DYNAMICS OF FLOWING POLYMER SOLUTIONS UNDER CONFINEMENT. Hongbo Ma

Transcription

1 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

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3 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.

4 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.

5 This work was supported through the NSF/NSEC program. iii

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7 iv TABLE OF CONTENTS Page LIST OF FIGURES vi ABSTRACT xiii 1 Introduction Problem Statement Molecular Interactions Chain Connectivity Brownian Force Hydrodynamic Interactions Excluded Volume Effect Migration Near Solid Surfaces Background Illustration of Migration Mechanism Kinetic Theory for a Dumbbell in Dilute Solution Steady State Depletion Layer near a Single Wall Temporal and Spatial Evolution of the Depletion Layer in a Semi-Infinite Domain Plane Couette Flow and Plane Poiseuille Flow Conclusion Brownian Dynamics Simulation Introduction Point-Dipole Theory of Polymer Migration Polymer Model and Simulation Method Results and Discussion Single Wall Migration in Simple Shear Slit Confinement: Shear Flow

8 v Page Highly Confined Polymer Chains General Flux Expression for Dumbbells Effect of Finite Reynolds Number on Wall-induced Hydrodynamic Migration Conclusion Simulating Polymer Solution Using Lattice-Boltzmann Method Introduction Lattice-Boltzmann Method Velocity Set Equilibrium Velocity Distribution Collision Operator External Force Boundary Conditions Polymer Chain Model The Bead-spring chain Model Coupling of the Polymer Chain and the Solvent Equation of Motion for Polymer Beads Simulation Parameters Chain Migration in Dilute Polymer Solution Flow in a Slit Complications of the Lattice-Boltzmann Method Fluid Relaxation Time Grid Size Effect Reynolds Number Effect Conclusion Polymer Chain Dynamics in a Grooved Channel Introduction Simulation Parameters Simulation Results Discussion Hydrodynamic Interactions Chain Connectivity Peclet Number Effect Conclusion Conclusion LIST OF REFERENCES

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10 vi LIST OF FIGURES Figure Page 1.1 Optical Mapping method developed by David Schwartz s group at the University of Wisconsin-Madison. YAC clone 5L5 derived from human chromosome 11, was digested with EagI and 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 for Molecular and Computational Genomics, University of Wisconsin-Madison 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. Fluorescent tagged DNA molecules are stretched at the entrance of a tapered channel due to the collision with the posts and the elongational flow. When the stretched DNA molecules travel through the narrow channel, the positions of the tag sites are read out by laser detectors. This method has a claimed resolution of ±0.8kb resolution and throughput of million bp/min Schematic of different regimes of confinement: (a) single wall confinement, (b) weak confinement: 2h R g, (c) strong confinement: 2h R g, and (d) extreme confinement: 2h L p 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 large amount of atoms within each segment into a bead which is 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 beads and rods by one larger bead and connecting those larger beads by elastic springs give the bead-spring model. This is the coarse-graining level we will work on Bead-spring chain model of a polymer moleclue. The springs account for the resistance to the stretch due to the entropic effect, and the beads represent the interaction sites along the chain contour

11 vii Figure Page 4.1 Illustration of the position vectors used for a point force above a plane wall Velocity field due to a point force in the x direction located at (x, y) = ( 5a, 5a), where a is the bead radius. The plane wall is at y = 0. The lines correspond to streamlines, while the light and dark area indicate regions where the wall-normal velocity is positive (away from the wall) and negative (towards the wall), respectively. Also shown is a bead of radius a located at (x, y) = (5a, 5a) - this can be thought of as the other end of a relaxing dumbbell oriented parallel to the wall Steady state concentration profiles scaled by the bulk value in uniform shear flow above a single wall at different Weissenberg numbers. The concentration profiles are calculated using a FENE-P dumbbell model with finite extensibility parameter b = 600 and hydrodynamic interaction parameter h = Depletion layer thickness vs. Weissenberg number in a uniform shear flow above a single wall for FENE-P dumbbell with finite extensibility parameter b = 600 and hydrodynamic interaction parameter h = The straight line is the high Weissenberg number asymptote, L d /R g Wi 2/ Temporal development of the concentration profile in uniform shear flow above a single wall at Wi = 10. A FENE-P dumbbell model with finite extensibility parameter b = 600 and hydrodynamic interaction parameter h = 0.25 is used Similarity solution for time evolution of the concentration profile in uniform shear flow above a single wall. The full numerical solutions including diffusion for Wi = 100 at two different times, t = 10λ H and t = 1000λ H, are also plotted for comparison. A FENE-P dumbbell with finite extensibility b = 600 and hydrodynamic interaction parameter h = 0.25 is used when solving for the numerical solutions Similarity solution for spatial development of the concentration profile in uniform shear flow above a single wall. The full numerical solutions including the diffusion for Wi = 100 at two different downstream positions, x = 10(k B /H) 1/2 and x = 10000(k B /H) 1/2, are also shown for comparison. A FENE-P dumbbell with finite extensibility b = 600 and hydrodynamic interaction parameter h = 0.25 is used when solving for the numerical solutions Steady state concentration profiles at Wi = 2, 10 and 100 in plane Couette flow in a slit with width 2h = 30 k B T/H. Length is scaled by k B T/H and concentration by its value at the centerline of the slit, n c. Migration effects due to the two walls of the slit are superimposed by taking the single-reflection approximation

12 viii Figure Page 4.9 Steady state concentration profiles at Wi = 2, 10, 100 in lane Poiseuille flow in a slit with width 2h = 30 k B T/H. Length is scaled by k B T/H and concentration by its value at the centerline of the slit, n c. Migration effects due to the two walls of the slit are superimposed by taking the single-reflection approximation Steady state concentration field for plane Poiseuille flow in the entrance region of a slit with width 2h = 300 k B T/H at Wi = 20. Only half of the slit is shown. The concentration is scaled by its bulk value n 0 before entering the slit. Migration contributions due to two walls of the slit are superimposed by taking the singlereflection approximation Schematic of different regimes of confinement: (a) Single wall confinement, (b) weak confinement: 2h R g, (c) strong confinement: 2h R g, and (d) extreme confinement: 2h L p Time evolution of axially averaged fluorescence intensity of fluorescent labeled T2- DNA solution as a function of cross-sectional position. The channel walls are at y = ±20µm. The solution is undergoing oscillatory pressure-driven flow at a maximum strain rate of 75 s 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 and the dark region represents the depletion layer near the channel walls Steady-state chain center-of-mass concentration profiles predicted by theory, using the Stokeslet-doublet (far-field) approximation, and the BD simulation at Wi = 0, 5, 10 and 20 in simple shear flow. The concentration is normalized using its value at y/(k B T/H) 1/2 = Migration velocity scaled with the point-dipole value for different dumbbell (forcedipole) sizes, as a function of distance from the wall Near-field center-of-mass steady-state concentration profiles predicted by theory, using the Stokeslet-doublet (far-field) approximation and finite-size dumbbells, and the BD simulation at Wi = 5 in simple shear flow Steady-state chain center-of-mass concentration profiles predicted by theory, using the Stokeslet-doublet (far-field) approximation, and the BD simulation of 10 springs chains, at Wi = 5 and 10 in simple shear flow

13 ix Figure Page 5.7 Steady-state chain center-of-mass concentration profiles predicted by theory, using far-field and single-reflection approximations, and the BD simulation at Wi = 0, 5 and 20 in shear flow Steady-state chain center-of-mass concentration profiles predicted by the BD simulation at Wi = 20 in shear flow, for different polymer discretizations: N s = 1, 5 and Schematic of two different discretization levels of a same molecule (a) dumbbell: the effect 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 with smaller separation Steady-state chain center-of-mass concentration profiles predicted by the theory, using far-field and single-reflection approximations, and the BD simulation at Wi = 20 in shear flow. The steady-state chain center-of-mass concentration profile at equilibrium (Wi = 0) and the bead-distribution from the simulation at Wi = 20 are also shown Steady-state chain center-of-mass concentration profiles predicted by the BD simulation of chains (N s = 10) for a highly confined polymer solution, 2h = 2.9R g Steady-state bead-concentration profiles predicted by the BD simulation of chains (N s = 10) for a highly confined polymer solution, 2h = 2.9R g Polymer stretch as a function of the wall-normal direction, y, for Wi = 0 (no flow); 2h = 2.9R g Polymer stretch in the flow direction, x, as a function of the wall normal direction, y; 2h = 2.9R g Polymer stretch in the confined direction, y, as a function of the wall normal direction, y; 2h = 2.9R g Schematic of the hydrodynamic migration mechanism (a) Re y 1: wall-induced migration momentum diffusion to the wall and back to the particle is fast; (b) Re y 1: No wall-induced migration the shear flow distorts the velocity perturbation due to the particle so that the particle is not affected by the presence of the wall

14 x Figure Page 6.1 The set of discrete velocities in a D3Q19 model shown in a lattice cube. The solid parallelogram represents the xy plane, the dashed rectangle the yz plane, and the dotted parallelogram the xz 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 cube faces, and 12 velocities with magnitude 2 represented by the arrows pointing to the cube-edge centers In the single-time-relaxation model, the velocity distribution at each site relaxes toward the equilibrium one at each time step. Without the external force, the equilibrium velocity distribution consists simply equal amount of fluid particles for each of the discretized velocities. The figure 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 neighboring sites stream along their directions of motion to that site. Second, the incoming distribution relaxes due to the particle collisions, according to the single-time-relaxation rule, towards the equilibrium distribution. (a) When τ s = 1, the incoming velocity distribution relaxes to the equilibrium distribution in one time step. (b) When τ s = 2, the post-relaxation distribution is halfway between the incoming and the equilibrium distributions Bounce-back rule for a solid-fluid interface. The arrows shows the velocity direction and 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 solid boundary Relaxation of a stretched polymer molecule in bulk solution. The mean square stretch of the chain < X 2 > is plotted against time for a chain of N s = 10 at two different temperatures k B T = 0.001, and An exponential decay fitting of < X(t) 2 >=< X( ) 2 > +X 0 exp(t/λ) gives the chain relaxation time as λ = 426 for k B T = and λ = 2037 for k B T = , in lattice units. X 0 and λ are the fitting parameters Mean square displacement of the center-of-mass of a polymer chain with N s = 10 as a function of time in bulk solution. The simulation parameters are µ = 0.2, ζ = 0.6, and k B T = A linear fitting to the diffusion equation < [(r(t) r(0)] 2 >= 6Dt gives the chain diffusion coefficient as D = , in lattice units Steady state chain center-of-mass distribution of a dilute polymer solution undergoing simple 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

15 xi Figure Page 6.7 Steady state chain center-of-mass distribution of a dilute polymer solution undergoing shear flow confined in a slit. The solid line is the equilibrium chain center-of-mass distribution, the dotted line is the chain center-of-mass distribution obtained from simulations with free draining model (FD) at Wi = 50, and the dashed line is the chain center-of-mass distribution obtained from simulations with hydrodynamic interactions (HI) at Wi = Steady state chain center-of-mass distribution of a dilute polymer solution undergoing shear flow confined confined in a slit at Wi = 50. The solid line is the chain center-ofmass distribution obtained from Lattice-Boltzmann Method, and the dashed line from Brownian Dynamics simulation with hydrodynamic interactions Viscous flow of a fluid near a wall suddenly sheared. At time t = 0, the bottom solid surface is set in motion in the positive x direction with velocity v Velocity profile in dimensionless form for flow near a wall suddenly sheared. (a) Results from Lattice-Boltzmann Method with τ s = 1.1. (b) Results from Lattice- Boltzmann Method with τ s = Contour plot of the wall normal component of the steady state flow field due to a stretched dumbbell (white beads connected by dotted line) confined in a slit. (a) Finite element solution. (b) Result from Lattice-Boltzmann Method with τ s = 1.1. (c) Result from Lattice-Boltzmann Method with τ = Comparison of the wall normal component of the steady state flow field due to a stretched dumbbell confined in a slit. (a) Slice of the flow field along wall-normal direction at x = 20. (b) Slice of the flow field along the wall-tangential direction at y = 5. The dotted lines in (b) indicates the positions of the two beads of the stretched dumbbell Steady state chain center-of-mass distribution of a dilute polymer solution undergoing shear flow confined in a slit at Wi=10. Line styles correspond to grid resolution of x = 1.0µm (dashed), x = 0.50µm (solid), and x = 0.25µm (dotted) Steady state chain center-of-mass distribution of a dilute polymer solution undergoing shear flow confined in a slit at Wi = 10. Line styles correspond to Reynolds numbers of Re = 10 (dotted), Re = 2 (dashed), Re = 0.4 (solid), and Re = 0.04 (dash-dotted) Schematic of a grooved channel. Shown in the figure is the xy plane cross-section. The simulation domain is periodic in x and z directions

16 xii Figure Page 7.2 Stream lines corresponding to the flow field generated by shearing the upper wall of the grooved channel in positive x direction. The contour variable is the velocity in x direction. Note that the magnitude of the velocity inside the groove is much smaller than outside Steady state chain center-of-mass distribution in a flowing polymer solution confined in 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 Slice of the two dimensional steady state chain center-of-mass distribution in flowing polymer solution confined in a grooved channel. The slice is taken along y direction at x = 20, which is the center of the channel in x direction. The vertical dotted line indicates the position of the groove top edge Steady state chain center-of-mass distribution in a dilute polymer solution confined in a grooved channel. The dash-dotted line is the distribution obtained from free draining (FD) simulation, and the solid line is the result from simulation with hydrodynamic interactions (HI). Both simulations are performed with Wi = Steady state center-of-mass distribution of isolated beads in shear flow in a grooved channel Snapshots of polymer chains in flowing solution confined in a grooved channel at time t = 711 t, 740 t, 756 t, and 766 t, chronologically from top to bottom. The arrows point to the polymer chain that approaches the corner Schematic of a chain crossing the boundary layer near the separatrix at the top edge of the groove

17 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 thickness is determined by the normal stresses that develop in flow and can be much larger than the size of the polymer molecule. Numerical and similarity solutions show that the developing concentration 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 that the kinetic theory captures the correct far-field (relative to the walls) behavior. Once a finite-size dipole is used, the theory improves its near-wall predictions. In the regime 2h L > R g, the results are significantly affected by the level of discretization of the polymer chain, because the 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

18 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 of increased 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.

