Dissertation SwapNil KoHale

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MOLECULAR HYDRODYNAMICS IN COMPLEX FLUIDS

by

Swapnil C. KohaleBachelor of Chemical Engineering

A Dissertation

In

CHEMICAL ENGINEERING

Submitted to the Graduate Faculty

of Texas Tech University inPartial Fulfillment ofthe Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

IN

CHEMICAL ENGINEERING

December, 2009


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Texas Tech University, Swapnil C. Kohale, December 2009

ii

TABLE OF CONTENTS

ABSTRACT .. v

LIST OF TABLES .. vi

LIST OF FIGURES vii

I. INTRODUCTION AND OVERVIEW ..................................................................... 1

Hydrodynamics in nanoscale systems .............................................................................. 1

Nanofluidics...................................................................................................................... 2

What is nanofluidics and what is its significance? ....................................................... 2

Applications of nanofluidic devices.............................................................................. 3

Experimental studies using nanofluidic devices........................................................... 6

Single molecule analysis........................................................................................... 6

Separation ................................................................................................................. 8

Molecule concentration............................................................................................. 9

Particle Micro and Nanorheology................................................................................... 10

Molecular dynamics simulations .................................................................................... 12

Organization.................................................................................................................... 13

II. CROSS-STREAM CHAIN MIGRATION IN NANOFLUIDIC CHANNELS... 15

Introduction..................................................................................................................... 15

Experimental findings................................................................................................. 15

Theoretical findings .................................................................................................... 16

Computational investigations...................................................................................... 16

Chain migration in nanofluidic channels .................................................................... 17

Simulation Method.......................................................................................................... 18

Simulation sets............................................................................................................ 21

Weissenberg number................................................................................................... 22

Results............................................................................................................................. 24

Effect of chain length.................................................................................................. 24

Effect of channel height .............................................................................................. 28

Effect of concentration................................................................................................ 31


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Effect of intermolecular interactions .......................................................................... 37

Summary and Discussion................................................................................................ 39

III. MOLECULAR HYDRODYNAMICS IN NANOPARTICLE SUSPENSIONS:

SMOOTH NANOPARTICLES....................................................................................... 42

Introduction..................................................................................................................... 42

Continuum treatment .................................................................................................. 43

Molecular simulation studies: Stokes-Einstein law .................................................... 43

Validity of Stokes law................................................................................................. 44

Cooperative hydrodynamic effects ............................................................................. 46


The smooth nanoparticle............................................................................................. 49

Simulation systems ..................................................................................................... 50

Fluid viscosity............................................................................................................. 52

Results............................................................................................................................. 53

Effect of cooperative hydrodynamic interactions....................................................... 58

Effect of slip at nanoparticle surface .......................................................................... 62

Effect of confining surfaces........................................................................................ 66


IV. MOLECULAR HYDRODYNAMICS IN NANOPARTICLE SUSPENSIONS:

FRICTION FORCE AND TORQUE ON A ROUGH NANOPARTICLE ................. 72

Introduction..................................................................................................................... 72


Rough nanoparticle generation ................................................................................... 74

Results............................................................................................................................. 79

Friction force and torque on a rough nanoparticle in monomeric solvent.................. 81

Friction force on a rough nanoparticle in a polymer melt .......................................... 88

Concentrated suspension of nanoparticles.................................................................. 91


V. ACTIVE NANORHEOLOGY IN POLYMER MELT......................................... 95

Introduction..................................................................................................................... 95

Microrheology............................................................................................................. 96


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Types of microrheological techniques.................................................................... 97

Passive Microrheology........................................................................................ 97

Active Microrheology......................................................................................... 98

Nanorheology............................................................................................................ 100

Simulation Method........................................................................................................ 101

System preparation: Isotropic orientation of polymer chains ................................... 103

Mathematical formulation for active nanorheology ..................................................... 104

Magnitude of inertial effects: Dimensional analysis .................................................... 107

Results........................................................................................................................... 109

Literature studies for comparison ............................................................................. 112

Results: Viscoelastic properties from Active Nanorheology simulations ................ 113

Viscoelastic moduli at different locations................................................................. 115

Slip at the nanoparticle surface................................................................................. 117

Active nanorheology simulations in the absence of slip........................................... 120

Summary and Discussion.............................................................................................. 125

Momentum diffusion time Vs. Period of oscillation ................................................ 126

Dependence on amplitude of oscillation................................................................... 127

VI. CONCLUSIONS AND FUTURE WORK............................................................ 129

VII.APPENDICES......................................................................................................... 133

APPENDIX A............................................................................................................... 133

APPENDIX B ............................................................................................................... 134

APPENDIX C ............................................................................................................... 139

APPENDIX D............................................................................................................... 144

VIII. BIBLIOGRAPHY ................................................................................................ 148


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ABSTRACT

Developments in the field of micro- and nanofluidics have renewed interest in the

study of molecular hydrodynamics in confined geometries. The conception and design of

these devices demand accurate knowledge of transport properties of fluids in these small

geometries. The continuum methodologies for the calculation of transport properties of

fluids in the bulk are well established. However, the assumptions made in the continuum

treatment need to be valid for the nanoconfined fluids. In this work molecular dynamics

simulation technique is used to study the hydrodynamic behavior of complex fluids in

nanoconfinement. The goal of the dissertation is to study the hydrodynamic behavior of

fluids in confinement using molecular simulations and to make a quantitative connection

with the continuum predictions.

Shear induced cross stream chain migration in a flowing polymeric solution is studied in

the first part of this dissertation. The role of hydrodynamic interactions and the effect of

chain conformational properties, interactions, channel geometry and chain concentration on

the chain migration phenomenon were studied. Next, molecular hydrodynamics in a

nanoparticle suspension were studied to quantify the effect of confinement, boundary

conditions at the nanoparticle surface and cooperative hydrodynamic interactions between

nanoparticles. Finally, a new technique, termed as active nanorheology, is also presented to

calculate the nanoscale local viscoelastic properties of the complex materials using molecular

dynamics simulations.


