kuhn length. Kuhn length is defined

as the the distance along the chain contour over which the chain orientation becomes statistically

uncorrelated. At equilibrium, the polymer chain takes the random coil configuration. In the limit

of Nk → ∞, the probability distribution function for the end-to-enddistance of the molecule is a

Gaussian with variance2b2kNk/3 [15]. Because of the huge amount of internal degrees of freedom

of the polymer chain, the resistance to deform from the equilibrium random coil configuration

when the chain is subjected to external force is dominated bythe entropic effect (i.e., bending,

rotational, or torsional resistance is negligible). For small deformations, the effective potential

between two ends of the chain can be shown to be a Hookean spring potential with spring constant

H = 3kBT/Nkb2k [15]. Extending this idea to our spring representation of the polymer segments,

14

Figure 3.1 Bead-spring chain model of a polymer moleclue. The springs account for the resistanceto the stretch due to the entropic effect, and the beads represent the interaction sites along the chaincontour.

15

we obtain an expression of the tension in theith spring:

Fsi =

3kBT

Nk,sb2k

qi, (3.1)

whereNk,s = Nk/Ns is the number of Kuhn segments per spring.

Hookean spring is simple and widely used. However, the Gaussian approximation is valid only

in the limit of infinite Nk and small stretchq. This approximation results in an obvious flaw: the

Hookean spring allows the chain to be infinitely stretched, which is not the case in a real polymer.

A real polymer chain is finite-stretchable. Various modifications have been made to take into

account the finite extensibility. Treloar [148] derived theinverse Langevin model by considering

the probability for the end-to-end distance of a freely jointed Kramer’s chain,

Fsi =

kBT

bkL−1

(qi

q0

)qi

qi, (3.2)

whereqi ≡ |qi|, q0 = Nk,sbk is the contour length of the spring, and

L(x) = coth x − 1

x, (3.3)

is known as the Langevin function. The involvement of the non-linear Langevin function is incon-

venient in numerical simulations. A popular alternative spring law to the inverse Langevin model

is the empirical FENE model [66],

Fsi =

3kBT

Nk,sb2k

qi

1 − (qi/q0)2. (3.4)

Both the inverse Langevin and FENE models account for the finite extensibility, and linearize to

the Hookean model in the limit of small stretch. However, thesingularity in the inverse Langevin

model can not be expressed as a polynomial while the FENE model has a single singularity of

1 − (qi/q0)2. As a result, FENE is much better suited for use in numerical simulations.

For semi-flexible DNA molecules, Marko and Siggia [96] derived a better spring model called

Worm-Like-Spring (WLS) model, based on the Porod-Kratky worm-like chain [127]. Unlike the

freely-jointed model, the worm-like chain model is a freelyrotating model in which the bending

angels are restricted to very small values. Fitting the experimental data [25] to match the asymp-

totics of the worm-like chain in both the small and large force limits, the resulting spring force law

16

is given by Marko and Siggia as

Fsi =

kBT

bk

[1

2(1 − qi/q0)2− 1

2+

qi

q0

]qi

qi

. (3.5)

We again note that the WLS model linearizes to the Hookean model in the limit of small extension,

with the singularity at large extension given by(1 − qi/q0)2.

3.2 Brownian Force

In solution, the long polymer chain immerses in an ocean of the tiny solvent molecules. Be-

cause of the thermal fluctuation, the solvent molecules constantly bump into the polymer chain.

The motion of polymer chain also affects the solvent in return. This interplay results in three

forces: the Brownian force, the hydrodynamic interactions, and the excluded volume force.

A direct consequence of the collision between the solvent molecules and the polymer chain

is the Brownian motion of the latter [43]. Because of the highly irregular and rapid nature of the

collision, the true force associated with the Brownian motion would be a rapid fluctuating function.

In kinetic theory, a statistically averaged force is used instead. Assuming the equilibration in

momentum space, the Brownian force takes a much simpler form,

Fbi = −kBT

∂ ln Ψ

∂ri. (3.6)

In stochastic simulation, the Brownian force is modeled by arandom variable with zero mean.

The magnitude of the Brownian force (or fluctuation) is related to the hydrodynamic friction (or

dissipation) according to the fluctuation-dissipation theorem:

〈Fα(t)〉 = 0 (3.7)

〈Fα(t)Fβ(t′)〉 = δ(t − t′)2δαβkBTζ, (3.8)

whereα, β represent the direction of the force,t, t′ mean different time, andζ is the bead friction

coefficient. The implementation of the fluctuation-dissipation in Brownian dynamics simulation

will be discussed in Chapter 5

17

3.3 Hydrodynamic Interactions

In solution, the motion of a segment of the polymer chain perturbs the externally imposed flow

field in the fluid, and thus influences the dynamics of the entire chain. This hydrodynamic coupling

between chain segments through the solvent is called hydrodynamic interaction. In kinetic theory,

the velocity perturbation due to the polymer molecule,v′, is generally taken to be due to a chain of

point forces acting on the fluid, and obtained by solving the incompressible Stokes flow problem,

η∇2v′ = ∇p −Nb∑

i

Fi δ(r − ri), (3.9)

∇ · v′ = 0, (3.10)

subject to appropriate boundary conditions. The velocity perturbation at the position of beadi due

to the beadj is represented in a Green’s function form as

v′i =

∑

i

Ωij · Fj , (3.11)

whereΩ is called hydrodynamic interaction tensor. The diffusion tensorD appearing in the diffu-

sion equation 5.6 is related toΩ as,

Dij = kBT

(1

ζIδij + Ωij

), (3.12)

whereζ = 6πηa is the bead friction coefficient. In the simplest case,Ωij is set to zero. Physically

this means that all the beads move independently without hydrodynamic coupling (free draining

model), resulting in the diffusivity of

D =kBT

6πηNba, (3.13)

wherea is the hydrodynamic radius of an individual bead. However, the free draining (FD) model

is in contrast to the experimental observations that the polymer coil diffuses through the fluid as if

it were actually a single large solid Brownian particle withdiffusivity given as,

D =kBT

6πηRH

, (3.14)

18

whereRH is the effective hydrodynamic radius of the chain, which is proportional to the size of the

polymer coil. As the size of the polymer scales withN0.588b in a good solvent, we can immediately

see that the free-draining model incorrectly predicts the diffusivity to scale withN−1b .

Far-field velocity perturbation due to a point force is givenby solving Equation 3.9 and Equa-

tion 3.10 in a infinite domain,

ΩOBii = 0 (3.15)

ΩOBij =

1

8πηrij

[I +

rijrij

r2ij

]i 6= j. (3.16)

This kind of hydrodynamic interaction form is called the Oseen-Burgers tensor. The diffusion ten-

sor obtained from using Oseen-Burgers hydrodynamics is notguaranteed to be positive-definite

for all chain configurations: when the bead separation is decreased, the diffusion tensor stemming

from the Oseen-Burgers tensor can have negative eigenvalues. The negative eigenvalues lead to

negative energy dissipation, which is clearly unphysical.The non-positive-definition of the diffu-

sion tensor in near-field is caused by the point-force assumption in the Oseen-Burgers treatment.

This assumption was eliminated to first order by Rotne and Prager [126] and Yamakawa [154].

They developed an expression for the hydrodynamic interaction tensor by considering the rate of

energy dissipation by the motion of the surrounding fluid. The Rotne-Prager-Yamakawa (RPY)

tensor regularizes the singularity ofrij = 0 in the Oseen-Burgers tensor and has the form

ΩRPYii = 0, (3.17)

ΩRPYij =

1

8πηrij

[(1 +

2a2

3r2ij

)I +

(1 − 2a2

r2ij

)rijrij

r2ij

]i 6= j andrij ≥ 2a, (3.18)

ΩRPYij =

1

6πηa

[(1 − 9rij

32a

)I +

3

32a

rijrij

r2ij

]i 6= j andrij < 2a. (3.19)

At large bead separationrij → ∞, the RPY tensor approaches the Oseen-Burgers tensor; while

at small bead separation, the correction ofrij < a takes hydrodynamic overlap of the beads into

account. The Oseen-Bugers tensor is relatively simple, andthus very useful in deriving tractable

analytical kinetic theory. On the other hand, the RPY tensorgreatly simplifies the computation and

is widely used in Brownian dynamics simulations.

19

Oseen-Burgers tensor and RPY tensor enable us to incorporate hydrodynamic interactions into

polymer kinetic theory in an infinite domain, where the characteristic length scale of the process or

confinement is much larger than the polymer dimension and thus irrelevant. They can be marked

as “free-space” hydrodynamic interactions. In cases wherethe characteristic length scale of the

process is not large enough and thus relevant, as in most microfluidic applications, one has to

take into account the no-slip boundary conditions at the confining surfaces when solving Equation

3.9 and Equation 3.10. This leads to modifications to the “free space” hydrodynamic interaction

tensors. In Chapter 4, we will introduce the modification to free-space hydrodynamic interaction

tensor due to a single-wall confinement. The modification dueto a slit geometry will be discussed

in Chapter 5

The Green’s function representation of the hydrodynamic interactions assumes that the beads

are coupled instantaneously. In other words, the characteristic time for the hydrodynamic interac-

tions to propagate over the distance of the characteristic length scale in the system should be much

smaller than the time scale of other signals. In free space, the relevant length scale is the radius of

gyration of the chainRg. The velocity perturbation caused by chain segment travelsthrough the

solvent by moment diffusion. The chain configuration relax on the scale of the relaxation time,

which is roughly speaking on the same order as the chain diffusion time over its own size. Thus,

the “instantaneity” means that the kinematic viscosity (orthe moment diffusivity),ν = η/ρ, is

much larger than the chain diffusivityD. In terms of Schmidt number (Sc), this means

Sc =ν

D≫ 1. (3.20)

Consider aλ-phage DNA in buffer solution with1.0cp viscosity [137, 136]. The chain diffusivity

is about0.5µm2/s, and the kinematic viscosity of water is around106µm2/s, thus the Schmidt

number of the system is on the order of106. Therefore, Equaqtion 3.20 is fulfilled for a dilute

free-space DNA solution, which justifies the approximationof the instantaneous hydrodynamic

coupling.

However, for a confined polymer solution, the characteristic length scale is the dimension of

the confinement. The instantaneous hydrodynamic coupling between the polymer chain and the

confining wall raises another requirement: the speed of the momentum diffusion over distance

20

between the polymer chain and the confining walls must be muchlarger than the chain convection.

In other words, the Reynolds number must be small. The effectof Reynolds number will be

discussed in Chapter 5 and Chapter 6.

3.4 Excluded Volume Effect

The choice of solvent can have a large impact on the configurational and rheological properties

of dilute polymer solutions. The quality of a solvent is measured by exclude volume effect. The

excluded volume effect is the result of the competition between the polymer-solvent interaction

and the polymer-polymer interaction. Solvents are typically grouped into three broad categories -

good solvents, theta solvents, and poor solvents - based on the energetic favorability of these two

interactions.

• In a good solvent, polymer-polymer contact leads to more free energy penalty than the

polymer-solvent contact. Thus, the polymer chain will be surrounded by solvent molecules

and swell. This class of solvents is modeled by a repulsive bead-bead potential.

• In a theta solvent, polymer-polymer interaction and polymer-solvent interaction are energet-

ically indistinguishable. The polymer chain behaves as a “phantom chain,” in which, at large

length scales, chain segments can penetrate each other. Theta solvents require no action in

kinetic theory since there is no effective force between chain segments.

• In a poor solvent, polymer-polymer interaction is energetically more favorable. The polymer

chain will take a globule configuration. Poor solvents are realized by attractive bead-bead

potentials.

In this work, we deal only with good solvents and theta solvents. Since no action is required

to accommodate theta solvents, we focus on the good solvent models. One common form of the

repulsive potential is the Lennard-Jones potential [88].

ULJij = 4ǫ

[(σ

rij

)12

−(

σ

rij

)6]

. (3.21)

21

Often, the attractive term is neglected, or the potential isshifted and truncated to give only repulsive

interactions.

In simulatingλ-phage DNA, Jendrejack et. al [71] derived an exponential form of the re-

pulsive excluded volume potential by considering the energy penalty due to the overlap of two

submolecules modeled as Gaussian coil,

Uij =1

2υkBTN2

k,s

(3

4πS2s

) 3

2

exp

[−

3r2ij

4S2s

], (3.22)

whereυ is the excluded volume parameter andS2s = Nk,sb

2k/6 is the mean square radius of gy-

ration of an ideal chain consisting ofNk,s Kuhn segments of lengthbk. The resulting expression

describing the force acting on beadi due to the the presence of beadj is then

Fυij = υkBTN2

k,sπ

(3

4πS2s

) 5

2

exp

[−

3r2ij

4S2s

rij

]. (3.23)

The choice of the excluded volume potential is arbitrary as long as the expected molecular

weight scaling of properties for good solvent conditions can be reproduced by fitting the parameters

in the model. The advantage of Jendrejack’s excluded volumemodel is that the dependence of the

the potential on the molecular discretization level (represented byNk,s) is known and explicit. In

our simulation, we adopt this form of the excluded volume potential.

22

Chapter 4

Migration Near Solid Surfaces

Experiments directly or indirectly indicate that in shear flow, flexible polymer molecules in

solution migrate away from solid boundaries, leading to theformation of depletion layers and ap-

parent slip at the boundaries [2]. These phenomena have obvious implications for adsorption and

desorption of macromolecules at solid surfaces, as molecules that tend to migrate away from walls

are unlikely to adsorb on them. Motivated by these considerations, the focus of this chapter is the

development of an analytical theory of dilute polymer solutions flowing near solid surfaces. With

this theory we derive a closed form expression for the steadystate depletion layer thickness. The

transient development of this depletion layer in uniform plane shear flow and the spatial develop-

ment of the depletion layer downstream of the entrance to a channel are described. Furthermore,

we extend this kinetic theory to slit geometry by using the single-reflection approximation. The

final result is a general framework for understanding the migration phenomena in dilute polymer

solutions.

4.1 Background

Molecular migration in flowing dilute polymeric solutions is a well-known phenomenon that

has received a significant amount of experimental and theoretical investigations. Much of this is

reviewed by Agarwalet al. [2], so we focus here on a few particularly relevant studies.A recent

experimental study was performed by Hornet al. [69], in which apparent slip in a “Boger fluid”

(a dilute solution of a high molecular weight polymer in a highly viscous solvent) was inferred

from measurements in a surface forces apparatus. The slip length Ls (the distance beyond the

23

solid surface at which the velocity extrapolates to zero) was estimated to be 3-5 times the radius

of gyration of the polymer (here polyisobutene). In a simplemodel of depletion layers, where the

layer consists of pure solvent with a sharp change to the bulkpolymer concentration at a distance

Ld from the wall, the relationship between slip lengthLs and depletion layer thicknessLd is simply

Ls = Ld(1 − β)/β, (4.1)

whereβ is the ratio of solvent viscosity to solution viscosity. Therefore, for a dilute solution, where

1 − β ≪ 1, the depletion layer thickness is expected to be significantly larger than the slip length.

As an example of a more classical study, Cohen and Metzner [37] performed careful capillary

flow experiments with nondilute polymer solutions, finding depletion layer thicknesses (using the

formula above) up to 8 times the polymer radius of gyration (other studies have found even larger

values [2]). Additionally, these authors observed a directcorrelation between the depletion layer

thickness and the degree of elasticity of the polymer solution. Finally, Fanget al. [47] have

recently reported direct observations, using fluorescencemicroscopy, of large DNA molecules in

shear flow near a solid surface. Their results clearly indicate the presence of a depletion layer,

whose thickness increases with increasing shear rate and can be more than 10 times the radius of

gyration of the molecule.

A number of researchers have performed computational and theoretical studies of flowing poly-

mer solutions near boundaries. In a nonhomogeneous flow, thedeformation and alignment of the

polymer molecules are position dependent. Garner and Nissan [57] proposed that the correspond-

ing spatial variation in free energy could drive cross-streamline migration. Later, Marrucci [97]

related the entropy change with the stress level for an Oldroyd-B liquid and Metzner [103] em-

ployed this result to analyze polymer retention in flows through porous media. Tirrell and Malone

[147] have made similar arguments. However, Aubertet al. [7] pointed out that it is not clear that

a spatial gradient in intramolecular free energy can resultin displacement of the center of mass.

Indeed, no such effect is found in first-principles kinetic theory developments for dilute solutions.

Phenomenological two-fluid models have also been widely used to study migration and concen-

tration fluctuations in polymer solutions at finite concentration [62, 42, 111, 106, 99, 107, 17]. In

these models, a contribution to the polymer mass flux proportional to∇ · τ p is found, whereτ p is

24

the polymer contribution to the stress tensor. Turning to the molecular kinetic theory for (infinitely)

dilute solutions, Aubert and Tirrell [9] modeled the polymer as a flexible dumbbell in a viscous

solvent and pointed out an effect in a nonhomogeneous flow field where the macromolecules lag

behind the solvent motion along the streamline. In some kinetic theory developments, a contribu-

tion to the polymer flux corresponding to the divergence of the stress is found [14, 112, 13, 38],

which is similar to the result from the two-fluid models [99].However, the above arguments only

lead to migration in a nonhomogeneous flow field. Furthermore, Curtiss and Bird [39] pointed out

that in the dilute solution kinetic theory results containing the divergence of the stress, the sum of

the mass fluxes over all species is not zero, violating mass conservation and thus indicating a flaw

in those developments.

In another approach to explaining the existence of depletion layers near confining surfaces, a

number of researchers have amended theories by incorporating boundary effects, specifically the

fact that polymer segments cannot pass through a solid wall.A typical method is treating the

wall effect on the polymer molecules as a short-range purelyrepulsive potential [10]. A refined

version of this wall exclusion effect is provided by Mavrantzas and Beris [100, 101, 102] and Woo

et al. [152, 153] where the change of the polymer chain statistics due to the wall is explicitly

considered. However, including this effect, the depletionlayer thickness is still only on the order

of the polymer molecule size, and would be insensitive to theflow strength, in contrast to the

experimental observations.

A significant limitation of all the aforecited dilute solution studies is the neglect of intramolec-

ular hydrodynamic interactions and the effect of the walls on the hydrodynamics of the solvent. If

hydrodynamic interactions (HI) between polymer segments are ignored entirely, then no migration

is found in shear flows without streamline curvature (plane shear flows, capillary flow . . . , etc.).

If HI are included, butnot their modifications due to presence of a wall, then migrationtoward

regions of higher shear rate is found; this is opposite to thetrend observed experimentally [74].

For example, Sekhonet al. [131] considered bulk hydrodynamic interactions in rectilinear slit flow

using kinetic theory for a bead-spring dumbbell model, and concluded that cross-stream migration

is possible with HI, and Brunn [21, 22] and Brunn and Chi [23] predicted migrationtowardsthe

25

walls using Oseen-Burgers free space hydrodynamic interactions for a bead-spring chain model.

To our knowledge, only two studies in the polymer literatureaside from our own (discussed below)

have addressed the effect of hydrodynamic interactions in wall-bounded flows of dilute polymer

solutions. Jhon and Freed [76] incorporated a highly approximate representation of the near-wall

hydrodynamics into a kinetic theory analysis for bead-spring polymer chains containing further

approximations, predicting (correctly) migration away from the wall in simple plane shear flow.

This prediction, however, is the result of cancelation of errors – the approximation to the hydro-

dynamics that they used would actually lead to a prediction of migration toward the wall if the

kinetic theory were done exactly. The other result that we are aware of is a direct simulation: Fan

et al. [46] used dissipative particle dynamics (DPD) [45, 59, 124]to study the behavior of flexible

polymers in rectilinear flow through microchannels, and predicted very weak migrationtowardthe

walls – a minimum at the centerline of the concentration distribution, in contrast to experimental

observations. However, in contrast to the experiments, both the particle and channel Reynolds

numbers in these simulations were much larger than 1. As discussed below, the hydrodynamic

interaction with the wall is the main driving force for migration, and if the Reynolds number is

not small that effect will be absent – a polymer molecule moves a significant distance down the

channel in the time it takes for hydrodynamic fluctuations topropagate to the channel walls, so

hydrodynamically, the polymer does not see the wall, and thus does not migrate.

Although motion of suspended droplets is not the focus of thepresent work, it is relevant to

note that their migration in flow has also received a fair amount of attention; the older literature

in this area is reviewed by Leal [86]. Starting with a rigid particle in a Newtonian fluid at zero

Reynolds number, Chan and Leal [27] perturbatively examined the effects of inertia, droplet de-

formability, and non-Newtonian fluid character. For a slightly deformable drop in a Newtonian

fluid in zero Reynolds number uniform shear flow near a wall, the wall modification to the hydro-

dynamic interaction is the sole contribution to droplet migration. Chan and Leal, using far-field

wall hydrodynamic interaction found that drift is always away from the wall (in agreement with

experiment). For pressure-driven flow, where the shear rateis nonuniform, they found that the first

order contribution to migration is due to the interaction with the gradient of the local shear rate,

26

provided that the shear rate changes significantly over the length scale of the droplet. They found

that the direction of migration dependes on the ratio of solvent and droplet viscosities; for the range

of viscosity ratios used in their experiments, migration was always toward regions of lower local

shear rate. For circular Couette flow, they found that the final position of the droplet is determined

by a competition between a streamline curvature effect and wall hydrodynamic interaction. An

important observation was made by Smart and Leighton [135],who pointed out that a droplet far

from a wall can be treated to leading order as a symmetric force dipole (stresslet) and that the

wall-induced migration effect is due to the flow induced by the image of the stresslet on the other

side of the wall. This result generalizes to any particle or macromolecule in flow above a wall and

plays an important role in the results described below.

The discussion of droplet dynamics makes clear the necessity to correctly account for hydrody-

namic effects in studying the motion of flexible particles ormacromolecules near solid boundaries.

In prior work, we have developed a coarse-grained (bead-spring chain) model of long (> 100

persistence length) double-stranded DNA, incorporating hydrodynamic interactions. The model

provides an accurate representation of experimental data (structural and dynamic) for DNA in bulk

solution [73, 71], and has been extended to capture the dynamics of DNA solutions in microchan-

nels, including hydrodynamic effects [75, 72, 74]. Relaxation and diffusion of chains in a channel

of square cross section [72, 75] follow the predictions of a simple scaling theory, due to Brochard

and de Gennes [19], that is based on the screening of segment-segment hydrodynamic interac-

tions by the confining walls. Furthermore, the simulation results for diffusion in a slit channel (i.e.

between parallel infinite walls) agree very well with experiments [32]. More interestingly, the sim-

ulations predict that during pressure-driven flow in a channel, the molecules will tend to migrate

toward the centerline, forming depletion layers that are much larger than the radius of gyration of

the molecules [72, 74, 31]. The goal of the present work is to complement those detailed sim-

ulations with theoretical results that provide a more fundamental understanding of the migration

phenomenon.

27

4.2 Illustration of Migration Mechanism

To illustrate the basic mechanism of hydrodynamic migration of a dissolved polymer molecule

in a confined geometry, we begin by considering a bead-springdumbbell model of the polymer

above a single wall. Hydrodynamically, each moving bead is treated as a point force acting on the

fluid; ignoring for the moment the Brownian forces, the hydrodynamic forces introduced by the

two beads must be equal and opposite, balancing the extension of the spring. The flow due to the

motion of each bead (i.e., the solution to Stokes’ equation)is available in simple analytical form

[119], thus allowing a complete description of the flow. In this section we will illustrate this flow;

below we will build it into an analytical theory allowing prediction of the dynamics of depletion

layer formation in flow.

Assuming the wall is aty = 0, let r0 = (x0, y0, z0) be the position of one bead, and denote the

distance vectors

r = r − r0, (4.2)

R = r − rIm0 , (4.3)

whererIm0 = (x0,−y0, z0) is the mirror image ofr0 with respect to the wall. The force exerted on

the fluid due to the motion of this bead isF. These vectors are shown in Fig. 4.1. The perturbation

flow at any other positionr(x, y, z) caused by the motion of the bead can be obtained by solving

the Stokes’ equation:

0 = −∇p + η∇2v + δ(r − r0)F (4.4)

subject to no-slip boundary condition at the wall:

v(x, y = 0, z; r0) = 0, (4.5)

whereη is the solvent viscosity,v is velocity, andp is pressure. The solution has the following

form:

v = Ω · F, (4.6)

Ω(r, r0) =1

8πη

[S(r) − S(R) + 2y2

0PD(R) − 2y0S

D(R)], (4.7)

28

wall

r

R

r0

rIm0

r

F

Figure 4.1 Illustration of the position vectors used for a point force above a plane wall.

29

whereS is the free-space Stokeslet,PD is the potential dipole andSD is the Stokeslet doublet [18].

These are given respectively as

Sij(r) =δij

r+

xixj

r3, (4.8)

P Dij (r) = ± ∂

∂xj

(xi

|r|3)

= ±(

δij

|r|3 − 3xixj

|r|5)

, (4.9)

SDij (r) = ±∂Si2

∂xj= x2P

Dij (r) ± δj2xi − δi2xj

|r|3 , (4.10)

with the minus sign forj = 2 corresponding to they direction, and the plus sign forj = 1, 3

corresponding to thex andz directions [119].