19 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-oilrecovery, 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 and electric 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 dyed DNA 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.

20 Figure 1.1 Optical Mapping method developed by David Schwartz s group at the University of Wisconsin-Madison. YAC clone 5L5 derived from human chromosome 11, was digested with EagI and 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 for Molecular and Computational Genomics, University of Wisconsin-Madison. 2

21 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. Fluorescent tagged DNA molecules are stretched at the entrance of a tapered channel due to the collision with the posts and the elongational flow. When the stretched DNA molecules travel through the narrow channel, the positions of the tag sites are read out by laser detectors. This method has a claimed resolution of ±0.8kb resolution and throughput of million bp/min. 3

22 4 The basic physics behind all the above processes or devices is the interactions between polymer 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 conception 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, The problem 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 hydrodynamic interactions 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 solutions in a slit. In Chapter 5, we introduce Brownian Dynamics simulation method with fluctuating hydrodynamic interactions for simulating the confined polymer solution flow. The results are compared 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 solutions flowing through a smooth slit, where the strength and complications 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 connectivity, 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.

23 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 distance 2h. When the slit width is much larger than the equilibrium polymer radius of gyration R g, 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 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 of R g, the free arrangement of the polymer chain is restricted by the walls and deviations from the 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 length L p, 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.

24 6 (a) Single wall oo (b) 2h >> R g R g 2h (c) 2h ~ R g R g 2h R g (d) 2h ~ L p L p 2h Figure 2.1 Schematic of different regimes of confinement: (a) single wall confinement, (b) weak confinement: 2h R g, (c) strong confinement: 2h R g, and (d) extreme confinement: 2h L p.

25 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 to the 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 computational demanding and accessible only for very small time and length 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 to each 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 circumstances. To reach time and length scale of seconds and µm, bead-rod model is still too expensive. Following the same logic, representing a group of beads and rods by one larger bead and connecting 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 simulation 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 of N b beads connected by N s = N b 1 springs as shown in Figure 2.2(c). When polymer chain is stretched, the entropy will

26 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 large amount of atoms within each segment into a bead which is 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 beads and rods by one larger bead and connecting those larger beads by elastic springs give the bead-spring model. This is the coarse-graining level we will work on. 8

27 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 by L = N k b k, where b k is the Kuhn length characterizing the stiffness of the chain and N k is the number of Kuhn segments in the molecule. A vector r with length of 3N b contains the Cartesian coordinates of the N b beads, with r i denoting the position vector of the i th bead. The core of the kinetic theory is the configurational distribution function Ψ(t, r), which gives a description of the probability of the polymer chain taking a given configuration r at a given time t. The vector r contains the 3N b Cartesian coordinates of N b beads. From this distribution, one can obtain all the structural information about the polymer chain and also the interplay between the polymer chain and the solvent. The configurational distribution function is governed by a diffusion equation, Ψ t = r ṙψ, (2.1) ṙ = v + 1 k B T D F D r ln Ψ, (2.2) where k B is Boltzmann s constant, T is the absolute temperature, and r / r is the gradient in configuration space. The v(r, t) denotes the unperturbed external imposed velocity field at each of the N b beads, which is the solution to the incompressible Navier-Stokes equations in the absence of the polymer molecules. The flow perturbation caused by the motion of the beads, v (r, t), enters the diffusion equation through the product of the 3N b 3N b diffusion tensor D. The vector F contains the 3N b 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.

28 10 Rather than using r, one may represent the position and configuration of the polymer molecule by the 3 coordinates of the center-of-mass, r c, and the 3N s = 3(N b 1) coordinates of the connector vectors, q, given as r c = 1 N b r i, (2.3) N b i=1 q i = r i+1 r i. (2.4) Expressed in terms of r c and q, the configurational probability distribution function, Ψ(t,r c,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 = r c r c Ψ 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 freedom of the molecules q n(r c, t) = Ψ(t,r c,q)dq, (2.6) Ψ(t,r c,q) = n(r c, t) Ψ(t,r ˆ c,q). (2.7) Accordingly, the governing equation of n(r, t) is obtained by integrating Equation 2.5 over q n t = r c r c n = rc j c, (2.8) where j c = r c n is the center-of-mass flux, and the angle brackets designate an ensemble average over the configuration variable q, A = AˆΨdq. (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) = τ p (r, t) + τ s (r, t). (2.10)

29 11 The polymer contribution to the stress tensor is given as following [15]: N b τ p (r, t) = n Ψ(r, t)[(r i r c )r i ]dr + n(n b 1)k B TI. (2.11) i=1 r 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-Stokes equation and the continuity equation with proper no-slip boundary conditions: ρ v t + v v = [pi + τ p ] + η 2 v, (2.12) v = 0, (2.13) where η is the solvent viscosity, and ρ 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 size R g. Polymer molecules lose the feeling of all other walls except the nearby one. This regime can also be characterized as single-wall confinement, and is discussed in Chapter 4 through analytical analysis and in Chapter 5 by Brownian dynamics simulations. Strong confinement where the characteristic length scale of the confinement is comparable to the polymer molecule size R g. This regime is treated in Chapter 5 using Brownian dynamics simulations.

30 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 of Ψ. One is generally concerned with ensemble averaged properties 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.

31 13 Chapter 3 Molecular Interactions The dynamics of polymer solutions in a process is determined by intramolecular and intermolecular interactions, interactions with solvent and confinement, and possibly other external fields. In this chapter, we discuss molecular connectivity (spring forces), Brownian force, excluded 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 Chain Connectivity Consider a linear flexible polymer molecule with contour length L = N k b k in a theta solvent, where N k is the number of Kuhn segments and b k 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 N k, the probability distribution function for the end-to-end distance of the molecule is a Gaussian with variance 2b 2 k N k/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 by the 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 = 3k B T/N k b 2 k [15]. Extending this idea to our spring representation of the polymer segments,

32 Figure 3.1 Bead-spring chain model of a polymer moleclue. The springs account for the resistance to the stretch due to the entropic effect, and the beads represent the interaction sites along the chain contour. 14

33 15 we obtain an expression of the tension in the i th spring: where N k,s = N k /N s is the number of Kuhn segments per spring. F s i = 3k BT q N k,s b 2 i, (3.1) k Hookean spring is simple and widely used. However, the Gaussian approximation is valid only in the limit of infinite N k and small stretch q. 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 the inverse Langevin model by considering the probability for the end-to-end distance of a freely jointed Kramer s chain, F s i = k BT b k L 1 ( qi where q i q i, q 0 = N k,s b k is the contour length of the spring, and q 0 ) qi q i, (3.2) L(x) = coth x 1 x, (3.3) is known as the Langevin function. The involvement of the non-linear Langevin function is inconvenient in numerical simulations. A popular alternative spring law to the inverse Langevin model is the empirical FENE model [66], F s i = 3k BT N k,s b 2 k q i 1 (q i /q 0 ) 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, the singularity in the inverse Langevin model can not be expressed as a polynomial while the FENE model has a single singularity of 1 (q i /q 0 ) 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 freely rotating model in which the bending angels are restricted to very small values. Fitting the experimental data [25] to match the asymptotics of the worm-like chain in both the small and large force limits, the resulting spring force law

34 16 is given by Marko and Siggia as F s i = k BT b k [ 1 2(1 q i /q 0 ) q ] i qi. (3.5) q 0 q i 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 q i /q 0 ) Brownian Force In solution, the long polymer chain immerses in an ocean of the tiny solvent molecules. Because 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, F b i = k B T ln Ψ r i. (3.6) In stochastic simulation, the Brownian force is modeled by a random 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: F α (t) = 0 (3.7) F α (t)f β (t ) = δ(t t )2δ αβ k B Tζ, (3.8) where α, β represent the direction of the force, t, t mean different time, and ζ is the bead friction coefficient. The implementation of the fluctuation-dissipation in Brownian dynamics simulation will be discussed in Chapter 5

35 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, N b η 2 v = p F i δ(r r i ), (3.9) v = 0, (3.10) subject to appropriate boundary conditions. The velocity perturbation at the position of bead i due to the bead j is represented in a Green s function form as i v i = i Ω ij F j, (3.11) where Ω is called hydrodynamic interaction tensor. The diffusion tensor D appearing in the diffusion equation 5.6 is related to Ω as, D ij = k B T ( ) 1 ζ Iδ ij + Ω ij, (3.12) where ζ = 6πηa is the bead friction coefficient. In the simplest case, Ω ij 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 = k BT 6πηN b a, (3.13) where a 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 with diffusivity given as, D = k BT 6πηR H, (3.14)

36 18 where R H 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 with N b in a good solvent, we can immediately see that the free-draining model incorrectly predicts the diffusivity to scale with N 1 b. Far-field velocity perturbation due to a point force is given by solving Equation 3.9 and Equation 3.10 in a infinite domain, Ω OB ii = 0 (3.15) Ω OB ij = 1 [ I + r ] ijr ij i j. (3.16) 8πηr ij rij 2 This kind of hydrodynamic interaction form is called the Oseen-Burgers tensor. The diffusion tensor obtained from using Oseen-Burgers hydrodynamics is not guaranteed 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 diffusion 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 of r ij = 0 in the Oseen-Burgers tensor and has the form Ω RPY ii = 0, Ω RPY ij = 1 8πηr ij Ω RPY ij = 1 6πηa [(1 + 2a2 3r 2 ij [( 1 9r ij 32a ) I + ) I a ) ] (1 2a2 rij r ij r 2 ij r ij r ij r 2 ij ] r 2 ij (3.17) i j and r ij 2a, (3.18) i j and r ij < 2a. (3.19) At large bead separation r ij, the RPY tensor approaches the Oseen-Burgers tensor; while at small bead separation, the correction of r ij < a takes hydrodynamic overlap of the beads into account. The Oseen-Bugers tensor is relatively simple, and thus very useful in deriving tractable analytical kinetic theory. On the other hand, the RPY tensor greatly simplifies the computation and is widely used in Brownian dynamics simulations.

37 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 where the 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 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 due to 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 interactions 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 chain R g. The velocity perturbation caused by chain segment travels through 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 (or the moment diffusivity), ν = η/ρ, is much larger than the chain diffusivity D. In terms of Schmidt number (Sc), this means Sc = ν D 1. (3.20) Consider a λ-phage DNA in buffer solution with 1.0cp viscosity [137, 136]. The chain diffusivity is about 0.5µm 2 /s, and the kinematic viscosity of water is around 10 6 µm 2 /s, thus the Schmidt number of the system is on the order of Therefore, Equaqtion 3.20 is fulfilled for a dilute free-space DNA solution, which justifies the approximation of 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

38 20 between the polymer chain and the confining walls must be much larger than the chain convection. In other words, the Reynolds number must be small. The effect of Reynolds number will be discussed in Chapter 5 and Chapter 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 energetically 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]. [ ( ) 12 ( ) ] 6 σ σ = 4ǫ. (3.21) U LJ ij r ij r ij

39 21 Often, the attractive term is neglected, or the potential is shifted and truncated to give only repulsive interactions. In simulating λ-phage DNA, Jendrejack et. al [71] derived an exponential form of the repulsive excluded volume potential by considering the energy penalty due to the overlap of two submolecules modeled as Gaussian coil, U ij = 1 2 υk BTN 2 k,s ( 3 4πS 2 s )3 [ ] 2 exp 3r2 ij, (3.22) 4Ss 2 where υ is the excluded volume parameter and Ss 2 = N k,sb 2 k /6 is the mean square radius of gyration of an ideal chain consisting of N k,s Kuhn segments of length b k. The resulting expression describing the force acting on bead i due to the the presence of bead j is then ( ) 5 [ 3 F υ ij = υk BTNk,s 2 π 2 exp 4πSs 2 3r2 ij 4S 2 s r ij ]. (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 volume model is that the dependence of the the potential on the molecular discretization level (represented by N k,s ) is known and explicit. In our simulation, we adopt this form of the excluded volume potential.