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LIST OF TABLES

Table 3.1: A matrix of simulation systems used in chapter 3 ................................................ 51

Table 3.2: Comparison of normalized friction force for WCA and LJ interactions. ............. 65Table 4.1: A table showing the simulation system sizes used in chapter 4............................ 83

Table 5.1: Table of relative magnitudes of the three ratios of coefficients for the frequencies

studied in chapter 5 ............................................................................................................... 109


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LIST OF FIGURES

Figure 1.1: A schematic of the cross section of a -hemolysin channel embedded in a lipidbilayer[49]................................................................................................................................. 7

Figure 2.1: A schematic of the simulation system showing polymer chain, fluid atoms andthe wall atoms. The Couette flow is generated by moving the two walls in opposite directionwith same velocity. ................................................................................................................. 19

Figure 2.2: Chain center of mass density profiles for chains of length 20 (lines with opensymbols) and 4 (filled symbols only) as a function of distance from the wall. Simulation

conditions are: Couette flow (shear rates = 0.05 and 0.075),= 0.8, Twall = 0.9 and WCAinteractions. The ratios of channel height (H) to the bulk chain radius of gyration (Rg) are:8.1 (N= 20) and 22.4 (N= 4). The Reynolds number (based on channel height) values are:5.1 for the shear rate of 0.05 and 7.7 for the shear rate of 0.075. ........................................... 24

Figure 2.3: Bead density profiles for chains of length 20 as a function of distance from thewall. Simulation conditions are same as those for Figure 2.2. .............................................. 25

Figure 2.4: Normalized root mean squared chain end-to-end distance (normalized by theequilibrium value i.e. at We = 0) as a function of shear rate (filled circles:N= 20 and opensquares:N= 4). Simulation conditions are the same as those for Figure 2.2. The averageerror bars on the absolute values of chain end-to-end distance are 0.21 for long chain (N=20) and 0.01 for short chain (N= 4). ...................................................................................... 26

Figure 2.5: Chain center of mass density profiles for chains of length 20 (lines with open

symbols) and 4 (filled symbols only) as a function of distance from the wall. Simulationconditions are: Couette flow (shear rates = 0.05 and 0.075),= 0.8, isothermal flowconditions with Tfluid= 0.9 and WCA interactions. The channel Reynolds number values are:4.8 for the shear rate of 0.05 and 7.2 for the shear rate of 0.075. The ratios of channel height(H) to the bulk chain radius of gyration (Rg) are: 8.1 (N= 20) and 22.4 (N= 4). .................. 27

Figure 2.6: Chain center of mass density profiles for chains of length 20 as a function ofdistance from the wall. Simulation conditions are: Couette flow (shear rates = 0.05 and

0.075),= 0.8, Twall = 0.9 and WCA interactions. Density profile is shown for a channelheight of 18. The ratio of channel height (H) to the bulk chain radius of gyration (Rg) for thiscase is 6.9. ............................................................................................................................... 28

Figure 2.7: Chain center of mass density profiles for chains of length 20 as a function ofdistance from the wall. Simulation conditions are identical to those for Figure 2.6. Densityprofile is shown for a channel height of 21. The ratio of channel height (H) to the bulk chainradius of gyration (Rg) for this case is 8.1............................................................................... 29

Figure 2.8: Chain center of mass density profiles for chains of length 20 as a function ofdistance from the wall. Simulation conditions are identical to those for Figure 2.6. Density


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profile is shown for a channel height of 27. The ratio of channel height (H) to the bulk chainradius of gyration (Rg) for this case is 10.4............................................................................. 29

Figure 2.9: Chain center of mass density profiles for chains of length 20 as a function ofdistance from the wall. Simulation conditions are identical to those for figure 2.6. Density

profile is shown for a channel height of 33. The ratio of channel height (H) to the bulk chainradius of gyration (Rg) for this case is 12.7............................................................................. 30

Figure 2.10: Normalized depletion layer thickness plotted as a function of normalizedchannel height for a chain of length 20. Simulation conditions are same as those for Figure2.6. Depletion layer thickness and channel height are normalized by the radius of gyration ofthe chain.................................................................................................................................. 30

Figure 2.11: Chain center of mass density profiles for chains of length 20. Simulation

conditions are: Couette flow,= 0.8, channel heightH= 21, Twall = 0.9 and WCAinteractions. Figures show density profiles at equilibrium and at shear rates of 0.05 and

0.075. Density profiles are shown for systems with concentrations of: (a) 0.11C* and (b)1.64 C*. ................................................................................................................................... 32

Figure 2.12: Normalized depletion layer thickness (normalized by chain radius of gyration)plotted as a function of concentration for a chain of length 20 at equilibrium and at shear

rates of 0.05 and 0.075. Simulation conditions: Couette flow,= 0.8, channel heightH= 21,Twall = 0.9 and WCA interactions............................................................................................ 33

Figure 2.13: Velocity profiles in the channel for solutions with concentrations 0.11C* and1.64 C*. Simulation conditions are the same as those for figure 2.12. The velocities of thetwo walls are 0.525. ............................................................................................................ 34

Figure 2.14: Chain longest semi-axis length profile for chain of length 20 at (a) equilibriumand at shear rates of (b) 0.05 and (c) 0.075. Simulation conditions are same as those forfigure 2.12. Profiles are plotted for lowest (C = 0.11C*) and the highest (C = 1.64 C*)values of concentrations studied in this set............................................................................. 36


symbols) and 3 (filled symbols only). Simulation conditions are: Couette flow, = 0.8, Twall= 0.9 and LJ interactions. The ratios of channel height (H) to the bulk chain radius ofgyration (Rg) are: 12.7 (N = 10) and 27.2 (N = 3). The Reynolds number values are: 4.2 forthe shear rate of 0.05 and 6.4 for the shear rate of 0.075........................................................ 38


symbols) and 3 (filled symbols only). Simulation conditions are: Couette flow, = 0.8, Tfluid= 0.9 and LJ interactions. The ratios of channel height (H) to the bulk chain radius ofgyration (Rg) are: 12.7 (N = 10) and 27.2 (N = 3). The Reynolds number values are: 4.2 forthe shear rate of 0.05 and 6.4 for the shear rate of 0.075........................................................ 38


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Figure 3.1: A schematic of the simulation system showing smooth nanoparticle, fluid andwall atoms. The nanoparticle is translated inx direction while the confining walls are normaltoz direction............................................................................................................................ 48

Figure 3.2: The pictorial representation of the simulation system showing the central

simulation box and its periodic images along thex andy directions. The spherical soluteparticle is pulled along thex direction. ................................................................................... 52

Figure 3.3: Solvent density profile near the channel walls for the system 45.5 x 45.5 x 30.Five significant layers of solvent were observed near either of the channel walls................. 54