Using this solution, we calculate the velocity field caused by a point force parallel to the wall,

which corresponds to one end of a relaxing dumbbell parallelto the wall. The flow field is shown in

Fig. 4.2. It can be seen that the flow induced by one bead of the relaxing dumbbell will be upward

at the position of the other bead, and vice versa. In other words, each bead will be convected away

from the wall by the velocity perturbation caused by its partner. As a whole, the center of mass of

the dumbbell migrates away from the wall. In contrast, a relaxing dumbbell perpendicular to the

wall will move toward the wall. A simple explanation of this result is that the mobility of the bead

closer to the wall is lower than that of its partner [74]. In shear flow, dumbbells are more likely to

be oriented parallel to the wall. Thus, migration away from the wall is dominant.

4.3 Kinetic Theory for a Dumbbell in Dilute Solution

The simple analysis in Sec. 4.2 predicts that a macromolecule near a wall will migrate due to

hydrodynamic interaction with the wall, providing a starting point to explore many interesting phe-

nomena. In this section, we will incorporate bead-wall hydrodynamic interactions in the polymer

kinetic theory for a bead-spring dumbbell in solution to investigate the formation of the depletion

layer in a flowing polymer solution near a solid wall.

Let r1 andr2 denote the position vectors of the two beads of a dumbbell. Then the position of

the center of mass isrc = (r1 + r2)/2, and the connector vector isq = r2 − r1. The quantities

rc andq give the rate of change of the center of mass and the connectorvector. The conservation

30

Figure 4.2 Velocity field due to a point force in thex direction located at(x, y) = (−5a, 5a),wherea is the bead radius. The plane wall is aty = 0. The lines correspond to streamlines, whilethe light and dark area indicate regions where the wall-normal velocity is positive (away from thewall) and negative (towards the wall), respectively. Also shown is a “bead” of radiusa located at(x, y) = (5a, 5a) - this can be thought of as the other end of a relaxing dumbbelloriented parallelto the wall.

31

equation for the probability distribution density function Ψ(rc,q, t) is [15]

∂Ψ

∂t= − ∂

∂rc· (rcΨ) − ∂

∂q· (qΨ) . (4.11)

Integrating the above equation overq and defining

Ψ(rc,q, t) = n(rc, t)Ψ(rc,q, t), (4.12)

n(rc, t) =

∫Ψ(rc,q, t)dq, (4.13)

gives the governing equation for the center of mass probability distribution (“concentration”),

n(rc, t),∂n

∂t= − ∂

∂rc

· jc, (4.14)

wherejc = 〈rc〉n is the center of mass flux integrated over the internal degrees of freedom of the

molecule, and the angle brackets designate an ensemble average over the configuration variableq,

〈A〉 =

∫AΨdq. (4.15)

The fluxesrcΨ andqΨ in Eq. (4.11) are determined by a balance between the spring forceFsi ,

hydrodynamic forceFhi , wall repulsion forceFw

i and Brownian forceFbi exerted on each bead:

Fhi + Fs

i + Fwi + Fb

i = 0 i = 1, 2. (4.16)

Assuming equilibrium in momentum space, the Brownian forceis given by [15]

Fbi = −kBT

∂

∂riln Ψ, (4.17)

wherekB is the Boltzmann constant andT is temperature. The hydrodynamic forceFhi is propor-

tional to the velocity difference between the beadi and fluid, as given by Stokes’ law. The actual

form of the spring force here is arbitrary. In other words, the analysis given here applies to any

spring law.

In our previous simulation work [74], which accounts for thewall exclusion forceFwi , we

found that in flow, this effect is generally small relative tothe hydrodynamic effect. In particular,

although the static exclusion force acts over a range of about the polymer radius of gyrationRg,

32

hydrodynamic effect on the chains in flow leads to depletion over length scales much larger than

Rg. Below, we further elucidate this phenomenon. This simulation result is consistent with many

experiments [47] that directly or indirectly indicate the existence of depletion layers with thick-

nesses much larger than the polymer molecule size, a result that cannot be accounted for by simple

wall exclusion arguments. Therefore, in the following analysis we setFwi = 0.

Using Eq. (4.16), the velocity of the center of mass,rc, is given by

rc =1

2

[2∑

i=1

v(ri) +1

kBT

2∑

i=1

2∑

j=1

Dij ·(Fs

j + Fbj

)]

. (4.18)

In this equation,

Dij = kBT

(1

6πηaIδij + Ωij

), (4.19)

a is the bead radius,v(ri) is the unperturbed flow velocity at the position of beadi, I is unit tensor,

δij is the Kronecker delta, andΩij denotes the hydrodynamic interaction tensor:

Ωij = Ω(ri, rj) −δij

8πηS(ri − rj), (4.20)

with Ω(ri, rj) given in Eq. (4.7). The rate of change of the connector vector, q, is given by

q = r2 − r1

= [v(r2) − v(r1)] +

2∑

j=1

(Ω2j − Ω1j) ·(Fs

j + Fbj

).

(4.21)

Defining the spring forceFs = Fs1 = −Fs

2 and using Eq. (4.17) along with

∂

∂r1

=1

2

∂

∂rc

− ∂

∂q, (4.22)

∂

∂r2=

1

2

∂

∂rc+

∂

∂q, (4.23)

the velocity of the center of mass of the dumbbell can be expressed as

rc = v +1

8qq : ∇∇v +

1

2Ω · Fs +

1

2D · ∂

∂qln Ψ − Dk ·

∂

∂rc

ln Ψ. (4.24)

33

where

Ω = (Ω11 −Ω22) + (Ω21 −Ω12) , (4.25)

D = kBT Ω, (4.26)

DK =1

4[(D11 + D22) + (D21 + D12)] . (4.27)

Herev is the unperturbed fluid velocity at the center of mass of the dumbbell,rc, and we have

Taylor-expandedv(r1) andv(r2) aroundrc and kept the terms up to second order. This accounts

for the difference, in a nonhomogeneous flow field, between the translational velocity of the center

of mass of the dumbbell and the unperturbed fluid velocity at the position of the center of mass.

The quantityDK is the so-called Kirkwood diffusivity for a dumbbell [15]. Multiplying Eq. (4.24)

by Ψ, integrating over the internal coordinateq and using incompressibility, one can arrive at the

mass flux expression:

jc =nv

+n

8〈qq〉 : ∇∇v

+1

2

⟨Ω ·

(Fs + kBT

∂

∂qln Ψ

)⟩n

−⟨

DK · ∂ ln Ψ

∂rc

⟩n

− 〈DK〉 · ∂n

∂rc

.

(4.28)

This expression is valid for an arbitrary flow geometry. A general discussion for the case of a

dumbbell near a single wall is given by Jendrejacket al. [74]. Here we simplify this expression

by considering the case where the extension of the dumbbell|q| is small compared to its distance

from the wally; i.e., we focus on the far field effects of the wall.

First, we define a reflection operatorT,

T = δ − 2eyey =

1 0 0

0 −1 0

0 0 1

. (4.29)

34

Then, the image positions of the two beads and the center of mass with respect to the wall are

rIm1 = T · r1, (4.30)

rIm2 = T · r2, (4.31)

rImc = T · rc. (4.32)

We also define a series of vectors:

Rc = rc − rImc = rc − T · rc, (4.33)

R11 = r1 − rIm1 = Rc −

1

2(q − T · q) , (4.34)

R22 = r2 − rIm2 = Rc +

1

2(q − T · q) , (4.35)

R12 = r1 − rIm2 = Rc −

1

2(q + T · q) , (4.36)

R21 = r2 − rIm1 = Rc +

1

2(q + T · q) , (4.37)

rαβ = rα − rβ. (4.38)

Using this notation and Eq. (4.20),Ω can be rewritten as following for flow above a single wall:

Ω =1

8πη

[−S(R11) + 2y2

1PD(R11) − 2y1S

D(R11)]

−[−S(R22) + 2y2

2PD(R22) − 2y2S

D(R22)]

+[S(r21) − S(R21) + 2y2

1PD(R21) − 2y1S

D(R21)]

−[S(r12) − S(R12) + 2y2

2PD(R12) − 2y2S

D(R12)]

.

(4.39)

For |q| ≪ |Rc|, we can Taylor expandΩ aroundRc. Keeping only leading terms yields:

Ω =1

8πη

−2 [T · q] · ∇S + 4y2 [T · q] · ∇PD

− 4y [T · q] · ∇SD − 8yqyPD(Rc) − 4qyS

D(Rc)

+ · · · .

(4.40)

35

The gradient terms are readily calculated from Eqs. (4.8), (4.9), and (4.10), allowing Eq. (4.40)

to be simplified further,

Ω =3

32πη

1

y2

−qy −qx 0

qx −2qy qz

0 −qz −qy

+ . . . . (4.41)

This can be rewritten compactly to leading order as

Ω =3

32πηy2M · q, (4.42)

whereM is a third order tensor with the following components:

M222 = −2, (4.43)

M211 = M233 = 1, (4.44)

M121 = M112 = M323 = M332 = −1, (4.45)

Mijk = 0 i, j, k = others. (4.46)

We will denote the tensorM = 3M/64πηy2 as themigration tensor. Finally we point out here

that this tensor can be defined for any geometry, given the point force solution for Stokes’ equation

in that geometry.

Similar toΩ, the leading orderDK is given by:

DK =kBT

12πηa

[I +

3a

4S(q)

]. (4.47)

Recalling that the polymer contribution to the stress tensor τ p [15] is:

τp = n 〈qFs〉 − nkBT I, (4.48)

and using Eqs. (4.42) and (4.47), we can simplify Eq. (4.28) at leading order to the following:

jc =nv +n

8〈qq〉 : ∇∇v + M : τ

p

− kBT

12πηa

⟨(I +

3a

4S(q)

)· ∂ ln Ψ

∂rc

⟩n

− kBT

12πηa

⟨I +

3a

4S(q)

⟩· ∂n

∂rc

.

(4.49)

36

Now we define

〈DK,b〉 =kBT

12πηa

⟨I +

3a

4S(q)

⟩. (4.50)

This is the bulk, ensemble averaged (but conformation dependent) Kirkwood diffusivity. Finally,

we use this to rewrite Eq. (4.49):

jc =nv +n

8〈qq〉 : ∇∇v + M : τ

p

− n∂

∂rc· 〈DK,b〉 − 〈DK,b〉 ·

∂n

∂rc.

(4.51)

The last term in this expression is the normal Fickian diffusion; the other terms lead to migra-

tion. Consider first the term containing the migration tensor and the stress tensor. Each dumbbell

induces a force dipole flow in the surrounding solvent - the stress tensor is the ensemble average

of this dipole. In the presence of a wall, the image of this force dipole induces a fluid velocity

M : τ p/n at the position of the dumbbell; migration arises from the convection of the dumbbell

due to this flow [135]. Note that the termM : τ p is generic for the flux ofanyflexible suspended

particle or molecule in a wall-bounded flow – in particular its validity is not restricted to the dumb-

bell model. This term is missing in previous theories of polymer migration. The term containing

the divergence of〈DK,b〉 can also lead to migration, but only in a flow where the conformation

distribution is spatially nonuniform (as in a pressure-driven flow) and only if the diffusivity of the

molecule depends on conformation. As mentioned in the Introduction, several previous studies

on the shear-induced migration in polymers focused on this term, but neglected the hydrodynamic

effect of the walls [54, 23, 132, 152]. In a pressure-driven flow, this term leads to a weak driving

force toward the wall, but except at the centerline of the channel where the hydrodynamic migra-

tion term vanishes by symmetry, our previous simulations show that this effect is small [74]. In

nonhomogeneous flow, the term containing∇∇v predicts the lag of a macromolecule behind the

solvent along the streamline [9] but no cross-streamline migration, and unless the nonhomogeneity

is so large that it cannot be ignored even on the length scale of the polymer molecule, this term is

small. Finally, the contribution to polymer flux proportional to∇ · τ p predicted by several models

37

[62, 42, 111, 106, 99, 107, 17, 14, 112, 13, 38] does not arise in the single-molecule limit analyzed

here.

4.4 Steady State Depletion Layer near a Single Wall

Now consider an initially homogeneous infinitely dilute polymer solution under uniform shear

flow in thex direction with constant shear rateγ above an infinite plane wall aty = 0. Due to

symmetry, no concentration variations will arise in thex andz directions; for they direction, using

Eq. (4.14) and Eq. (4.51), we have that

∂n

∂t= −∂jc,y

∂y= − ∂

∂y

(K(y)

y2n − D

∂n

∂y

), (4.52)

where

K =3

64πηn

[M : τ

p]

y=

3

64πηn(N1 − N2), (4.53)

D =kBT

12πηa. (4.54)

HereN1 andN2 are the first and second normal stress differences, defined by

N1 = τpxx − τp

yy, N2 = τpyy − τp

zz. (4.55)

In addition, for simplicity and because it is a good approximation for highly stretched dumbbells,

we have replaced the conformation dependent diffusivity byits free draining valueD. Note that in

Eq. (4.52) the migration velocity in the wall normal direction is given by

vmig =K

y2=

3

64πηny2(N1 − N2). (4.56)

This result is identical to that derived by Smart and Leighton for a suspended droplet [135].

If we make the further assumption thatK is independent of the position, again a good as-

sumption for dilute solution in uniform shear, then Eq. (4.52) can be solved for the steady-state

concentration profile:

n = nb exp

(−Ld

y

), (4.57)

38

wherenb is bulk concentration andLd is the depletion layer thickness, defined by

Ld =K

D=

9

16

N1 − N2

nkBTa. (4.58)

This quantity characterizes the length scale of the steady state depletion layer in a semi-infinite

domain. Note that for a long flexible molecule,(N1 − N2)/nkBT can be much greater than unity.

Since for a dumbbell model, the hydrodynamic bead radiusa is proportional to the molecular size,

this result shows that depletion layers much thicker than the molecular size should be expected to

arise in flows of dilute polymer solutions.

Eq. (4.58) for the depletion layer thickness applies to any force law chosen for bead-spring

model, since the spring force has been automatically built into the polymer contribution to the

stress tensorτ p. Ideally, the evolution equation of the stress tensor wouldarise from the theory

presented in Section 4.3, but the presence of the hydrodynamic interactions precludes development

of a closed form equation [15]. Therefore, to proceed with the analysis, we will use the FENE-P

dumbbell model, which is simple, theoretically well-understood and widely used in simulations

[66]. The FENE-P spring force is given as follows:

Fs =Hq

1 − 〈q/q0〉2, (4.59)

whereq = |q|, H is the spring constant, andq0 is the maximum extended length of the dumbbell.

The stress relaxation time isλH = ζ/4H, whereζ = 6πηa is the bead friction coefficient. For

Hq20/kBT ≫ 1, the radius of gyration is given byRg =

√kBT/2H.

Introducing the length unit√

kBT/H and time unitλH = ζ/4H, dimensionless quantities can

be defined:

q = q/√

kBT/H, t = t/λH ,

v = vλH/√

kBT/H), b = Hq20/kBT.

(4.60)

For the FENE-P model, the dimensionless stress tensor then is:

τ p

nkBT=

〈qq〉1 − 〈q2〉 /b

− I. (4.61)

39

The evolution equation for the structure tensorα(t) = 〈qq〉 is [66]:

dα

dt= ∇vT · α + α · ∇v − α

1 − tr(α)/b+ I. (4.62)

In addition to the Peterlin closure for the spring law, thereare two approximations involved

in using this equation for the stress. The first is the neglectof hydrodynamic interactions, either

between the beads or between the dumbbell and the wall. The second is the neglect of transport

of conformation due to diffusion and migration. The first approximation is necessary because it is

impossible to get a closed form evolution equation for the structure tensor if full hydrodynamic in-

teractions are included. The effect of this approximation is primarily to ignore a weak dependence

of relaxation time with distance from the wall. We would liketo point out that this approximation

is invoked only when evaluating the numerical value of the depletion layer thickness where the

stress value is needed. So the main physics (e.g., the migration mechanism, the expression for the

center of mass flux, and the expression for depletion layer thickness) is free from this approxima-

tion. The effect of the second approximation will be negligible unless the velocity gradient varies

on the scale of the molecular size, as migration and diffusion occur on a time scale much larger

than the relaxation time.

Having specified the polymer model, we now return to the expression for the depletion layer

thickness, Eq. (4.58). By introducing the hydrodynamic interaction parameterh∗ [15],

h∗ =ζ

ηs

√H

36π3kBT, (4.63)

the depletion layer thickness can be expressed as:

Ld =9√

π

128

N1 − N2

nkBTh∗Rg. (4.64)

Now we define the Weissenberg numberWi = λγ. The polymer contribution to the stress at

different Weissenberg numbers can be calculated using FENE-P model Eq. (4.62). Figure 4.3

shows the steady state concentration (probability) profilefor different Weissenberg numbers when

b = 600 andh∗ = 0.25 (These parameters will be used throughout this paper). The vertical axis is

the concentration scaled by the bulk concentrationnb and the horizontal axis is the distance from

40

y/(kBT/H)1/2

n/n b

0 1000 2000 30000

0.2

0.4

0.6

0.8

1

Wi= 20Wi = 40Wi = 60Wi = 80Wi = 100

Figure 4.3 Steady state concentration profiles scaled by thebulk value in uniform shear flow abovea single wall at different Weissenberg numbers. The concentration profiles are calculated using aFENE-P dumbbell model with finite extensibility parameterb = 600 and hydrodynamic interactionparameterh∗ = 0.25.

41

the wall scaled by√

kBT/H. Note that the depletion layer extends a very large distancefrom the

wall, much larger than polymer molecule size.

Finally, for a FENE-P dumbbell in shear flow,(N1 − N2)/nkBT scales asWi2/3 at highWi

[66] (This scaling also holds for the FENE dumbbell model without the closure approximation.).

Therefore, the depletion layer thickness scales as

Ld ∼ Wi2/3Rg. (4.65)

Figure 4.4 showsLd vs. Wi on a log-log scale forb = 600 andh∗ = 0.25. The two-thirds power

law at highWi is evident. The result thatLd ≫√

kBT/H for Wi ≫ 1 justifies our neglect of the

wall exclusion in the model.

4.5 Temporal and Spatial Evolution of the Depletion Layer ina Semi-InfiniteDomain

The above results show that at steady state, the hydrodynamic effect of a polymer with a wall

leads to concentration variations on scales that can be orders of magnitude larger than the size of the

polymer. We now turn to the temporal and spatial developmentof the depletion layer, beginning

with the transient evolution of the concentration field in fluid above an infinite plane wall. At

time t = 0, the fluid begins to undergo uniform shear with shear rateγ. The transient process is

governed (under the same approximations as used above) by

∂n

∂t= −∂jc,y

∂y= − ∂

∂y

(K

y2n

)+ D

∂2n

∂y2, (4.66)

initial condition : n(y, 0) = nb,

boundary conditions:

jc,y(0, t) = 0,

n → nb as y → ∞.

The time evolution of the concentration profile has been obtained by numerically solving this

equation coupled with the stress evolution equation (Eq. (4.62)) using the FENE-P model, which

determinesK(t). Figure 4.5 shows the concentration profile at different times forWi = 10, b =

42

Figure 4.4 Depletion layer thickness vs. Weissenberg number in a uniform shear flow above asingle wall for FENE-P dumbbell with finite extensibility parameterb = 600 and hydrodynamicinteraction parameterh∗ = 0.25. The straight line is the high Weissenberg number asymptote,Ld/Rg ∼ Wi2/3.

43

600, h∗ = 0.25. This figure shows clearly that immediately after inceptionof the shear flow, a peak

appears in concentration. This is simple to understand in terms of the dependence of migration rate

on the distance from the wall. The migration rate (Eq. 4.56) is larger near the wall than far from

it. So the polymer molecules will “pile up.” This effect is sodominant at short times that diffusion

cannot smooth out the spike. At long times, however, as shownin the figure, the spike is smoothed

out. The calculation of Hudson [70] on the wall migration of fluid droplets in emulsions illustrated

similar results.

The time scale involved in this process is remarkably large.Even after104 relaxation times,

the steady state is still not reached. A simple estimate of the time required to reach steady state is

given by the timetmig it takes for a molecule to migrate fromy = 0 to y = Ld:

tmig ∼∫ Ld

0

dy

vmig∼ L2

d

D. (4.67)

By this estimate, the migration time scale is on the same order as that of the diffusion time over

distanceLd. For the computation shown in Fig. 4.5,L2d/D ≈ 7 · 103; the results show that the

simple estimate dramatically underpredicts the actual time required to approach steady state. This

discrepancy arises because, as pointed out above, the depletion region, though characterized byLd,

is extremely broad – aty = Ld, the steady state concentration is only about37% of nb, the bulk

concentration.

Further insight into the transient development of the depletion layer can be gained by consid-

ering the behavior at times much shorter than the diffusion time over the distanceLd but long

compared toλH , soK in Eq. (4.66) can be treated as time-independent. Introducing a transient

depletion layer thicknessδy(t), an order-of-magnitude analysis of Eq. (4.66) shows that atthese

short times the dominant balance is between the time-derivative term and the migration term, and

that the depletion layer thickness scales as follows:

δy(t) ∼ (Kt)1/3. (4.68)

Neglecting diffusion in Eq. (4.66) and defining the variable,

ω =y

(3Kt)1/3, (4.69)

44

y/(kBT/H)1/2

n/n b

50 100 150 200 250 3000

0.5

1

1.5

2

t = 0t = 10λH

t = 100λH

t = 1000λH

t = 10000λH

steady state

Figure 4.5 Temporal development of the concentration profile in uniform shear flow above asingle wall atWi = 10. A FENE-P dumbbell model with finite extensibility parameter b = 600and hydrodynamic interaction parameterh∗ = 0.25 is used.

45

a similarity solution can be found:

n(ω)

nb

=

0, if 0 ≤ ω ≤ 1.

ω2

(ω3−1)2/3, if ω > 1.

(4.70)

This solution is shown in Fig. 4.6. Interestingly, it has an integrable singularity and discontinuity at

ω = 1. The singularity arises because diffusion has been neglected completely, on the assumption

that it is not important over the length scaleδy(t). Moreover, in the absence of the diffusion all

molecules escape from inside the depletion layer (0 ≤ ω ≤ 1), giving rise to the discontinuity

at the frontier of the depletion layer (ω = 1). Very near the singular point diffusion will become

important at leading order, smearing out the singularity. Using the similarity solution to solve forω

values at which the migration contribution to the flux is comparable to the diffusion contribution,

we found that the width of this region is proportional toδy(t)/Ld. To illustrate this better, we

calculate the full numerical solution to Eq. (4.66) by usinga FENE-P dumbbell model with finite

extensibilityb = 600 and hydrodynamic interaction parameterh∗ = 0.25 at We = 100. The

full numerical solutions att = 10λH and1000λH are plotted in Fig. 4.6 against the similarity

solution. We see that the “pile up” phenomenon that occurs inthe full numerical solution appears

in idealized form in the similarity solution, showing its origin in the balance between migration

and accumulation of the polymer, as described qualitatively above. Considering the large time

difference (two orders of magnitude) between the two numerical solutions, the similarity solution

captures the transient evolution of the depletion layer remarkably well. Finally, we note that the

time scale for development of the steady state profile can be estimated from the scaling analysis

by determining the time at whichδy(t) = Ld. This estimate recovers our earlier prediction that

tmig ∼ L2d/D.

Another important process is the spatial development of theconcentration field near the en-

trance to a channel: we will address this situation here by considering the migration analogue of

the Graetz-Leveque problem [87]. At low Reynolds number, the velocity field near the entrance

to the channel becomes fully developed over a length scale comparable to the height of the chan-

nel. Considering the region sufficiently near the channel entrance that the depletion layer is thin

compared to the channel heightB, we can treat the domain as semi-infinite in they-direction and

46

ω

n/n b

0 1 2 3 4 50

1

2

3

4Similarity Solutiont = 10λH

t = 1000λH

Figure 4.6 Similarity solution for time evolution of the concentration profile in uniform shear flowabove a single wall. The full numerical solutions includingdiffusion forWi = 100 at two differenttimes,t = 10λH andt = 1000λH, are also plotted for comparison. A FENE-P dumbbell with finiteextensibilityb = 600 and hydrodynamic interaction parameterh∗ = 0.25 is used when solving forthe numerical solutions.

47

treat the velocity field as a simple shear flow. As above, flow isin thex-direction, and the wall is

aty = 0. The conservation equation becomes

γy∂n

∂x= − ∂

∂y

(K

y2n

)+ D

(∂2n

∂x2+

∂2n

∂y2

). (4.71)

Introducing a spatially varying depletion layer thicknessδy(x), an order-of-magnitude analysis

shows that very near the wall, they-migration term andx-convection terms balance, and the scaling

of the depletion layer thickness is given by:

δy(x) ∼(

Kx

γ

)1/4

. (4.72)

Based on this scaling, we neglect the diffusion terms in Eq. (4.71) and seek a similarity solution

n(σ), where

σ =y

(4Kx/γ)1/4. (4.73)

The solution is:

n(σ)

nb

=

0, if 0 ≤ σ ≤ 1.