40 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 the formation of depletion layers and apparent 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 steady state depletion layer thickness. The transient development of this depletion layer in uniform plane shear flow and the spatial development 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 Agarwal et al. [2], so we focus here on a few particularly relevant studies. A recent experimental study was performed by Horn et 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 L s (the distance beyond the

41 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 simple model of depletion layers, where the layer consists of pure solvent with a sharp change to the bulk polymer concentration at a distance L d from the wall, the relationship between slip length L s and depletion layer thickness L d is simply L s = L d (1 β)/β, (4.1) where β 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 direct correlation between the depletion layer thickness and the degree of elasticity of the polymer solution. Finally, Fang et al. [47] have recently reported direct observations, using fluorescence microscopy, 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 polymer solutions near boundaries. In a nonhomogeneous flow, the deformation and alignment of the polymer molecules are position dependent. Garner and Nissan [57] proposed that the corresponding 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] employed this result to analyze polymer retention in flows through porous media. Tirrell and Malone [147] have made similar arguments. However, Aubert et al. [7] pointed out that it is not clear that a spatial gradient in intramolecular free energy can result in 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 concentration 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, where τ p is

42 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 contribution 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 purely repulsive 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 depletion layer thickness is still only on the order of the polymer molecule size, and would be insensitive to the flow strength, in contrast to the experimental observations. A significant limitation of all the aforecited dilute solution studies is the neglect of intramolecular 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, but not their modifications due to presence of a wall, then migration toward regions of higher shear rate is found; this is opposite to the trend observed experimentally [74]. For example, Sekhon et 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 migration towards the

43 25 walls using Oseen-Burgers free space hydrodynamic interactions for a bead-spring chain model. To our knowledge, only two studies in the polymer literature aside 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 hydrodynamics 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 migration toward the 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 to propagate 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 the present 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 deformability, 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 hydrodynamic 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 rate is nonuniform, they found that the first order contribution to migration is due to the interaction with the gradient of the local shear rate,

44 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 hydrodynamic effects in studying the motion of flexible particles or macromolecules 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 microchannels, 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 interactions 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 simulations 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 simulations with theoretical results that provide a more fundamental understanding of the migration phenomenon.

45 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-spring dumbbell 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 at y = 0, let r 0 = (x 0, y 0, z 0 ) be the position of one bead, and denote the distance vectors ˆr = r r 0, (4.2) ˆR = r r Im 0, (4.3) where r Im 0 = (x 0, y 0, z 0 ) is the mirror image of r 0 with respect to the wall. The force exerted on the fluid due to the motion of this bead is F. These vectors are shown in Fig The perturbation flow at any other position r(x, y, z) caused by the motion of the bead can be obtained by solving the Stokes equation: subject to no-slip boundary condition at the wall: 0 = p + η 2 v + δ(r r 0 )F (4.4) v(x, y = 0, z;r 0 ) = 0, (4.5) where η is the solvent viscosity, v is velocity, and p is pressure. The solution has the following form: v = Ω F, (4.6) Ω(r,r 0 ) = 1 [ ] S(ˆr) S(ˆR) + 2y0 2 8πη (ˆR) 2y 0 S D (ˆR), (4.7)

46 28 F ˆr r 0 r ˆR wall r Im 0 Figure 4.1 Illustration of the position vectors used for a point force above a plane wall.

47 29 where S is the free-space Stokeslet, P D is the potential dipole and S D is the Stokeslet doublet [18]. These are given respectively as S ij (r) = δ ij r + x ix j r, (4.8) 3 Pij D (r) = ± ( ) ( ) xi δij = ± x j r 3 r 3x ix j, (4.9) 3 r 5 S D ij (r) = ± S i2 x j = x 2 P D ij (r) ± δ j2x i δ i2 x j r 3, (4.10) with the minus sign for j = 2 corresponding to the y direction, and the plus sign for j = 1, 3 corresponding to the x and z 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 parallel to the wall. The flow field is shown in Fig 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 phenomena. 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 r 1 and r 2 denote the position vectors of the two beads of a dumbbell. Then the position of the center of mass is r c = (r 1 + r 2 )/2, and the connector vector is q = r 2 r 1. The quantities ṙ c and q give the rate of change of the center of mass and the connector vector. The conservation

48 Figure 4.2 Velocity field due to a point force in the x direction located at (x, y) = ( 5a, 5a), where a is the bead radius. The plane wall is at y = 0. The lines correspond to streamlines, while the light and dark area indicate regions where the wall-normal velocity is positive (away from the wall) and negative (towards the wall), respectively. Also shown is a bead of radius a located at (x, y) = (5a, 5a) - this can be thought of as the other end of a relaxing dumbbell oriented parallel to the wall. 30

49 31 equation for the probability distribution density function Ψ(r c,q, t) is [15] Integrating the above equation over q and defining Ψ t = (ṙ c Ψ) ( qψ). (4.11) r c q Ψ(r c,q, t) = n(r c, t)ˆψ(r c,q, t), (4.12) n(r c, t) = Ψ(r c,q, t)dq, (4.13) gives the governing equation for the center of mass probability distribution ( concentration ), n(r c, t), n t = j c, (4.14) r c where j c = ṙ c 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 variable q, A = AˆΨdq. (4.15) The fluxes ṙ c Ψ and qψ in Eq. (4.11) are determined by a balance between the spring force F s i, hydrodynamic force F h i, wall repulsion force Fw i and Brownian force F b i exerted on each bead: F h i + Fs i + Fw i + F b i = 0 i = 1, 2. (4.16) Assuming equilibrium in momentum space, the Brownian force is given by [15] F b i = k B T r i ln Ψ, (4.17) where k B is the Boltzmann constant and T is temperature. The hydrodynamic force F h i is proportional to the velocity difference between the bead i 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 the wall exclusion force F w i, we found that in flow, this effect is generally small relative to the hydrodynamic effect. In particular, although the static exclusion force acts over a range of about the polymer radius of gyration R g,

50 32 hydrodynamic effect on the chains in flow leads to depletion over length scales much larger than R g. 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 thicknesses 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 set F w i = 0. Using Eq. (4.16), the velocity of the center of mass, ṙ c, is given by [ 2 ṙ c = 1 v(r i ) D ij (F s j + F b 2 k B T j) ]. (4.18) In this equation, i=1 D ij = k B T i=1 j=1 ( ) 1 6πηa Iδ ij + Ω ij, (4.19) a is the bead radius, v(r i ) is the unperturbed flow velocity at the position of bead i, I is unit tensor, δ ij is the Kronecker delta, and Ω ij denotes the hydrodynamic interaction tensor: Ω ij = Ω(r i,r j ) δ ij 8πη S(r i r j ), (4.20) with Ω(r i,r j ) given in Eq. (4.7). The rate of change of the connector vector, q, is given by q = ṙ 2 ṙ 1 = [v(r 2 ) v(r 1 )] + 2 (Ω 2j Ω 1j ) (F ) (4.21) s j + F b j. Defining the spring force F s = F s 1 = Fs 2 and using Eq. (4.17) along with j=1 = 1 r 1 2 r c q, (4.22) = 1 + r 2 2 r c q, (4.23) the velocity of the center of mass of the dumbbell can be expressed as ṙ c = v qq : v Ω F s D q ln Ψ D k ln Ψ. (4.24) r c

51 33 where Ω = (Ω 11 Ω 22 ) + (Ω 21 Ω 12 ), (4.25) D = k B T Ω, (4.26) D K = 1 4 [(D 11 + D 22 ) + (D 21 + D 12 )]. (4.27) Here v is the unperturbed fluid velocity at the center of mass of the dumbbell, r c, and we have Taylor-expanded v(r 1 ) and v(r 2 ) around r c 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 quantity D K is the so-called Kirkwood diffusivity for a dumbbell [15]. Multiplying Eq. (4.24) by Ψ, integrating over the internal coordinate q and using incompressibility, one can arrive at the mass flux expression: j c =nv + n qq : v 8 Ω D K ( F s + k B T q ln ˆΨ r c D K n r c. n ) ln ˆΨ n (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 Jendrejack et 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 wall y; i.e., we focus on the far field effects of the wall. First, we define a reflection operator T, T = δ 2e y e y = (4.29) 0 0 1

52 34 Then, the image positions of the two beads and the center of mass with respect to the wall are We also define a series of vectors: r Im 1 = T r 1, (4.30) r Im 2 = T r 2, (4.31) r Im c = T r c. (4.32) ˆR c = r c r Im c = r c T r c, (4.33) ˆR 11 = r 1 r Im 1 = ˆR c 1 (q T q), (4.34) 2 ˆR 22 = r 2 r Im 2 = ˆR c + 1 (q T q), (4.35) 2 ˆR 12 = r 1 r Im 2 = ˆR c 1 (q + T q), (4.36) 2 ˆR 21 = r 2 r Im 1 = ˆR c + 1 (q + T q), (4.37) 2 ˆr αβ = r α r β. (4.38) Using this notation and Eq. (4.20), Ω can be rewritten as following for flow above a single wall: Ω = 1 {[ ] S(ˆR 11 ) + 2y1 2 8πη PD (ˆR 11 ) 2y 1 S D (ˆR 11 ) [ ] S(ˆR 22 ) + 2y2P 2 D (ˆR 22 ) 2y 2 S D (ˆR 22 ) [ ] + S(ˆr 21 ) S(ˆR 21 ) + 2y1 2 PD (ˆR 21 ) 2y 1 S D (ˆR 21 ) [ ]} S(ˆr 12 ) S(ˆR 12 ) + 2y2 2 PD (ˆR 12 ) 2y 2 S D (ˆR 12 ). For q ˆR c, we can Taylor expand Ω around ˆR c. Keeping only leading terms yields: (4.39) Ω = 1 { 2 [T q] S + 4y 2 [T q] P D 8πη 4y [T q] S D 8yq y P D (ˆR c ) 4q y S D (ˆR c ) (4.40) + }.

53 35 The gradient terms are readily calculated from Eqs. (4.8), (4.9), and (4.10), allowing Eq. (4.40) to be simplified further, q Ω = 3 1 y q x 0 32πη y 2 q x 2q y q z (4.41) 0 q z q y This can be rewritten compactly to leading order as where Ω = M is a third order tensor with the following components: 3 32πηy 2 M q, (4.42) M 222 = 2, (4.43) M 211 = M 233 = 1, (4.44) M 121 = M 112 = M 323 = M 332 = 1, (4.45) M ijk = 0 i, j, k = others. (4.46) We will denote the tensor M = 3 M/64πηy 2 as the migration 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 to Ω, the leading order D K is given by: D K = k BT 12πηa Recalling that the polymer contribution to the stress tensor τ p [15] is: [I + 3a4 S(q) ]. (4.47) τ p = n qf s nk B TI, (4.48) and using Eqs. (4.42) and (4.47), we can simplify Eq. (4.28) at leading order to the following: j c =nv + n p qq : v + M : τ 8 ( k BT I + 3a ) 12πηa 4 S(q) ln ˆΨ n r c k BT I + 3a4 12πηa S(q) n. r c (4.49)

54 36 Now we define D K,b = k BT I + 3a4 12πηa S(q). (4.50) This is the bulk, ensemble averaged (but conformation dependent) Kirkwood diffusivity. Finally, we use this to rewrite Eq. (4.49): j c =nv + n p qq : v + M : τ 8 n D K,b D K,b n. r c r c (4.51) The last term in this expression is the normal Fickian diffusion; the other terms lead to migration. 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 term M : τ p is generic for the flux of any flexible suspended particle or molecule in a wall-bounded flow in particular its validity is not restricted to the dumbbell model. This term is missing in previous theories of polymer migration. The term containing the divergence of D K,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 migration 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

55 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 the x direction with constant shear rate γ above an infinite plane wall at y = 0. Due to symmetry, no concentration variations will arise in the x and z directions; for the y direction, using Eq. (4.14) and Eq. (4.51), we have that n t = j c,y y = ( K(y) n D n ), (4.52) y y 2 y where K = 3 [ ] M : τ p = 3 64πηn y 64πηn (N 1 N 2 ), (4.53) D = k BT 12πηa. (4.54) Here N 1 and N 2 are the first and second normal stress differences, defined by N 1 = τ p xx τ p yy, N 2 = τ p yy τ p 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 by its free draining value D. Note that in Eq. (4.52) the migration velocity in the wall normal direction is given by v mig = K y 2 = 3 64πηny 2(N 1 N 2 ). (4.56) This result is identical to that derived by Smart and Leighton for a suspended droplet [135]. If we make the further assumption that K is independent of the position, again a good assumption for dilute solution in uniform shear, then Eq. (4.52) can be solved for the steady-state concentration profile: ( n = n b exp L ) d, (4.57) y

56 38 where n b is bulk concentration and L d is the depletion layer thickness, defined by L d = K D = 9 16 N 1 N 2 a. (4.58) nk B T This quantity characterizes the length scale of the steady state depletion layer in a semi-infinite domain. Note that for a long flexible molecule, (N 1 N 2 )/nk B T can be much greater than unity. Since for a dumbbell model, the hydrodynamic bead radius a 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 tensor τ p. Ideally, the evolution equation of the stress tensor would arise 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: F s = Hq 1 q/q 0 2, (4.59) where q = q, H is the spring constant, and q 0 is the maximum extended length of the dumbbell. The stress relaxation time is λ H = ζ/4h, where ζ = 6πηa is the bead friction coefficient. For Hq0 2/k BT 1, the radius of gyration is given by R g = k B T/2H. Introducing the length unit k B T/H and time unit λ H = ζ/4h, dimensionless quantities can be defined: ˆq = q/ k B T/H, ˆt = t/λ H, ˆv = vλ H / k B T/H), b = Hq 2 0 /k BT. (4.60) For the FENE-P model, the dimensionless stress tensor then is: τ p nk B T = ˆqˆq I. (4.61) 1 ˆq 2 /b

57 39 The evolution equation for the structure tensor α(ˆt) = ˆqˆq is [66]: dα dˆt = α ˆvT α + α ˆv + I. (4.62) 1 tr(α)/b In addition to the Peterlin closure for the spring law, there are two approximations involved in using this equation for the stress. The first is the neglect of 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 interactions are included. The effect of this approximation is primarily to ignore a weak dependence of relaxation time with distance from the wall. We would like to 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 approximation. 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 parameter h [15], h = ζ H η s 36π 3 k B T, (4.63) the depletion layer thickness can be expressed as: L d = 9 π 128 N 1 N 2 nk B T h R g. (4.64) Now we define the Weissenberg number Wi = λ γ. 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) profile for different Weissenberg numbers when b = 600 and h = 0.25 (These parameters will be used throughout this paper). The vertical axis is the concentration scaled by the bulk concentration n b and the horizontal axis is the distance from

58 n/n b Wi= 20 Wi = 40 Wi = 60 Wi = 80 Wi = y/(k B T/H) 1/2 Figure 4.3 Steady state concentration profiles scaled by the bulk value in uniform shear flow above a single wall at different Weissenberg numbers. The concentration profiles are calculated using a FENE-P dumbbell model with finite extensibility parameter b = 600 and hydrodynamic interaction parameter h = 0.25.