Figure 3.4: Density profile of the solvent atoms as a function of the distance from thetranslating sphere. Results are shown for sphere pulling velocity of 0.02 in a system withbox dimensions 45.5 x 45.5 x 30. ........................................................................................... 54

Figure 3.5: A schematic showing the primary quantity of interest in this work the friction

force coefficient. ..................................................................................................................... 55

Figure 3.6: Friction force experienced by the sphere as a function of the sphere velocity forthe system with box dimensions 45.5 x 45.5 x 30. ................................................................. 56

Figure 3.7: Normalized friction force as a function of the normalized channel length forthree different channel widths calculated using bare radius. Channel height isH= 30.Squares represent data forLy = 23.4, triangles forLy = 33.8 and circles forLy = 45.5. Errorbars on the data points are of the same size as the symbols. .................................................. 58

Figure 3.8: Normalized friction force as a function of the normalized channel length forthree different channel widths calculated using effective hydrodynamic radius. Channelheight isH= 30. Squares represent data forLy = 23.4, triangles forLy = 33.8 and circles forLy = 45.5. Error bars on the data points are of the same size as the symbols. ....................... 60

Figure 3.9: Normalized friction force as a function of the normalized channel width for threedifferent channel lengths calculated using bare radius. Channel height isH= 30. Squaresrepresent data forLx = 23.4, triangles forLx = 33.8 and circles forLx = 45.5. Error bars onthe data points are of the same size as the symbols. ............................................................... 60

Figure 3.10: Normalized friction force as a function of the normalized channel width forthree different channel lengths calculated using effective hydrodynamic radius. Channelheight isH= 30. Squares represent data forLx = 23.4, triangles forLx = 33.8 and circles forLx = 45.5. Error bars on the data points are of the same size as the symbols. ....................... 61

Figure 3.11: Fluid velocity profile (only the x component of fluid velocity is plotted for thecubes residing along the axes) in theyz plane for the system with dimensions 33.8 x 33.8 x30. The sphere resides at the center of the figure and is moving with velocity V = 0.02 alongthex direction. The uncertainties in the numbers range from 0.002 to 0.006, with the averageuncertainty being 0.004........................................................................................................... 63


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Figure 3.12: Fluid velocity profile in theyz plane for the system with dimensions 33.8 x 33.8x 30. All conditions are the same as those for Figure 3.11 except that sphere-fluid interactionis modeled using the full LJ potential. X-components of the fluid velocity are shown for the

cases of (a) sphere-fluid = 4.0 and (b) sphere-fluid = 7.0. ............................................................... 64

Figure 3.13: Normalized friction force on a translating sphere as a function of the distancedfrom the nearest surface. The data points are for the system with dimensions 33.8 x 33.8 x30............................................................................................................................................. 66

Figure 3.14: Normalized friction force on a translating sphere as a function of normalizedchannel length for two different distances from the nearest surface. The data points are forthe system with dimensionsLy = 33.8 and H = 30.................................................................. 67

Figure 3.15: Normalized friction force on a translating sphere as a function of normalizedchannel width for two different distances from the nearest surface. The data points are forthe system with dimensionsLy = 33.8 and H = 30.................................................................. 68

Figure 4.1: A schematic of the simulation system showing rough nanoparticle, fluid and thewall atoms. .............................................................................................................................. 74

Figure 4.2: A schematic of the rough nanoparticle made up of beads of the size same as thesolvent atoms and arranged in FCC fashion. The beads of the nanoparticle are connectedtogether using a simple harmonic potential. ........................................................................... 75

Figure 4.3: Friction force experienced by the translating nanoparticle plotted versus the forceconstant for the harmonic potential between the beads of the nanoparticle. The system sizewas 33.8 x 23.4 x 30.0. The nanoparticle pulling force constant of 100 was used................ 77

Figure 4.4: Friction force on the translating nanoparticle plotted for three different values offorce constant for the harmonic potential used for pulling the nanoparticle. The forceconstant for harmonic potential between beads of the nanoparticle was kept constant at 500.................................................................................................................................................. 78

Figure 4.5: Density profile of solvent atoms as a function of distance from the center of massof the nanoparticle for a translating nanoparticle with a velocity of 0.02 (circles) and for arotating nanoparticle with angular velocity of 0.008 (dotted line) in a simulation box ofdimensions 0.308.338.33 . The average error bars on density values are of size 0.001... 80

Figure 4.6: Normalized friction force (squares) and normalized torque (triangles) as afunction of normalized channel length. The channel height isH= 30.0. Error bars on the datapoints are of the same size as the size of symbols used.......................................................... 84

Figure 4.7: Normalized friction force (squares) and normalized torque (triangles) as afunction of normalized channel width. The channel height isH= 30.0. Error bars on the datapoints are of the same size as the size of symbols used.......................................................... 85


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Figure 4.8: Fluid velocity profile (only thex component of fluid velocity is plotted for thecubes residing along the axes) in theyz plane for the system with dimensions

0.308.338.33 . The sphere resides at the center of the figure and is translating withvelocity V = 0.02 along the x direction. The uncertainties in the data range from 0.002 to0.007, with the average uncertainty being 0.004. ................................................................... 86

Figure 4.9: Fluid velocity profile (x component of fluid velocity is plotted for the cubesresiding along thez axes, whilez component of velocity is plotted for cubes residing alongxaxes) in thexz plane for the system with dimensions 0.308.338.33 . The sphere resides atthe center of the figure and is rotating with velocity = 0.008 (corresponds to velocity of0.02) about the y axis. The uncertainties in the data range from 0.002 to 0.007, with theaverage uncertainty being 0.004. ............................................................................................ 87

Figure 4.10: Normalized friction force on a nanoparticle translating in a polymer meltplotted as a function of normalized channel length. The channel height is H= 30.0. Error barson the data points are of the same size as the size of symbols used. ...................................... 89

Figure 4.11: Normalized friction force on a nanoparticle translating in a polymer meltplotted as a function of normalized channel width. The channel height is H= 30.0. Error barson the data points are of the same size as the size of symbols used. ...................................... 90

Figure 4.12: Friction force on a nanoparticle translating in a concentrated suspension ofnanoparticles plotted as a function of normalized channel length. The channel height isH=30.0. Error bars on the data points are of the same size as the size of symbols used. Thesimulation conditions are: =0.844, T=1.0 and WCA interactions. ....................................... 92