σ2√|σ4−1|

, if σ > 1.(4.74)

This solution is plotted in Fig.4.7; for a channel with height B it will be valid in the caseδh ≪Ld ≪ B. Again, there is a weak singularity in this solution (which will be regularized by dif-

fusion), showing that a “pile up” similar to that found in thetransient development appears here

too. So we expect that near the entrance to a channel the concentration distribution of polymer

chains will display a peak near each wall. The assumption of negligible diffusion breaks down

in a region around the singularity pointσ = 1 with width proportional toδy(x)/Ld. The nu-

merical solution without neglecting they-diffusion term is solved by using a FENE-P dumbbell

model with finite extensibilityb = 600 and hydrodynamic interaction parameterh∗ = 0.25 at

Wi = 100, and the result is shown in Fig. 4.7 for downstream positionsx = 10(kBT/H)1/2 and

x = 10000(kBT/H)1/2. It is clear from the figure that the similarity solution captures the spatial

development of the concentration field very well over a largelength scale.

With the knowledge of the depletion layer thicknessLd = K/D in the fully developed region

(i.e., where convection is negligible and diffusion and migration balance), a scaling estimate of the

48

σ

n/n b

0 1 2 3 40

1

2

3

4

Similarity Solutionx = 10 (kBT/h)1/2

x = 10000 (kBT/H)1/2

Figure 4.7 Similarity solution for spatial development of the concentration profile in uniform shearflow above a single wall. The full numerical solutions including the diffusion forWi = 100 at twodifferent downstream positions,x = 10(kB/H)1/2 andx = 10000(kB/H)1/2, are also shown forcomparison. A FENE-P dumbbell with finite extensibilityb = 600 and hydrodynamic interactionparameterh∗ = 0.25 is used when solving for the numerical solutions.

49

entrance lengthLx for the depletion layer can be obtained by setting the boundary layer thickness

δy equal toLd:

δy =

(KLx

γ

)1/4

=K

D. (4.75)

Therefore the entrance length is given by

Lx =K3γ

D4= L3

d

γ

D. (4.76)

Combining with Eq. (4.65) and using the scaling relationλH ∼ R2g/D, we can rewriteLx in terms

of Wi andRg for FENE dumbbells at highWi as follows:

Lx ∼ Wi3Rg. (4.77)

This result shows that the entrance length is very sensitiveto the Weissenberg number – a large

entrance length should be expected at high Weissenberg number.

Finally, we address the issue of what residence time the fluidshould have in the channel before

the depletion layer can be considered to be fully developed.For the caseLd ≪ B, this time can be

estimated as the travel time fromx = 0 to x = Lx for a fluid element at a distance ofLd from the

wall:

ttravel ∼Lx

γLd∼ L2

d

D(4.78)

So roughly speaking, the residence time required for establishment of the fully developed concen-

tration profile is the diffusion time over the distanceLd. Based on the transient results presented

above, however, we expect this estimate to underpredict theactual time required, because of the

broad structure of the steady state depletion layer. The experimentally obtained concentration pro-

files of Fanget al. [47] for DNA in a microchannel show a weak maximum, suggesting that they

are not fully developed.

4.6 Plane Couette Flow and Plane Poiseuille Flow

The above analysis of shear-induced depletion in a semi-infinite domain reveals the basic mech-

anism of molecular migration and the time and length scales involved. In this section, we extend

50

our discussion to a slit geometry, and consider plane Couette flow and plane Poiseuille flow, which

are very common flow types in experiments.

Consider the gap between two parallel plates separated by a distance2h, and filled with poly-

mer solution. As a first approximation, we can calculate the migration effects due to each wall

in a semi-infinite domain with the other wall ignored, and then superimpose the results. In this

“single-reflection” [70] approximation, the total mass fluxin the wall normal direction will be:

jc,y =K(y)

y2n − K(2h − y)

(2h − y)2n − D

∂n

∂y. (4.79)

The dependence ofK on position arises indirectly as a result of the position dependence of the

shear rate. In plane Couette flow, whereK is position independent, the steady state concentration

profile under this approximation is:

n

nc= exp

[−Ld

(1

y+

1

B − y− 4

B

)], (4.80)

whereLd = K/D is the depletion layer thickness for an unbounded domain,y is the distance from

one wall, andnc is the concentration at the centerline of the slit. Figure 4.8 shows the solutions for

Wi = 2, 10, 100, which correspond toLd = 2√

kBT/H, 33√

kBT/H, and387√

kBT/H. The pa-

rameters used areb = 600, h∗ = 0.25, 2h = 30√

kBT/H. As the Weissenberg number increases,

the concentration profile becomes sharper and sharper, which indicates a stronger migration effect

at higherWi.

We now present results for plane Poiseuille flow. Here the velocity profile is parabolic:

vx(y) = Um

[1 −

(1 − y

h

)2]

, (4.81)

whereUm is the velocity at the center of the slit. The steady state concentration profiles in plane

Poiseuille flow are shown in Fig. 4.9. HereWi is defined based on the wall shear rate. In the middle

of the slit, the concentration profile for plane Couette flow is steeper than that for plane Poiseuille

flow. This is because the the shear rate in the middle region ofthe plane Couette flow is larger than

that of plane Poiseuille flow.

51

y/(kbT/H)1/2

n/n c

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Wi = 2Ld = 2

Wi = 10Ld = 33

Wi=100Ld = 387

Figure 4.8 Steady state concentration profiles atWi = 2, 10 and100 in plane Couette flow in a slitwith width 2h = 30


√kBT/H and concentration by its value at

the centerline of the slit,nc. Migration effects due to the two walls of the slit are superimposed bytaking the “single-reflection” approximation.

52

y/(kBT/H)1/2

n/n c

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Wi = 2Ld = 2

Wi = 10Ld = 33

Wi = 100Ld = 387

Figure 4.9 Steady state concentration profiles atWi = 2, 10, 100 in lane Poiseuille flow in a slitwith width 2h = 30


√kBT/H and concentration by its value at

the centerline of the slit,nc. Migration effects due to the two walls of the slit are superimposed bytaking the “single-reflection” approximation.

53

A case of particular interest is the spatial development of the depletion layer downstream from

the entrance to a slit, as we first discussed in Section 4.5. This situation is governed by the follow-

ing equation:

vx(y)∂n

∂x= − ∂

∂y

(K(y)

y2n

)+

∂

∂y

(K(2h − y)

(2h − y)2n

)

+ D

(∂2n

∂y2+

∂2n

∂x2

).

(4.82)

The numerical solution to this equation is shown in Fig. 4.10for 2h = 300√

kBT/H, b = 600,

h∗ = 0.25, andWi = 20. Concentration is scaled by its bulk valuen0 before entering the slit. It can

be seen that the distance over which the concentration evolves into the fully developed profile is

remarkably large: even forx = 105√

kBT/H, the fully developed region is still not reached. This

is consistent with the result from the similarity solution found in Section 4.5. Therefore, in order

to measure the fully developed concentration in experiment, the residence time should be much

larger than the diffusion time over the distance of the depletion layer thickness. The concentration

field also shows clearly the “pile up” phenomenon, consistent with the similarity solution described

above.

Finally, we note that there is one qualitative feature that is found in our detailed simulations

of confined chains [74] and predicted by Eq. (4.51), but not reproduced by the analysis presented

in this section. This is the dip in the concentration profile at the center of the channel, which

arises from the fourth term of Eq. (4.51), the migration toward regions of lower diffusivity that

can arise in situations with a conformation-dependent diffusivity. This feature was lost due to

our assumption of constant diffusivity. It would appear were we to use, for example, a model of

diffusivity based on a deformation drag coefficient (see, e.g. [2, 22, 131, 23]. However, the effect

is small in pressure-driven flow [74], and in uniform shear its only effect is to makeD dependent

on Weissenberg number .

4.7 Conclusion

In this paper, we developed a kinetic theory that describes migration phenomenon in flowing

dilute polymer solutions near solid surfaces. The theory, which is based on a bead-spring dumbbell

54

Figure 4.10 Steady state concentration field for plane Poiseuille flow in the entrance region of aslit with width 2h = 300

√kBT/H at Wi = 20. Only half of the slit is shown. The concentration

is scaled by its bulk valuen0 before entering the slit. Migration contributions due to two walls ofthe slit are superimposed by taking the “single-reflection”approximation.

55

model of the polymer molecules, shows that the migration comes from two contributions: one that

arises from the hydrodynamic interaction between the polymer molecule and the wall, and another

that arises from intra-chain hydrodynamic interactions. The first of these effects is generic for a

flexible particle, droplet or polymer molecule above a wall.Relaxation of the particle against the

flow generates a force dipole, and if the particle is stretched and aligned parallel to the wall, the

wall-normal flow induced by this force dipole convects the dumbbell away from the wall. The

second effect is Brownian and drives the polymer molecules to regions of lower mobility. The

latter effect is small in homogeneous shear flow, in which case the mobility of the polymer is

virtually independent of position.

With this theory, we predict the steady state concentrationprofile in uniform shear flow above

an infinite plane wall. The profile, determined by the balanceof migration and diffusion, has the

form of a Boltzmann distribution and is characterized by a length scaleLd, the depletion layer

thickness. The depletion layer thickness is proportional to the normal stress differences and the

size of the polymer molecule. For FENE dumbbells at high Weissenberg number,Ld ∼ Wi2/3Rg,

which can be much larger than the molecular size. In the transient development of the depletion

layer, numerical simulations using the theory predict a spike on the concentration profile, which

is corroborated by a similarity solution and can be explained by the dependence of the migration

rate on the distance from the wall. The time scale for this transient process is shown to scale as the

polymer diffusion time over the distanceLd. However, because of the extremely large breadth of

the steady state depletion layer, this estimate significantly underpredicts the actual time required to

reach steady state. Using similar arguments, the entrance lengthLx for the concentration evolution

in a channel is estimated (for FENE dumbbells) to scale withWi3Rg. A spike in the spatially

developing concentration profile also appears, as shown by numerical and similarity solutions. By

taking the “single-reflection” approximation, the concentration profiles for plane Couette flow and

plane Poiseuille flow are obtained.

The theory in its present form is only strictly valid for infinitely dilute solutions of dumbbells,

though the dominant migration effect, the force-dipole interaction with the wall, is not restricted to

the dumbbell model. At finite concentration, some hydrodynamic screening of the wall effect will

56

occur, tending to weaken the migration effect, but on the other hand, because of the lower polymer

concentration (and thus viscosity) near the wall, the shearrate will be higher there than in the

bulk, tending to enhance migration. The balance of these effects will determine the concentration

dependence of the depletion layer thickness and apparent slip velocity in dilute polymer solutions.

The results presented here provide a starting point for addressing these issues.

57

Chapter 5

Brownian Dynamics Simulation

In the field of dynamics of polymeric liquids, computer simulation has been playing an impor-

tant role since the very beginning. Various simulation methods have been developed or tailored to

polymer system, including Molecular Dynamics (MD), Brownian Dynamics (BD), Monte Carlo

(MC), and more recently, the Lattice-Boltzmann Method (LBM) which we will discuss in Chapter

6. These simulation techniques have drawn attention of researchers because the analytical theory

relies on assumptions which can be examined by simulations.Furthermore, the simulation en-

ables us to solve more complicated problems inaccessible totheoretical analysis yet. The interplay

of analytical theory, simulation, and experiment has been proven to be a powerful combination

in understanding the behavior of the dissolved polymer chains. In this chapter, we study cross-

streamline migration in flow of individual flexible polymer molecules in solution using Brownian

Dynamics simulation. The primary goals of the work are as following: (1) characterize migration

in the regimesRg ≪ h andRg ∼ h; (2) evaluate the analytical theory developed in previous

chapter in the regimeRg ≪ h; (3) examine the issue of coarse-graining of chains into dumbbells

in confined geometries, especially in the regimeRg ≪ h ∼ L, where separation of scales between

molecule and geometry begins to fail. The simulation methodused here is based on a Green’s

function description of Stokes flow.

5.1 Introduction

Rapid advances in photo-lithography and soft lithography have greatly facilitated the design

and fabrication of novel microfluidic devices working on thelength scale of micron and smaller

58

[110, 53]. These devices are now used in diverse applications such as DNA sequencing and map-

ping, clinical diagnosis, and environmental monitoring [125, 79]. A particular example that has

been extensively used in genomics is the optical mapping method of Schwartz and coworkers

[78, 155, 41], where DNA stretching by flow and deposition onto an absorbing surface in a mi-

crofluidic device has been used to enable subsequent gene mapping by restriction digestion or

hybridization. A related example is a microfluidic system that can directly read out the positions

of fluorescently tagged sites on a linear DNA molecule stretched by flow, with a throughput of

thousands of molecules per minute [26]. Because of the largesurface to volume ratio of such small

devices, their design requires a good understanding of the interaction of the target molecules (e.g.

DNA, viruses, other analytes), or macromolecules in general, with the microfluidic confinement.

There are several primary regimes of confinement, dependingon the ratio of the slit width

and the characteristic length scale of the polymer molecule. Consider a flexible polymer chain at

equilibrium in solution, confined between two infinite wallsseparated by a distance2h. When the

slit width is much larger than the equilibrium polymer radius of gyrationRg, the chain adopt its

unperturbed isotropic coil conformation at equilibrium. We call this the weak confinement regime;

it is illustrated in Figs. 5.1a and b. We note that during flow another length scale, the contour length

L of the molecule, can become comparable to the degree of confinement. When the slit width is

reduced to about the unperturbed chain dimension ofRg, the free arrangement of the polymer

chain is restricted by the walls and deviations from the bulkequilibrium coil conformation are

expected. This regime is called strong or high confinement, and is shown in Fig. 5.1c. If the slit

width is reduced further to the order of the chain persistence lengthLp, then the chain dynamics

is extremely restricted [145, 123], as shown in Fig. 5.1c. Inthis paper, we will explore dynamics

of dilute polymer solutions, and especially center-of-mass distribution, in the weakly and highly

confined regimes.

In the weakly confined regime, an important and long-recognized result of the interaction be-

tween the polymer chain and the confinement is the formation of depletion layers during flow [2].

Chenet al. [31] recently presented direct visualizations of the depletion layer in flow of DNA

59

solutions in a channel., Fig. 5.2 shows their experimental measurements of axially averaged fluo-

rescent intensity in the cross section of a 40µm × 40µm microchannel as a function of time for

fluorescently-labeled T2-DNA solution undergoing oscillatory pressure-driven flow. The dark re-

gions near the walls indicate depletion layers with thickness of about 10jm, which is much larger

than the radius of gyration of the T2-DNA molecule (about 1.6µm). Note that it takes more than

a minute for the depletion layer to fully develop. In relatedwork, Fanget al. [47] found that in

channel flow of diluteλ-phage DNA solution, inside a region extending from a glass surface in a

micro-channel to about one third of the contour length ofλ-phage DNA molecule, the stretch and

concentration of the DNA molecules was considerably smaller than in the bulk. Similar results

were found in a steady torsional shear flow [89]. These observations have obvious implications

for surface-based DNA analysis methods, since the development of a depletion layer significantly

decreases the probability of adsorption during flow. Another consequence of depletion layers is

“apparent slip”: inside the depletion layer, the fluid viscosity is lower than that in the bulk and thus

the velocity gradient higher. Macroscopically, this apparent slip can be measured in terms of the

enhancement of the flow rate in pipe flow of dilute polymer solution under a given pressure drop

[37, 69].

Despite the important practical implications, our understanding of the migration process in

dilute polymer solution flow that results in the depletion layer and the apparent-slip is still very

limited. Researchers have proposed a number of arguments toexplain these phenomena including

thermodynamic models [103, 147], two-fluid models [62, 42, 111, 106, 99, 107, 17], molecular

kinetic theories [9, 14, 112, 13, 38, 39, 131, 21, 22, 23, 76, 46], and simply wall excluded volume

effects [10, 100, 40, 101, 102, 152, 153]. However, the predictions of those theories are contro-

versial, even with regard to the direction of the chain migration in simple flows [2]. A significant

limitation of these previous studies lies in the fact they did not include the hydrodynamic effect of

the confining walls on the polymer molecules, or did it incorrectly [76].

To address the role of hydrodynamics in confinement, Jendrejacket al. [72, 75, 74] performed

Brownian dynamics simulations of pressure-driven flow of a dilute λ-phage DNA solution in a

square micro-channel accounting for hydrodynamic interactions both between chain segments and

60

(b) 2h >> Rg

Rg

2h

(c) 2h ~ Rg

Rg

2h

Lp

(d) 2h ~ Lp

2h

Rg

(a) Single wall oo

Figure 5.1 Schematic of different regimes of confinement: (a) Single wall confinement, (b) weakconfinement:2h ≫ Rg, (c) strong confinement:2h ∼ Rg, and (d) extreme confinement:2h ∼ Lp.

61

Figure 5.2 Time evolution of axially averaged fluorescence intensity of fluorescent labeled T2-DNA solution as a function of cross-sectional position. Thechannel walls are aty = ±20µm. Thesolution is undergoing oscillatory pressure-driven flow ata maximum strain rate of 75s−1 and afrequency of 0.25Hz in a 40µm× 40µm microchannel [31]. The bright band at the center indicateshigher concentration of T2-DNA molecule and the dark regionrepresents the depletion layer nearthe channel walls.

62

between chains and the channel walls. They predicted that, when the channel width2h is much

larger than the equilibrium chain radius of gyrationRg, the DNA molecules migrate toward the

channel center during flow. More importantly, they demonstrated that the migration phenomenon

is due to chain-wall hydrodynamic interactions, in a mannersimilar to that found for suspensions

of deformable droplets [70, 27, 135]. Santillanet al. [128] have performed related simulations for

bead-rod chains.

Building on the work of Jendrejacket al. [74], Ma and Graham [94] developed an analytical

expression for the polymer flux in an infinitely dilute solution in a semi-infinite domain bounded

by a flat no-slip wall. This result was based on kinetic theoryfor a bead-spring dumbbell polymer

model; an assumption that the polymer extension was small compared to the distance of the poly-

mer from the wall enabled derivation of relatively simple closed form results. Subsequent approx-

imations led to explicit expressions for the steady state depletion layer thickness in homogeneous

shear flow, as well as a scaling estimates of the spatial and temporal scales for the depletion layer

to become fully developed.

Turning to the highly confined regimeRg ∼ h, a very interesting phenomenon observed by

Jendrejack and coworkers [74] is that when the channel size is very small, the concentration near

the channel wall islarger than that at equilibrium, indicating migration toward the wall, in contrast

to the behavior in a large channel. This effect was also observed in a recent Lattice Boltzmann

simulation by Ustaet al. [149]. The physical origin of this reversal is addressed below.

In the present work, we study cross-stream migration duringflow of individual flexible polymer

molecules in solution using Brownian dynamics simulations. The primary goals of the work are

as follows: (1) characterize migration in the regimesRg ≪ h and Rg ∼ h; (2) evaluate the

analytical theory of Ma and Graham in the regimeRg ≪ h for which it was derived; (3) examine

the issue of coarse-graining of chains into dumbbells in confined geometries, especially in the

regimeRg ≪ h ∼ L, where separation of scales between molecule and geometry begins to fail.

The simulation method used here is based on a Green’s function description of Stokes flow. Other

simulation approaches such as lattice Boltzmann and dissipative particle dynamics do not explicitly

63

enforce low-Reynolds number flow, so in the Section 5.5 we examine with scaling arguments the

effect of Reynolds number on wall-induced hydrodynamic migration.

5.2 Point-Dipole Theory of Polymer Migration

A point-force dipole, or Stokeslet doublet,d suspended influid in a confined domain will drive

a flow that in general will lead to a nonzero migration velocity vmig at the position of the dipole:

vmig = M : d, (5.1)

where the third-order tensorM is determined by solution of Stokes equations in the relevant

geometry. For a dilute polymer solution confined by a single wall, Ma and Graham [94] used this

result in a kinetic theory for a bead-spring dumbbell in solution to find the following expression

for the center-of-mass flux,jc,

jc =nv +n

8〈qq〉 : ∇∇v + M : τ

p

− n∂

∂rc· 〈DK,b〉 − 〈DK,b〉 ·

∂n

∂rc,

(5.2)

wheren(rc, t) is the center of mass probability distribution function (i.e. “concentration”),v is

the imposed velocity field evaluated at the center of massrc of the dumbbell,q is the end-to-

end vector of the dumbbell,τ p is the polymer contribution to the stress tensor and〈DK,b〉 is the

ensemble average bulk Kirkwood diffusivity of the dumbbell. Angle brackets denote ensemble

averaging over the wall normal direction. To reach Eq. (5.2)the point-dipole (far-field) limit is

been used; in Section 5.4.4 we show a more general expressionthat incorporates wall-excluded

volume effects and does not use the point-dipole approximation.

The last term in Eq. (5.2) is normal Fickian diffusion. In rectilinear flow, the term containing

∇∇u only gives the lag of a macromolecule behind the solvent along the streamline [9] but no

cross-streamline migration, although in flow with curvature cross-streamline migration is possible.

The term containing the migration tensor,M, and the stress tensor,τp, arises from the presence of

walls. For single wall confinement,M is given by Ma and Graham [94]; ify is the distance from

the wall, it decays as1/y2. Note that this term is generic for the flux ofany flexible suspended

64

particle or molecule in a wall-bounded flow− in particular its validity is not restricted to the

dumbbell model. The term containing the divergence of〈DK,b〉 can also lead to migration if the

diffusivity of the molecule depends on conformation (whichin general it does), but only in a flow

where the conformation distribution is spatially nonuniform (as in a pressure-driven flow). In a

pressure-driven flow, this term leads to a weak driving forcetoward the wall, because the mobility

of a stretched chain is lower than the mobility of a coiled one[94, 74].

At steady state, the migration due to the hydrodynamic interactions is balanced by diffusion.

With some simplifying assumptions, an analytical expression for the resulting concentration profile

can be obtained. The depletion layer thicknessLd is determined primarily by the first normal stress

difference in the flowing solution. In a flow where the chain isstrongly stretched,Ld becomes much

larger than the equilibrium size of the polymer chain. For the finitely extensible dumbbell model,

the analysis predicts thatLd/Rg ∼ Wi2/3, whereWi = γλ is the Weissenberg number withγ shear

rate andλ the longest relaxation time of the chain. The model also predicts that the chain density

profile reaches steady state over a time scale ofL2d/D, whereD is the molecular diffusivity of the

stretched chain. Finally, this analysis can be extended to aslit geometry, using a single-reflection

approximation for the hydrodynamics.

5.3 Polymer Model and Simulation Method

In the present work, a linear polymer molecule dissolved in aviscous solvent is represented

by a freely jointed bead-spring chain, i.e.,Nb beads connected throughNs = Nb − 1 springs.

Neglecting inertia, on each bead the force balance requires

Fhi + Fs

i + Fvi + Fw

i + Fbi = 0, for i = 1, ..., Nb, (5.3)

where, for beadi, Fhi is the hydrodynamic force,Fv

i is the bead-to-bead excluded volume force,

Fwi the bead-wall excluded volume force,Fb

i is the Brownian force andFsi is the spring force. The

characteristic variables are the bead hydrodynamic radius, a, for distance,ζa2/kBT for time and

kBT/a for force, wherekB is the Boltzmann’s constant,T the absolute temperature, andζ the

65

bead friction coefficient, which is related to the solvent viscosity,η, anda through Stokes’ law, i.e.

ζ = 6πηa [85, 118].

A finitely extensible nonlinear (FENE) spring defined by the following dimensionless potential

energy [15]

φsij =

1

2b ln

[1 −

r2ij

b

], (5.4)

is used. Hererij = |ri − rj| is the distance between beadsi and j, andb is the extensibility

parameter, andb = Hsq20/kBT , whereHs is the spring constant per spring,H = Hs/Ns is the

total spring constant for the molecule andq0 = L/Ns the maximum stretch of each spring. For the

special case of the dumbbell model,H = Hs andq0 = L.

The force balance Eq. (5.3) can be written as the following system of stochastic differential

equations of the motion for the bead positions [71, 74]

dr =

[u0 + D · F +

∂

∂r· D

]dt +

√2B · dw. (5.5)

Herer is a vector containing the3Nb coordinates of the beads that constitute the polymer chain,

with ri denoting the Cartesian coordinates of beadi. The vectorv0 of length3Nb represents the

unperturbed velocity field, i.e. the velocity field in the absence of any polymer molecule. The

vectorF has length3Nb, with Fi denoting the total non-Brownian, non-hydrodynamic force acting

on beadi. Finally, the independent components ofdw are obtained from a real-valued Gaussian

distribution with mean zero and variancedt.