59 41 the wall scaled by k B T/H. Note that the depletion layer extends a very large distance from the wall, much larger than polymer molecule size. Finally, for a FENE-P dumbbell in shear flow, (N 1 N 2 )/nk B T scales as Wi 2/3 at high Wi [66] (This scaling also holds for the FENE dumbbell model without the closure approximation.). Therefore, the depletion layer thickness scales as L d Wi 2/3 R g. (4.65) Figure 4.4 shows L d vs. Wi on a log-log scale for b = 600 and h = The two-thirds power law at high Wi is evident. The result that L d k B T/H for Wi 1 justifies our neglect of the wall exclusion in the model. 4.5 Temporal and Spatial Evolution of the Depletion Layer in a Semi-Infinite Domain 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 development of 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 rate γ. The transient process is governed (under the same approximations as used above) by n t = j c,y y = ( ) K y y 2n + D 2 n y, (4.66) 2 initial condition : n(y, 0) = n b, { jc,y (0, t) = 0, boundary conditions : n n b 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 determines K(t). Figure 4.5 shows the concentration profile at different times for Wi = 10, b =

60 Figure 4.4 Depletion layer thickness vs. Weissenberg number in a uniform shear flow above a single wall for FENE-P dumbbell with finite extensibility parameter b = 600 and hydrodynamic interaction parameter h = The straight line is the high Weissenberg number asymptote, L d /R g Wi 2/3. 42

61 43 600, h = This figure shows clearly that immediately after inception of 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 so dominant at short times that diffusion cannot smooth out the spike. At long times, however, as shown in 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 after 10 4 relaxation times, the steady state is still not reached. A simple estimate of the time required to reach steady state is given by the time t mig it takes for a molecule to migrate from y = 0 to y = L d : t mig Ld 0 dy v mig L2 d D. (4.67) By this estimate, the migration time scale is on the same order as that of the diffusion time over distance L d. For the computation shown in Fig. 4.5, L 2 d /D ; 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 by L d, is extremely broad at y = L d, the steady state concentration is only about 37% of n b, the bulk concentration. Further insight into the transient development of the depletion layer can be gained by considering the behavior at times much shorter than the diffusion time over the distance L d but long compared to λ H, so K in Eq. (4.66) can be treated as time-independent. Introducing a transient depletion layer thickness δ y (t), an order-of-magnitude analysis of Eq. (4.66) shows that at these short times the dominant balance is between the time-derivative term and the migration term, and that the depletion layer thickness scales as follows: Neglecting diffusion in Eq. (4.66) and defining the variable, δ y (t) (Kt) 1/3. (4.68) ω = y (3Kt) 1/3, (4.69)

62 t = 0 t = 10 λ H t = 100 λ H t = 1000 λ H t = λ H steady state n/n b y/(k B T/H) 1/2 Figure 4.5 Temporal development of the concentration profile in uniform shear flow above a single wall at Wi = 10. A FENE-P dumbbell model with finite extensibility parameter b = 600 and hydrodynamic interaction parameter h = 0.25 is used.

63 45 a similarity solution can be found: { n(ω) 0, if 0 ω 1. = n b ω 2, if ω > 1. (ω 3 1) 2/3 (4.70) This solution is shown in Fig 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 scale δ y (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 for ω 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 to δ y (t)/l d. To illustrate this better, we calculate the full numerical solution to Eq. (4.66) by using a FENE-P dumbbell model with finite extensibility b = 600 and hydrodynamic interaction parameter h = 0.25 at We = 100. The full numerical solutions at t = 10λ H and 1000λ H are plotted in Fig. 4.6 against the similarity solution. We see that the pile up phenomenon that occurs in the 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 which δ y (t) = L d. This estimate recovers our earlier prediction that t mig L 2 d /D. Another important process is the spatial development of the concentration field near the entrance 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 channel. Considering the region sufficiently near the channel entrance that the depletion layer is thin compared to the channel height B, we can treat the domain as semi-infinite in the y-direction and

64 Similarity Solution t = 10 λ H t = 1000 λ H n/n b ω Figure 4.6 Similarity solution for time evolution of the concentration profile in uniform shear flow above a single wall. The full numerical solutions including diffusion for Wi = 100 at two different times, t = 10λ H and t = 1000λ H, are also plotted for comparison. A FENE-P dumbbell with finite extensibility b = 600 and hydrodynamic interaction parameter h = 0.25 is used when solving for the numerical solutions.

65 47 treat the velocity field as a simple shear flow. As above, flow is in the x-direction, and the wall is at y = 0. The conservation equation becomes γy n x = y ( K y 2n ) ( ) 2 n + D x + 2 n. (4.71) 2 y 2 Introducing a spatially varying depletion layer thickness δ y (x), an order-of-magnitude analysis shows that very near the wall, the y-migration term and x-convection terms balance, and the scaling of the depletion layer thickness is given by: ( ) 1/4 Kx δ y (x). (4.72) γ Based on this scaling, we neglect the diffusion terms in Eq. (4.71) and seek a similarity solution n(σ), where The solution is: y σ = (4Kx/ γ) 1/4. (4.73) { n(σ) 0, if 0 σ 1. = n σ 2 b, if σ > 1. (4.74) σ 4 1 This solution is plotted in Fig.4.7; for a channel with height B it will be valid in the case δ h L d B. Again, there is a weak singularity in this solution (which will be regularized by diffusion), showing that a pile up similar to that found in the transient 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 point σ = 1 with width proportional to δ y (x)/l d. The numerical solution without neglecting the y-diffusion term is solved by using a FENE-P dumbbell model with finite extensibility b = 600 and hydrodynamic interaction parameter h = 0.25 at Wi = 100, and the result is shown in Fig. 4.7 for downstream positions x = 10(k B T/H) 1/2 and x = 10000(k B T/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 large length scale. With the knowledge of the depletion layer thickness L d = K/D in the fully developed region (i.e., where convection is negligible and diffusion and migration balance), a scaling estimate of the

66 Similarity Solution x = 10 (k B T/h) 1/2 x = (k B T/H) 1/2 n/n b σ Figure 4.7 Similarity solution for spatial development of the concentration profile in uniform shear flow above a single wall. The full numerical solutions including the diffusion for Wi = 100 at two different downstream positions, x = 10(k B /H) 1/2 and x = 10000(k B /H) 1/2, are also shown for comparison. A FENE-P dumbbell with finite extensibility b = 600 and hydrodynamic interaction parameter h = 0.25 is used when solving for the numerical solutions.

67 49 entrance length L x for the depletion layer can be obtained by setting the boundary layer thickness δ y equal to L d : Therefore the entrance length is given by ( ) 1/4 KLx δ y = = K γ D. (4.75) L x = K3 γ D 4 = L 3 γ d D. (4.76) Combining with Eq. (4.65) and using the scaling relation λ H R 2 g/d, we can rewrite L x in terms of Wi and R g for FENE dumbbells at high Wi as follows: L x Wi 3 R g. (4.77) This result shows that the entrance length is very sensitive to the Weissenberg number a large entrance length should be expected at high Weissenberg number. Finally, we address the issue of what residence time the fluid should have in the channel before the depletion layer can be considered to be fully developed. For the case L d B, this time can be estimated as the travel time from x = 0 to x = L x for a fluid element at a distance of L d from the wall: t travel L x L2 d γl d D (4.78) So roughly speaking, the residence time required for establishment of the fully developed concentration profile is the diffusion time over the distance L d. Based on the transient results presented above, however, we expect this estimate to underpredict the actual time required, because of the broad structure of the steady state depletion layer. The experimentally obtained concentration profiles of Fang et 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 mechanism of molecular migration and the time and length scales involved. In this section, we extend

68 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 distance 2h, and filled with polymer 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 flux in the wall normal direction will be: j c,y = K(y) K(2h y) n y 2 (2h y) n D n 2 y. (4.79) The dependence of K on position arises indirectly as a result of the position dependence of the shear rate. In plane Couette flow, where K is position independent, the steady state concentration profile under this approximation is: [ ( n 1 = exp L d n c y + 1 B y 4 )], (4.80) B where L d = K/D is the depletion layer thickness for an unbounded domain, y is the distance from one wall, and n c is the concentration at the centerline of the slit. Figure 4.8 shows the solutions for Wi = 2, 10, 100, which correspond to L d = 2 k B T/H, 33 k B T/H, and 387 k B T/H. The parameters used are b = 600, h = 0.25, 2h = 30 k B T/H. As the Weissenberg number increases, the concentration profile becomes sharper and sharper, which indicates a stronger migration effect at higher Wi. We now present results for plane Poiseuille flow. Here the velocity profile is parabolic: v x (y) = U m [1 ( 1 y ) ] 2, (4.81) h where U m is the velocity at the center of the slit. The steady state concentration profiles in plane Poiseuille flow are shown in Fig Here Wi 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 of the plane Couette flow is larger than that of plane Poiseuille flow.

69 Wi = 2 L d = 2 n/n c 0.4 Wi = 10 L d = Wi=100 L d = y/(k b T/H) 1/2 Figure 4.8 Steady state concentration profiles at Wi = 2, 10 and 100 in plane Couette flow in a slit with width 2h = 30 k B T/H. Length is scaled by k B T/H and concentration by its value at the centerline of the slit, n c. Migration effects due to the two walls of the slit are superimposed by taking the single-reflection approximation.

70 Wi = 2 L d = 2 n/n c Wi = 10 L d = Wi = 100 L d = y/(k B T/H) 1/2 Figure 4.9 Steady state concentration profiles at Wi = 2, 10, 100 in lane Poiseuille flow in a slit with width 2h = 30 k B T/H. Length is scaled by k B T/H and concentration by its value at the centerline of the slit, n c. Migration effects due to the two walls of the slit are superimposed by taking the single-reflection approximation.

71 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 following equation: v x (y) n x = y + D ( K(y) y 2 ) n ( 2 n y + 2 n 2 x 2 + y ). ( ) K(2h y) (2h y) n 2 (4.82) The numerical solution to this equation is shown in Fig for 2h = 300 k B T/H, b = 600, h = 0.25, and Wi = 20. Concentration is scaled by its bulk value n 0 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 for x = 10 5 k B T/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 make D 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

72 Figure 4.10 Steady state concentration field for plane Poiseuille flow in the entrance region of a slit with width 2h = 300 k B T/H at Wi = 20. Only half of the slit is shown. The concentration is scaled by its bulk value n 0 before entering the slit. Migration contributions due to two walls of the slit are superimposed by taking the single-reflection approximation. 54

73 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 concentration profile in uniform shear flow above an infinite plane wall. The profile, determined by the balance of migration and diffusion, has the form of a Boltzmann distribution and is characterized by a length scale L d, 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, L d Wi 2/3 R g, 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 distance L d. 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 length L x for the concentration evolution in a channel is estimated (for FENE dumbbells) to scale with Wi 3 R g. 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

74 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 shear rate 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.

75 57 Chapter 5 Brownian Dynamics Simulation In the field of dynamics of polymeric liquids, computer simulation has been playing an important 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 enables us to solve more complicated problems inaccessible to theoretical 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 crossstreamline 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 regimes R g h and R g h; (2) evaluate the analytical theory developed in previous chapter in the regime R g h; (3) examine the issue of coarse-graining of chains into dumbbells in confined geometries, especially in the regime R g 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. 5.1 Introduction Rapid advances in photo-lithography and soft lithography have greatly facilitated the design and fabrication of novel microfluidic devices working on the length scale of micron and smaller

76 58 [110, 53]. These devices are now used in diverse applications such as DNA sequencing and mapping, 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 microfluidic 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 large surface 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, depending on 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 walls separated by a distance 2h. When the slit width is much larger than the equilibrium polymer radius of gyration R g, 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 of R g, the free arrangement of the polymer chain is restricted by the walls and deviations from the bulk equilibrium 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 length L p, then the chain dynamics is extremely restricted [145, 123], as shown in Fig. 5.1c. In this 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 between the polymer chain and the confinement is the formation of depletion layers during flow [2]. Chen et al. [31] recently presented direct visualizations of the depletion layer in flow of DNA

77 59 solutions in a channel., Fig. 5.2 shows their experimental measurements of axially averaged fluorescent intensity in the cross section of a 40 µm 40µm microchannel as a function of time for fluorescently-labeled T2-DNA solution undergoing oscillatory pressure-driven flow. The dark regions near the walls indicate depletion layers with thickness of about 10 jm, which is much larger than the radius of gyration of the T2-DNA molecule (about 1.6 µm). Note that it takes more than a minute for the depletion layer to fully develop. In related work, Fang et al. [47] found that in channel flow of dilute λ-phage DNA solution, inside a region extending from a glass surface in a micro-channel to about one third of the contour length of λ-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 to explain 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 controversial, 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, Jendrejack et 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

78 60 (a) Single wall oo (b) 2h >> R g R g 2h (c) 2h ~ R g R g 2h R g (d) 2h ~ L p L p 2h Figure 5.1 Schematic of different regimes of confinement: (a) Single wall confinement, (b) weak confinement: 2h R g, (c) strong confinement: 2h R g, and (d) extreme confinement: 2h L p.