Figure 4.13: Friction force on a nanoparticle translating in a concentrated suspension of

nanoparticles plotted as a function of normalized channel width. The channel height isH=30.0. Error bars on the data points are of the same size as the size of symbols used. Thesimulation conditions are: =0.844, T=1.0 and WCA interactions. ....................................... 92

Figure 5.1: A schematic of the simulation system showing a rough nanoparticle in a polymermelt confined between walls of the nanochannel. ................................................................ 101

Figure 5.2: A schematic showing the rough nanoparticle subjected to an input oscillatingforce oscillating in a harmonic trap. ..................................................................................... 104

Figure 5.3: Measured displacement of a nanoparticle oscillating with a frequency of 0.02 in apolymer melt in a simulation system of size 33.8x33.8x30. All the interactions are WCA.111

Figure 5.4: Fourier transform of the displacement of the oscillating nanoparticle in polymermelt for the simulation conditions of figure 5.3. .................................................................. 112

Figure 5.5: Comparison of the storage modulus (G) for the polymer melt (N=20) calculatedusing active nanorheology simulations with the literature values for a simulation system ofsize 33.8x33.8x30. All the interactions in the simulations are of type WCA...................... 114


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Figure 5.6: Comparison of the loss modulus (G) for the polymer melt (N=20) calculatedusing active nanorheology simulations with the literature values for a simulation system ofsize 33.8x33.8x30. All the interactions in the simulations are of type WCA...................... 114

Figure 5.7: A schematic showing the meaning of the parameter d, i.e. the distance of the

nanoparticle center of mass from the channel surface. ......................................................... 115

Figure 5.8: Comparison of the storage modulus (G) for the polymer melt (N=20) calculatedusing active nanorheology simulations at two different d locations with the bulk literaturevalues for a simulation system of size 33.8x33.8x30. All the interactions in the simulationsare of type WCA. .................................................................................................................. 116

Figure 5.9: Comparison of the loss modulus (G) for the polymer melt (N=20) calculatedusing active nanorheology simulations at two different d locations with the bulk literaturevalues for a simulation system of size 33.8x33.8x30. All the interactions in the simulationsare of type WCA. .................................................................................................................. 116

Figure 5.10: Velocity field around a nanoparticle translating in a polymer melt with velocityof 0.039 for the simulation conditions identical to those in figure 5.5. ................................ 118

Figure 5.11: Velocity field around a nanoparticle translating in a polymer melt with velocityof 0.169 for the simulation conditions identical to those in figure 5.5. ................................ 118

Figure 5.12: Velocity field around a nanoparticle translating in a polymer melt with velocityof 0.149 for the simulation conditions identical to those in figure 5.5, but with LJ interactionsbetween the nanoparticle and the medium............................................................................ 119

Figure 5.13: Comparison of the storage modulus (G) for the polymer melt (N=20)calculated using active nanorheology simulations with the literature values for a simulationsystem of size 33.8x33.8x30. Nanoparticle-medium interactions are LJ while all otherinteractions in the simulations are of type WCA. ................................................................. 121

Figure 5.14: Comparison of the loss modulus (G) for the polymer melt (N=20) calculatedusing active nanorheology simulations with the literature values for a simulation system ofsize 33.8x33.8x30. Nanoparticle-medium interactions are LJ while all other interactions inthe simulations are of type WCA.......................................................................................... 121

Figure 5.15: Comparison of the storage modulus (G) for the polymer melt (N=20)calculated using active nanorheology simulations at two different d locations with the bulkliterature values for a simulation system of size 33.8x33.8x30. Nanoparticle-mediuminteractions are of type LJ while all other interactions in the simulations are of type WCA................................................................................................................................................ 122

Figure 5.16: Comparison of the loss modulus (G) for the polymer melt (N=20) calculatedusing active nanorheology simulations at two different d locations with the bulk literature


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values for a simulation system of size 33.8x33.8x30. Nanoparticle-medium interactions areof type LJ while all other interactions in the simulations are of type WCA......................... 122

Figure 5.17: Comparison of the storage modulus (G) for a different polymer melt (N=10)calculated using active nanorheology simulations with the bulk literature values for a

simulation system of size 33.8x33.8x30. Nanoparticle-medium interaction are of type LJwhile all other interactions in the simulations are of type WCA.......................................... 124

Figure 5.18: Comparison of the loss modulus (G) for a different polymer melt (N=10)calculated using active nanorheology simulations with the bulk literature values for asimulation system of size 33.8x33.8x30. Nanoparticle-medium interaction are of type LJwhile all other interactions in the simulations are of type WCA.......................................... 124


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1. CHAPTER 1

I. INTRODUCTION AND OVERVIEW

Hydrodynamics in nanoscale systems

About half a century ago in his much celebrated talk titled Theres plenty of room

at the bottom, in the annual meeting of American Physical Society, Richard Feynman had

called upon the scientific community to explore the opportunities that exist at the smallest

possible scales and to challenge the definition of smallest possible itself. Science has come

a long way since then and has been in a pursuit of going to increasingly smaller length scales

to achieve bigger and bigger goals. The never-ending quest for miniaturization and the

tremendous potential of small scale devices in scientific development has inspired a

tremendous surge in the conception and development of micro- and nanoscale devices used

for various applications.

The conception and design of these micro- and nanoscale devices demands accurate

knowledge of the transport properties of the fluids involved in these small systems.

Technological advances in techniques of fabricating nanoscale structures,[1-6] have renewed

interest in understanding the transport processes at the nanometer length scales. The

continuum mechanics based methodologies for the calculation of transport properties of

fluids on macroscale are very well established. However, the transport properties of fluids in

micro- and nanofluidic devices could be significantly different from those in the macroscale

devices. The assumptions underlying continuum transport equations, such as of the

homogeneity of the system or the boundary conditions at surfaces that are applicable for

macroscale systems may not hold at the micro- and nanoscales and hence need to be verified.


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Several studies, that have used both experimental and simulation techniques have

focused in past on studying the hydrodynamics in micro- and nanoscale systems.[7-13] In

these small systems, a large fraction of the fluid in the system is in contact with the surface of

the channel and it was observed that the fluid-surface interactions have a significant impact

on the transport properties of fluid. In simulation studies, the systems usually consisted of

three components solute, solvent and the surface. In most of these studies, while the solute

was treated explicitly, the solvent was assumed to be implicit and bulk continuum

assumptions for the solvent were used to describe the solvent, as will be discussed in

subsequent chapters in detail. One of the primary objectives of this thesis is to treat the

solvent explicitly so as to study the hydrodynamic interactions between the solute particles

that propagate through the solvent and to study their effect on the transport properties of the

fluid. In this approach, diffusion and hydrodynamic interactions are completely determined

by the intermolecular interactions in the system and no prior assumptions (e.g. Oseen

hydrodynamics) need to be made. Furthermore, the molecular nature of the solvent allows

for explicit treatment of the structural heterogeneities in the system which is an essential

feature of the nanoscale systems.