The motion of a bead of the chain perturbs the entire flow field,which in turns affects the

motion of the other beads. These hydrodynamic interactions(HI) enter the polymer chain dynamics

through the3 × 3 block components (Dij) of the3Nb × 3Nb diffusion tensor,D, which may be

separated into the bead Stokes drag and the hydrodynamic interaction tensor,Ω, [15, 113]

D = [I + Ω] , (5.6)

whereI is the identity matrix. Computation ofΩ · f will be discussed below. The Brownian

perturbation,dw, is coupled to the hydrodynamic interactions through the fluctuation-dissipation

theorem [114, 122, 157]

D = B ·BT . (5.7)

66

For excluded volume a Gaussian potential is assumed for bead-to-bead interactions:

φvij = Ab exp

[−αr2

ij

], for rij ≤ 3, (5.8)

while a repulsive potential is used for wall-bead interactions as follows

φvwi =

Aw

3(riw − 2)3 , for riw ≤ 2, (5.9)

whereriw represents the distance of beadi from the wall in the wall-normal direction.

The HI are included by assuming that each bead is a point-force and the velocity perturbation

is the solution of the fundamental singular solution of Stokes equations

η∂2vj

i (r, r0)

∂xk∂xk− ∂pj (r, r0)

∂xi= −δ (r − r0) δij ,

∂vji (r, r0)

∂xi= 0,

(5.10)

whereη is the solvent viscosity,δ (r) is the Dirac delta function,δij is the Kronecker delta and

vji (r, r0) is the fundamental singular solution or Green’s function ofthe Stokes equations, known

as a Stokeslet, located at the pointr0 and oriented in thej-th direction [81, 118, 117]. In order to

make the expression compact, we usexj to represent the three Cartesian coordinates of position

vectorr, with j = 1, 2, 3 corresponding to thex, y andz directions. In an infinite domain (no

confinement), the free-space Green’s function is

vji (x,x0) =

1

8πηr

[δij +

(xi − x0i) (xj − x0j)

r2

], (5.11)

sometimes also called the Oseen-Burgers (OB) tensor. For free-space simulations, with the point

force formalism, the3× 3 non-diagonal sub-matrices of the3Nb × 3Nb hydrodynamic interaction

tensor,Ωij , are Stokeslets, as follows

Ωνµζ−1 = (1 − δνµ) vj

i (rν , rµ) , (5.12)

whereν andµ are polymer beads andi, j represent Cartesian coordinates.

In confinement, the Stokeslet must be modified to account for the boundary conditions. In

1971, Blake showed that in the presence of a rigid wall atx2 = w, the Green’s function, that

67

satisfy the non-slip at the wall,wvji (r), may be expressed in terms of a free-space Stokeslet and a

finite collection of image singularities, including an image free-space Stokeslet, a potential dipole

and a Stokeslet-doublet as follows [18]

wvji (r, r0) =vj

i (r, r0) − vji

(r, rI

0

)+

2 [x02 − w]2 UDij

(r, rI

0

)−

2 [x02 − w]USDij

(r, rI

0

),

(5.13)

whererI0 = (x01, 2w − x02, x03) is the image ofr0 with respect to the wall. The tensorsUD(r)

andUSD(r) represent potential dipoles and Stokes-doublets. For three-dimensional domains, they

are given by

UDij (r) = ± ∂

∂xj

(xi

|r|3)

= ±(

δij

|r|3 − 3xixj

|r|5)

, (5.14)

USDij (r) = ±∂v3

i

∂xj= x2U

Dij (r) ± δj2xi − δi2xj

|r|3 , (5.15)

with a plus sign forj = 2, in they-direction, and a minus sign forj = 1, 3, corresponding to the

x- andz-directions.

For single-wall confinement BD simulations the3 × 3 non-diagonal submatrices of the HI

tensor,Ωij , are calculated using the Stokeslet in Eq. (5.13). In addition, the self-induced HI, due to

each bead image, must be included in the diagonal3 × 3 sub-matrices. The single-wall HI tensor

is then calculated as follows

Ωνµζ−1 =(1 − δνµ) vj

i (rν , rµ) − vji

(rν , r

Iµ

)+

2 [xµ2 − w]2 UDij

(rν , r

Iµ

)−

2 [xµ2 − w]USDij

(rν , r

Iµ

)(5.16)

The Stokeslet in Eq. (5.13) is singular when the bead is at thewall, and in fact the BD-point-

force model will break when the distance between the beads and the wall is less than a bead

hydrodynamic radius,a. However, in practice an excluded volume force at the wall preventing the

beads from getting to near the wall is used so the probabilityof finding a bead at the wall is zero.

For Stokes flow between two parallel plates, Liron and Mochonfound two alternative expres-

sions for the Stokeslet, one in terms of infinite integrals and the other in terms of infinite series [92].

68

These solutions have been used in the past to investigate themotion of particles and droplets be-

tween parallel walls using boundary integral techniques [138, 91], theoretical approaches [55, 56],

etc. The use of these solutions for computational multi-particle systems is expensive, with compu-

tation time scaling asO(N3), whereN is the number of particles. There are different approaches

to calculate the Green’s function due to a force of arbitraryorientation between two walls. One

approach was introduced by Jendrejacket. al [74], where a finite element method was used to

find the collection of image singularities for an internal mesh, and whenever needed interpolation

was used to calculate the complete Green’s function. This method in combination with Fixman’s

method for computing the Brownian fluctuations created a BD simulation method that scales as

O(N2.25) [73, 74, 32, 31].

Recently Muchaet al. developed anO(N log N) method for computing HI in a slit geometry

[108]. Based on this method Hernandez-Ortizet. al developed a BD simulation algorithm that

scales asO(N1.25 log N) [63]. Here, we are interested in a system confined between twoinfinite

walls separated a distance2h along thez (or x3) -coordinate, with periodic boundary conditions

in the other two directions,x (or x1) andy (or x2), of periodic lengthW andL, respectively. The

basic outline of the method is as follows. It starts by splitting the slit Stokeslet,Svji (r, r0), into

three column vectors, i.e,

Svji (r, r0) = [S1,S2,S3] , (5.17)

whereSj (r, r0) = (uj, vj, wj) represents the velocity perturbation due to a point force inthe

j-direction at positionx0, with the corresponding pressure

pj (r, r0) = p1, p2, p3 . (5.18)

The calculation of each piece of the Green’s function,(Sj , pj), proceeds by Fourier series expan-

sion in the two periodic dimensions

Sj (r, r0) =∑

k‖

vj

(k‖, x2, x02

)e[ik‖·(r−r0)‖], (5.19)

pj (r, r0) =∑

k‖

pj(k‖, x2, x02

)e[ik‖·(r−r0)‖], (5.20)

69

with the summation over two-dimensional wave vectorsk‖ = (k1, k3). Here the subscript‖ indi-

cates the two periodic directionsx1 andx3 andvj (r, r0) = (vj , vj, wj).

Inserting Eqns. (5.17) and (5.18) into the Stokes equations, Eq. (5.10), a set of ordinary differ-

ential equations for the Fourier coefficients,vj (r, r0), is obtained. The solution for these coeffi-

cients has the following from

vj (r, r0) = vj

(k‖, ajn

(k‖, x02

), x2

), (5.21)

whereajn

(k‖, x02

)for j = 1, 2, 3 andn = 1, ..., 6 are a set of field coefficients. These coef-

ficients are function ofr0 but not of r, so they can be calculated only once per configuration.

Muchaet. alrealized that after a sorting of the particles with respect to x2-direction the calculation

of the Green’s functions can be performed inO(N log N) calculations [108]. Details of the BD

implementation can be found in Hernandez-Ortizet. al [63].

There are two Brownian based terms in Eq. (5.5), the random vectorB · dw and the divergence

of the diffusion tensor. The most common method in the literature to find the matrixB for the

fluctuating terms, in a way such that the fluctuation dissipation theorem is satisfied, is to perform

the Cholesky decomposition of the diffusion tensorD [120, 20],

D = S · S with S = ST , (5.22)

which typically means to calculate directly the diffusion tensor, an operation that scales asO (N2),

and to use a regular method to do the Cholesky decomposition,O (N3), which will imply long

computational times even for dilute systems. Instead, the method described by Fixman [52, 51]

and the algorithm of Jendrejacket. al [73] can be used in order to obtain the needed terms for the

fluctuating force in an algorithm which, combined with Mucha’s method for HI, scales roughly as

O (N1.25 log N) [63].

The polymer chain represented by the bead-spring model is fully characterized by 4 parameters:

Ns, bk, nk, h∗, namely the number of springs, Kuhn length, number of Kuhn segments and the

hydrodynamic interaction parameter [15, 94],

h∗ =ζ

η

√H

36π3kBT. (5.23)

70

The polymer selected for the results presented in this paperis similar to aλ-phage DNA (we

are using a FENE spring instead of a worm-like spring) wherebk = 0.106µm, nk = 198 and

h∗ = 0.25.

The strength of the flow field is characterized by the Weissenberg number,Wi, representing the

ratio between the time scale of molecular relaxation to thatof the solvent relaxation. For shear

flows is defined byWi = λγ, whereλ is the longest relaxation time of the molecule. Due to the

fact that in this work we need only an estimate of the longest relaxation time we used the relaxation

time from Rouse theory (Hookean springs, free-draining, theta solvent) [15]

λ =ζ

2H

1

4 sin2 (π/2Nb).

The excluded volume parameters wereAb = 2 andα = 0.5 for the bead-to-bead andAw = 3

for the wall. An adaptive time step was selected in a way that it was lower than 10% of both the

bead diffusion time and the bead convection time for the far-wall region. For the near-wall region

it was selected to be 0.5% of the bead diffusion time to prevent the beads from touching the walls.

For the single wall simulations the box is infinite in the directions parallel to the wall. Molecules

that moved beyondy = 100 (kBT/H)1/2 were reflected back into the domain, a procedure that

has no effect on the steady state chain distribution. For theslit simulations the domain sizeW in

the wall-parallel directions was always set to be the largerof three times the total wall separation,

(3 (2h)) and three times the contour length of the molecule, (3 (nkbk/ a)).

5.4 Results and Discussion

5.4.1 Single Wall Migration in Simple Shear

To address cross-stream migration in the caseRg ≪ L ≪ h we take the situationh → ∞, i.e.

a semi-infinite domain. Consider a infinitely dilute solution of dumbbells (Nb = 2) under uniform

shear flow in thex-direction with constant shear rateγ above a plane wall aty = 0. Using the

point dipole theory described above, the center-of-mass flux, jc, reduces to

jc = M : τp − 〈DK,b〉 ·

∂n

∂rc. (5.24)

71

With an additional assumption of constant diffusivity, Ma and Graham found that the steady-state

concentration is given by [94]

n = nb exp

(−Ld

y

), (5.25)

wherenb is the bulk concentration andLd is the depletion layer thickness,

Ld =9

32

N1 − N2

D, (5.26)

whereN1 = τpxx − τp

yy andN2 = τpyy − τp

zz are the first and second normal stress differences

andD is the Kirkwood diffusivity. Equation (5.26) for the depletion layer thickness applies to

any force law chosen for the bead-spring model. For the theoretical approach, these values are

determined by the solution to the governing equation for dumbbells. However, the values forN1,

N2 andD can be obtained from an experimental setup or from simulations. In particular, for the

BD simulations described below, the polymer contribution to the stress tensor is calculated using

the Kramers-Kirkwood equation [15]

τp = −

Nb∑

i=1

〈(ri − rc)Fi〉 , (5.27)

while the diffusivity can be determined using the Kirkwood formula,

DK =1

3tr

⟨1

N2b

Nb∑

i=1

Nb∑

j=1

Dij

⟩. (5.28)

Figure 5.3 shows the steady-state concentration profiles predicted by Eq. (5.25) and a BD

simulation of dumbbells, withh∗ = 0.25, b = 594. Position is scaled with(kBT/H)1/2 − for

a dumbbell model withb ≫ 1, Rg = (3kBT/H)1/2. The depletion layer thickness,Ld, was

calculated using the values from the simulation. Both theory and simulation predict the migration

of the polymers away from the wall due to hydrodynamic interactions. Far from the walls, the

theory with the Stokeslet-doublet approximation, which implicitly assumes that the only length

scale is the distance of the polymer to the wall, agrees well with the BD results.

When the polymer is close to the wall, its size is an additional length scale and the far-field

approximation overpredicts the near-field concentration.This near-field behavior is closely re-

lated to the migration velocity,vmig. Relaxing the point-dipole approximation, this becomes for a

72

0 10 20 300

0.2

0.4

0.6

0.8

1

y/(kBT/H)1/2

n/n [y

=90

(kBT

/H)1/

2 ]

Theory: Stokeslet−doublet

Theory: finite−size−dumbbell

BD simulation (dumbbells)

Figure 5.3 Steady-state chain center-of-mass concentration profiles predicted by theory, usingthe Stokeslet-doublet (far-field) approximation, and the BD simulation atWi = 0, 5, 10 and20 insimple shear flow. The concentration is normalized using itsvalue aty/(kBT/H)1/2 = 90.

73

dumbbell,

vmig,y =[Ω12 · Fs

2]y + [Ω21 · Fs1]y

2. (5.29)

This exact expression, unlike the far field expression Eq. (5.1), approaches zero as the dumbbell

approaches the wall, leading to a smaller migration velocity than that predicted by the far-field

theory, as shown in Fig. 5.4 for various fractional extensions of the dumbbell, whereq = |q|.To include the finite-size effect in the theory, we calculated the average end-to-end distance at

Wi = 5. Using this distance, Eq. (5.29) is used to calculate the migration velocity. This velocity

was incorporated in the migration term of the equation for the center-of-mass flux. The theoretical

finite-size steady-state concentration is shown in Fig. 5.5and compared with the simulation results.

As can be seen, the near-field is improved and a finite concentration at the excluded volume cut-off

distance from the wall is predicted, also present in the simulation. Inside the excluded volume

range (hard sphere for the theory), the concentration goes rapidly to zero.

We now extend the comparison to simulations of chains. Figure 5.6 shows a comparison

between the concentration profiles predicted by the theory with the Stokes-doublet (far-field) ap-

proximation, Eq. (5.25), and BD simulation of chains withNs = 10. The values ofN1, N2 andD

in Eq. (5.26) were calculated from the values from the simulation. The agreement between both

theory and simulation is satisfactory. Once spring resolution is improved, i.e. using 10 springs to

represent the same molecule that only dumbbells were being used (Figs. 5.3 and 5.5), the near-wall

region is improved because the finite-size effect given by the “large” force-dipole of the stretched

dumbbell is reduced into ten “smaller” force-dipoles in thechains; we elaborate on this observation

below.

5.4.2 Slit Confinement: Shear Flow

For a slit geometry, Ma and Graham [94] used as a first approximation a single-reflection, where

the effects of the migration due to each wall are calculated separately in a semi-infinite domain and

then the results are superimposed. They found that the steady-state concentration profile is given

74

0 25 50 75 1000

0.2

0.4

0.6

0.8

1

y/(kBT/H)1/2

v mig

,y/v

mig

−S

toke

slet

−do

uble

t,y

q/q0=0.2

q/q0=0.3

q/q0=0.4

Figure 5.4 Migration velocity scaled with the point-dipolevalue for different dumbbell (force-dipole) sizes, as a function of distance from the wall.

75

0 10 20 300

0.2

0.4

0.6

0.8

1

x3/(k

BT/H)1/2

n/n [x

3=90

(kBT

/H)1/

2 ]


Theory: finite−size−dumbbell

BD simulation (dumbbells)

Figure 5.5 Near-field center-of-mass steady-state concentration profiles predicted by theory, usingthe Stokeslet-doublet (far-field) approximation and finite-size dumbbells, and the BD simulationat Wi = 5 in simple shear flow.

76

0 25 500

0.2

0.4

0.6

0.8

1

y/(kBT/H)1/2

n/n [y

=50

(kBT

/H)1/

2 ]

BD simulation (chains)


Wi=5

Wi=10

Figure 5.6 Steady-state chain center-of-mass concentration profiles predicted by theory, using theStokeslet-doublet (far-field) approximation, and the BD simulation of 10 springs chains, atWi = 5and10 in simple shear flow.

77

by

n = nc exp

[−Ld

(1

y+

1

2h − y− 2

h

)], (5.30)

whereLd is the depletion layer thickness for an unbounded domain given in Eq. (5.26) andnc

is the concentration at the centerline of the slit. Note thatin Eq. (5.30) there are two main ap-

proximations: the Stokes-doublet (point-force-dipole orfar-field) and the single-reflection for the

wall-chain hydrodynamic interactions.

Figure 5.7 shows the steady-state concentration profiles for Wi = 5 and20 calculated with

the theory with the Stokeslet-doublet and the single-reflection approximations and BD simulations

with dumbbells. Here,2h/Rg = 56.4 and2h/L = 1.6 where the contour length of the poly-

mer,L, is equivalent to the maximum spring length,q0, because the polymer is represented by a

dumbbell. As the Weissenberg number increases the concentration profile becomes sharper and

sharper. However, and similar to the single wall, the degreeof migration from the near-wall region

is overpredicted by the theory. The dumbbell simulations predict a finite concentration near the

walls indicated by the “shoulders” in the near-wall region.

Figure 5.8 shows the steady-state concentration profiles for dumbbells (Ns = 1) and two chain

models (Ns = 5 and 10) and constant contour length, for a wall separation of20(kBT/H)1/2

(2h/Rg = 28.2 and2h/L = 0.8). As the figure indicates, the shoulders disappear for the chain

simulations and the results forNs = 5 and10 are virtually indistinguishable. To understand the dis-

crepancies between the dumbbell and chain results, consider Fig. 5.9. For a highly stretched dumb-

bell, the distance between beads can be comparable to the wall separation (2h ∼ L) (Fig. 5.9a).

In this situation all the force exerted by the polymer on the fluid is concentrated at two points

whose distance is comparable to the length scale of the confinement. With this extremely coarse

discretization of the force distribution, there is significant (artificial) screening by the walls of the

hydrodynamic interactions between beads. This results in an underprediction of the extent of mi-

gration. A better discretization is given by the chain modelillustrated in Fig. 5.9b. This model

provides a more uniform force distribution that is not susceptible to artificial screening.

Using the stress and diffusivity data from the chain simulation with Ns = 5, we calculate the

predictions by the theory using the Stokeslet-doublet and single-reflection approximation. Figure

78

0 10 20 30 400

0.2

0.4

0.6

0.8

1

1.2

1.4

y/(kBT/H)1/2

n/n c

BD simulation

Theory: far−field + single−reflection

Wi=5

Wi=20

Wi=0

Figure 5.7 Steady-state chain center-of-mass concentration profiles predicted by theory, using far-field and single-reflection approximations, and the BD simulation atWi = 0, 5 and20 in shearflow.

79

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

y/(kBT/H)1/2

n/n c

Wi=20Ns=1

q0/2h=1.24

Ns=5

q0/2h=0.24

Ns=10

q0/2h=0.12

Figure 5.8 Steady-state chain center-of-mass concentration profiles predicted by the BD simulationat Wi = 20 in shear flow, for different polymer discretizations:Ns = 1, 5 and10.

80

(a) dumbbell

(b) chain

q ~ h

q << h

Figure 5.9 Schematic of two different discretization levels of a same molecule (a) dumbbell: theeffect of the molecule on the solvent is approximated as two point forces with large separation;(b) chain: the effect of the molecule on the solvent is approximated as several point forces withsmaller separation.

81

5.10 illustrates the agreement between these results. Interestingly, the Stokeslet-doublet theory is a

better model for a confined chain than a confined dumbbell. This is because the chain gives a more

compact force distribution than does the dumbbell model. Similar to the single wall confinement

there is a small discrepancy for the near-wall region where the theory overpredicts the migration.

5.4.3 Highly Confined Polymer Chains

Finally we consider highly confined systems:2h ∼ Rg. In particular, we perform simulations

of chains (Ns = 10) undergoing Couette flow, in a slit with a wall separation2h = 2.9Rg.

Figure 5.11 shows probability densities as a function of position at Wi = 0 (equilibrium) and

Wi = 20 for cases where HI are included and neglected (the so-calledfree-draining (FD) case).

These simulations were performed over 65 molecular diffusion times across the slit width; the error

bars are smaller than the symbols.

In weakly confined systems, i.e.2h ≫ Rg, the free-draining model leads to no migration away

from the walls. For the highly confined systems, on the other hand, there is migration toward the

walls for both the HI and FD cases, as also observed by Jendrejacket al. [74] and by Ustaet al.

[149]. The fact that the HI and FD models give the same resultsimplies that hydrodynamic effects

in the highly confined case are less important than simple steric effects, as we now demonstrate.

Figure 5.13 shows the degree on chain stretch in all three directions at equilibrium under these

confinement conditions. Here the molecule stretch is definedby

〈R〉c =⟨rmax− rmin

⟩c, (5.31)

where the subscriptc denotes a conditional average: i.e. given a chain at a particular wall-normal

positiony, 〈R〉c is the expected value of its stretch. Note that the chains found near the wall are

more stretched in the two periodic directionsx andz (i.e. parallel to the walls) than chains in the

center of the slit. In addition, the chains are less extendedin the wall normal direction,y.

This effect becomes more pronounced in flow. Figures 5.14 and5.15 show that in flow the

chains extend in the flow direction and are correspondingly less extended in the wall-normal di-

rection. Because the geometry is so confined, the wall-chainhydrodynamic interactions from each

82

0 5 10 15 200

0.2

0.4

0.6

0.8

1

y /(kBT/H)1/2

n/n c

Theory

BD:center−of−mass

BD:bead−distribution

Wi=20

BD:Wi=0

Figure 5.10 Steady-state chain center-of-mass concentration profiles predicted by the theory, usingfar-field and single-reflection approximations, and the BD simulation atWi = 20 in shear flow.The steady-state chain center-of-mass concentration profile at equilibrium (Wi = 0) and the bead-distribution from the simulation atWi = 20 are also shown.

83

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y/(kBT/H)1/2

n/n c

Wi=0

FD: Wi=20

HI: Wi=20

Figure 5.11 Steady-state chain center-of-mass concentration profiles predicted by the BD simula-tion of chains (Ns = 10) for a highly confined polymer solution,2h = 2.9Rg.

84

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y/(kBT/H)1/2

n b/nb,

c

Wi=0

FD: Wi=20

HI: Wi=20

Figure 5.12 Steady-state bead-concentration profiles predicted by the BD simulation of chains(Ns = 10) for a highly confined polymer solution,2h = 2.9Rg.

85

0 0.5 1 1.5 20

0.5

1

1.5

2

y/(kBT/H)1/2

< R

>c/(

k BT

/H)1/

2

<x>c

<y>c

<z>c

Figure 5.13 Polymer stretch as a function of the wall-normaldirection,y, for Wi = 0 (no flow);2h = 2.9Rg.

86

wall cancel one another out so hydrodynamic migration away from the walls is suppressed. Simple

steric effects thus dominate− since chains in flow take up less room in the wall-normal direction

than they do at equilibrium, they can more easily sample the regions near the wall, so there is a

weak net migration toward the wall [40].

5.4.4 General Flux Expression for Dumbbells

To conclude the discussion, we revisit the theoretical expression, Eq. (4.51). That expression

was derived in the point-dipole limit, where there are no steric effects and no screening via the

walls of hydrodynamic interactions between different parts of the chain. Therefore, it cannot be

expected to be predictive in the case whereh ∼ Rg. Based on the theoretical framework of Ma

and Graham [94], it is straightforward to develop a more general theoretical expression (still in the

context of a dumbbell model) that does not make the point-particle approximation. LetΨ(q, rc)

be the conformational probability distribution of a dumbbell with connector vectorq and center-

of-mass positionrc. At any positionrc,∫

Ψdq = 1. If Fsi (q) is the connector force andFw

i (q, rc)

is the excluded volume force between beadi and the walls, then the flux expression is

jc =nv +n

8〈qq〉 : ∇∇v +

1

2

⟨Ω ·

(Fs + kBT

∂

∂qln Ψ

)⟩n

−⟨

DK · ∂ ln Ψ

∂rc

⟩

n − 〈DK〉 · ∂n

∂rc

+1

2kBT〈(D11 + D21) · Fw

1 + (D12 + D22) · Fw2 〉n,

(5.32)

where

Ω = (Ω11 − Ω22) + (Ω21 − Ω12) ,

DK =1

4[(D11 + D22) + (D21 + D12)] .

(5.33)

The first four terms in this expression correspond directly to those in Eq. (4.51), while the last term

represents the wall exclusion effect, which leads to a static depletion layer of thickness∼ Rg. In

a single-wall domain or when2h ≫ Rg, a flexible molecule in flow withWi ≫ 1 will exhibit

a depletion layer of thickness≫ Rg, as discussed in Sections 5.4.1 and 5.4.2, making the steric

wall effect largely irrelevant. In contrast, when2h ∼ Rg, hydrodynamic wall effects become

87

0 0.5 1 1.5 21

2

3

4

5

6

7

8

y/(kBT/H)1/2

<x>

c/(k B

T/H

)1/2

Wi=0

FD: Wi=20

HI: Wi=20

Figure 5.14 Polymer stretch in the flow direction,x, as a function of the wall normal direction,y;2h = 2.9Rg.