79 Figure 5.2 Time evolution of axially averaged fluorescence intensity of fluorescent labeled T2- DNA solution as a function of cross-sectional position. The channel walls are at y = ±20µm. The solution is undergoing oscillatory pressure-driven flow at a maximum strain rate of 75 s 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 and the dark region represents the depletion layer near the channel walls. 61

80 62 between chains and the channel walls. They predicted that, when the channel width 2h is much larger than the equilibrium chain radius of gyration R g, 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 manner similar to that found for suspensions of deformable droplets [70, 27, 135]. Santillan et al. [128] have performed related simulations for bead-rod chains. Building on the work of Jendrejack et 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 theory for a bead-spring dumbbell polymer model; an assumption that the polymer extension was small compared to the distance of the polymer from the wall enabled derivation of relatively simple closed form results. Subsequent approximations 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 regime R g 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 is larger 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 Usta et al. [149]. The physical origin of this reversal is addressed below. In the present work, we study cross-stream migration during flow 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 regimes R g h and R g h; (2) evaluate the analytical theory of Ma and Graham in the regime R g h for which it was derived; (3) examine the issue of coarse-graining of chains into dumbbells in confined geometries, especially in the regime R g 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

81 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 v mig at the position of the dipole: v mig = M : d, (5.1) where the third-order tensor M 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, j c, j c =nv + n p qq : v + M : τ 8 n D K,b D K,b n, r c r c (5.2) where n(r c, t) is the center of mass probability distribution function (i.e. concentration ), v is the imposed velocity field evaluated at the center of mass r c of the dumbbell, q is the end-toend vector of the dumbbell, τ p is the polymer contribution to the stress tensor and D K,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 we show a more general expression that 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, τ p, arises from the presence of walls. For single wall confinement, M is given by Ma and Graham [94]; if y is the distance from the wall, it decays as 1/y 2. Note that this term is generic for the flux of any flexible suspended

82 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 D K,b can also lead to migration if the diffusivity of the molecule depends on conformation (which in 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 force toward 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 thickness L d is determined primarily by the first normal stress difference in the flowing solution. In a flow where the chain is strongly stretched, L d becomes much larger than the equilibrium size of the polymer chain. For the finitely extensible dumbbell model, the analysis predicts that L d /R g Wi 2/3, where Wi = γλ is the Weissenberg number with γ shear rate and λ the longest relaxation time of the chain. The model also predicts that the chain density profile reaches steady state over a time scale of L 2 d /D, where D is the molecular diffusivity of the stretched chain. Finally, this analysis can be extended to a slit 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 a viscous solvent is represented by a freely jointed bead-spring chain, i.e., N b beads connected through N s = N b 1 springs. Neglecting inertia, on each bead the force balance requires F h i + Fs i + Fv i + Fw i + F b i = 0, for i = 1,..., N b, (5.3) where, for bead i, F h i is the hydrodynamic force, F v i is the bead-to-bead excluded volume force, F w i the bead-wall excluded volume force, F b i is the Brownian force and F s i is the spring force. The characteristic variables are the bead hydrodynamic radius, a, for distance, ζa 2 /k B T for time and k B T/a for force, where k B is the Boltzmann s constant, T the absolute temperature, and ζ the

83 65 bead friction coefficient, which is related to the solvent viscosity, η, and a through Stokes law, i.e. ζ = 6πηa [85, 118]. A finitely extensible nonlinear (FENE) spring defined by the following dimensionless potential energy [15] φ s ij = 1 [ ] 2 b ln 1 r2 ij, (5.4) b is used. Here r ij = r i r j is the distance between beads i and j, and b is the extensibility parameter, and b = H s q 2 0 /k BT, where H s is the spring constant per spring, H = H s /N s is the total spring constant for the molecule and q 0 = L/N s the maximum stretch of each spring. For the special case of the dumbbell model, H = H s and q 0 = 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 = [u 0 + D F + r ] D dt + 2B dw. (5.5) Here r is a vector containing the 3N b coordinates of the beads that constitute the polymer chain, with r i denoting the Cartesian coordinates of bead i. The vector v 0 of length 3N b represents the unperturbed velocity field, i.e. the velocity field in the absence of any polymer molecule. The vector F has length 3N b, with F i denoting the total non-brownian, non-hydrodynamic force acting on bead i. Finally, the independent components of dw are obtained from a real-valued Gaussian distribution with mean zero and variance dt. 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 the 3 3 block components (D ij ) of the 3N b 3N b 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) where I is the identity matrix. Computation of Ω f will be discussed below. The Brownian perturbation, d w, is coupled to the hydrodynamic interactions through the fluctuation-dissipation theorem [114, 122, 157] D = B B T. (5.7)

84 66 For excluded volume a Gaussian potential is assumed for bead-to-bead interactions: φ v ij = A b exp [ αr 2 ij], for rij 3, (5.8) while a repulsive potential is used for wall-bead interactions as follows φ vw i = A w 3 (r iw 2) 3, for r iw 2, (5.9) where r iw represents the distance of bead i 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 2 v j i (r,r 0) x k x k pj (r,r 0 ) x i = δ (r r 0 )δ ij, v j i (r,r 0) x i = 0, (5.10) where η is the solvent viscosity, δ (r) is the Dirac delta function, δ ij is the Kronecker delta and v j i (r,r 0) is the fundamental singular solution or Green s function of the Stokes equations, known as a Stokeslet, located at the point r 0 and oriented in the j-th direction [81, 118, 117]. In order to make the expression compact, we use x j to represent the three Cartesian coordinates of position vector r, with j = 1, 2, 3 corresponding to the x, y and z directions. In an infinite domain (no confinement), the free-space Green s function is v j i (x,x 0) = 1 8πηr [ δ ij + (x i x 0i )(x j x 0j ) r 2 ], (5.11) sometimes also called the Oseen-Burgers (OB) tensor. For free-space simulations, with the point force formalism, the 3 3 non-diagonal sub-matrices of the 3N b 3N b hydrodynamic interaction tensor, Ω ij, are Stokeslets, as follows Ω νµ ζ 1 = (1 δ νµ ) v j i (r ν,r µ ), (5.12) where ν and µ are polymer beads and i, 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 at x 2 = w, the Green s function, that

85 67 satisfy the non-slip at the wall, w v j i (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] w v j i (r,r 0) =v j i (r,r ( ) 0) v j i r,r I [x 02 w] 2 ( ) Uij D r,r I 0 (5.13) ( ) 2 [x 02 w]uij SD r,r I 0, where r I 0 = (x 01, 2w x 02, x 03 ) is the image of r 0 with respect to the wall. The tensors U D (r) and U SD (r) represent potential dipoles and Stokes-doublets. For three-dimensional domains, they are given by Uij D (r) = ± ( ) ( ) xi δij = ± x j r 3 r 3x ix j, (5.14) 3 r 5 U SD ij (r) = ± v3 i x j = x 2 U D ij (r) ± δ j2x i δ i2 x j r 3, (5.15) with a plus sign for j = 2, in the y-direction, and a minus sign for j = 1, 3, corresponding to the x- and z-directions. For single-wall confinement BD simulations the 3 3 non-diagonal submatrices of the HI tensor, Ω ij, 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 diagonal 3 3 sub-matrices. The single-wall HI tensor is then calculated as follows Ω νµ ζ 1 =(1 δ νµ ) v j i (r ν,r µ ) v j i ( ) rν,r I µ + 2 [x µ2 w] 2 ( ) Uij D rν,r I µ ( 2 [x µ2 w]uij SD rν,rµ) I (5.16) The Stokeslet in Eq. (5.13) is singular when the bead is at the wall, and in fact the BD-pointforce 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 probability of finding a bead at the wall is zero. For Stokes flow between two parallel plates, Liron and Mochon found two alternative expressions for the Stokeslet, one in terms of infinite integrals and the other in terms of infinite series [92].

86 68 These solutions have been used in the past to investigate the motion of particles and droplets between 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 computation time scaling as O(N 3 ), where N is the number of particles. There are different approaches to calculate the Green s function due to a force of arbitrary orientation between two walls. One approach was introduced by Jendrejack et. 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(N 2.25 ) [73, 74, 32, 31]. Recently Mucha et al. developed an O(N log N) method for computing HI in a slit geometry [108]. Based on this method Hernández-Ortiz et. al developed a BD simulation algorithm that scales as O(N 1.25 log N) [63]. Here, we are interested in a system confined between two infinite walls separated a distance 2h along the z (or x 3 ) -coordinate, with periodic boundary conditions in the other two directions, x (or x 1 ) and y (or x 2 ), of periodic length W and L, respectively. The basic outline of the method is as follows. It starts by splitting the slit Stokeslet, S v j i (r,r 0), into three column vectors, i.e, S v j i (r,r 0) = [S 1,S 2,S 3 ], (5.17) where S j (r,r 0 ) = (u j, v j, w j ) represents the velocity perturbation due to a point force in the j-direction at position x 0, with the corresponding pressure p j (r,r 0 ) = {p 1, p 2, p 3 }. (5.18) The calculation of each piece of the Green s function, (S j, p j ), proceeds by Fourier series expansion in the two periodic dimensions S j (r,r 0 ) = k v j ( k, x 2, x 02 ) e [ik (r r 0 ) ], (5.19) p j (r,r 0 ) = k p j ( k, x 2, x 02 ) e [ik (r r 0 ) ], (5.20)

87 69 with the summation over two-dimensional wave vectors k = (k 1, k 3 ). Here the subscript indicates the two periodic directions x 1 and x 3 and v j (r,r 0 ) = ( v j, v j, ŵ j ). Inserting Eqns. (5.17) and (5.18) into the Stokes equations, Eq. (5.10), a set of ordinary differential equations for the Fourier coefficients, v j (r,r 0 ), is obtained. The solution for these coefficients has the following from v j (r,r 0 ) = v j ( k, a jn ( k, x 02 ), x2 ), (5.21) where a jn ( k, x 02 ) for j = 1, 2, 3 and n = 1,..., 6 are a set of field coefficients. These coefficients are function of r 0 but not of r, so they can be calculated only once per configuration. Mucha et. al realized that after a sorting of the particles with respect to x 2 -direction the calculation of the Green s functions can be performed in O(N log N) calculations [108]. Details of the BD implementation can be found in Hernández-Ortiz et. al [63]. There are two Brownian based terms in Eq. (5.5), the random vector B dw and the divergence of the diffusion tensor. The most common method in the literature to find the matrix B for the fluctuating terms, in a way such that the fluctuation dissipation theorem is satisfied, is to perform the Cholesky decomposition of the diffusion tensor D [120, 20], D = S S with S = S T, (5.22) which typically means to calculate directly the diffusion tensor, an operation that scales as O (N 2 ), and to use a regular method to do the Cholesky decomposition, O (N 3 ), which will imply long computational times even for dilute systems. Instead, the method described by Fixman [52, 51] and the algorithm of Jendrejack et. 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 (N 1.25 log N) [63]. The polymer chain represented by the bead-spring model is fully characterized by 4 parameters: {N s, b k, n k, h }, namely the number of springs, Kuhn length, number of Kuhn segments and the hydrodynamic interaction parameter [15, 94], h = ζ H η 36π 3 k B T. (5.23)

88 70 The polymer selected for the results presented in this paper is similar to a λ-phage DNA (we are using a FENE spring instead of a worm-like spring) where b k = 0.106µm, n k = 198 and h = The strength of the flow field is characterized by the Weissenberg number, Wi, representing the ratio between the time scale of molecular relaxation to that of the solvent relaxation. For shear flows is defined by Wi = λ γ, where λ 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 1 2H 4 sin 2 (π/2n b ). The excluded volume parameters were A b = 2 and α = 0.5 for the bead-to-bead and A w = 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 beyond y = 100 (k B T/H) 1/2 were reflected back into the domain, a procedure that has no effect on the steady state chain distribution. For the slit simulations the domain size W in the wall-parallel directions was always set to be the larger of three times the total wall separation, (3 (2h)) and three times the contour length of the molecule, (3 (n k b k /a)). 5.4 Results and Discussion Single Wall Migration in Simple Shear To address cross-stream migration in the case R g L h we take the situation h, i.e. a semi-infinite domain. Consider a infinitely dilute solution of dumbbells (N b = 2) under uniform shear flow in the x-direction with constant shear rate γ above a plane wall at y = 0. Using the point dipole theory described above, the center-of-mass flux, j c, reduces to j c = M : τ p D K,b n r c. (5.24)

89 71 With an additional assumption of constant diffusivity, Ma and Graham found that the steady-state concentration is given by [94] ( n = n b exp L ) d, (5.25) y where n b is the bulk concentration and L d is the depletion layer thickness, where N 1 = τ p xx τp yy and N 2 = τ p yy τp zz L d = 9 N 1 N 2 32 D, (5.26) are the first and second normal stress differences and D 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 for N 1, N 2 and D 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] N b τ p = (r i r c )F i, (5.27) i=1 while the diffusivity can be determined using the Kirkwood formula, D K = 1 N 3 tr 1 b N b D ij. (5.28) N 2 b i=1 Figure 5.3 shows the steady-state concentration profiles predicted by Eq. (5.25) and a BD simulation of dumbbells, with h = 0.25, b = 594. Position is scaled with (k B T/H) 1/2 for a dumbbell model with b 1, R g = (3k B T/H) 1/2. The depletion layer thickness, L d, 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 related to the migration velocity, v mig. Relaxing the point-dipole approximation, this becomes for a j=1

90 n/n [y=90(kb T/H) 1/2 ] Theory: Stokeslet doublet Theory: finite size dumbbell BD simulation (dumbbells) y/(k B T/H) 1/2 Figure 5.3 Steady-state chain center-of-mass concentration profiles predicted by theory, using the Stokeslet-doublet (far-field) approximation, and the BD simulation at Wi = 0, 5, 10 and 20 in simple shear flow. The concentration is normalized using its value at y/(k B T/H) 1/2 = 90.