Nanofluidics

What is nanofluidics and what is its significance?

Nanofluidics is a discipline of science that deals with the study of fluid flow in the

geometries where one of the dimensions is less than 100 nm.[14] It is basically the study of

fluid flow in nanoscopic channels or around nanometer sized objects. This form of scientific

study has been around since long time in various branches of science such as biology,

chemistry, physiology etc. although the name Nanofluidics has attracted attention for the


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past decade due to the growing interest and rapid advances made using the nanoscale

systems. Naturally, nanoscale systems are characterized by nanoscale forces. The theory

developed by Derjaguin, Verwey, Landau and Overbeek (DLVO) in 1940s accounts for

electrostatic and van der Walls forces that act in these geometries.[15, 16] With the advent of

and subsequent improvements in the surface forces apparatus (SFA)[17, 18] and the more

recently developed atomic force microscope (AFM)[19] it was possible to measure these and

other surface forces such as hydration forces, salvation forces and capillary forces. The fluid

behavior in nanoscale systems is governed by these forces. The gravitational and inertial

forces play a significant role in macroscale systems have a negligible effect in nanoscale

systems. In nanoscale systems, the surface area to volume ratio is very high thus resulting in

an increased contact of the fluid molecules with the nanochannel surface. As a result, the

thermodynamic and transport properties of fluids in these small geometries can be

significantly different from their bulk properties, where a relatively smaller proportion of the

fluid is in contact with the channel surface.

Applications of nanofluidic devices

The current and potential applications of nanofluidic devices can primarily be divided

into two broad categories: Single molecule analysis and separations.

A primary application of nanofluidic devices is in single molecule analysis, especially

for biomolecules. As the size of the channel becomes small the volume available for an

individual molecule reduces. The confinement in nanofluidic devices is used to elongate the

biomolecules and high resolution detection techniques are used to analyze the structure,

composition and configuration of the biomolecules.[20, 21] Each molecule in the

nanochannel can be analyzed individually with greater resolution as opposed to the average


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methods used in bulk measurements. Nanofluidic devices have primarily been used to

perform single molecule analysis of DNA by stretching the DNA molecule and using optical

or electric detection techniques. Several studies performed on DNA molecules such as their

motion in porous media, length of fragments, restriction mapping and their interactions with

proteins have helped in obtaining a deeper insight into the functional properties of the

molecules.[22-24] Excellent reviews dedicated to fabrication of nanochannels can be found

in the literature.[25, 26]. Nanofluidic devices exhibit great potential in biological

applications such as in biomolecule separation, analysis, concentration etc. and have been

looked upon as an effective mean to analyze the Deoxyribonucleic acid (DNA) molecule. In

studying DNA-protein interactions, the critical factors are to determine the number of

proteins attached and the binding sites. To address these issues the DNA molecule has to be

stretched and then the sequence should be read using an analytical technique. Nanochannels,

both naturally and artificially synthesized, with width in nanometers range have been used

for this purpose. The ultimate goal in DNA sequencing research is to reduce the cost of

sequencing the DNA molecule and much of the biological research in the past decade or so

has been dedicated towards developing better sequencing strategies. Nanofluidic devices

with sizes on the order of size of these molecules have been a huge contributor in this

research.

In addition to the development of single molecule analysis techniques, another critical

application is the development of high throughput systems which can perform functions,

such as analysis and separation in a commercially viable way. Lab-on-chip devices or the

Micro Total Analysis Systems offer exciting prospects for creating these high throughput

systems.[27] Several functions are integrated on a small chip in these devices that are a few


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millimeters in size. These devices hold nanolitres of fluid volumes in nanoscale geometries.

Accurate knowledge of the fluid transport properties on nanoscale is critical in the efficient

design of these systems.

Theoretical models to better understand and mimic the naturally occurring sieving

and separation systems, such as human kidney[28] are difficult to construct due to the

random nature of the structures, such as the pore shape and size, involved in naturally

occurring systems. These systems are mimicked by constructing nanoscale porous devices

and studying the separation efficiency with respect to the pore geometry. Design of such

devices will require a thorough understanding of the nanoscale transport properties.[29]

The comparable sizes of the nanofluidic channel dimensions and the molecules in the

solution under study provide an opportunity to perform separation of solute molecules based

on size considerations. Randomly structured nanoporous materials are used for separation in

techniques such as size exclusion chromatography and high performance liquid

chromatography, while specifically manufactured regular arrays fabricated using

micromachining are also used for constructing various lab-on-chip devices to perform

separation.[30, 31] Flowing solution of biomolecules of different sizes through these

nanostructured arrays causes the size based separation. The shape and structure of these

arrays are tailored based on specific applications and intended separation mechanism such as

sieving, routing, fractionation, shear-flow driven, batch or continuous flow and entropy

driven.[14] These separation techniques are commonly used currently for separation of

various biomolecules such as DNA or proteins and are also popular in genomics and

proteomics.[32]


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6

Experimental studies using nanofluidic devices

The developments in photolithographic techniques to construct nanofluidic devices

have enabled researchers to investigate the transport properties in nanoscale geometries. The

majority of this research was targeted towards biomedical systems and can be categorized

into various groups such as single molecule analysis, biomolecule separation, biomolecule

concentration as described in what follows.

Single molecule analysis: Single stranded RNA and DNA were driven through a 2.6nm

channel in a lipid bilayer membrane using an electric field.[33-39] Since the nanochannel

dimensions were just large enough to accommodate only a single strand of DNA the

molecule had to travel through the channel as a stretched molecule. The polymer molecule

translocating through the nanochannel would cause partial blocking of the nanochannel

resulting in reduction in the ionic currents proportional to the length of the molecule. The

technique was used in calculating the polynucleotide length and the characteristics of the

base. Improvements in nanopore detection techniques aided in extension of this study to

identify homopolymers of different polyacids in the solutions based on blockade amplitudes

and kinetics.[35, 36, 40]

Various forms of single molecule detection techniques involved analyte molecules

binding to the channel surface [41-43] with the recognition sites located inside[44, 45] or at

the opening[46] of the channel. Multianalyte detection was made possible by attaching

recognition sites to the polymers which modifies the translocation process of each analyte

differently and hence results in detection.[47]


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7

Voltage driven DNA translocation through an -hemolysin pore was carried out to

study the dynamics of the DNA molecule.[35, 48] A schematic of the cross section of a -

hemolysin channel embedded in a lipid bilayer is shown in Figure 1.1.