88

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

y/(kBT/H)1/2

<y>

c/(k B

T/H

)1/2

Wi=0

FD: Wi=20

HI: Wi=20

Figure 5.15 Polymer stretch in the confined direction,y, as a function of the wall normal direction,y; 2h = 2.9Rg.

89

negligible and the final term in Eq. (5.32) becomes significant. Indeed, as shear rate increases

and the dumbbell begins to align parallel to the wall, this term changes accordingly, allowing the

dumbbell to sample closer to the wall, as shown in the chain simulations of Section 5.4.3.

5.5 Effect of Finite Reynolds Number on Wall-induced Hydrodynamic Migra-tion

Theory and simulation approaches to confined polymer hydrodynamics that use Stokeslet-

based methods explicitly enforce the condition that the Reynolds number is small [73, 71, 72,

75, 74, 94, 140, 63]. Other simulation approaches such as molecular dynamics (MD) [80], dissi-

pative particle dynamics (DPD) [46] and lattice Boltzmann (LB) [149] do not explicitly impose

this condition. In particular the DPD results of Fanet al. [46] were obtained at Reynolds num-

bers based on the channel height on the order of102, and migration away from the walls was not

observed. It is therefore important to understand the effect of Reynolds number on wall-induced

migration.

Consider for definiteness the single wall case, with the center of mass of the dissolved polymer

chain a distancey from the wall. Let us impose a flow with uniform shear rateγ, in which the

relevant Reynolds number is

Rey =γy2

ν. (5.34)

whereν is the kinematic viscosity of the solvent. The time scale fordiffusion of momentum

between chain and wall istd = y2/ν, while the velocity of the chain isγy. So the distance traveled

by a chain in the timetd is γytd = yRey. If Rey is small, then the chain moves only a small distance

in the timetd that is required by the flow perturbation induced by the chain’s stress to propagate

to the wall and back− it is the effect of the wall on this perturbation that drives the migration

mechanism we consider here [94]. OnceRey ≫ 1, however, the chain moves downstream a

distance≫ y in the timetd − by the time the perturbation induced by the chain propagatesto the

wall and back to the original position occupied by chain, thechain is no longer at that position,

but is downstream a significant distance, and thus the wall will not have a significant effect on the

chain’s behavior.

90

Another view of this phenomenon is illustrated in Fig 5.16, which shows schematically the

shape of the flow perturbation due to a stretched chain and itsimage, in a reference frame moving

with the chain, in the casesRey ≪ 1 andRey ≫ 1. WhenRey ≪ 1, the shear flow does not

significantly distort the perturbation, but whenRey ≫ 1 the shear convects the perturbation into a

highly anisotropic shape and the perturbation due to the wall (or equivalently due to the image of

the chain’s stresslet) does not affect the chain itself.

This simple argument probably explains the absence of migration in the DPD simulations of

Fan and coworkers [46] and demonstrates the qualitative importance of the Reynolds number for

migration of suspended polymer chains or other deformable particles near walls. Finally, asRe

increases and the purely viscous effect studied here becomes less important, lift effects arise [130,

151, 86, 48], further complicating the interpretation of finite Resimulation results.

5.6 Conclusion

This paper examines cross-stream migration due to hydrodynamic wall effects in dilute flowing

polymer solutions in the regimesRg ≪ L ≪ h (e.g. flow in a half-plane),Rg ≪ h ∼ L and

Rg ∼ h. In the former two cases simulations are compared to a previously developed theory

for point-dipole molecules. Both simulations and theory indicate strong migration away from

the confining walls. Whenh ∼ L the standard dumbbell model breaks down because of the

coarse discretization of the force distribution. In highlyconfined domains,Rg ∼ h, hydrodynamic

migration effects are overwhelmed by steric effects which lead chains to migrate toward walls in

flow rather than away.

91

(a) Reh<<1

Wall Flow perturbationdue to image

Flow perturbationdue to particle

Wall

(b) Reh>1

Flow perturbationdue to image

Flow perturbationdue to particle

Figure 5.16 Schematic of the hydrodynamic migration mechanism (a)Rey ≪ 1: wall-inducedmigration – momentum diffusion to the wall and back to the particle is fast; (b)Rey ≫ 1: Nowall-induced migration – the shear flow distorts the velocity perturbation due to the particle so thatthe particle is not affected by the presence of the wall.

92

Chapter 6

Simulating Polymer Solution Using Lattice-Boltzmann Method

In previous chapters, the dynamics of dilute polymer solutions in simple geometries, single-

wall confinement and slit, are discussed. The rest of this thesis is devoted to the behavior of

polymer solutions flowing through more complex geometries.A simulation method which couples

a bead-spring chain model of the polymer molecule to a Lattice-Boltzmann fluid is implemented.

The strengths and complications of this method are discussed. In Chapter 7, we use this method to

investigate the transport and dynamics of flowing dilute polymer solutions in a grooved channel.

6.1 Introduction

Modeling of hydrodynamic interactions in flowing polymer solutions confined in complex ge-

ometry or with high concentration remains a challenge for both theories and simulations. The

commonly used methods including Brownian Dynamics(BD), Molecular Dynamics (MD), and

Dissipative Particle Dynamics (DPD) have inherent strengths, but also some disadvantages. Al-

though elegant and well-understood in simulating dynamicsof dilute polymer solutions in free

space or relatively simple geometry, the Brownian Dynamicsmethod with fluctuating hydrody-

namic interactions is prohibitively expensive when dealing with polymer solutions in complex

geometry or with high concentration. It has an unfavorable scaling with the number of the inter-

action sites in the system: the time to calculate a single step for a chain ofN segments scales as

N2.25c [73], with the computational cost dominated by the factorization of the diffusion tensor.

For the complex geometry, the challenge lies in the construction of the diffusion tensor (or mo-

bility matrix) with correct no-slip boundary conditions atconfining surfaces. On the other hand,

93

Molecular Dynamics (MD) and Dissipative Particle Dynamics(DPD) [77] treat the solvent parti-

cles explicitly, leading to CPU intensive simulations of several thousand particles even for a single

chain of 30 monomers. Tremendous efforts are being devoted to developing novel or improved

efficient simulation methods. For example, there is an on-going project in our group incorporating

the hydrodynamic interactions in Brownian Dynamics for a general geometry [63, 65].

The direction we take in this thesis is a simulation method that couples a bead-spring chain

model for the flexible polymer with a Lattice-Boltzmann Method (LBM) for the surrounding sol-

vent [3, 4, 5]. The Lattice-Boltzmann Method is an alternative way to solve flow problems gov-

erned by the Navier-Stokes equation. It is based on the microscopic Boltzmann equation for the

particle distribution function, in contrast to the traditional numerical methods which focus on the

macroscopic variables, such as velocity and pressure. The Lattice-Boltzmann Method has been

successfully applied to a variety of flow problems [84, 4, 115, 98, 60], and offers an easy and fast

way to resolve the hydrodynamics in complex geometry because of the straightforward implemen-

tation of boundary conditions. A bead-spring chain model ofthe polymer molecule can be coupled

to the Lattice-Boltzmann model of the solvent to simulate the dynamics of polymer solutions [3].

The fluid exerts a hydrodynamic friction force on each polymer bead proportional to the difference

between the bead velocity and the local fluid velocity at the bead position. In return, the force by

each polymer bead is redistributed back to the fluid. In otherwords, the polymer chain and the

solvent exchange momentum through the friction forces. This method provides a straightforward

and computationally efficient alternative to Brownian Dynamics, incorporating the same level of

description of the hydrodynamic and thermodynamic forces.

In Section 6.2, we outline the essential ingredients of the Lattice-Boltzmann Method. In Sec-

tion 6.3, the bead-spring chain model of the polymer molecule is explained. The simulation param-

eters are discussed in Section 6.4. In Section 6.5, the simulation results on the chain center-of-mass

distribution in shear flow confined in a slit is presented. Thecomplications of the LBM is discussed

in Section 6.6. Finally, we give a conclusion in Section 6.7.

94

6.2 Lattice-Boltzmann Method

The Lattice-Boltzmann Method discretizes the Boltzmann equation in space, velocity and time

[28], leading to a simple equation for a discrete velocity distribution functionfi(r, t) describing

the number of particles at lattice siter at time t with velocity ci. All the details of molecular

motion in the Boltzmann kinetic equation are smeared out except those that are really strictly

needed to recover the macroscopic hydrodynamic behavior ofthe fluid - mass, momentum, and

energy conservation. A simple convection (or streaming) combined with a relaxation (or collision)

process allows the recovery of the non-linear macroscopic advection. Boundary conditions are

implemented using a bounce-back rule which is also very simple, compared with other numerical

schemes, suggesting the Lattice-Boltzmann Method as a promising model for hydrodynamics in

complex geometries. A particular Lattice-Boltzmann modelis specified by a set of discrete veloc-

ities ci, equilibrium velocity distributionf eqi , boundary treatment, and a collision operator which

advances the velocity distribution with time.

6.2.1 Velocity Set

Theci must be chosen so that in one time step, a particle beginning at one lattice site ends up

on a neighboring one. Furthermore, to recover faithful fluiddynamics, the discrete velocities must

guarantee mass, momentum, and energy conservation as well as rotational invariance. In3D space,

one commonly used model, the D3Q19 model (19 velocities in 3 dimensions) [84], which satisfies

these conditions, is shown in Fig. 6.1. The D3Q19 model consists of one zero velocity (or (0, 0,

0)), six velocities with speed 1 (connecting a cube’s centerto its nearest-neighbor face centers, or

(±1, 0, 0), (0,±1, 0), and (0, 0,±1)), and 12 velocities with speed√

2 (connecting the cube’s

center to its edge centers, or (±1, ±1, 0), (0,±1,±1), and (±1, 0,±1)). Here speed is treated in

unit of ∆x/∆t where∆x is the lattice spacing and∆t the time step size. There are other velocity

models, like 14-velocity, 18-velocity and 27-velocity models. The 14-velocity model suffers from

“checkboard” invariants [82]. Ladd and Verberg [84] pointed out that 19-velocity model (D3Q19)

leads to substantial improvements over the 14 velocity model in the equipartition of energy between

95

x

y

z

Figure 6.1 The set of discrete velocities in a D3Q19 model shown in a lattice cube. The solidparallelogram represents thexy plane, the dashed rectangle theyz plane, and the dotted parallelo-gram thexz plane. The D3Q19 model consists of a zero velocity represented by the cube center,six velocities with magnitude unity represented by the arrows pointing to the centers of the cubefaces, and12 velocities with magnitude

√2 represented by the arrows pointing to the cube-edge

centers.

96

particles and fluid in simulations of Brownian suspensions,but that no additional improvement in

accuracy was found when simulating incompressible flows with a more complex model involving

27 velocities. Therefore, the D3Q19 model is utilized in ourstudies.

6.2.2 Equilibrium Velocity Distribution

In steady uniform flow with velocityv, the velocity distribution function of the D3Q19 model

can be represented as a second order expansion of the Maxwell-Boltzmann distribution in Mach

number. In the Lattice-Boltzmann literature this is often called the “equilibrium” distribution and

is given by [143, 121]:

f eqi = aci

[ρ +

ρv · ci

c2s

+ρvv : (cici − c2

sI)

2c4s

], (i = 0, 1, · · · , 18) (6.1)

whereρ is the fluid density,v is the local velocity,cs is the speed of sound

c2s =

1

3

(∆x

∆t

)2

(6.2)

with ∆x the lattice grid size,∆t the time step, andaci is a normalized weight that describes the

fraction of particles with velocityci at thermodynamic equilibrium (ie. whenv = 0). When all the

nodes are at their so called “local” equilibrium state, the global flow is actually at steady state, not

necessarily at rest. In order for the viscous stresses to be independent of direction, the velocities

and the weight must also satisfy the isotropy condition:

∑

i

aciciαciβciγciν = C4c4 (δαβδγν + δαγδβν + δανδβγ) , (6.3)

wherec = ∆x/∆t, andC4 is a numerical coefficient depending on the choice of weights. The

optimum choice of weights for D3Q19 model is [84]

a0 =1

3, a1 =

1

18, a

√2 =

1

36. (6.4)

In this case the coefficient isC4 = (cs/c)4.

In the D3Q19 model, 19 moments of the velocity distribution function can be defined. The first

ten moments give the densityρ, the momentum densityj = ρv, and the momentum flux tensor

97

Π = ρvv:

ρ =∑

i

f eqi , (6.5)

j = ρv =∑

i

f eqi ci, (6.6)

Π = ρvv =∑

i

ficici. (6.7)

The equilibrium distribution is used in Equation (6.5) and (6.6) because mass and momentum are

conserved during the collision process. The equilibrium momentum flux is given as

Πeq =∑

i

f eqi cici = ρc2

sI + ρvv. (6.8)

The remaining 9 moments refer to kinetic energy fluxes, whichconserve energy. They are non-

hydrodynamic modes and irrelevant to the Navier-Stokes equations.

6.2.3 Collision Operator

The velocity distribution function evolves with time according to a discrete analogue of the

Boltzmann equation,

fi(r + ci∆t, t + ∆t) = fi(r, t) −∑

j

Lij

[fj(r, t) − f eq

j (r, t)]∆t (6.9)

whereLij are the matrix elements of the linearized collision operator L for dissipation due to fluid

particle collisions. The collisions relax the fluid towardsthe local equilibrium. Local relaxation is

justified given the particle mean free path is much shorter than the lattice size, i.e.Kn ≪ 1. It has

been shown that Equation (6.9) recovers the Navier-Stokes equations at the low Mach number and

low Knudsen number limit by means of a Chapman-Enskog expansion [12, 28].

Governing the relaxation of the velocity distribution function fi, the collision operatorL is a

19 × 19 matrix in the D3Q19 model. The 19 eigenvalues ofL, (τ−10 , τ−1

1 , · · · , τ−118 ), characterize

the relaxation time of the 19 moments. For the conserved moments (ρ andρv), the relaxation

time is infinite andτ−1i = 0. In the Bhatnagar-Gross-Krook (BGK) [29, 121] collision operator,

which is the most popular one because of its simplicity and computational efficiency, the relaxation

98

time for the momentum flux momentsΠ are set to a single constantτi = τs. In general,τi for

other moments irrelevant to the Navier-Stokes equation areset to∆t, which both simplifies the

simulation and ensures a rapid relaxation of the non-hydrodynamic modes [83]. Following the

simulation algorithm of Ladd [83], the post-collision velocity distributionf ∗i is written as

f ∗i = aci

[ρ +

j · ci

c2s

+(ρvv + Πneq,∗) : (cici − c2

sI)

2c4s

], (6.10)

where

Πneq,∗ = (1 +1

τs)Πneq +

1

3(1 +

1

τs)(Πneq : 1)1, (6.11)

with Πneq = Π − Πeq. Πeq is the traceless part ofΠneq. The kinematic viscosityν of the fluid is

determined by the relaxation timeτs as:

ν = c2s (τs − 0.5) . (6.12)

Figure 6.2 illustrates one Lattice-Boltzmann step consisting of a streaming process and a single-

time-relaxation process in 2D sapce. Without the external force, the equilibrium velocity distri-

bution consists simply equal amount of fluid particles for each of the discretized velocities. A

non-equilibrium velocity distribution has more particlesfor some velocities while less particles for

other velocities. In the streaming process, particles convect to the neighboring lattice sites accord-

ing to the direction of their velocities. In the relaxation process, the collision rules force the larger

fi’s at the site to decrease and the smallerfi’s to increase so that the velocity distribution relax

toward the equilibrium one.

6.2.4 External Force

In the presence of an external field, such as a pressure gradient or a gravitational field, a force

densityF on the fluid needs to be included in the model. The force altersthe velocity distribution

functionfi such that velocity grows in the direction of the force and shrinks in the opposite direc-

tion, and thus generates net flow in the fluid. With the body force in the system, the time evolution

equation of the Lattice-Boltzmann model, Equation (6.9), is modified by an additional contribution

99

streaming relaxation

τs = 1

equilibrium distribution

(a) single-time-relaxation with τs = 1

streaming relaxation

τs = 2

halfway distribution

(b) single-time-relaxation with τs = 2

Figure 6.2 In the single-time-relaxation model, the velocity distribution at each site relaxes towardthe equilibrium one at each time step. Without the external force, the equilibrium velocity distri-bution consists simply equal amount of fluid particles for each of the discretized velocities. Thefigure shows the two processes that occur during each time step: the streaming and the relaxation.First, the incoming velocity distribution assembles at a lattice site as the particles in the neigh-boring sites stream along their directions of motion to thatsite. Second, the incoming distributionrelaxes due to the particle collisions, according to the single-time-relaxation rule, towards the equi-librium distribution. (a) Whenτs = 1, the incoming velocity distribution relaxes to the equilibriumdistribution in one time step. (b) Whenτs = 2, the post-relaxation distribution is halfway betweenthe incoming and the equilibrium distributions.

100

f fi (r, t) [84]

fi(r + ci∆x, t + ∆t) = fi(r, t) −∑

j

Lij

[fj(r, t) − f eq

j (r, t)]∆t + f f

i (r, t). (6.13)

The forcing termf fi (r, t) is given by

f fi = aci

[A +

B · ci

c2s

+C : (cici − c2

sI)

2c4s

]∆t, (6.14)

whereA, B, andC are determined by requiring that the moments off fi are consistent with the

hydrodynamic equations:

A =∑

i

f fi = 0, (6.15)

B∆t =∑

i

f fi = F∆t, (6.16)

C = vF + Fv. (6.17)

The second momentC is usually neglected. In that case, more accurate solutionsto the velocity

field are obtained if an additional momentum is added to each node [93],

ρv′ =∑

i

fici +1

2F∆t. (6.18)

However, if the conventional definition of the momentum flux is retained, the expression forC

needs to be modified to account for discrete lattice effects.Nevertheless, for a spatially uniform

force, numerical simulations show that variations inC have a negligible effect on the flow [84].

6.2.5 Boundary Conditions

For confined polymer solutions, an appropriate boundary treatment must be adopted in the

Lattice-Boltzmann simulation to incorporate the no-slip boundary conditions imposed by the con-

fining surfaces. The boundary treatment also influences the accuracy and the stability of the

Lattice-Boltzmann Method. The no-slip boundary conditions at the confining surfaces are real-

ized by a “bounce-back” scheme [84, 28].

101

In a typical confined Lattice-Boltzmann fluid, there will be fluid nodes, on which the flow

collision operator is applied, and solid nodes, which represent the walls. The node type can be

marked by a Boolean marker. At the fluid-solid interface, there are fluid nodes where flows impinge

on at least one solid node. during the streaming step, the component of the distribution function

that would stream into the solid node is bounced back and endsup back at the fluid node, but

pointing in the opposite direction. This means that incoming particles are reflected back towards

the nodes they came from. Assuming thatci is the velocity towards the wall at one fluid node, and

c′i is the opposite velocity (ri = −ri′), the velocity distribution is changed as following:

fi′(r, t + ∆t) = fi(r, t). (6.19)

The logic of using the bounce-back rule to achieve zero velocity at the wall can be argued as fol-

lowing: at the wall node, corresponding to the incoming particle, we can imagine there is another

particle that moves in the opposite direction on the other side of the wall. The bounce-back rule

ensures the zero velocity by simply sending the imaginary particle to the fluid to cancel the incom-

ing momentum. The straightforward implementation of the no-slip boundary by the bounce-back

scheme makes LBM an promising method for simulating fluid flowin complex geometries, as

demonstrated in flow through sandstones [24, 49].

For a node near a boundary, some of its neighboring nodes are solid and lie outside the simu-

lation domain. The bounce-back scheme is a simple way to fix the unknown distributions on the

solid wall nodes, restricting the accuracy of the Lattice-Boltzmann method to only first-order on

the boundary. Ziegler [156] has shown that if the fluid-solidboundary is shifted one half lattice

spacing into the fluid along the link vector joining the solidand fluid nodes, then the bounce-back

rule gives second-order accuracy. There are cases where thesecond-order accuracy is desired and

the zero-velocity plane must be located exactly on the solidboundary nodes rather than being

shifted from the location of the solid boundary nodes half-way into the fluid. A great deal of ef-

forts have been made to maintain the second-order accuracy in these cases, such as using velocity

gradients or a pressure constraint at the wall nodes [30, 28,134, 109]. In the present work those

more complex approaches are not implemented; Ziegler’s treatment is used.

102

For a moving solid boundary like that in a plane Couette flow, the fluid gains momentum from

the wall. Accordingly, the incoming velocity distributionat the boundary fluid nodes is altered in

proportion to the velocity of the wallvb:

fi′(r, t + ∆t) = fi(r, t) −2aciρvb · ci

c2s

. (6.20)

The bounce-back rule for the stationary and moving boundaries are illustrated in Figure 6.3.

6.3 Polymer Chain Model

6.3.1 The Bead-spring chain Model

We model a linear flexible polymer molecule dissolved in a good solvent as a bead-spring chain

model. The whole chain is discretized intoNs units, and each unit is represented by an elastic

“spring”. The mass of the segment is concentrated to a “bead”which is also the interaction site;

there will beNb = Ns + 1 beads. For the spring force, we adopt the Finite Extensible Non-linear

Elastic (FENE) spring model:

Fsi =

3kBT

Nk,sb2k

qi

1 − (qi/q0)2, i = 1, . . . , Ns, (6.21)

wherekB is the Boltzmann constant,T is the absolute temperature,qi is the stretch of theith

spring,Nk,s is the number of Kuhn segments per spring, andq0 = Nk,sbk is the contour length

of that spring. For most of the results reported here, we chooseNs = 10 and each spring has

Nk,s = 10 Kuhn segments with length ofbk = 0.106µm, corresponding to a halfλ-phage DNA

molecule [72].

A Gaussian excluded volume potential between any two beads of the chain is employed [71],

Uevij =

1

2υkBTN2

k,s

(3

4πS2s

) 3

2

exp

[−

3r2ij

4S2s

], (6.22)

whereυ = b3k is the excluded volume parameter,rij is the distance between beadi and beadj,

andS2s = Nk,sb

2k/6 is the radius of gyration of an ideal chain consisting ofNk,s Kuhn segments of

lengthbk.

103

(a)

solid fluid

t

bounce

backsolid fluid

t + ∆t

(b)

solid

vw

fluid

t

bounce

backsolid

vw

fluid

t + ∆t

Figure 6.3 Bounce-back rule for a solid-fluid interface. Thearrows shows the velocity directionand their lengths are proportional to the magnitude of the velocity distribution in that direction.(a) Bounce-back rule for a stationary solid boundary. (b) Bounce-back rule for a moving solidboundary.

104

6.3.2 Coupling of the Polymer Chain and the Solvent

The bead-spring chain and the Lattice-Boltzmann fluid are coupled together though a friction

force and by random Brownian fluctuations. It is assumed thatthe drag force exerted by the fluid

on one bead is proportional to the velocity difference between the bead and the fluid at the bead’s

position:

Fhi = ζ(ri − v(ri)), (6.23)

whereζ = 6πρνa is the bead friction coefficient with the bead radiusa = 0.08µm in our simula-

tions,ri is the velocity of theith bead, andv(ri) is the local fluid velocity at theith bead position.

v(ri) is determined by a trilinear interpolation of the fluid velocity v(nn)j at the neighboring lattice

sites (n.n.):

v(ri) =∑

j∈(n.n)

wjvnnj . (6.24)

The weighting functionswj of the bead’s neighboring lattice sitej are normalized.vnnj is the

fluid velocity at the neighboring lattice sitej. The momentum exchange between the fluid and

the bead,je = −Fhi ∆t/∆x3, is distributed back to the neighboring lattice sites with the same

weighting functionswj used in the trilinear interpolation. For velocitycq on the neighbor sitej,

the momentum exchange is given by

Fj,q = wjje · cq. (6.25)

In principle, thermal fluctuations can be incorporated intothe Lattice-Boltzmann fluid via the

addition of a random stress in the momentum flux during the collision process [83, 1]. In the sim-

ulation of suspension systems, the fluid thermal fluctuations affect the motion of particles through

the no-slip boundary conditions at the particle-fluid interface. Thus the Brownian motion for col-

loidal particles can be captured with this approach. Unfortunately, that is not the case in the usual

(and present) approach to simulation of polymer solutions,where the beads of the polymer chain

are treated as point forces. In this case, the random fluctuations in the fluid are not properly trans-

mitted to the polymer beads – the fluid fluctuations exist onlyon wavelengths larger than the lattice

size∆x, while the polymer beads have a scale smaller than it. Therefore, to generate proper fluc-

tuations of the polymer beads, we directly add a Brownian force (Gaussian, with zero mean and

105

variance2kBT/∆t) to the equation of motion for each polymer bead [3, 150]. Details of that is

explained in Chapter 3

6.3.3 Equation of Motion for Polymer Beads

In our simulation, each polymer bead has mass ofm. The position and velocity of the individual

bead are updated using the explicit Euler method:

ri(t + δt) = ri(t) + Fiδt/m, (6.26)

ri(t + δt) = ri(t) + ri(t)δt, (6.27)

whereδt is the integration time step (we callδt polymer time step to distinguish from the Lattice-

Boltzmann fluid time step∆t), andFi denotes the total force acting on beadi:

Fi = Fsi + Fev

i + Fbi + Fh

i + Fwalli . (6.28)

Fwalli is the wall excluded volume force defined by a cubic bead-wallrepulsive potential of the

form

Uwalli =

Awall

δ3wall

(h − δwall)3 for h < δwall (6.29)

= 0 for h ≥ δwall, (6.30)

whereh represents the distance of beadi from the wall in the wall-normal direction (into the fluid).