91 73 dumbbell, v mig,y = [Ω 12 F s 2] y + [Ω 21 F s 1] y. (5.29) 2 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, where q = 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. 5.5 and 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) approximation, Eq. (5.25), and BD simulation of chains with N s = 10. The values of N 1, N 2 and D 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 the chains; we elaborate on this observation below 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

92 74 1 v mig,y /v mig Stokeslet doublet,y q/q 0 =0.2 q/q 0 =0.3 q/q 0 = y/(k B T/H) 1/2 Figure 5.4 Migration velocity scaled with the point-dipole value for different dumbbell (forcedipole) sizes, as a function of distance from the wall.

93 n/n [x3 =90(k B T/H) 1/2 ] Theory: Stokeslet doublet Theory: finite size dumbbell BD simulation (dumbbells) x /(k T/H) 1/2 3 B Figure 5.5 Near-field center-of-mass steady-state concentration profiles predicted by theory, using the Stokeslet-doublet (far-field) approximation and finite-size dumbbells, and the BD simulation at Wi = 5 in simple shear flow.

94 76 1 BD simulation (chains) Theory: Stokeslet doublet 0.8 n/n [y=50(kb T/H) 1/2 ] Wi=5 Wi= y/(k B T/H) 1/2 Figure 5.6 Steady-state chain center-of-mass concentration profiles predicted by theory, using the Stokeslet-doublet (far-field) approximation, and the BD simulation of 10 springs chains, at Wi = 5 and 10 in simple shear flow.

95 77 by [ ( 1 n = n c exp L d y + 1 2h y 2 )], (5.30) h where L d is the depletion layer thickness for an unbounded domain given in Eq. (5.26) and n c is the concentration at the centerline of the slit. Note that in Eq. (5.30) there are two main approximations: the Stokes-doublet (point-force-dipole or far-field) and the single-reflection for the wall-chain hydrodynamic interactions. Figure 5.7 shows the steady-state concentration profiles for Wi = 5 and 20 calculated with the theory with the Stokeslet-doublet and the single-reflection approximations and BD simulations with dumbbells. Here, 2h/R g = 56.4 and 2h/L = 1.6 where the contour length of the polymer, L, is equivalent to the maximum spring length, q 0, 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 degree of 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 (N s = 1) and two chain models (N s = 5 and 10) and constant contour length, for a wall separation of 20(k B T/H) 1/2 (2h/R g = 28.2 and 2h/L = 0.8). As the figure indicates, the shoulders disappear for the chain simulations and the results for N s = 5 and 10 are virtually indistinguishable. To understand the discrepancies between the dumbbell and chain results, consider Fig For a highly stretched dumbbell, 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 migration. A better discretization is given by the chain model illustrated 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 N s = 5, we calculate the predictions by the theory using the Stokeslet-doublet and single-reflection approximation. Figure

96 78 n/n c Wi=0 Wi=5 BD simulation Theory: far field + single reflection Wi= y/(k B T/H) 1/2 Figure 5.7 Steady-state chain center-of-mass concentration profiles predicted by theory, using farfield and single-reflection approximations, and the BD simulation at Wi = 0, 5 and 20 in shear flow.

97 N s =1 q 0 /2h=1.24 Wi= n/n c N s =5 q 0 /2h=0.24 N s =10 q 0 /2h= y/(k B T/H) 1/2 Figure 5.8 Steady-state chain center-of-mass concentration profiles predicted by the BD simulation at Wi = 20 in shear flow, for different polymer discretizations: N s = 1, 5 and 10.

98 80 (a) dumbbell q ~ h (b) chain q << h Figure 5.9 Schematic of two different discretization levels of a same molecule (a) dumbbell: the effect 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 with smaller separation.

99 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 Highly Confined Polymer Chains Finally we consider highly confined systems: 2h R g. In particular, we perform simulations of chains (N s = 10) undergoing Couette flow, in a slit with a wall separation 2h = 2.9R g. 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-called free-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 R g, 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 Jendrejack et al. [74] and by Usta et al. [149]. The fact that the HI and FD models give the same results implies 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 defined by R c = r max r min, (5.31) c where the subscript c denotes a conditional average: i.e. given a chain at a particular wall-normal position y, R c is the expected value of its stretch. Note that the chains found near the wall are more stretched in the two periodic directions x and z (i.e. parallel to the walls) than chains in the center of the slit. In addition, the chains are less extended in the wall normal direction, y. This effect becomes more pronounced in flow. Figures 5.14 and 5.15 show that in flow the chains extend in the flow direction and are correspondingly less extended in the wall-normal direction. Because the geometry is so confined, the wall-chain hydrodynamic interactions from each

100 82 1 BD:Wi=0 0.8 n/n c Wi=20 BD:bead distribution BD:center of mass Theory y/(k B T/H) 1/2 Figure 5.10 Steady-state chain center-of-mass concentration profiles predicted by the theory, using far-field and single-reflection approximations, and the BD simulation at Wi = 20 in shear flow. The steady-state chain center-of-mass concentration profile at equilibrium (Wi = 0) and the beaddistribution from the simulation at Wi = 20 are also shown.

101 Wi=0 FD: Wi=20 HI: Wi= n/n c y/(k T/H) 1/2 B Figure 5.11 Steady-state chain center-of-mass concentration profiles predicted by the BD simulation of chains (N s = 10) for a highly confined polymer solution, 2h = 2.9R g.

102 Wi=0 FD: Wi=20 HI: Wi=20 n b /n b,c y/(k T/H) 1/2 B Figure 5.12 Steady-state bead-concentration profiles predicted by the BD simulation of chains (N s = 10) for a highly confined polymer solution, 2h = 2.9R g.

103 85 2 < R> c /(k B T/H) 1/ <x> c <y> c <z> c y/(k T/H) 1/2 B Figure 5.13 Polymer stretch as a function of the wall-normal direction, y, for Wi = 0 (no flow); 2h = 2.9R g.

104 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] 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 where h R g. 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. Let ˆΨ(q,r c ) be the conformational probability distribution of a dumbbell with connector vector q and centerof-mass position r c. At any position r c, ˆΨdq = 1. If F s i (q) is the connector force and F w i (q,r c) is the excluded volume force between bead i and the walls, then the flux expression is j c =nv + n 8 qq : v + 1 ( Ω F s + k B T ) ln ˆΨ n 2 q ln ˆΨ D K n D K n r c r c where + 1 2k B T (D 11 + D 21 ) F w 1 + (D 12 + D 22 ) F w 2 n, (5.32) Ω = (Ω 11 Ω 22 ) + (Ω 21 Ω 12 ), D K = 1 4 [(D 11 + D 22 ) + (D 21 + D 12 )]. (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 R g. In a single-wall domain or when 2h R g, a flexible molecule in flow with Wi 1 will exhibit a depletion layer of thickness R g, as discussed in Sections and 5.4.2, making the steric wall effect largely irrelevant. In contrast, when 2h R g, hydrodynamic wall effects become

105 87 <x> c /(k B T/H) 1/ Wi=0 FD: Wi=20 HI: Wi= y/(k B T/H) 1/2 Figure 5.14 Polymer stretch in the flow direction, x, as a function of the wall normal direction, y; 2h = 2.9R g.

106 <y> c /(k B T/H) 1/ Wi=0 FD: Wi=20 HI: Wi= y/(k B T/H) 1/2 Figure 5.15 Polymer stretch in the confined direction, y, as a function of the wall normal direction, y; 2h = 2.9R g.

107 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 Effect of Finite Reynolds Number on Wall-induced Hydrodynamic Migration Theory and simulation approaches to confined polymer hydrodynamics that use Stokesletbased 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], dissipative particle dynamics (DPD) [46] and lattice Boltzmann (LB) [149] do not explicitly impose this condition. In particular the DPD results of Fan et al. [46] were obtained at Reynolds numbers based on the channel height on the order of 10 2, 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 distance y from the wall. Let us impose a flow with uniform shear rate γ, in which the relevant Reynolds number is Re y = γy2 ν. (5.34) where ν is the kinematic viscosity of the solvent. The time scale for diffusion of momentum between chain and wall is t d = y 2 /ν, while the velocity of the chain is γy. So the distance traveled by a chain in the time t d is γyt d = yre y. If Re y is small, then the chain moves only a small distance in the time t d 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]. Once Re y 1, however, the chain moves downstream a distance y in the time t d by the time the perturbation induced by the chain propagates to the wall and back to the original position occupied by chain, the chain 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.

108 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 its image, in a reference frame moving with the chain, in the cases Re y 1 and Re y 1. When Re y 1, the shear flow does not significantly distort the perturbation, but when Re y 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, as Re increases and the purely viscous effect studied here becomes less important, lift effects arise [130, 151, 86, 48], further complicating the interpretation of finite Re simulation results. 5.6 Conclusion This paper examines cross-stream migration due to hydrodynamic wall effects in dilute flowing polymer solutions in the regimes R g L h (e.g. flow in a half-plane), R g h L and R g 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. When h L the standard dumbbell model breaks down because of the coarse discretization of the force distribution. In highly confined domains, R g h, hydrodynamic migration effects are overwhelmed by steric effects which lead chains to migrate toward walls in flow rather than away.

109 91 (a) Re h <<1 Flow perturbation due to particle Wall Flow perturbation due to image (b) Re h >1 Flow perturbation due to particle Wall Flow perturbation due to image Figure 5.16 Schematic of the hydrodynamic migration mechanism (a) Re y 1: wall-induced migration momentum diffusion to the wall and back to the particle is fast; (b) Re y 1: No wall-induced migration the shear flow distorts the velocity perturbation due to the particle so that the particle is not affected by the presence of the wall.

110 92 Chapter 6 Simulating Polymer Solution Using Lattice-Boltzmann Method In previous chapters, the dynamics of dilute polymer solutions in simple geometries, singlewall 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 geometry 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. Although elegant and well-understood in simulating dynamics of dilute polymer solutions in free space or relatively simple geometry, the Brownian Dynamics method with fluctuating hydrodynamic 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 interaction sites in the system: the time to calculate a single step for a chain of N segments scales as N 2.25 c [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 mobility matrix) with correct no-slip boundary conditions at confining surfaces. On the other hand,

111 93 Molecular Dynamics (MD) and Dissipative Particle Dynamics (DPD) [77] treat the solvent particles 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 solvent [3, 4, 5]. The Lattice-Boltzmann Method is an alternative way to solve flow problems governed 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 implementation of boundary conditions. A bead-spring chain model of the 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 other words, 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 Section 6.3, the bead-spring chain model of the polymer molecule is explained. The simulation parameters 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. The complications of the LBM is discussed in Section 6.6. Finally, we give a conclusion in Section 6.7.

112 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 function f i (r, t) describing the number of particles at lattice site r at time t with velocity c i. 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 of the 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 model is specified by a set of discrete velocities c i, equilibrium velocity distribution f eq i, boundary treatment, and a collision operator which advances the velocity distribution with time Velocity Set The c i 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 fluid dynamics, the discrete velocities must guarantee mass, momentum, and energy conservation as well as rotational invariance. In 3D space, one commonly used model, the D3Q19 model (19 velocities in 3 dimensions) [84], which satisfies these conditions, is shown in Fig The D3Q19 model consists of one zero velocity (or (0, 0, 0)), six velocities with speed 1 (connecting a cube s center to 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

113 95 z y x Figure 6.1 The set of discrete velocities in a D3Q19 model shown in a lattice cube. The solid parallelogram represents the xy plane, the dashed rectangle the yz plane, and the dotted parallelogram the xz 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 cube faces, and 12 velocities with magnitude 2 represented by the arrows pointing to the cube-edge centers.