Figure 1.1: A schematic of the cross section of a -hemolysin channel embedded in a lipid bilayer[49]

The translocation velocity was calculated as a function of polymer length. It was observed

that the molecules longer than the nanopore translocated with the same velocity while, for the

polymers shorter than the nanopore length the velocity increased with decrease in polymer

length. It was proposed that the confinement produces strong drag on the polymer molecules

which can not be approximated from the bulk hydrodynamics.

Statics and dynamics of single DNA molecule studied by electrophoretically

stretching the molecule in a 30 400 nm wide nanochannel revealed a deviation from the

deGennes scaling theory for average extension of the confined self-avoiding polymer

molecule.[50] A crossover in polymer physics was observed with crossover scale to be

roughly twice the persistence length of the molecule.


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8

Pressure driven transport of DNA molecule in a nanofluidic channel showed two

distinct transport regimes.[51] The pressure driven mobility increased with the molecular

length for the nanochannel with width greater than few times the radius of gyration of the

molecule whereas the mobility was independent of molecular length for the thinner

nanochannels.

A single molecule barcoding system for DNA molecule analysis was developed using

enzymatic labeling technique that tags specific sequences on the electrokinetically stretched

DNA molecule.[52] Effect of buffer solution strength and confinement was studied on the

molecule stretching. The polymer elongation was observed to increase with decrease in

buffer strength. Polymer elongations studied as a function of confinement yielded deviations

from de Gennes scaling theory as was previously observed.[50]

Separation: Nanofluidic devices prepared using various lithographic techniques, such as

electron, x-beam and ion beam lithographs, have been shown capable of performing DNA

electrophoresis. The polymer length based difference in mobility combined with controlled

electric field was used to perform fractionation.[2, 53] Entropic trapping of DNA molecule

in microfabricated arrays with nanoscale constrictions was demonstrated for separation

purpose.[49] Interesting dynamics were observed with longer polymer molecules escaping

the entropic traps faster than the shorter ones. A model based on the trapping lifetime of

molecules was proposed to explain the observed escape behavior.

Nanofilter array chip for gel free biomolecule separation was proposed by Fu and

coworkers.[54] The proposed technique using Ogston sieving[55] mechanism, in which the

separation is caused by creating energy barriers using deep and shallow regions in the flow

geometry, was described as an alternative to the conventionally used gel electrophoresis


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9

technique for which limited knowledge about the sieving mechanism is available. The

technique resulted in effective and efficient separation of sodium dodecyl sulfate-protein

complexes and small DNA molecules. Anisotropic nanofluidic sieving structure for

continuous-flow separation of DNA and proteins was used for a size based and charge based

separation of biomolecules.[56] These devices with their separation efficiency and generality

are expected to be potentially used as a generic molecular sieving structure for an integrated

biomolecule sampler preparation and analysis system.

Molecule concentration: The detection mechanisms used in analysis equipments function

better if the concentration of species to be detected is higher. Nanofluidic sieving structures

could be used as preconcentration devices to increase the concentration of selected species.

Several such techniques have been developed using nanofluidic filters in microfluidic

devices,[57] and using nanolabeling techniques for detection purposes. These systems can be

used in fabrication of microsystems for chemical-biological agent detection. Another such

preconcentration device used a nanofluidic channel created between weak reversibly bonded

glass and polydimethylsiloxane (PDMS).[58-60] The electric field applied across the

channel causes selective movement of ions and results in increased concentration of charged

proteins near the channel opening. These preconcentration devices also help in facilitating

enzymatic reactions due to increased concentrations of reacting species.[61]

Other experimental techniques using nanofluidic devices include diffraction gradient

lithography[62] which uses continuous spatial gradient structures to obtain a smooth

transition of DNA molecules from microchannel region to the nanochannel regions in micro-

nanofluidic devices. Usually the nanofluidic devices are a part of a bigger microfluidic

device, and the biomolecule has to overcome a huge entropic barrier while moving from the


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micron to nanoscale geometries. The entropic barrier is gradually reduced using this

technique by prestretching the DNA molecules using micropost arrays.

Restriction mapping of individual -DNA molecules were carried out in a 100 200nm

nanochannel by Riehn and coworkers.[63] Two microchannels were connected together

using array of nanochannels and the restriction reactions were carried out in the nanochannel

by electrophoretically driving the DNA in the nanochannel. A study of conformational

response of single DNA molecule to changes in ionic environment showed a tremendous

increase in DNA extension with decrease in ionic strength.[64] It was shown that the

decrease in ionic strength results in a reduced screening of electrostatic interactions leading

to increased self avoidance and hence increased stretching in DNA molecule. The authors

proposed an additional parameter in deGennes theory[65] of average extension of confined

self avoiding molecule an effective DNA width that gives the increase in excluded volume

due to electrostatic repulsions.

Particle Micro and Nanorheology

Calculation of rheological properties of a fluid has traditionally been carried out using

laboratory rheometers. The rheological properties such as the complex viscoelastic moduli

are determined by studying the mechanical response of the fluid to the applied shear. A

typical rheological experiment requires about a milliliter quantity of sample and probes the

sample viscoelastic properties over a limited frequency range. These techniques pose serious

limitations in case of fluids which are precarious in nature, for biological fluids or for fluids

that are expensive to procure. Moreover, the fluid under study is always deformed in the

rheological measurements. In rheological experiments, the fluid under study is assumed to

be homogeneous and the measured response the average, bulk response of the fluid. In many


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applications that involve biological materials such as cells, the mechanical properties vary

spatially and hence the measurement of local viscoelastic properties of interest.