Throughout this work, we takeAwall = 25kBT andδwall = 3a wherea is the bead radius.

This inertial form of the equation of motion needs to be integrated at time scale given bym/ζ .

In our simulation,m = 1 and ζ = 0.6. To resolve the inertial time scale ofm/ζ = 1.7, we

choose the polymer time stepδt = 0.1. One might think of eliminating this tiny time scale by

ignoring the bead mass in the equation of motion. However, the inertialess limit of the equation of

motion is a singular one. In Brownian dynamics, this singular limit is reflected in the presence of

the divergence of the bead mobility tensor in the stochasticdifferential equation for bead positions

[44, 50, 58]. When the Lattice-Boltzmann scheme is used to evolve the fluid velocity, there is no

straightforward way to compute this divergence; thus the bead inertia is retained (If it is not, the

concentration distribution at equilibrium will be artificially nonuniform in a complex geometry.).

106

6.4 Simulation Parameters

In applying Lattice-Boltzmann Method to complex fluid modeling, it is important to be con-

scious of the wide spectrum of length and time scales in the real system. LBM, and in general any

multi-scale simulation method, cannot fully resolve the hierarchy of length and time-scales present

in complex fluids. Thus, there persists the question of how the time and length of the lattice fluid

relate to the scales of the physical phenomena being studied[68].

In order to capture the intra-chain hydrodynamic interactions in polymer solutions, the grid

size in the Lattice-Boltzmann Method should ideally be smaller than the average spring length.

However, strictly satisfying this condition leads to very small grid spacings and correspondingly

large computation times - for a given flow domain size, the computation time scales as(1/∆x)3.

With the Kuhn length of 0.106µm for λ-DNA molecule, our chain model withNs = 10 and

Nk,s = 10 for a halfλ-DNA would correspond to an average spring length of 0.34µm and radius

of gyration ofRg = 0.5µm. We thus choose a lattice spacing of∆x = Rg = 0.5µm, a compro-

mise between quantitative accuracy and computational feasibility. With this lattice spacing, the

hydrodynamic radius of the coarse-grained bead isa = 0.159∆x. Another important free param-

eter is the fluid relaxation timeτs which determines the viscosityν according to Equation (6.12).

Choosingτs = 1.1 and matching the viscosity to that of water, we get the Lattice-Boltzmann time

step asδt = 8 × 10−5s.

A multi-scale simulation model should be able to separate the time and length scales of interest

from those of not. The extent of the separation is measured byinsightful dimensionless numbers.

Specifically, in simulating the polymer solution using Lattice-Boltzmann Method, the important

numbers are: Reynolds numberRe, Mach numberMa, Schmidt numberSc, and Weissenberg

107

numberWi. They are defined as:

Re =vl

ν, (6.31)

Ma =v

cs, (6.32)

Sc =ν

D, (6.33)

Wi = γλ, (6.34)

wherev andl are the characteristic velocity and length scale respectively, cs is the speed of sound,

ν is the kinematic viscosity,D is the polymer diffusivity,λ is the polymer chain relaxation time,

andγ is the characteristic shear ratev/l. To achieve good separation of the length and time scales in

simulating micofluidic flow of polymer solution, a near-zeroReynolds number and Mach number

and a large Schmidt number are desired. The dependence of thepolymer dynamics onWi under

these conditions is the issue of primary interest. Given thegeometry and the fluid viscosity, the

only way to decrease the Reynolds number is to decrease the characteristic fluid velocity, which

will decrease theMa number as well. However, one realizes that doing this will decrease the shear

rate in the system, and thus decrease the Weissenberg number. Maintaining the same Weissenberg

number requires increasing the chain relaxation time by decreasing the temperature. As a side

effect, the Schmidt number also increases because the chaindiffusivity decreases as temperature

decreases. The price we pay here is a longer simulation time (which grows linearly with the chain

relaxation time).

As we can see from the above discussion, there are many degrees of freedom in the Lattice-

Boltzmann Method: the grid resolution, the fluid relaxationtime, the Reynolds number, etc. On

one hand, these factors provide a sophisticated method to model complicated flow problems; on the

other hand, a lot of subtleties are introduced. One needs to be very careful in choosing simulation

parameters. A discussion on the effect of the grid size, the fluid relaxation time and the Reynolds

number on applying LBM to polymer solution is given in Section 6.6.

In this section, we determine the chain relaxation time and diffusivity for the choice of param-

eters used in the following studies. The LBM parameters areτs = 1.1, andζ = 0.6 in lattice units.

ThekBT will be tuned between1 × 10−3 and5 × 10−5 to obtain desired Weissenberg numbers.

108

The smallest Schmidt number isSc = 1.3×103 corresponding tokBT = 0.001. For the Brownian

motion of the coarse-grained beads, a integration time stepof deltat = 0.1 is chosen, which is one

order of magnitude smaller than the mass-relaxation timem/ζ = 1.7 and three order of magnitude

smaller than the bead diffusion time over its own size in all the following simulations.

Figure 6.4 shows the mean square chain stretch ,< X2 > (X = maximum dimension of the

chain), of initially stretched DNA chains as the chains are allowed to relax in a periodic simulation

box of size 40x40x40. The chain contour length is 21.9 lattice spacings. The chain relaxation

timescan be extracted by fitting the last 30% of the curve to exponential decay function. The

relaxation time is found to beλ = 426 for kBT = 0.001, andλ = 2037 for kBT = 0.0002, in

lattice units. The chain relaxation time is inversely proportional to the temperature as expected. In

our following simulations, the relaxation time will be tuned by changing temperature to obtain the

desired Weissenberg numbers.

We also performed simulations of DNA chains to determine thechain diffusivity. A chain is

released in the simulation domain to diffuse. The mean-square-displacement of the chain center-

of-mass,< [r(t) − r(0)]2 >, is tracked as a function of timet. Figure 6.5 shows the result for

a chain withNs = 10 at temperature ofkBT = 0.001. A linear fitting to the diffusion equation

< [r(t)− r(0)]2 >= 6Dt gives the chain diffusion coefficient asD = 1.73× 10−4, in lattice units.

Based on the chain diffusivity, we can estimate the chain diffusion time across the channel in the

normal direction. In all our following simulations, the simulation time is at least three channel

diffusion times to ensure steady state.

6.5 Chain Migration in Dilute Polymer Solution Flow in a Slit

In Chapter 4 and Chapter 5, we studied chain migration in shear flow of dilute polymer solu-

tions confined in a slit, using kinetic theory and Brownian Dynamics simulation respectively. The

results corroborate with each other and predict a depletionlayer near the wall due to the hydrody-

namic interactions between chain segments and the walls. The depletion layer thickens when the

109

t

<X

2 >

0 2000 40000

100

200

300

400

kBT=0.0010:λ = 426kBT=0.0002:λ = 2034

Figure 6.4 Relaxation of a stretched polymer molecule in bulk solution. The mean square stretchof the chain< X2 > is plotted against time for a chain ofNs = 10 at two different temper-atureskBT = 0.001, and0.0002. An exponential decay fitting of< X(t)2 >=< X(∞)2 >+X0 exp(t/λ) gives the chain relaxation time asλ = 426 for kBT = 0.001 andλ = 2037 forkBT = 0.0002, in lattice units.X0 andλ are the fitting parameters.

110

t

<[r

c(t)

-rc(

0)]2 >

0 500000 1E+060

200

400

600

800

1000

D = 1.73x10-4

Figure 6.5 Mean square displacement of the center-of-mass of a polymer chain withNs = 10as a function of time in bulk solution. The simulation parameters areµ = 0.2, ζ = 0.6, andkBT = 0.001. A linear fitting to the diffusion equation< [(r(t) − r(0)]2 >= 6Dt gives the chaindiffusion coefficient asD = 1.73 × 10−4, in lattice units.

111

Weissenberg number increases. In this section, we use this problem as a benchmark for the Lattice-

Boltzmann Method. The goal here is to evaluate the method andreveal the inherent subtleties in

LBM.

Consider a dilute solution of halfλ-phage DNA confined in a slit. The DNA molecule is

modeled as a bead-spring chain withNb = 11 beads andNs = 10 springs. Each spring contains

Nk,s = 10 Kuhn segments. The slit height isL = 10Rg. The two walls of the slit slide in

opposite directions to generate simple shear flow. In our Lattice-Boltzmann simulation, periodic

boundary conditions are utilized in flow and neutral directions, and the simulation box size in

these two directions are both20Rg. With the wall velocity ofvw = 0.1, the shear rate isγ =

0.02. The corresponding Mach number isMa = 0.17, smaller than0.3 which is suggested by

Ladd and Verberg as the upper limit [84]. The Schmidt number is at least1300 in all of our

simulations. Here we fix the Reynolds number at 2 (which is practical in terms of the simulation

time) for all the Weissenberg numbers, so we can focus on the effect of Weissenberg number and

hydrodynamic interactions. Later in Section 6.6, simulations with different Reynolds number but

the same Weissenberg number will be performed and the effectof Reynolds number is clarified.

Other LBM parameters areτs = 1.1, bead friction coefficientζ = 0.6. We perform simulations at

different temperatures:kBT = 10−3, 10−4, 5× 10−5, corresponding to Weissenberg number of 10,

100, and 200.

The steady state distribution of the chain center-of-massn is plotted in Figure 6.6 as a function

of the positiony in slit normal direction. Note thatn is normalized such that the area under each

curve is unity, and the positiony is scaled by the chain radius of gyrationRg. We observe chain

migration away from both slit walls and towards the slit center at all finite Weissenberg numbers,

which is in agreement with our kinetic theory model [94] and Brownian Dynamics simulation [64].

Moreover, a consistent trend of stronger migration at higher Weissenberg numbers is observed,

again in line with prior studies.

To check the chain-confinement hydrodynamic interactions in LBM simulation, we conduct

simulations of polymer solutions in shear flow confined in a slit at Wi = 50 with and without hy-

drodynamic interactions (HI). Simulation without hydrodynamic interactions is referred to as free

112

y/Rg

n

0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

EquilibriumWi = 10Wi = 100Wi = 200

Figure 6.6 Steady state chain center-of-mass distributionof a dilute polymer solution undergoingsimple shear flow confined in a slit at Weissenberg number of 0,10, 100, and 200. The center-of-mass distributions are normalized such that the area under the curves are all unity.

113

draining (FD) simulation. The free draining model is implemented by eliminating the momentum

transfer from polymer chains to the solvent, i. e. the chainsstill feel the hydrodynamic drag force

but they do not perturb the solvent. The steady state chain center-of-mass distributions from HI and

FD simulations at Weissenberg number of 50 are compared in Figure 6.7, as well as the equilibrium

distribution. As we can see, the chain center-of-mass distribution from the free draining simulation

is flat except in the region very close to the slit walls due to the excluded volume effect. Statisti-

cally, the FD chain center-of mass distribution is undistinguishable from the equilibrium one. On

the other hand, the chain center-of-mass distribution fromsimulation with hydrodynamic interac-

tions displays a peak at the slit center, indicating chain migration. This result supports the idea

that the cross-streamline chain migration is due to the hydrodynamic interactions. It also shows

that the hydrodynamic interactions are correctly resolvedqualitatively in our Lattice-Boltzmann

Method simulation.

To perform a quantitative comparison, we also conducted Brownian Dynamics (BD) simulation

with the same bead-spring chain model atWi = 50 andRe = 0, using the method described in

a previous paper [64]. The result from BD and LBM are plotted in Figure 6.8. At the same

Weissenberg number, Lattice-Boltzmann Method gives a muchweaker migration compared to the

BD simulation. We note that the weaker migration is also observed in LBM simulations by other

researchers [149, 33], where chain migration is weak for simulations with Weissenberg less than

50 in both studies. Although might be attributed to the finiteReynolds number in LBM [33], this

discrepancy remains an open question. In Section 6.6, some of the complications of the Lattice-

Boltzmann Method are discussed, aiming to clarify this discrepancy.

6.6 Complications of the Lattice-Boltzmann Method

As we mentioned in Section 6.4, there are many degrees of freedom in choosing Lattice-

Boltzmann simulation parameters. This is particularly true when simulating polymer solutions. As

a result, Lattice-Boltzmann Method is quite prone to many subtleties, and close examination for

systematic errors is required. In this section, we address the complications of Lattice-Boltzmann

114

y/Rg

n

0 2 4 6 80

0.05

0.1

0.15

0.2

EquilibriumWi = 50 FDWi = 50 HI

Figure 6.7 Steady state chain center-of-mass distributionof a dilute polymer solution undergoingshear flow confined in a slit. The solid line is the equilibriumchain center-of-mass distribution, thedotted line is the chain center-of-mass distribution obtained from simulations with free drainingmodel (FD) atWi = 50, and the dashed line is the chain center-of-mass distribution obtained fromsimulations with hydrodynamic interactions (HI) atWi = 50.

115

y/Rg

n

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2 BD-HILBM

Figure 6.8 Steady state chain center-of-mass distributionof a dilute polymer solution undergoingshear flow confined confined in a slit atWi = 50. The solid line is the chain center-of-mass dis-tribution obtained from Lattice-Boltzmann Method, and thedashed line from Brownian Dynamicssimulation with hydrodynamic interactions.

116

Method in simulating polymer solutions in terms of fluid relaxation time, the lattice resolution, and

the finite Reynolds number.

6.6.1 Fluid Relaxation Time

In the single-time-relaxation model of Lattice-BoltzmannMethod, the fluid relaxation timeτs

is a crucial free parameter. Microscopically, it characterizes how fast the velocity distribution on a

lattice site relaxes to equilibrium. Macroscopically,τs determines the fluid viscosity according to

Equation (6.12): the larger theτs, the longer the local non-equilibrium persists, the more efficient

the momentum transfer to neighboring sites, and thus the larger the viscosity. For numerical sta-

bility, τs should be larger than 0.5, while the upper limit is not clearly specified [143]. However, in

microfluidic flow of polymer solutions, the Reynolds number is typically much smaller than 1, and

a largeτs is desired. Practically, in our simulation, we found thatτs beyond 12 can cause instability

in complex geometry. Even in simple geometry, the transientflow field can be distorted at largeτs.

Consider a semi-infinite body of fluid bounded below by a horizontal surface. Initially the fluid

and the solid are at rest. Then at timet = 0, the solid surface is set in motion in the positivex

direction with velocityv0 as shown in Figure 6.9. The fluid velocityvx as a function of distance

from the wally and timet is known to be

vx(y, t)

v0= 1 − erf

y√4νt

, (6.35)

where erf is the error function [16].

Using Lattice-Boltzmann Method, we simulate the evolutionof this flow field with different

values ofτs, and plot the velocity field in dimensionless form in Figure 6.10. Ideally, the curves

obtained from different time should all collapse into one single master curve corresponding to

Equation (6.35). The velocity profile forτs = 1.1 at different time collapse nicely onto the master

curve, while the results forτs = 10.5 deviate from the master curve significantly. This indicates

that the Lattice-Boltzmann fluid has limitations on how fastthe momentum can be transferred while

maintaning proper Navier-Stokes behavior. One can imaginethat due to LBM’s discretization

nature, the momentum at a given lattice site should not be expected to diffuse beyond one lattice

117

y

vx(y,t)

v0

Figure 6.9 Viscous flow of a fluid near a wall suddenly sheared.At time t = 0, the bottom solidsurface is set in motion in the positivex direction with velocityv0

118

unit in one time step. In other words, the kinematic viscosity can not be larger than 1. Inserting

this bound into Equation (6.12), it is seen that anyτs value larger than 3.5 is questionable. We

conclude thatτs has to fall in the range of(0.5, 3.5].

In the chain cross-streamline migration mechanism, the hydrodynamic coupling between the

polymer beads and the confining walls is very crucial [74, 72,94]. Lattice-Boltzmann must capture

this coupling correctly in order to obtain the chain migration. Thus, we now examine the steady

state flow field due to a stretched dumbbell confined in a slit. In Figure 6.11, the results from the

Lattice-Boltzmann Method with differentτs values are compared to finite element solution of the

corresponding Stokes equation. The finite element solutionis assumed to be the exact solution

since a high resolution is chosen in the calculation. The flowfield obtained fromτs = 1.1 agrees

with the the exact solution very well as shown in the figure. However, the flow field corresponding

to τs = 3.5 is significantly different. We take the slices of the flow fields in Figure 6.11 alongx

andy direction and put the results together in Figure 6.12 for closer examination. The flow field

obtained fromτs = 3.5 is quite far from the exact solution in both cases. Thus,in all of simulation,

we chooseτs = 1.1.

6.6.2 Grid Size Effect

In Lattice-Boltzmann Method, the grid resolution should bechosen according to the charac-

teristic length scale of interest in the system. Generally speaking, the hydrodynamics of the fluid

is resolved only down to length scale of the grid size in any discretized method. In order to re-

solve the intra-chain hydrodynamic interactions in polymer solution, the lattice spacing is set as

∆x = 0.5µm in our LBM simulation, close to the average spring length ofthe chain0.34µm.

However, considering that the chain radius of gyrationRg = 0.5µm is only slightly larger than

∆x, the grid resolution might not be fine enough. To address thisissue, simulations with grid size

of ∆x = 0.25µm and∆x = 1.0µm are also performed. In these simulations, all other parame-

ters are kept the same as in Section 6.5 except for the grid resolution. Particularly, the Reynolds

number is 2. Note that even the smallest lattice spacing∆x = 0.25µm is still more than 3 times

119

y/(4ν)0.5

v x/v 0

0 1 2 3 40

0.01

0.02

0.03t=20t=40t=60t=80t=100Exact solution

(a)

y/(4ν)0.5

v x/v 0

0 0.5 1 1.5 20

0.01

0.02

0.03t=4t=8t=12t=16t=20Exact solution

(b)

Figure 6.10 Velocity profile in dimensionless form for flow near a wall suddenly sheared. (a)Results from Lattice-Boltzmann Method withτs= 1.1. (b) Results from Lattice-Boltzmann Methodwith τs= 10.5.

120

Figure 6.11 Contour plot of the wall normal component of the steady state flow field due to astretched dumbbell (white beads connected by dotted line) confined in a slit. (a) Finite elementsolution. (b) Result from Lattice-Boltzmann Method withτs = 1.1. (c) Result from Lattice-Boltzmann Method withτ = 3.5.

121

y

v y

0 2 4 6 8 10-1.5

-1

-0.5

0

0.5

1

1.5Exact Solutionτs = 1.1τs = 3.5

(a)

x

v y

0 10 20 30 40

-0.04

-0.02

0

0.02

0.04

Exact Solutionτs = 1.1τs = 3.5

(b)

Figure 6.12 Comparison of the wall normal component of the steady state flow field due to astretched dumbbell confined in a slit. (a) Slice of the flow field along wall-normal direction atx = 20. (b) Slice of the flow field along the wall-tangential direction aty = 5. The dotted lines in(b) indicates the positions of the two beads of the stretcheddumbbell.

122

larger than the polymer bead sizea = 0.0795µm, which justifies the point force assumption in the

model.

The steady state chain center-of-mass distribution for dilute polymer solution in shear flow

confined in a slit obtained from LBM simulation with three different grid resolutions are plotted

in Figure 6.13. When the grid resolution increases from∆x = 1.0µm to∆x = 0.50µm, the chain

center-of-mass distribution becomes sharper, and therefore closer to kinetic theory and Brownian

Dynamics simulation results. However, further increasingthe grid resolution to∆x = 0.25µm

leads to only a slight change. The chain center-of-mass distribution obtained from grid resolution

of ∆x = 0.25µm, represented as dotted line in Figure 6.13, is statistically undistinguishable from

the distribution corresponding to∆x = 0.5µm, the solid line in Figure 6.13.

6.6.3 Reynolds Number Effect

Another degree of freedom we check with our Lattice-Boltzmann simulation is the Reynolds

number. Although a nearly zero Reynolds number is desired for simulating microfluidic flow, in

practice the Reynolds number is related to the simulation time if we want to keep Weissenberg

number the same. The lower the Reynolds number, the longer the simulation time. A combination

of low Reynolds number and high Weissenberg number is the most computational demanding one.

In our previous simulations, the Reynolds number is2, which allows us to investigate the chain

migration at Weissenberg number as high as 200. Here we fix Weissenberg number as 10, and

change the Reynolds number to reveal the effect of the Reynolds number on our results.

Lattice-Boltzmann simulations with the same parameters asin Section 6.5 except shear rate

and temperature are performed. Temperature is changed according to the shear rate to maintain the

same Weissenberg number. Steady state chain center-of-mass distributions from LBM simulations

with Weissenberg number 10 but different Reynolds numbers of 10, 2, 0.4, and 0.04 are plotted in

Figure 6.14. The steady state chain center-of-mass distributions obtained from different Reynolds

number are statistically undistinguishable. Note that theReynolds numbers cover more than two

orders of magnitude around unity. Within this range of Reynolds number, we find no evidence that

the steady state chain center-of-mass distribution is affected in a significant way by the Reynolds

123

y/Rg

n

0 2 4 6 8 100

0.05

0.1

0.15

dx = 1.0µmdx = 0.5µmdx = 0.25µm

Figure 6.13 Steady state chain center-of-mass distribution of a dilute polymer solution undergoingshear flow confined in a slit atWi=10. Line styles correspond to grid resolution of∆x = 1.0µm(dashed),∆x = 0.50µm (solid), and∆x = 0.25µm (dotted).

124

number. The quantitative discrepancy between LBM and BD thus remains unexplained. It would

be desirable to perform comparison in the doulbe limit∆x → 0, Re→ 0, but LBM computation

in that regime are extremely expensive.

6.7 Conclusion

In this chapter, we performed Lattice-Boltzmann simulation of a dilute polymer solution in

shear flow confined in a slit. As with other simulation methodsthat attempt to span a large sep-

aration of length and time scales, the parameters chosen in our simulations necessarily focus on

long range, large time scale effects such as the flow-inducedchain migration phenomenon. The

chain cross-streamline migration predicted by previous kinetic theory is observed in our Lattice-

Boltzmann simulation. The migration becomes stronger as the Weissenberg number increases,

which is also in agreement with the theory and the Brownian dynamics simulation. However, a

close comparison of the steady state center-of-mass distributions obtained from LBM and BD re-

veals that quantitatively, the migration effect is under-predicted by LBM at the same Weissenberg

number.

A key difference between the Brownian Dynamics simulation and the LBM calculation is that

the inertial effects are neglected in Brownian Dynamics. InBD, the microfluidic flow is always

considered to haveRe= 0, and hydrodynamic interactions between beads are assumed to propagate

instantly. Neglecting inertial effects in microfluidic flowis generally justified becauseRe ≪ 1.

The Reynolds number in our LBM simulation is finite, about 2. However, the steady state chain

center-of-mass distributions obtained from different Reynolds number between 10 and 0.04 show

no evidence that the weaker migration in LBM is due to the finite Reynolds number, at least within

this range of Reynolds number. A comparison between the simulation results using three different

lattice resolution is carried out. It is shown that when the lattice spacing is larger than the polymer

radius of gyration, the hydrodynamic interactions are compromised, leading to a weaker migration.

Lattice resolution beyond the polymer radius of gyration does not improve the result further.

125

y/Rg

n

0 2 4 6 80

0.05

0.1

0.15

0.2

Re = 10Re = 2Re = 0.4Re = 0.04

Figure 6.14 Steady state chain center-of-mass distribution of a dilute polymer solution undergoingshear flow confined in a slit atWi = 10. Line styles correspond to Reynolds numbers ofRe= 10(dotted),Re= 2 (dashed),Re= 0.4 (solid), andRe= 0.04 (dash-dotted).

126

Chapter 7

Polymer Chain Dynamics in a Grooved Channel

In Chapter 6, we investigated the chain cross-streamline migration in smooth slit using Latttice-

Boltzmann Method (LBM). The LBM method is shown to be able to resolve the hydrodynamics

in confined geometry with efficiency. In this Chapter, we willutilize this weapon to explore the

dynamics of polymer solution confined in a non-smooth structured channel.