114 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 our studies Equilibrium Velocity Distribution In steady uniform flow with velocity v, 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 eq i = a c i [ ρ + ρv c i + ρvv : (c ic i c 2 s I) ], (i = 0, 1,, 18) (6.1) c 2 s 2c 4 s where ρ is the fluid density, v is the local velocity, c s is the speed of sound c 2 s = 1 3 ( ) 2 x (6.2) t with x the lattice grid size, t the time step, and a c i is a normalized weight that describes the fraction of particles with velocity c i at thermodynamic equilibrium (ie. when v = 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: a c i c iα c iβ c iγ c iν = C 4 c 4 (δ αβ δ γν + δ αγ δ βν + δ αν δ βγ ), (6.3) i where c = x/ t, and C 4 is a numerical coefficient depending on the choice of weights. The optimum choice of weights for D3Q19 model is [84] In this case the coefficient is C 4 = (c s /c) 4. a 0 = 1 3, a1 = 1 18, a 2 = (6.4) In the D3Q19 model, 19 moments of the velocity distribution function can be defined. The first ten moments give the density ρ, the momentum density j = ρv, and the momentum flux tensor

115 97 Π = ρvv: ρ = i f eq i, (6.5) j = ρv = i Π = ρvv = i f eq i c i, (6.6) f i c i c i. (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 Π eq = i f eq i c i c i = ρc 2 si + ρvv. (6.8) The remaining 9 moments refer to kinetic energy fluxes, which conserve energy. They are nonhydrodynamic modes and irrelevant to the Navier-Stokes equations Collision Operator The velocity distribution function evolves with time according to a discrete analogue of the Boltzmann equation, f i (r + c i t, t + t) = f i (r, t) j L ij [ fj (r, t) f eq j (r, t)] t (6.9) where L ij are the matrix elements of the linearized collision operator L for dissipation due to fluid particle collisions. The collisions relax the fluid towards the 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 f i, the collision operator L is a matrix in the D3Q19 model. The 19 eigenvalues of L, (τ 1 0, τ 1 1,, τ 1 18 ), characterize the relaxation time of the 19 moments. For the conserved moments (ρ and ρv), the relaxation time is infinite and τ 1 i = 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

116 98 time for the momentum flux moments Π are set to a single constant τ i = τ s. In general, τ i for other moments irrelevant to the Navier-Stokes equation are set 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 distribution fi is written as [ fi = a c i ρ + j c i + (ρvv + ] Πneq, ) : (c i c i c 2 si), (6.10) c 2 s 2c 4 s where Π neq, = (1 + 1 τ s ) Π neq (1 + 1 τ s )(Π neq : 1)1, (6.11) with Π neq = Π Π eq. Πeq is the traceless part of Π neq. The kinematic viscosity ν of the fluid is determined by the relaxation time τ s as: ν = c 2 s (τ s 0.5). (6.12) Figure 6.2 illustrates one Lattice-Boltzmann step consisting of a streaming process and a singletime-relaxation process in 2D sapce. Without the external force, the equilibrium velocity distribution consists simply equal amount of fluid particles for each of the discretized velocities. A non-equilibrium velocity distribution has more particles for some velocities while less particles for other velocities. In the streaming process, particles convect to the neighboring lattice sites according to the direction of their velocities. In the relaxation process, the collision rules force the larger f i s at the site to decrease and the smaller f i s to increase so that the velocity distribution relax toward the equilibrium one External Force In the presence of an external field, such as a pressure gradient or a gravitational field, a force density F on the fluid needs to be included in the model. The force alters the velocity distribution function f i such that velocity grows in the direction of the force and shrinks in the opposite direction, 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

117 99 (a) single-time-relaxation with τ s = 1 streaming relaxation τ s = 1 equilibrium distribution (b) single-time-relaxation with τ s = 2 streaming relaxation τ s = 2 halfway distribution Figure 6.2 In the single-time-relaxation model, the velocity distribution at each site relaxes toward the equilibrium one at each time step. Without the external force, the equilibrium velocity distribution consists simply equal amount of fluid particles for each of the discretized velocities. The figure 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 neighboring sites stream along their directions of motion to that site. Second, the incoming distribution relaxes due to the particle collisions, according to the single-time-relaxation rule, towards the equilibrium distribution. (a) When τ s = 1, the incoming velocity distribution relaxes to the equilibrium distribution in one time step. (b) When τ s = 2, the post-relaxation distribution is halfway between the incoming and the equilibrium distributions.

118 100 f f i (r, t) [84] f i (r + c i x, t + t) = f i (r, t) j [ L ij fj (r, t) f eq j (r, t)] t + f f i (r, t). (6.13) The forcing term f f i (r, t) is given by f f i = a c i [ A + B c i + C : (c ic i c 2 s I) c 2 s 2c 4 s ] t, (6.14) where A, B, and C are determined by requiring that the moments of f f i are consistent with the hydrodynamic equations: A = i B t = i f f i = 0, (6.15) f f i = F t, (6.16) C = vf + Fv. (6.17) The second moment C is usually neglected. In that case, more accurate solutions to the velocity field are obtained if an additional momentum is added to each node [93], ρv = i f i c i + 1 F t. (6.18) 2 However, if the conventional definition of the momentum flux is retained, the expression for C needs to be modified to account for discrete lattice effects. Nevertheless, for a spatially uniform force, numerical simulations show that variations in C have a negligible effect on the flow [84] 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 confining 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 realized by a bounce-back scheme [84, 28].

119 101 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 ends up 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 that c i is the velocity towards the wall at one fluid node, and c i is the opposite velocity (r i = r i ), the velocity distribution is changed as following: f i (r, t + t) = f i (r, t). (6.19) The logic of using the bounce-back rule to achieve zero velocity at the wall can be argued as following: 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 incoming momentum. The straightforward implementation of the no-slip boundary by the bounce-back scheme makes LBM an promising method for simulating fluid flow in 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 simulation 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-solid boundary is shifted one half lattice spacing into the fluid along the link vector joining the solid and fluid nodes, then the bounce-back rule gives second-order accuracy. There are cases where the second-order accuracy is desired and the zero-velocity plane must be located exactly on the solid boundary nodes rather than being shifted from the location of the solid boundary nodes half-way into the fluid. A great deal of efforts 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.

120 102 For a moving solid boundary like that in a plane Couette flow, the fluid gains momentum from the wall. Accordingly, the incoming velocity distribution at the boundary fluid nodes is altered in proportion to the velocity of the wall v b : f i (r, t + t) = f i (r, t) 2ac i ρv b c i. (6.20) c 2 s The bounce-back rule for the stationary and moving boundaries are illustrated in Figure Polymer Chain Model 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 into N s 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 be N b = N s + 1 beads. For the spring force, we adopt the Finite Extensible Non-linear Elastic (FENE) spring model: F s i = 3k BT N k,s b 2 k q i 1 (q i /q 0 ) 2, i = 1,..., N s, (6.21) where k B is the Boltzmann constant, T is the absolute temperature, q i is the stretch of the i th spring, N k,s is the number of Kuhn segments per spring, and q 0 = N k,s b k is the contour length of that spring. For most of the results reported here, we choose N s = 10 and each spring has N k,s = 10 Kuhn segments with length of b k = 0.106µm, corresponding to a half λ-phage DNA molecule [72]. A Gaussian excluded volume potential between any two beads of the chain is employed [71], Uij ev = 1 ( )3 [ ] 3 2 υk BTNk,s 2 2 exp 3r2 ij, (6.22) 4πSs 2 4Ss 2 where υ = b 3 k is the excluded volume parameter, r ij is the distance between bead i and bead j, and S 2 s = N k,s b 2 k /6 is the radius of gyration of an ideal chain consisting of N k,s Kuhn segments of length b k.

121 103 (a) solid t fluid bounce back solid t + t fluid v w (b) solid t fluid bounce back v w solid t + t fluid Figure 6.3 Bounce-back rule for a solid-fluid interface. The arrows shows the velocity direction and 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 solid boundary.

122 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 that the 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: F h i = ζ(ṙ i v(r i )), (6.23) where ζ = 6πρνa is the bead friction coefficient with the bead radius a = 0.08µm in our simulations, ṙ i is the velocity of the i th bead, and v(r i ) is the local fluid velocity at the i th bead position. v(r i ) is determined by a trilinear interpolation of the fluid velocity v (nn) j sites (n.n.): v(r i ) = j (n.n) at the neighboring lattice w j v nn j. (6.24) The weighting functions w j of the bead s neighboring lattice site j are normalized. v nn j is the fluid velocity at the neighboring lattice site j. The momentum exchange between the fluid and the bead, j e = F h i t/ x3, is distributed back to the neighboring lattice sites with the same weighting functions w j used in the trilinear interpolation. For velocity c q on the neighbor site j, the momentum exchange is given by F j,q = w j j e c q. (6.25) In principle, thermal fluctuations can be incorporated into the Lattice-Boltzmann fluid via the addition of a random stress in the momentum flux during the collision process [83, 1]. In the simulation 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 colloidal 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 transmitted to the polymer beads the fluid fluctuations exist only on wavelengths larger than the lattice size x, while the polymer beads have a scale smaller than it. Therefore, to generate proper fluctuations of the polymer beads, we directly add a Brownian force (Gaussian, with zero mean and

123 105 variance 2k B T/ t) to the equation of motion for each polymer bead [3, 150]. Details of that is explained in Chapter Equation of Motion for Polymer Beads In our simulation, each polymer bead has mass of m. The position and velocity of the individual bead are updated using the explicit Euler method: ṙ i (t + δt) = ṙ i (t) + F i δt/m, (6.26) r i (t + δt) = r i (t) + ṙ i (t)δt, (6.27) where δt is the integration time step (we call δt polymer time step to distinguish from the Lattice- Boltzmann fluid time step t), and F i denotes the total force acting on bead i: F i = F s i + F ev i + F b i + F h i + F wall i. (6.28) F wall i form is the wall excluded volume force defined by a cubic bead-wall repulsive potential of the U wall i = A wall (h δ δwall 3 wall ) 3 for h < δ wall (6.29) = 0 for h δ wall, (6.30) where h represents the distance of bead i from the wall in the wall-normal direction (into the fluid). Throughout this work, we take A wall = 25k B T and δ wall = 3a where a is the bead radius. This inertial form of the equation of motion needs to be integrated at time scale given by m/ζ. In our simulation, m = 1 and ζ = 0.6. To resolve the inertial time scale of m/ζ = 1.7, we choose the polymer time step δt = 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 stochastic differential 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.).

124 Simulation Parameters In applying Lattice-Boltzmann Method to complex fluid modeling, it is important to be conscious 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 µm for λ-dna molecule, our chain model with N s = 10 and N k,s = 10 for a half λ-dna would correspond to an average spring length of 0.34 µm and radius of gyration of R g = 0.5µm. We thus choose a lattice spacing of x = R g = 0.5µm, a compromise between quantitative accuracy and computational feasibility. With this lattice spacing, the hydrodynamic radius of the coarse-grained bead is a = x. Another important free parameter is the fluid relaxation time τ s which determines the viscosity ν according to Equation (6.12). Choosing τ s = 1.1 and matching the viscosity to that of water, we get the Lattice-Boltzmann time step as δt = s. 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 by insightful dimensionless numbers. Specifically, in simulating the polymer solution using Lattice-Boltzmann Method, the important numbers are: Reynolds number Re, Mach number Ma, Schmidt number Sc, and Weissenberg

125 107 number Wi. They are defined as: Re = vl ν, (6.31) Ma = v c s, (6.32) Sc = ν D, (6.33) Wi = γλ, (6.34) where v and l are the characteristic velocity and length scale respectively, c s is the speed of sound, ν is the kinematic viscosity, D is the polymer diffusivity, λ is the polymer chain relaxation time, and γ is the characteristic shear rate v/l. To achieve good separation of the length and time scales in simulating micofluidic flow of polymer solution, a near-zero Reynolds number and Mach number and a large Schmidt number are desired. The dependence of the polymer dynamics on Wi under these conditions is the issue of primary interest. Given the geometry and the fluid viscosity, the only way to decrease the Reynolds number is to decrease the characteristic fluid velocity, which will decrease the Ma 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 chain diffusivity 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 relaxation time, 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 parameters used in the following studies. The LBM parameters are τ s = 1.1, and ζ = 0.6 in lattice units. The k B T will be tuned between and to obtain desired Weissenberg numbers.

126 108 The smallest Schmidt number is Sc = corresponding to k B T = For the Brownian motion of the coarse-grained beads, a integration time step of deltat = 0.1 is chosen, which is one order of magnitude smaller than the mass-relaxation time m/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, < X 2 > (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 be λ = 426 for k B T = 0.001, and λ = 2037 for k B T = , 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 the chain diffusivity. A chain is released in the simulation domain to diffuse. The mean-square-displacement of the chain centerof-mass, < [r(t) r(0)] 2 >, is tracked as a function of time t. Figure 6.5 shows the result for a chain with N s = 10 at temperature of k B T = A linear fitting to the diffusion equation < [r(t) r(0)] 2 >= 6Dt gives the chain diffusion coefficient as D = , 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 solutions confined in a slit, using kinetic theory and Brownian Dynamics simulation respectively. The results corroborate with each other and predict a depletion layer near the wall due to the hydrodynamic interactions between chain segments and the walls. The depletion layer thickens when the

127 k B T=0.0010: λ = 426 k B T=0.0002: λ = 2034 <X 2 > t Figure 6.4 Relaxation of a stretched polymer molecule in bulk solution. The mean square stretch of the chain < X 2 > is plotted against time for a chain of N s = 10 at two different temperatures k B T = 0.001, and An exponential decay fitting of < X(t) 2 >=< X( ) 2 > +X 0 exp(t/λ) gives the chain relaxation time as λ = 426 for k B T = and λ = 2037 for k B T = , in lattice units. X 0 and λ are the fitting parameters.

128 <[r c (t)-r c (0)] 2 > D = 1.73x E+06 t Figure 6.5 Mean square displacement of the center-of-mass of a polymer chain with N s = 10 as a function of time in bulk solution. The simulation parameters are µ = 0.2, ζ = 0.6, and k B T = A linear fitting to the diffusion equation < [(r(t) r(0)] 2 >= 6Dt gives the chain diffusion coefficient as D = , in lattice units.