Developments in the optical techniques of particle manipulation and single particle

tracking have enabled rheologists to devise new methodologies for calculation of local

mechanical properties on micrometer length scales and to resolve the issue of local

heterogeneities. In these techniques, commonly known as Microrheology, micrometer

sized probe particles are embedded in the fluid to be studied. The embedded probe particles

are used to locally deform the sample and optical techniques are used to track the motion of

the particles. The measured response of the fluid on micrometer length scales is used to

determine the local material properties and the technique is called as Microrheology.[66]

Microrheological experiments are typically divided into two broad catagories active and

passive. In active microrheology, the probe particle is actively manipulated by the local

application of force and the material response is studied to the motion of the probe or the

correlated motion of two probes.[67, 68] On the other hand, in passive microrheology, the

passive motion of the probe particle due to thermal or Brownian fluctuations is tracked to

obtain the material response.[69-73] Only microliter quantities of sample are needed in these

experiments thus providing a huge advantage over macrorheological experimental

techniques.

Many systems such as polymer nanocomposites and polymer thin films show

nanoscale structural heterogeneities. The mechanical properties of these systems are thus

expected to show nanoscale variation; these local viscoelastic properties could be

determined by particle nanorheology. Molecular simulation techniques with capabilities to

mimic the systems on the length scales of size of the molecule can be very effective in


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exploring the physics in nanoscale geometries and to calculate the transport, mechanical as

well as the thermodynamic properties of these systems.

Molecular dynamics simulations

Molecular dynamics (MD) simulation technique is an atomistic simulation method

where each atom is treated as a point mass. MD simulation numerically solves the Newtons

equation of motion for the system to obtain information about its time dependent properties,

dx

dUF

dt

xdm ==

2

2

(1.1)

where m is the mass of the atom,x is position of the atom, tis time, Fis the force on the atom

and U is the potential between the atoms. Starting with an initial configuration of atoms,

various atoms of the system interact via a chosen potential form which is used to calculate

the forces on each atom center. These forces are then used to advance the particle in time

with chosen time step to obtain the new positions of the atoms using one of the different

finite different methods available such as: predictor-corrector, Verlet, leap-frog, velocity-

Verlet etc. In the simulations described in all of the following chapters we have used

velocity-Verlet algorithm[74] described below.

Ifr(t), v(t) and a(t) are respectively the position, velocity and acceleration of an atom in the

system at time t, then the position, velocity and acceleration of the atom at time t + t ,

represented by r(t+ t), v(t+ t) and a(t+ t) respectively, is determined using the velocity-

Verlet algorithm as,


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13

)(.2

1)(.)()( 2 tattvttrttr ++=+ (1.2)

))()(.(2

1)()( ttatattvttv +++=+ (1.3)

The time step trepresents advancement in time in each simulation step. The time step must

be small enough to avoid discretization errors in the calculations but at the same time large

enough to capture the effect being modeled without taking an extraordinary period of time.

Hence the positions, velocities, forces etc. on each atom at each time step are obtained and

can be used to calculate various properties of the system using statistical mechanics methods.

The desirable qualities of a molecular simulation algorithm are: it should be fast, it should

take as little memory as possible, it should permit the use of long time step, it should

duplicate the classical trajectories as closely as possible, it should satisfy the known

conservation laws for energy and momentum, it should be time reversible, and it should be

simple in form and easy to program.

Organization

A large number of experiments have captured the features of flow of polymeric

solutions, mainly DNA, in nanoscale channels. The mechanisms underlying these

observations have been hypothesized from the experiments; these assumptions can be

validated by using molecular simulations.

A specific example that of cross stream migration of a polymer chain in a shear

flow in a nanochannel has been selected for detailed study, presented in chapter 2. The

shear induced polymer migration mechanism holds potential for effective separation and pre-

concentration of molecules of interest from a mixture. A detailed study of the cross stream

chain migration phenomenon and the role of hydrodynamic interactions in this process is


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14

presented. Furthermore, the effect of parameters such as polymer chain length, nanochannel

dimensions, intermolecular interactions and polymer concentration is studied in detail.

Next, molecular hydrodynamics in confined nanoparticle suspensions are studied

using molecular simulations of smooth (Chapter 3) and rough (Chapter 4) particle translation

and rotation. Simulations are used to quantify three effects: effect of confining channel

surface, effect of slip at the nanoparticle surface and cooperative hydrodynamic interactions

between the particles. It is demonstrated that simulation results can be quantitatively

described by continuum mechanics if these effects are explicitly accounted for in the

continuum treatment. Based on these principles, an Active Nanorheology technique is

presented for the calculation of viscoelastic properties of complex fluids using molecular

dynamics simulations in Chapter 5. The technique provides a very effective way for

calculating the local mechanical properties on nanometer length scales and will serve as a

method to probe structural behavior of heterogeneous materials such as polymer

nanocomposites. The technique is validated by calculating the viscoelastic properties of a

polymer melt and comparing these results with literature values that were obtained using

different simulation techniques.


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2. CHAPTER 2

II. CROSS-STREAM CHAIN MIGRATION IN NANOFLUIDICCHANNELS

Introduction

Flow behavior of dilute polymeric solutions in micro- or nanofluidic channels has

been experimentally studied because of the potential applications of the process for

manipulation of DNA and other biological molecules.[1, 51, 63, 75-80] Cross stream

migration of chains plays an important role in some of these applications; the literature on the

flow behavior and cross-stream chain migration phenomenon in dilute polymer solutions has

been captured in an older as well as a recent review.[81, 82] A major part of this chapter is

taken from our recently published work.[83]

Experimental findings

Fluorescence microscopy experiments on DNA solutions in microfluidic channels

have shown that the chains migrate away from the channel walls when subjected to flow.[84-

86] For the very dilute DNA solutions, the thickness of the chain depletion layer near the

channel walls was found to increase with shear rate in these experiments leading to depletion

layer thicknesses that were several times the size (radius of gyration) of the chains.[84]

These observations were attributed to the hydrodynamic interactions (HI) between the chains

stretched by the flow and the channel walls as was also asserted in Brownian dynamics

simulations.[87] More recent experiments showed that the amount of chain migration due to

wall HI reduced as the solution concentration increased from 0.1C* to 1.0C* (where C* is


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16

the chain overlap concentration); the HI effects were almost completely screened out for the

solution of concentration 3.0C* leading to a very small degree of chain migration at this

concentration.[86]

Theoretical findings

The cross-stream migration of chains in flowing polymer solutions has also been

predicted by kinetic theory.[88, 89] The theoretical development presented by Ma and

Graham[89] showed that for polymer solutions undergoing pressure driven flow, cross

stream chain migration can occur both by wall HI and the gradient in chain mobility (which

might arise if chain mobility is a function of chain conformations). Effect of temperature on

cross stream polymer chain migration was also studied in detail using kinetic theory.[90]