7.1 Introduction

The dynamics of polymer solutions driven by flow or electric fields in a confined geometry

is a fundamental research topic underlying many practical applications, including enhanced oil

recovery from porous media and separation of synthetic or biological molecules using various

chromatography methods. With recent advances in design andfabrication of novel microfluidic

devices for gene mapping [26, 78, 79], DNA separation and hybridization [125, 61, 142, 141],

this long-standing research topic has attracted renewed interests. In some approaches to this set of

problems, simple devices such as channels with rectangularcross-section are used. Other studies,

however, have begun to examine the behavior of chains in morecomplex geometries. A particular

geometry of interest for recent DNA electrophoresis studies is a slit whose wall contains grooves

or corrugations - quite interesting behavior has been observed during DNA electrophoresis in this

geometry [26, 61, 142]. In the present work we address the related problem of shear flow in the

same kind of geometry.

It has been known that when polymer solutions flow through a porous media, retention of poly-

mer molecules inside the pores occurs. Early studies indicate that when polymer solution flow

127

through porous sandstone cores, the concentration inside the cores is higher than the steady state

inlet-outlet concentration [95, 103]. Aubert and Tirrel [8] reported when dilute polystyrene solu-

tions flow through packed chromatographic column, the polymer retention in the column increased

with increasing shear rate and with polymer molecular weight. Flow rate dependent diffusion of

macromolecules into the pores is attributed to be the main reason. However, the contribution of

adsorption of the macromolecules in the porous media is not entirely clear. Metzner et. al [104]

reported that in channel flow of polyacrylamide solution (0.05-0.2 wt%), high concentration is

observed in the stagnant liquid within the cavity in the channel wall. These earlier studies reveal

that when polymer solution flow through complex geometry, re-partition of the polymer happens

between bulk flow region and relatively stagnant region. Recently, Han et al. [61] reported en-

tropic trap separation of DNA molecules in a grooved channelwith contractions comparable to

DNA persistence length. Further experimental and simulation studies by Streek et al. [142] re-

vealed higher concentration bands close to the groove and upper wall when DNA molecules are

electrically driven through a grooved channel with contraction comparable to DNA radius of gy-

ration. Besides its technological importance on chromatography and electrophresis analysis, these

phenomena also challenge our fundamental understanding ofthe polymer dynamics in complex

geometries.

Previous studies have shown that during flow in a smooth-walled channel, flexible polymer

molecules in solution will migrate towards the channel center, due to the hydrodynamic interac-

tions of the chain segments and the channel walls [74, 72, 94,64]. This migration phenomena

has obvious implications for the surface-adsorption basedchemical and biological applications

[78, 41], as molecules that tend to migrate away from the walls are unlikely to adsorb on them.

Thus, improvement on the channel design is desired to control the chain distribution in the mi-

crochannels. In the present paper, we investigate the cross-streamline migration of chains in dilute

solution during flow in a simple or structured channel, as shown in Figure 7.1. The simulation

method we use couples a bead-spring chain model for dissolved linear flexible polymer molecule

[71] with a Lattice-Boltzmann Method (LBM) for the surrounding solvent [3, 4, 5]. The Lattice-

Boltzmann Method is an alternative way to solve flow problemsgoverned by the Navier-Stokes

128

Figure 7.1 Schematic of a grooved channel. Shown in the figureis thexy plane cross-section. Thesimulation domain is periodic inx andz directions.

129

equation. It is based on the microscopic Boltzmann equationfor the particle distribution function,

in contrast to the traditional Navier-Stokes-based numerical methods which directly solve for the

velocity and pressure. The Lattice-Boltzmann Method has been successfully applied to a variety

of flow problems [84, 4, 115, 98, 60], and is particularly attractive for flows in complex geometries

because of the straightforward implementation of boundaryconditions. A bead-spring chain model

of the polymer molecule can be coupled to the Lattice-Boltzmann model of the solvent to simu-

late the dynamics of polymer solutions [3, 150]. The fluid exerts a hydrodynamic friction force

on each polymer bead proportional to the difference betweenthe bead velocity and the local fluid

velocity at the bead position. In return, the force by each polymer bead is redistributed back to the

fluid. In other words, the polymer chain and the solvent exchange momentum through the friction

forces. This method provides an alternative to Brownian Dynamics, incorporating the same level

of description of the hydrodynamic and thermodynamic forces.

7.2 Simulation Parameters

In our simulation, the polymer molecule is modeled as a bead-spring chain withNs = 10

springs and each spring containsNk,s = 10 Kuhn segments. The radius of gyration of the chain is

aboutRg = 0.5µm, which is also the lattice spacing in our simulation. Random forces are intro-

duced to account for the Brownian motion of the beads. The solvent hydrodynamics is resolved by

the Lattice-Boltzmann model which leads to Navier-Stokes equation in low Mach number limit.

The details of the simulation method is outlined in Chapter 6. We use the same fluid relaxation

timeτs = 1.1 in this chapter.

Consider a grooved channel as shown in Figure 7.1: the bulk channel has the height ofLy =

19Rg and length ofLx = 40Rg, the length of the groove isLa = 20Rg, and the depth of the groove

is Lb = 9Rg unless otherwise specified. The simulation box has the dimension ofLz = 19Rg in

the neutralz direction. In our simulation, five chains are put in the simulation box, with a chain

concentration at least three orders of magnitude lower thanthe overlap concentration. Note for the

half λ-phage DNA chain model we are using, chain radius of gyrationRg = ∆x = 0.5µm. The

upper wall is moved in the positivex direction with speedvw to shear the fluid in the channel. The

130

Figure 7.2 Stream lines corresponding to the flow field generated by shearing the upper wall of thegrooved channel in positivex direction. The contour variable is the velocity inx direction. Notethat the magnitude of the velocity inside the groove is much smaller than outside.

strength of the flow field is characterized by Weisenberg number defined as

Wi = λvw

Ly − Lb, (7.1)

whereλ is the polymer molecule relaxation time. Figure 7.2 is a plotof the resulting flow field

with streamlines. We note that the flow field outside the groove is much stronger than that inside.

The velocity is about one order of magnitude larger.

7.3 Simulation Results

To investigate the concentration variation when polymer solution flows through a grooved chan-

nel, we performed Lattice-Boltzmann simulations at three different Weissenberg numbers: 0, 5,

and 10. The steady state chain center-of-mass distributionis plotted in Figure 7.3.

At equilibrium, the chain center-of-mass distribution is uniform as shown in Figure 7.3(a).

When the upper wall is sheared to generate the flow field as shown in Figure 7.2, three interesting

phenomena arise. First, the chain center-of-mass distribution inside the groove is significantly

reduced atWi = 5 as shown in Figure 7.3(b), indicating that the polymer chainis being depleted

out of the groove. This depletion effect becomes stronger asthe flow strength increases: the

131

Figure 7.3 Steady state chain center-of-mass distributionin a flowing polymer solution confinedin a grooved channel at effective Weissenberg number of (a)Wi = 0, (b) Wi = 5, and (c)Wi = 10.Note the strong depletion downstream of upstream horizontal wall, which is clearly related to thesteric depletion layer near the walls.

132

chain center-of-mass distribution inside the groove is even lower atWi = 10 as shown in Figure

7.3(c). Second, inside the groove, the concentration is notonly lower in general, it is also non-

uniform. Note the strong depletion downstream of upstream horizontal wall, which is clearly

related to the steric depletion layer near the walls. The concentration field displays a circular

pattern: corresponding to the circulating flow in the cavityshown in Figure 7.2, the concentration

is slightly higher along the outer streamline than that along the inner ones. Third, at the edge of

the groove, a bright band is visible in both Figure 7.3(b) and7.3(c), indicating a relatively higher

concentration region. Moreover, the concentration insidethis region increases as the Weissenberg

number increases.

To better illustrate the above observations, we take slicesfrom the two dimensional chain

center-of-mass distribution in Figure 7.3 alongy direction atx = 20, and plot them together

in Figure 7.4. The dotted vertical line indicates the position of the groove top edge. Now we can

clearly see that the chain center-of-mass distribution is significantly lower inside the groove and

higher outside at bothWi = 5 andWi = 10. Close to the groove bottom wall, the concentration

is higher corresponding to the outer streamlines of the circumfluence inside the groove. As the

Weissenberg number increases, the concentration close to the groove bottom wall decreases. At

the same time, near the groove top edge, the concentration band grows and shifts closer to the

groove edge. The curve labeled “bead distribution” will be discussed below in Section 7.4.

7.4 Discussion

The above results show two phenomena that are unexpected andpotentially important: (1)

the depletion of polymer chains from the cavity, and (2) the peak in concentration near the wall

containing the cavity. We now turn to some investigations that shed light on how these phenomena

arise.

7.4.1 Hydrodynamic Interactions

In Chapter 6 Section 6.5 and also our previous work [94, 64], we showed that concentration

variation arises in a channel flow of dilute polymer solutionat the length scale of the channel width.

133

y/Rg

n

0 5 10 150

0.001

0.002

0.003

0.004EquilibriumWi = 5Wi = 10Bead distribution

Groove Edge

Figure 7.4 Slice of the two dimensional steady state chain center-of-mass distribution in flowingpolymer solution confined in a grooved channel. The slice is taken alongy direction atx = 20,which is the center of the channel inx direction. The vertical dotted line indicates the positionofthe groove top edge.

134

Hydrodynamic interactions between chain segments and the channel walls push the polymer chain

away from the walls, and thus are responsible for the concentration gradient. It is of interest to

know the role of the hydrodynamic interactions in the grooved channel.

Therefore we performed LBM simulation with free draining (FD) model of the bead-spring

chain atWi = 10 to compare with the simulation with hydrodynamic interactions (HI). In flow,

the polymer beads still sample the velocity field of the solvent. However, in FD simulation, the

momentum is not redistributed back to the solvent, which means the polymer beads do not perturb

the solvent. In Figure 7.5, the chain center-of-mass distribution obtained from FD simulation

is compared with that from HI simulation. First, the concentration inside the groove is lower

in HI simulation. The concentration peak close to the groovebottom wall located aty = 0 is

reduced. This is not surprising, since we know that the hydrodynamic interactions between the

chain segments and the wall push chains away from the wall, and thus contribute to the reduction

of the peak and the lower concentration inside the groove. This effect also shows up at the top edge

of the groove: the concentration peak there is reduced also.However, comparing to the slit upper

wall atx = 19, the groove top edge wall atx = 9 is partially missing due to the groove, leading to

less migration in the grooved channel compared to that in a smooth channel.

In summary, the fingerprint of the complex concentration pattern in the grooved channel qual-

itatively remains in the free draining simulation. Although the hydrodynamic interactions are not

the leading reason for the complex concentration pattern, they do affect it in a way consistent with

our kinetic theory predictions [94].

7.4.2 Chain Connectivity

Consider the difference between a bead-spring chain and a group of unconnected beads. Each

of the unconnected beads has the same size as the bead in the bead-spring chain. While the uncon-

nected beads travel independently in flow, the beads on a chain must move collectively because of

the spring connectors. Moreover, the chain can be deformed by flow or by the interactions with

the confinement. All these lead to very different dynamics. To figure out the effect of the chain

connectivity, we eliminate the springs in the bead-spring chain model of the polymer chains, and

135

y/Rg

n

0 5 10 150

0.001

0.002

0.003

0.004

Wi = 10 HIWi = 10 FD

Groove Edge

Figure 7.5 Steady state chain center-of-mass distributionin a dilute polymer solution confinedin a grooved channel. The dash-dotted line is the distribution obtained from free draining (FD)simulation, and the solid line is the result from simulationwith hydrodynamic interactions (HI).Both simulations are performed withWi = 10.

136

Figure 7.6 Steady state center-of-mass distribution of isolated beads in shear flow in a groovedchannel.

track the bead distribution in the flow field corresponding toWi = 10 for polymer chain. Figure

7.6 shows the bead distribution.

Comparing to the center-of-mass distribution of the chain in Figure 7.3(c), the bead distribution

in Figure 7.6 is pretty much uniform in most of the region. However, inside the groove the bead

distribution is slightly lower than that outside, because the concave streamlines convect the lower

concentration fluid in the upstream wall excluded volume region into the groove, as evident from

the concentration pattern near the left corner of the groovein Figure 7.6. We also notice that close

to the right corner of the groove, there is a bright region stretched down into the groove. This is

because in that region, the competition between hydrodynamic drag force and the wall excluded

volume force results in a longer residence time near the right corner, and thus higher probability

of finding beads there. Moreover, because of the closed streamlines inside the groove as shown

in Figure 7.2, this high concentration region is convected down into the bottom of the groove.

Eventually, it fades out because of the Brownian diffusion.A slice from the two dimensional bead

distribution in Figure 7.6 is taken along they direction atx = 20 and shown in Figure 7.4 to

compare with polymer cases.

137

With the isolated bead distribution in mind, we now revisit the chain center-of-mass distri-

bution. Obviously, the above mentioned mechanisms also apply to a chain. But different from

individual beads, polymer chains can deform and dangle around the corner. Effectively, chains

will be stuck there for a while before they can rearrange the configuration to release themselves

either downstream along the channel or down into the groove.To confirm this idea, snapshots of

the chains taken from a simulation are shown in Figure 7.7, chronologically from top to bottom.

We can clearly see the whole process of a chain approaching the right (downstream) corner, dan-

gling, rearranging, and eventually escaping. As a result, ahigher concentration pattern near the

right corner is anticipated, and it is more profound than that in the case of individual beads. How-

ever, different from individual beads, chains are more likely to escape downstream along the bulk

channel than down into the groove. Consider a chain danglingaround the right corner. Because

the flow outside the groove is stronger than that inside, the portion of chain outside the groove

experiences more drag. Thus, the whole chain is more likely to be pulled downstream along the

bulk channel, resulting in the bright band along the groove top edge in the chain center-of-mass

distribution.

7.4.3 Peclet Number Effect

We now explain the depletion of polymer chains from the groove. Because of the wall excluded

volume effect, polymer chain center of mass can not move to solid walls closer than the polymer

molecule size, resulting in a steric depletion layer next toeach confining surface. The steric deple-

tion layers with the thickness of the polymer radius of gyration are shown as grey regions in Figure

7.8. The steric depletion layer above the upstream wall is convected across the top of the groove,

which gives rise to a boundary layer of thicknessRg that polymer chains need to diffuse across in

order to cross the separatix streamline. In other words a chain at the upstream edge of the cavity is

at least a distanceRg from the separatrix, which it must cross to enter the cavity.

One mechanism for the chain to cross the boundary layer is by diffusion. However, the chain

only has limited time to diffuse which is the convective timealong the separatrix. We define the

cavity Peclet numberPec as the ratio of the diffusion time of a chain over the boundarylayer to its

138

0 10 20 30 40

Y

0

10

20

0510

0 10 20 30 40

Y

0

10

20

0510

0 10 20 30 40

Y

0

10

20

0510

0 10 20 30 40

Y

0

10

20

0510

Figure 7.7 Snapshots of polymer chains in flowing solution confined in a grooved channel at timet = 711∆t, 740∆t, 756∆t, and766∆t, chronologically from top to bottom. The arrows point tothe polymer chain that approaches the corner.

139

Figure 7.8 Schematic of a chain crossing the boundary layer near the separatrix at the top edge ofthe groove.

140

convective time along the separatrix,

Pec =R2

g/D

La/v. (7.2)

Note the Peclet number by this definition is proportional to Weissenberg number in shear flow. The

Peclet number in our simulation is about 5 at Weissenberg number of10. In other words, the chain

only has 1/5 of the expected time to diffuse across the boundary layer. The other mechanism for

the chain to cross the boundary layer lies in the fact that theflow is stronger outside the groove than

inside. Due to the Brownian motion, a few beads of a chain originally inside the groove diffuse out

of the groove, as shown in Figure 7.8. Since the flow field outside the groove is stronger, the two

beads outside feel larger drag forces and they will eventually pull the remaining three beads out

of the groove. On the other hand, if a chain originally outside groove wants to enter, it has to rely

on the Brownian motion to bring most of its beads into the groove to overcome the stronger drag

force outside. Otherwise, the chain will be kept out of the groove. As a result, the chain center-

of-mass distribution is lower inside the groove than that outside, and it decreases as Peclet number

(or Weissenberg number) increases, as we observed in our simulation results shown in Figure 7.3.

7.5 Conclusion

In our Lattice-Boltzmann simulation of polymer solutions flowing through a grooved channel,

we observed the depletion of polymer chains from the cavity,and a concentration band formed

near the wall containing the grooves, which can be explainedin terms of chain connectivity, defor-

mation, Peclet number and hydrodynamic interactions.

A comparison between simulations with unconnected beads and polymer chains reveals that

the depletion effect is due to the chain connectivity. Chainsegments outside the groove feel much

stronger hydrodynamic drag forces than those inside. Therefore, the whole chain will be pulled

out of the groove. At the same time, the relatively high Peclet number prevents the chain from

diffusing back into the groove. Snapshots taken from the simulation show the detailed dynamics

of the chain in the grooved channel. Chains around the stagnation point close to the downstream

groove corner are deformed by the flow, and dangling for a longtime at the corner. Combining

with the convection in the flow direction, this dangling effect gives rise to the concentration band

141

at the top edge of the groove. Finally, comparison of the chain center-of-mass from free draining

simulation and simulation with hydrodynamic interactionsreveals that although the fingerprint of

the chain center-of-mass distribution is determined by thechain connectivity and deformation,

hydrodynamic interactions quantitatively changes the distribution: the probability of chain staying

near the groove bottom wall and the top edge of the groove is reduced.

These observations are of particular interest for surface-based analyses like Optical Mapping

[116]. Our simulation shows that the grooved channel provides a way to control the DNA chain

migration and brings the DNA molecules closer to the reactive surface, thus facilitating the DNA

analysis.

142

Chapter 8

Conclusion

In this thesis, we have presented a systematic investigation into the transport and dynamics of

flowing polymer solutions in confined geometry. Our comprehensive approach consists of kinetic

theory, Brownian Dynamics simulation and Lattice-Boltzmann Method.

A kinetic theory based on a dumbbell model of the polymer chain is proposed to explain the

shear-induced cross-streamline migration in confined flowing polymer solutions. We showed that

polymer chains in flow migrate away from the confining walls due to the hydrodynamic interactions

between the polymer segments and the confining walls. An expression for the thickness of the

resulting depletion layer near the wall is given. Further studies reveal that the length and time scale

at which the migration happens are quite large comparing to the chain characteristics. This theory

solves a long-standing puzzle on the behavior of confined polymer solutions, and successfully

explains a lot of experimental observations.

Furthermore, Brownian Dynamics simulation methods with fluctuating hydrodynamic interac-

tions in a single-wall confinement and a slit are developed. The simulation results quantitatively

confirmed the kinetic theory predictions. We find that when the confinement is comparable with

the polymer radius of gyration, the chain migration in flow istowards the walls. This is caused by

the extension of the chain in the flow direction and the corresponding shrinkage of the chains in the

confined direction. The discretization level of the polymerchain is shown to affect the simulation

results in this highly confined regime.

Finally, Lattice-Boltzmann Method is introduced. A close examination of the newly developed

method in terms of the chain migration in a smooth slit reveals the strengths and subtleties of the

LBM. Making use of its strength in dealing with complex geometry, Lattice-Boltzmann Method is

143

used to study the dynamics of polymer chain in a grooved channel. The results show that chains

will be depleted out of the groove in flow. More interestingly, a concentration band appears near

the top edge of the groove, which is explained in terms of chain connectivity, hydrodynamics

interactions and finite Peclet number.

Through this study, we have arrived at a general framework combining kinetic theory and

new simulation algorithms, leading to a better understanding of the structure, transport, and flow

characteristics of macromolecules in confinement. The knowledge we gained from this study offers

guidelines to the design and optimization of novel processes and devices in a very broad area.

144

LIST OF REFERENCES

[1] R. Adhikari, K. Stratford, M. E. Cates, and A. J. Wagner. Fluctuating lattice Boltzmann.Europhys. Lett., 71:473, 2005.

[2] U. S. Agarwal, A. Dutta, and R. A. Mashelkar. Migration ofmacromolecules under flow:The physical origin and engineering implications.Chem. Eng. Sci., 49:1693, 1994.

[3] P. Ahlrichs and B. Dunweg. Lattice-Boltzmann simulation of polymer-solvent systems.Int.J. Mod. Phys. C, 9:1429, 1998.

[4] P. Ahlrichs and B. Dunweg. Simulation of a single polymerchain in solution by combininglattice Boltzmann and molecular dynamics.J. Chem. Phys., 111:8225, 1999.

[5] P. Ahlrichs, R. Everaers, and B. Dunweg. Screening of hydrodynamic interactions insemidilute polymer solutions: A computer simulation study. Phys. Rev. E, 64:040501, 2001.

[6] O. S. Andersen. Sequencing and the single channel.Biophys. J., 77:2899, 1999.

[7] J. H. Aubert, S. Prager, and M. Tirrell. Macromolecules in nonhomogeneous velocity gra-dient fields. 2.J. Chem. Phys., 73:4103, 1980.

[8] J. H. Aubert and M. Tirrell. Flows of dilute polymer-solutions through packed porous chro-matographic columns.Rheologica Acta, 19:452, 1980.

[9] J. H. Aubert and M. Tirrell. Macromolecules in nonhomogeneous velocity gradient fields.J. Chem. Phys., 72:2694, 1980.

[10] J. H. Aubert and M. Tirrell. Effective viscosity of dilute polymer-solutions near confiningboundaries.J. Chem. Phys., 77:553, 1982.

[11] O. B. Bakajin, T. A. J. Duke, C. F. Chou, S. S. Chan, R. H. Austin, and E. C. Cox. Elec-trohydrodynamic stretching of DNA in confined environments. Phys. Rev. Lett., 80:2737,1998.

[12] R. Benzi, S. Succi, and M. Vergassola. The Lattice Boltzmann-equation - Theory and ap-plications.Phys. Rep.-Rev. Sec. Phys. Lett., 222:145, 1992.

145

[13] A. N. Beris and V. G. Mavrantzas. On the compatibility between various macroscopicformalisms for the concentration and flow of dilute polymer solutions. J. Rheol., 38:1235,1994.

[14] A. V. Bhave, R. C. Armstrong, and R. A. Brown. Kinetic theory and rheology of dilute,nonhomogeneous polymer solutions.J. Chem. Phys., 95:2988, 1991.

[15] R. B. Bird, C. F. Curtiss, R. C. Armstrong, and O. Hassager. Dynamics of Polymeric Liquids,volume 2. Wiley, New York, 2nd edition, 1987.

[16] R. Byron Bird, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena. Wiley,New York, 2nd edition, 2001.

[17] W. B. Black and M. D. Graham. Slip, concentration fluctuations, and flow instability insheared polymer solutions.Macromolecules, 34:5731, 2001.

[18] J. R. Blake. A note on the image system for a Stokeslet in ano-slip boundary.Proc. Camb.Phil. Soc., 70:303, 1971.

[19] F. Brochard and P. G. de Gennes. Dynamics of confined polymer chains.J. Chem. Phys.,67:52, 1977.

[20] I. N. Bronstein, K. A. Semendjajew, G. Musiol, and H. Muehlig. Taschenbuch der Mathe-matik. Verlag Harri-Deutsch, Frankfurt am Main, 2001.

[21] P. O. Brunn. Hydrodynamically induced cross-stream migration of dissolved macro-molecules (modeled as nonlinearly elastic dumbbells).Int. J. Multiphase Flow, 9:187, 1983.

[22] P. O. Brunn. Polymer migration phenomena based on the general bead-spring model forflexible polymers.J. Chem. Phys., 80:5821, 1984.

[23] P. O. Brunn and S. Chi. Macromolecules in non-homogeneous flow fields: A general studyfor dumbbell model macromolecules.Rheol. Acta, 23:163, 1984.

[24] J. Buckles, R. Hazlett, S. Chen, K. G. Eggert, D. W. Grunau, and W. E. Soll. Flow throughporous media using lattice boltzmann method.Los Alamos Sci., 22:112, 1994.

[25] C. Bustamante, J. F. Marko, E. D. Siggia, and S. Smith. Entropic elasticity of lambda-phagedna.Science, 265:1599, 1994.

[26] E. Y. Chan, N. M. Goncalves, R. A. Haeusler, A. J. Hatch, J. W. Larson, A. M. Maletta,G. R. Yantz, E. D. Carstea, M. Fuchs, G. G. Wong, S. R. Gullans,and R. Gilmanshin.DNA mapping using microfluidic stretching and single-molecule detection of fluorescentsite-specific tags.Genome Res., 14:1137, 2004.

[27] P. C. H. Chan and L. G. Leal. The motion of a deformable drop in a second-order fluid.J.Fluid Mech., 92:131, 1979.

146

[28] S. Chen and G. D. Doolen. Lattice Boltzmann method for fluid flows. Ann. Rev. FluidMech., 30:329, 1998.

[29] S. Y. Chen, H. D. Chen, D. Martinez, and W. Matthaeus. Lattice Boltzmann model forsimulation of magnetohydrodynamics.Phys. Rev. Lett., 67:3776, 1991.