129 111 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 and reveal the inherent subtleties in LBM. Consider a dilute solution of half λ-phage DNA confined in a slit. The DNA molecule is modeled as a bead-spring chain with N b = 11 beads and N s = 10 springs. Each spring contains N k,s = 10 Kuhn segments. The slit height is L = 10R g. 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 both 20R g. With the wall velocity of v w = 0.1, the shear rate is γ = The corresponding Mach number is Ma = 0.17, smaller than 0.3 which is suggested by Ladd and Verberg as the upper limit [84]. The Schmidt number is at least 1300 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 effect of Reynolds number is clarified. Other LBM parameters are τ s = 1.1, bead friction coefficient ζ = 0.6. We perform simulations at different temperatures: k B T = 10 3, 10 4, , corresponding to Weissenberg number of 10, 100, and 200. The steady state distribution of the chain center-of-mass n is plotted in Figure 6.6 as a function of the position y in slit normal direction. Note that n is normalized such that the area under each curve is unity, and the position y is scaled by the chain radius of gyration R g. 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 hydrodynamic interactions (HI). Simulation without hydrodynamic interactions is referred to as free

130 n Equilibrium Wi = 10 Wi = 100 Wi = y/r g Figure 6.6 Steady state chain center-of-mass distribution of a dilute polymer solution undergoing simple shear flow confined in a slit at Weissenberg number of 0, 10, 100, and 200. The center-ofmass distributions are normalized such that the area under the curves are all unity.

131 113 draining (FD) simulation. The free draining model is implemented by eliminating the momentum transfer from polymer chains to the solvent, i. e. the chains still 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. Statistically, the FD chain center-of mass distribution is undistinguishable from the equilibrium one. On the other hand, the chain center-of-mass distribution from simulation with hydrodynamic interactions 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 resolved qualitatively 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 at Wi = 50 and Re = 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 much weaker 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 finite Reynolds 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

132 n Equilibrium Wi = 50 FD Wi = 50 HI y/r g Figure 6.7 Steady state chain center-of-mass distribution of a dilute polymer solution undergoing shear flow confined in a slit. The solid line is the equilibrium chain center-of-mass distribution, the dotted line is the chain center-of-mass distribution obtained from simulations with free draining model (FD) at Wi = 50, and the dashed line is the chain center-of-mass distribution obtained from simulations with hydrodynamic interactions (HI) at Wi = 50.

133 BD-HI LBM n y/r g Figure 6.8 Steady state chain center-of-mass distribution of a dilute polymer solution undergoing shear flow confined confined in a slit at Wi = 50. The solid line is the chain center-of-mass distribution obtained from Lattice-Boltzmann Method, and the dashed line from Brownian Dynamics simulation with hydrodynamic interactions.

134 116 Method in simulating polymer solutions in terms of fluid relaxation time, the lattice resolution, and the finite Reynolds number Fluid Relaxation Time In the single-time-relaxation model of Lattice-Boltzmann Method, the fluid relaxation time τ s is a crucial free parameter. Microscopically, it characterizes how fast the velocity distribution on a lattice site relaxes to equilibrium. Macroscopically, τ s determines the fluid viscosity according to Equation (6.12): the larger the τ s, the longer the local non-equilibrium persists, the more efficient the momentum transfer to neighboring sites, and thus the larger the viscosity. For numerical stability, τ s 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 large τ s is desired. Practically, in our simulation, we found that τ s beyond 12 can cause instability in complex geometry. Even in simple geometry, the transient flow field can be distorted at large τ s. Consider a semi-infinite body of fluid bounded below by a horizontal surface. Initially the fluid and the solid are at rest. Then at time t = 0, the solid surface is set in motion in the positive x direction with velocity v 0 as shown in Figure 6.9. The fluid velocity v x as a function of distance from the wall y and time t is known to be v x (y, t) v 0 y = 1 erf, (6.35) 4νt where erf is the error function [16]. Using Lattice-Boltzmann Method, we simulate the evolution of this flow field with different values of τ s, and plot the velocity field in dimensionless form in Figure Ideally, the curves obtained from different time should all collapse into one single master curve corresponding to Equation (6.35). The velocity profile for τ s = 1.1 at different time collapse nicely onto the master curve, while the results for τ s = 10.5 deviate from the master curve significantly. This indicates that the Lattice-Boltzmann fluid has limitations on how fast the momentum can be transferred while maintaning proper Navier-Stokes behavior. One can imagine that due to LBM s discretization nature, the momentum at a given lattice site should not be expected to diffuse beyond one lattice

135 117 y v x (y,t) v 0 Figure 6.9 Viscous flow of a fluid near a wall suddenly sheared. At time t = 0, the bottom solid surface is set in motion in the positive x direction with velocity v 0

136 118 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 any τ s value larger than 3.5 is questionable. We conclude that τ s 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 different τ s values are compared to finite element solution of the corresponding Stokes equation. The finite element solution is assumed to be the exact solution since a high resolution is chosen in the calculation. The flow field obtained from τ s = 1.1 agrees with the the exact solution very well as shown in the figure. However, the flow field corresponding to τ s = 3.5 is significantly different. We take the slices of the flow fields in Figure 6.11 along x and y direction and put the results together in Figure 6.12 for closer examination. The flow field obtained from τ s = 3.5 is quite far from the exact solution in both cases. Thus, in all of simulation, we choose τ s = Grid Size Effect In Lattice-Boltzmann Method, the grid resolution should be chosen according to the characteristic 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 resolve 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 of the chain 0.34µm. However, considering that the chain radius of gyration R g = 0.5µm is only slightly larger than x, the grid resolution might not be fine enough. To address this issue, simulations with grid size of x = 0.25µm and x = 1.0µm are also performed. In these simulations, all other parameters 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

137 119 (a) v x /v t=20 t=40 t=60 t=80 t=100 Exact solution y/(4ν) 0.5 (b) v x /v t=4 t=8 t=12 t=16 t=20 Exact solution y/(4ν) 0.5 Figure 6.10 Velocity profile in dimensionless form for flow near a wall suddenly sheared. (a) Results from Lattice-Boltzmann Method with τ s = 1.1. (b) Results from Lattice-Boltzmann Method with τ s = 10.5.

138 Figure 6.11 Contour plot of the wall normal component of the steady state flow field due to a stretched dumbbell (white beads connected by dotted line) confined in a slit. (a) Finite element solution. (b) Result from Lattice-Boltzmann Method with τ s = 1.1. (c) Result from Lattice- Boltzmann Method with τ =

139 121 (a) Exact Solution τ s = 1.1 τ s = 3.5 v y y (b) v y Exact Solution τ s = 1.1 τ s = x Figure 6.12 Comparison of the wall normal component of the steady state flow field due to a stretched dumbbell confined in a slit. (a) Slice of the flow field along wall-normal direction at x = 20. (b) Slice of the flow field along the wall-tangential direction at y = 5. The dotted lines in (b) indicates the positions of the two beads of the stretched dumbbell.

140 122 larger than the polymer bead size a = µm, 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 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 increasing the 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 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 is 2, 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 as in 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 The steady state chain center-of-mass distributions obtained from different Reynolds number are statistically undistinguishable. Note that the Reynolds 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

141 n 0.05 dx = 1.0 µm dx = 0.5 µm dx = 0.25 µm y/r g Figure 6.13 Steady state chain center-of-mass distribution of a dilute polymer solution undergoing shear flow confined in a slit at Wi=10. Line styles correspond to grid resolution of x = 1.0µm (dashed), x = 0.50µm (solid), and x = 0.25µm (dotted).

142 124 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 methods that attempt to span a large separation of length and time scales, the parameters chosen in our simulations necessarily focus on long range, large time scale effects such as the flow-induced chain 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 reveals 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. In BD, the microfluidic flow is always considered to have Re = 0, and hydrodynamic interactions between beads are assumed to propagate instantly. Neglecting inertial effects in microfluidic flow is generally justified because Re 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.

143 n Re = 10 Re = 2 Re = 0.4 Re = y/r g Figure 6.14 Steady state chain center-of-mass distribution of a dilute polymer solution undergoing shear flow confined in a slit at Wi = 10. Line styles correspond to Reynolds numbers of Re = 10 (dotted), Re = 2 (dashed), Re = 0.4 (solid), and Re = 0.04 (dash-dotted).

144 126 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 will utilize 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 and fabrication 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 rectangular cross-section are used. Other studies, however, have begun to examine the behavior of chains in more complex 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 polymer molecules inside the pores occurs. Early studies indicate that when polymer solution flow

145 127 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 solutions 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 ( 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 entropic trap separation of DNA molecules in a grooved channel with contractions comparable to DNA persistence length. Further experimental and simulation studies by Streek et al. [142] revealed 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 gyration. Besides its technological importance on chromatography and electrophresis analysis, these phenomena also challenge our fundamental understanding of the 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 interactions of the chain segments and the channel walls [74, 72, 94, 64]. This migration phenomena has obvious implications for the surface-adsorption based chemical 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 microchannels. 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 problems governed by the Navier-Stokes

146 Figure 7.1 Schematic of a grooved channel. Shown in the figure is the xy plane cross-section. The simulation domain is periodic in x and z directions. 128

147 129 equation. It is based on the microscopic Boltzmann equation for 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 boundary conditions. A bead-spring chain model of the polymer molecule can be coupled to the Lattice-Boltzmann model of the solvent to simulate the dynamics of polymer solutions [3, 150]. 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 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 with N s = 10 springs and each spring contains N k,s = 10 Kuhn segments. The radius of gyration of the chain is about R g = 0.5µm, which is also the lattice spacing in our simulation. Random forces are introduced 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 time τ s = 1.1 in this chapter. Consider a grooved channel as shown in Figure 7.1: the bulk channel has the height of L y = 19R g and length of L x = 40R g, the length of the groove is L a = 20R g, and the depth of the groove is L b = 9R g unless otherwise specified. The simulation box has the dimension of L z = 19R g in the neutral z direction. In our simulation, five chains are put in the simulation box, with a chain concentration at least three orders of magnitude lower than the overlap concentration. Note for the half λ-phage DNA chain model we are using, chain radius of gyration R g = x = 0.5µm. The upper wall is moved in the positive x direction with speed v w to shear the fluid in the channel. The

148 130 Figure 7.2 Stream lines corresponding to the flow field generated by shearing the upper wall of the grooved channel in positive x direction. The contour variable is the velocity in x direction. Note that 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 v w Wi = λ, (7.1) L y L b where λ is the polymer molecule relaxation time. Figure 7.2 is a plot of 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 channel, we performed Lattice-Boltzmann simulations at three different Weissenberg numbers: 0, 5, and 10. The steady state chain center-of-mass distribution is 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 at Wi = 5 as shown in Figure 7.3(b), indicating that the polymer chain is being depleted out of the groove. This depletion effect becomes stronger as the flow strength increases: the

149 Figure 7.3 Steady state chain center-of-mass distribution in a flowing polymer solution confined in 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

150 132 chain center-of-mass distribution inside the groove is even lower at Wi = 10 as shown in Figure 7.3(c). Second, inside the groove, the concentration is not only lower in general, it is also nonuniform. 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 cavity shown 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) and 7.3(c), indicating a relatively higher concentration region. Moreover, the concentration inside this region increases as the Weissenberg number increases. To better illustrate the above observations, we take slices from the two dimensional chain center-of-mass distribution in Figure 7.3 along y direction at x = 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 both Wi = 5 and Wi = 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 Discussion The above results show two phenomena that are unexpected and potentially 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 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 solution at the length scale of the channel width.

151 Equilibrium Wi = 5 Wi = 10 Bead distribution n Groove Edge y/r g Figure 7.4 Slice of the two dimensional steady state chain center-of-mass distribution in flowing polymer solution confined in a grooved channel. The slice is taken along y direction at x = 20, which is the center of the channel in x direction. The vertical dotted line indicates the position of the groove top edge.

152 134 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 at Wi = 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 groove bottom wall located at y = 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 at x = 19, the groove top edge wall at x = 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 qualitatively 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] 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 unconnected 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

153 n Groove Edge Wi = 10 HI Wi = 10 FD y/r g Figure 7.5 Steady state chain center-of-mass distribution in a dilute polymer solution confined in a grooved channel. The dash-dotted line is the distribution obtained from free draining (FD) simulation, and the solid line is the result from simulation with hydrodynamic interactions (HI). Both simulations are performed with Wi = 10.

154 136 Figure 7.6 Steady state center-of-mass distribution of isolated beads in shear flow in a grooved channel. track the bead distribution in the flow field corresponding to Wi = 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 groove in 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 the y direction at x = 20 and shown in Figure 7.4 to compare with polymer cases.

155 137 With the isolated bead distribution in mind, we now revisit the chain center-of-mass distribution. 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, dangling, rearranging, and eventually escaping. As a result, a higher concentration pattern near the right corner is anticipated, and it is more profound than that in the case of individual beads. However, different from individual beads, chains are more likely to escape downstream along the bulk channel than down into the groove. Consider a chain dangling around 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 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 to each confining surface. The steric depletion 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 thickness R g 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 distance R g 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 time along the separatrix. We define the cavity Peclet number Pe c as the ratio of the diffusion time of a chain over the boundary layer to its

156 Y Y Y Y Figure 7.7 Snapshots of polymer chains in flowing solution confined in a grooved channel at time t = 711 t, 740 t, 756 t, and 766 t, chronologically from top to bottom. The arrows point to the polymer chain that approaches the corner.

157 Figure 7.8 Schematic of a chain crossing the boundary layer near the separatrix at the top edge of the groove. 139