[30] S. Y. Chen, D. Martinez, and R. W. Mei. On boundary conditions in lattice Boltzmannmethods.Phys. Fluids, 8:2527, 1996.

[31] Y. L. Chen, M. D. Graham, J. J. de Pablo, K. Jo, and D. C. Schwartz. DNA molecules inmicrofluidic oscillatory flow.Macromolecules, 38:6680, 2005.

[32] Y. L. Chen, M. D. Graham, J. J. de Pablo, G. C. Randall, M. Gupta, and P. S. Doyle.Conformation and dynamics of single DNA in parallel-plate slit microchannels.Phys. Rev.E, 70:060901, 2004.

[33] Y.-L. Chen, H. B. Ma, M. D. Graham, and J. J. de Pablo. Modeling DNA in confinement: Acomparison between brownian dynamics and lattice Boltzmann method.Macromolecules,to appear.

[34] Curtis D. Chin, Vincent Linder, and Samuel K. Sia. Lab-on-a-chip devices for global health:Past studies and future opportunities.Lab Chip, 7:41, 2007.

[35] M. Chopra and R. G. Larson. Brownian dynamics simulations of isolated polymermolecules in shear flow near adsorbing and nonadsorbing surfaces.J. Rheol., 46:831, 2002.

[36] C. F. Chou, R. H. Austin, O. Bakajin, J. O. Tegenfeldt, J.A. Castelino, S. S. Chan, E. C.Cox, H. Craighead, N. Darnton, T. Duke, J. Han, and S. Turner.Sorting biomolecules withmicrodevices.Electrophoresis, 21:81, 2000.

[37] Y. Cohen and A. B. Metzner. Apparent slip flow of polymer solutions. J. Rheol., 29:67,1985.

[38] C. F. Curtiss and R. B. Bird. Statistical mechanics of transport phenomena: Polymericliquid mixtures.Adv. Polym. Sci., 125:1, 1996.

[39] C. F. Curtiss and R. Byron Bird. Diffusion-stress relations in polymer mixtures.J. Chem.Phys., 111:10362, 1999.

[40] J. J. de Pablo, H. C.Ottinger, and Y. Rabin. Hydrodynamic changes of the depletion layerof dilute polymer solutions near a wall.AIChE J., 38:273, 1992.

[41] E. T. Dimalanta, A. Lim, R. Runnheim, C. Lamers, C. Churas, D. K. Forrest, J. J. de Pablo,M. D. Graham, S. N. Coppersmith, S. Goldstein, and D. C. Schwartz. A microfluidic systemfor large DNA molecule arrays.Anal. Chem., 76:5293, 2004.

147

[42] M. Doi. Effects of viscoelasticity on polymer diffusion. In A. Onuki and K. Kawasaki,editors,Dynamics and patterns in complex fluids: New aspects of physics and chemistry ofinterfaces, page 100. Springer Verlag, Berlin, 1990.

[43] Albert Einstein. On the motion - required by the molecular kinetic theory of heat - of smallparticles suspended in a stationary liquid.Annalen der Physik, 17:549, 1905.

[44] D. L. Ermak and J. A. Mccammon. Brownian dynamics with hydrodynamic interactions.J.Chem. Phys., 69:1352, 1978.

[45] P. Espanol and P. Warren. Statistical mechanics of Dissipative Particle Dynamics.Euro-phys. Lett., 30:191, 1995.

[46] X. Fan, N. Phan-Thien, N. T. Yong, X. Wu, and D. Xu. Microchannel flow of a macro-molecular suspension.Phys. Fluids, 15:11, 2003.

[47] L. Fang, H. Hu, and R. G. Larson. DNA configurations and concentrations in shearing flownear a glass surface in a microchannel.J. Rheol., 49:127, 2005.

[48] J. Feng, H. H. Hu, and D. D. Joseph. Direct simulation of initial-value problems for themotion of solid bodies in a Newtonian fluid .2. Couette and Poiseuille flows.J. Fluid Mech.,277:271, 1994.

[49] B. Ferreol and D. H. Rothman. Lattice-boltzmann simulations of flow-throughfontainebleau sandstone.Trans. Porous Media, 20:3, 1995.

[50] M. Fixman. Simulation of polymer dynamics .1. general theory. Journal of ChemicalPhysics, 69:1527, 1978.

[51] M. Fixman. Construction of Langevin forces in the simulation of hydrodynamic interaction.Macromolecules, 19:1204, 1986.

[52] M. Fixman. Implicit algorithm for Brownian dynamics ofpolymers. Macromolecules,19:1195, 1986.

[53] C. K. Fredrickson and Z. H. Fan. Macro-to-micro interfaces for microfluidic devices.LabChip, 4:526, 2004.

[54] G. G. Fuller and L. G. Leal. The effects of conformation-dependent friction and internalviscosity on the dynamics of the nonlinear dumbbell model for a dilute polymer solution.J.Non-Newtonian Fluid Mech., 8:271, 1981.

[55] P. Ganatos, R. Pfeffer, and S. Weinbaum. A strong interaction theory for the creepingmotion of a sphere between plane parallel boundaries. 2. Parallel motion. J. Fluid Mech.,99:755, 1980.

148

[56] P. Ganatos, S. Weinbaum, and R. Pfeffer. A strong interaction theory for the creepingmotion of a sphere between plane parallel boundaries. 1. Perpendicular motion.J. FluidMech., 99:739, 1980.

[57] F. H. Garner and A. H. Nissan. Rheological properties ofhigh viscosity solutions of longmolecules.Nature, 158:634, 1946.

[58] P. S. Grassia, E. J. Hinch, and L. C. Nitsche. Computer-simulations of brownian-motion ofcomplex-systems.Journal of Fluid Mechanics, 282:373, 1995.

[59] R. D. Groot and P. B. Warren. Dissipative Particle Dynamics: Bridging the gap betweenatomic and mesoscopic simulation.J. Chem. Phys., 107:4423, 1997.

[60] Z. L. Guo, T. S. Zhao, and Y. Shi. A lattice Boltzmann algorithm for electro-osmotic flowsin microfluidic devices.J. Chem. Phys., 122:144907, 2005.

[61] J. Han and H. G. Craighead. Separation of long DNA molecules in a microfabricated en-tropic trap array.Science, 288:1026, 2000.

[62] E. Helfand and G. H. Fredrickson. Large fluctuations in polymer solutions under shear.Phys. Rev. Lett., 62:2468, 1989.

[63] J. P. Hernandez-Ortiz, J. J. de Pablo, and M. D. Graham.NLogN method for hydrodynamicinteractions of confined flowing polymer systems: Brownian dynamics. J. Chem. Phys.,125:164906, 2006.

[64] J. P. Hernandez-Ortiz, H. B. Ma, J. J. de Pablo, and M. D.Graham. Cross-stream-linemigration in confined flowing polymer solutions: Theory and simulation. J. Chem. Phys.,18:123101, 2006.

[65] Juan P. Hernandez-Ortiz, Juan J. de Pablo, and MichaelD. Graham. Fast computation ofmany-particle hydrodynamic and electrostatic interactions in a confined geometry.Phys.Rev. Lett., 98:140602, 2007.

[66] M. Herrchen and H. C.’Ottinger. A detailed comparison of various FENE dumbbell models.J. Non-Newtonian Fluid Mech., 68:17, 1997.

[67] M. Hinz, S. Gura, B. Nitzan, S. Margel, and H. Seliger. Polymer support for exonucleolyticsequencing.J. Biotech., 86:281, 2001.

[68] J. Horbach and S. Succi. Lattice Boltzmann versus molecular dynamics simulation ofnanoscale hydrodynamic flows.Phys. Rev. Lett., 96:224503, 2006.

[69] R. G. Horn, O. I. Vinogradova, M. E. Mackay, and N. Phan-Thien. Hydrodynamic slippageinferred from thin-film drainage measurements in a solutionof nonadsorbing polymer.J.Chem. Phys., 112:6424, 2000.

149

[70] S. D. Hudson. Wall migration and shear-induced diffusion of fluid droplets in emulsions.Phys. Fluids, 15:1106, 2003.

[71] R. M. Jendrejack, J. J. de Pablo, and M. D. Graham. Stochastic simulations of DNA in flow:dynamics and the effects of hydrodynamic interactions.J. Chem. Phys., 116:7752, 2002.

[72] R. M. Jendrejack, E. T. Dimalanta, D. C. Schwartz, M. D. Graham, and J. J. de Pablo. DNAdynamics in a microchannel.Phys. Rev. Lett., 91:038102, 2003.

[73] R. M. Jendrejack, M. D. Graham, and J. J. de Pablo. Hydrodynamic interactions in longchain polymers: Application of the Chebyshev polynomial approximation in stochastic sim-ulations.J. Chem. Phys., 113:2894, 2000.

[74] R. M. Jendrejack, D. C. Schwartz, J. J. de Pablo, and M. D.Graham. Shear-induced mi-gration in flowing polymer solutions: simulation of long-chain DNA in microchannels.J.Chem. Phys., 120:2513, 2004.

[75] R. M. Jendrejack, D. C. Schwartz, M. D. Graham, and J. J. de Pablo. Effect of confinementon DNA dynamics in microfluidic devices.J. Chem. Phys., 119:1165, 2003.

[76] M. S. Jhon and K. F. Freed. Polymer migration in Newtonian fluids. J. Polym. Sci. Polym.Phys. Ed., 23:955, 1985.

[77] Wenhua Jiang, Jianhua Huang, Yongmei Wang, and MohamedLaradji. Hydrodynamicinteraction in polymer solutions simulated with Dissipative Particle Dynamics.J. Chem.Phys., 126:044901, 2007.

[78] J. Jing, J. Reed, J. Huang, X. Hu, V. Clarke, J. Edington,D. Housman, T. S. Anantharaman,E. J. Huff, B. Mishra, B. Porter, A. Shenker, E. Wolfson, C. Hiort, R. Kantor, C. Aston,and D. C. Schwartz. Automated high resolution optical mapping using arrayed, fluid-fixedDNA molecules.Proc. Natl. Acad. Sci. U.S.A., 95:8046, 1998.

[79] C. W. Kan, C. P. Fredlake, E. A. S. Doherty, and A. E. Barron. DNA sequencing andgenotyping in miniaturized electrophoresis sytems.Electrophoresis, 25:3564, 2004.

[80] R. Khare, M. D. Graham, and J. J. de Pablo. Cross-stream migration of flexible moleculesin a nanochannel.Phys. Rev. Lett., 2006.

[81] S. Kim and S. Karrila. Microhydrodynamics: Principles and selected applications.Butterworth-Heinemann, 1991.

[82] A. Koponen. Simulations of Fluid Flow in Porous Media by Lattice-Gas andLattice-Boltzmann Methods. PhD thesis, University of Jyvakyla, 1998.

[83] A. J. C. Ladd. Numerical simulations of particulate suspensions via a discretizedboltzmann-equation .1. theoretical foundation.J. Fluid Mech., 271:285, 1994.

150

[84] A. J. C. Ladd and R. Verberg. Lattice-Boltzmann simulations of particle-fluid suspensions.J. Stat. Phys., 104:1191, 2001.

[85] L. D. Landau and E. M. Lifshitz.Fluid Mechanics. Butterworth-Heinemann, Oxford, 2ndedition, 1987.

[86] L. G. Leal. Particle motions in a viscous fluid.Ann. Rev. Fluid Mech., 12:435, 1980.

[87] L. G. Leal. Laminar Flow and Convective Transport Processes. Butterworth-Heinemann,Boston, 1992.

[88] J. E. Lennard-Jones. Cohesion.Proc. Phys. Soc., 43:461, 1931.

[89] L. Li, H. Hu, and R. G. Larson. DNA molecular configurations in flows near adsorbing andnonadsorbing surfaces.Rheol. Acta, 44:38, 2004.

[90] A. Lim, E. T. Dimalanta, K. D. Potamousis, G. Yen, J. Apodoca, C. Tao, J. Lin, R. Qi,J. Skiadas, A. Ramanathan, N. T. Perna, G. Plunkett III, V. Burland, B. Mau, J. Hackett,F. R. Blattner, T. S. Anantharaman, B. Mishra, and D. C. Schwartz. Shotgun optical mapsof the whole Escherichia coli O157 : H7 genome.Genome Res., 11:1584, 2001.

[91] N. Liron and E. Barta. Motion of a rigid particle in Stokes-flow- A new 2nd-kind boundary-integral equation formulation.J. Fluid Mech., 238:579, 1992.

[92] N. Liron and S. Mochon. Stokes flow for a stokeslet between two parallel flat plates.J.Engr. Math., 10:287, 1976.

[93] L. S. Luo. Unified theory of lattice Boltzmann models fornonideal gases.Phys. Rev. Lett.,81:1618, 1998.

[94] H. B. Ma and M. D. Graham. Theory of shear-induced migration in dilute polymer solutionsnear solid boundaries.Phys. Fluids, 17:083103, 2005.

[95] J. M. Maerker. Dependence of polymer retention on flow-rate. Journal of Petroleum Tech-nology, 25:1307, 1973.

[96] J. F. Marko and E. D. Siggia. Stretching DNA.Macromolecules, 28:8759, 1995.

[97] G. Marrucci. The free energy constitutive equation forpolymer solutions from the dumbbellmodel.Tran. Soc. Rheol., 16:321, 1972.

[98] N. S. Martys and J. F. Douglas. Critical properties and phase separation in lattice Boltzmannfluid mixtures.Phys. Rev. E, 6303:031205, 2001.

[99] V. G. Mavrantzas and A. N. Beris. Modeling of the rheology and flow-induced concentra-tion changes in polymer solutions.Phys. Rev. Lett., 69:273, 1992.

151

[100] V. G. Mavrantzas and A. N. Beris. Theoretical study of wall effects on the rheology of dilutepolymer solutions.J. Rheol., 36:175, 1992.

[101] V. G. Mavrantzas and A. N. Beris. A hierarchical model for surface effects on chain confor-mation and rheology of poymer solutions. I. General formulation. J. Chem. Phys., 110:616,1999.

[102] V. G. Mavrantzas and A. N. Beris. A hierarchical model for surface effects on chain con-formation and rheology of poymer solutions. II. Application to a neutral surface.J. Chem.Phys., 110:628, 1999.

[103] A. B. Metzner. Flow of polymeric solutions and emulsions through porous media – currentstatus. In D. O. Shah and R. S. Schechter, editors,Improved Oil Recovery by Surfactant andPolymer Flooding, page 439. Academic Press, New York, 1977.

[104] A. B. Metzner, Y. Cohen, and C. Rangelnafaile. Inhomogeneous flows of non-newtonianfluids - generation of spatial concentration gradients.Journal of Non-Newtonian Fluid Me-chanics, 5:449, 1979.

[105] Jaime A. Millan, Wenhua Jiang, Mohamed Laradji, and Yongmei Wang. Pressure drivenflow of polymer solutions in nanoscale slit pores.J. Chem. Phys., 126:124905, 2007.

[106] S. T. Milner. Hydrodynamics of semidilute polymer solutions. Phys. Rev. Lett., 66:1477,1991.

[107] S. T. Milner. Dynamical theory of concentration fluctuations in polymer solutions undershear.Phys. Rev. E, 48:3674, 1993.

[108] P. J. Mucha, S. Y. Tee, D. A. Weitz, B. I. Shraiman, and M.P. Brenner. A model for velocityfluctuations in sedimentation.J. Fluid Mech., 501:71, 2004.

[109] D. R. Noble, S. Y. Chen, J. G. Georgiadis, and R. O. Buckius. A consistent hydrodynamicboundary-condition for the lattice Boltzmann method.Phys. Fluids, 7:203, 1995.

[110] T. W. Odom, V. R. Thalladi, J. C. Love, and G. M. Whitesides. Generation of 30-50 nmstructures using easily fabricated, composite PDMS masks.J. Am. Chem. Soc., 124:12112,2002.

[111] A. Onuki. Dynamic equations of polymers with deformations in semidilute regions.J.Phys. Soc. Jpn., 59:3423, 1990.

[112] H. C. Ottinger. Incorporation of polymer diffusivity and migration into constitutive equa-tions. Rheol. Acta, 31:14, 1992.

[113] H. C.Ottinger.Stochastic Processes in Polymeric Fluids. Springer, 1996.

[114] H. C.Ottinger.Beyond Equilibrium Thermodynamics. Wiler-Interscience, 2005.

152

[115] I. Pagonabarraga, F. Capuani, and D. Frenkel. Mesoscopic lattice modeling of electrokineticphenomena.Comput. Phys. Comm, 169:192, 2005.

[116] N. Perna, G. Plunkett III, V. Burland, B. Mau, J. Glasner, D. Rose, G. Mayhew, P. Evans,J. Gregor, H. Kirkpatrick, G. Posfai, J. Hackett, S. Klink, A. Boutin, Y. Shao, L. Miller,E. Grotbeck, N. Davis, A. Lim, E. Dimalanta, K. Potamousis, J. Apodaca, T. Anantharaman,J. Lin, G. Yen, D. Schwartz, R. Welch, and F. Blattner. Genomesequence of enterohaemor-rhagic Escherichia coli O157 : H7.Nature, 409:529, 2001.

[117] H. Power and L. C. Wrobel.Boundary Integral Methods in Fluid Mechanics. Computa-tional Mechanics Publications, 1995.

[118] C. Pozrikidis. Boundary Integral and Singularity Methods for Linearized Viscous Flow.Cambridge University, 1992.

[119] C. Pozrikidis. Introduction to Theoretical and Computational Fluid Dynamics. OxfordUniversity Press, New York, 1997.

[120] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P.Flannery. Numerical Recipes inFortran 77. Cambridge, 2nd edition, 1992.

[121] Y. H. Qian, D. Dhumieres, and P. Lallemand. Lattice BGKmodels for Navier-Stokes equa-tion. Europhys. Lett., 17:479, 1992.

[122] L. Reichl.A Modern Course in Statistical Physics. Wiley-Interscience, 2nd edition, 1998.

[123] W. Reisner, K. J. Morton, R. Riehn, Y. M. Wang, Z. Yu, M. Rosen, J. C. Sturn, S. Y.Chou, E. Frey, and R. H. Austin. Statics and dynamics of singeDNA molecules confined innanochannels.Phys. Rev. Lett., 94:196101, 2005.

[124] M. Ripoll, M. H. Ernst, and P. Espanol. Large scale and mesoscopic hydrodynamics forDissipative Particle Dynamics.J. Chem. Phys., 115:7271, 2001.

[125] M. G. Roper, C. J. Easley, and J. P. Landers. Advances inpolymerase chain reaction onmicrofluidic chips.Anal. Chem., 77:3887, 2005.

[126] J. Rotne and S. Prager. Variational treatment of hydrodynamic interaction in polymers.J.Chem. Phys., 50:4831, 1969.

[127] M. Rubinstein and R. H. Colby.Polymer Physics. Oxford University Press, 2003.

[128] D. Saintillan, E. Shaqfeh, and E. Darve. Effect of flexibility on the shear-induced migrationof short-chain polymers in parabolic channel flow.J. Fluid Mech., 557:297, 2006.

[129] M. Sauer, B. Angerer, W. Ankenbauer, Z. F. o ldes Papp, bel F. Go, K. T. Han, R. Rigler,A. Schultz, J. Wolfrum, and C. Zander. Single molecule DNA sequencing in submicrometerchannels: state of the art and future prospects.J. Biotech., 86:181, 2001.

153

[130] G. Segre and A. Silberberg. Behaviour of macroscopicrigid spheres in Poiseuille flow. IIexperimental results and interpretation.J. Fluid. Mech., 14:136, 1962.

[131] G. Sekhon, R. C. Armstrong, and M. S. Jhon. The origin ofpolymer migration in a non-homogeneous flow field.J. Polym. Sci. Polym. Phys. Ed., 20:947, 1982.

[132] Y. H. Seo, O. O. Park, and M. S. Chun. The behavior of velocity enhancement in microcap-illary flows of flexible water-soluble polymers.J. Chem. Eng. Jpn., 29:611, 1996.

[133] P. J. Shrewsbury, D. Liepmann, and S. J. Muller. Concentration effects of a biopolymer in amicrofluidic device.Biomed. Microdevices, 4:17, 2002.

[134] P. A. Skordos. Initial and boundary-conditions for the lattice Boltzmann method.Phys. Rev.E, 48:4823, 1993.

[135] J. R. Smart and D. T. Leighton. Measurement of the driftof a droplet due to the presence ofa plane.Phys. Fluids A, 3:21, 1991.

[136] D. E. Smith and S. Chu. Response of flexible polymers to asudden elongational flow.Science, 281:1335, 1998.

[137] D. E. Smith, T. T. Perkins, and S. Chu. Dynamical scaling of DNA diffusion coefficients.Macromolecules, 29:1372, 1996.

[138] M. E. Staben, A. Z. Zinchenko, and R. H. Davis. Motion ofa particle between two parallelplane walls in low-Reynolds-number Poiseuille flow.J. Fluid Mech., 15:1711, 2003.

[139] D. Stein, F. H. J. van der Heyden, W. J. A. Koopmans, and C. Dekker. Pressure-driven trans-port of confined dna polymers in fluidic channels.Proc. Natl. Acad. Sci, U S A., 103:15853,2006.

[140] C. Stoltz, J. J. de Pablo, and M. D. Graham. Concentration dependence of shear and exten-sional rheology of polymer solutions: Brownian dynamics simulations.J. Rheol., 50:137,2006.

[141] A. J. Storm, J. H. Chen, H. W. Zandbergen, and C. Dekker.Translocation of double-strandDNA through a silicon oxide nanopore.Phys. Rev. E, 71:051903, 2005.

[142] M. Streek, F. Schmid, T. T. Duong, D. Anselmetti, and A.Ros. Two-state migration of DNAin a structured microchannel.Phys. Rev. E, 71:011905, 2005.

[143] S. Succi.The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford Univer-sity Press, Oxford, 1st edition, 2001.

[144] Nerayo P. Teclemariam, Victor A. Beck, Eric S. G. Shaqfeh, and Susan J. Muller. Dy-namics of DNA polymers in post arrays: Comparison of single molecule experiments andsimulations.Macromolecules, 40:3848, 2007.

154

[145] J. O. Tegenfeldt, C. Prinz, H. Cao, S. Chou, W. W. Reisner, R. Riehn, Y. M. Wang, E. C.Cox, J. C. Sturm, P. Silberzan, and R. H. Austin. The dynamicsof genomic-length DNAmolecules in 100-nm channels.Proc. Natl. Acad. Sci. U. S. A., 101:10979, 2004.

[146] J. O. Tegenfeldt, C. Prinz, H. Cao, R. L. Huang, R. H. Austin, S. Y. Chou, E. C. Cox, andJ. C. Sturm. Micro- and nanofluidics for DNA analysis.Anal. Bioanal. Chem., 378:1678,2004.

[147] M. Tirrell and M. F. Malone. Stress induced diffusion of macromolecules.J. Polym. Sci.Polym. Phys. Ed., 15:1569, 1977.

[148] L. R. G. Treloar.The Physics of Rubber Elasticity. Oxford University Press, London, 3rdedition, 1975.

[149] O. B. Usta, J. E. Butler, and A. J. C. Ladd. Flow-inducedmigration of polymers in dilutesolution.Phys. Fluids, 18:031703, 2006.

[150] O. B. Usta, A. J. C. Ladd, and J. E. Butler. Lattice-boltzmann simulations of the dynamicsof polymer solutions in periodic and confined geometries.J. Chem. Phys., 122:094902,2005.

[151] P. Vasseur and R. G. Cox. Lateral migration of a spherical-particle in 2-dimensional shearflows. J. Fluid Mech., 78:385, 1976.

[152] N. J. Woo, E. S. G. Shaqfeh, and B. Khomami. The effect ofconfinement on dynamicsand rheology of dilute DNA solutions. I. Entropic spring force under confinement and anumerical algorithm.J. Rheol., 48:281, 2004.

[153] N. J. Woo, E. S. G. Shaqfeh, and B. Khomami. The effect ofconfinement on dynamicsand rheology of dilute dna solutions. II. Effective rheology and single chain dynamics.J.Rheol., 48:299, 2004.

[154] H. Yamakawa. Transport properties of polymer chains in dilute solution - hydrodynamicinteraction.J. Chem. Phys., 53:436, 1970.

[155] S. Zhou, E. Kvikstad, A. Kile, J. Severin, D. Forrest, R. Runnheim, C. Churas, J. W. Hick-man, C. Mackenzie, M. Choudhary, T. Donohue, S. Kaplan, and D. C. Schwartz. Whole-genome shotgun optical mapping of rhodobacter sphaeroidesstrain 2.4.1. and its use forwhole-genome shotgun sequence assembly.Genome Res., 13:2142, 2003.

[156] D. P. Ziegler. Boundary-conditions for lattice Boltzmann simulations. J. Stat. Phys.,71:1171, 1993.

[157] R. Zwanzig.Nonequilibrium Statistical Mechanics. Oxford University Press, 2001.

dynamics of flowing polymer solutions …dynamics of flowing polymer solutions under confinement by...

Documents