Computational investigations

The phenomenon of cross-stream chain migration in confined channels has also been

studied by modeling; these studies have primarily used the mesoscopic modeling techniques

of Brownian dynamics (BD), dissipative particle dynamics (DPD) or Lattice Boltzmann (LB)

simulations.[87, 91-98] BD simulations of Jendrejacket al.[87] showed that for the same

flow rate, the depletion layer thickness increased with an increase in chain length. In another

study, BD simulations carried out using a chain consisting of freely jointed rigid rods showed

that the depletion layer thickness is insensitive to the chain flexibility.[91] The direction of

chain migration (either towards or away from the walls) was found to be governed by the

degree of chain confinement in these simulations. Specifically, for weakly confined chains


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17

( 5>

gR

H, where H is the channel height and

gR is the chain radius of gyration), chain

migration was noted to be away from the walls; on the other hand, for strongly confined

chains ( 3>

1, shear flow stretches the chains on a time scale that is much faster than their natural

relaxation time scale and the chains acquire a stretched configuration. The calculation ofWe

thus requires determination of the chain longest relaxation time. Here, the chain relaxation

time is determined from equilibrium MD simulations in conjunction with the Zimm model,

as described in previous work.[99] In brief, the chain center of mass diffusion coefficient


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23

was obtained by measuring the mean squared displacement of the chain in an equilibrium

MD simulation of a bulk (unconfined) system. In equilibrium simulations a 3-D periodic box

is used and the chain is allowed to move freely, and the mean-squared displacement of the

chain is monitored. The mean squared displacement when plotted against the time gives a

straight line passing through the origin, slope of which yields the diffusion coefficient. The

chain relaxation time was then obtained by using the Zimm model expression that relates the

chain diffusion coefficient with the relaxation time: [105]

D

R 20637.0 = (2.3)

whereR is the chain end-to-end distance andD is the diffusion coefficient calculated from

the equilibrium molecular dynamics simulations.

This procedure is based on the assumptions that the excluded volume interactions

between the beads can be neglected and that the mobility matrix can be represented using the

Oseen tensor. Although based on these approximations, we consider this to be adequate for

our purposes since our only interest is to use the relaxation time value so obtained to

determine the shear rates for which We > 1. It has been pointed out in the literature that the

cross stream chain migration phenomenon due to wall hydrodynamic interactions is only

observed at small values of the Reynolds number.[92] Calculation of the Reynolds number

for the simulated systems necessitates the value of the viscosity. The viscosity of each of the

systems is determined by measuring the shear stress on the channel walls during the Couette

flow simulations and then dividing the wall stress by the shear rate.


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24

Results

Effect of chain length

The principal quantity of interest in this work is the distribution of the chain centers

of mass in the channel. Figure 2.2 shows the chain center of mass density profiles for the

longer and the shorter chains (N= 20 andN= 4) obtained from the Couette flow simulation.

0 1 2 3 4 5 6 7 8 9 10z

0

0.5

1

1.5

2

2.5

Chaincenterofmassdensity

Equilibrium

Equilibrium

We = 48.3We = 1.0We = 72.4We = 1.6

Figure 2.2: Chain center of mass density profiles for chains of length 20 (lines with open symbols) and 4

(filled symbols only) as a function of distance from the wall. Simulation conditions are: Couette flow

(shear rates = 0.05 and 0.075),= 0.8, Twall= 0.9 and WCA interactions. The ratios of channel height (H)to the bulk chain radius of gyration (Rg) are: 8.1 (N= 20) and 22.4 (N= 4). The Reynolds number (based

on channel height) values are: 5.1 for the shear rate of 0.05 and 7.7 for the shear rate of 0.075.

For the sake of clarity, the uncertainties as calculated from the technique of block

averaging[106] are shown for only one shear rate; these are of similar magnitude for all other

profiles and are not shown in the rest of the figures. The figure compares the distribution of

chain centers of mass at equilibrium (no flow) with those at two different shear rates. Very

different behavior is exhibited by the two chains: the longer chain (N= 20) shows a strong


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25

tendency to migrate away from the channel wall (atz = 0) with an increase in the shear rate.

On the other hand, the concentration profiles for the shorter chain (N= 4) at the shear rates

studied are virtually indistinguishable from the profile at equilibrium. A more detailed

picture of the effect of shear flow on the distribution of the longer chain in the channel can be

obtained by focusing on the density of the individual beads in the channel (Figure 2.3). For

the longer chain in this dilute solution at equilibrium, a weak tendency for the beads to layer

against the walls is observed, followed by a bulk-like flat region in the density profile away

from the walls. As the shear rate increases, the beads of the longer chain start migrating

away from the channel walls and the tendency for layering against the wall is further

weakened.

0 1 2 3 4 5 6 7 8 9 10

z

0

0.5

1

1.5

2

2.5

Beaddensity

Equilibrium

We = 48.3We = 72.4

Figure 2.3: Bead density profiles for chains of length 20 as a function of distance from the wall.

Simulation conditions are same as those for Figure 2.2.

The origin of the different migration behavior shown by the two chains in Figure 2.2

can be deduced by focusing on the stretching behavior of the two chains (Figure 2.4). As


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26

determined from the root mean squared end-to-end distance, at the highest shear rate, the

longer chain stretches by about 54% compared to its equilibrium size, whereas the shorter

chain hardly stretches at all. This behavior is to be expected from the values ofWe for the

two chains: We for the longer chain is much higher than 1 (~ 72) whereas for the shorter

chain, it is barely greater than 1 (~ 2), at the highest shear rate studied.

0 0.02 0.04 0.06 0.08Shear rate

0.8

1

1.2

1.4

1.6

Normalizedchainend-to-end

distance

Figure 2.4: Normalized root mean squared chain end-to-end distance (normalized by the equilibrium

value i.e. at We = 0) as a function of shear rate (filled circles:N= 20 and open squares:N= 4).

Simulation conditions are the same as those for Figure 2.2. The average error bars on the absolute values

of chain end-to-end distance are 0.21 for long chain (N= 20) and 0.01 for short chain (N= 4).

These results indicate that the longer chains that are stretched by the flow exhibit cross-

stream chain migration due to hydrodynamic interactions with the channel walls[87] whereas

the shorter chains do not get stretched by the flow, and hence do not exhibit cross-stream

chain migration.

The simulations reported above were carried out with only the channel walls being

maintained at a constant temperature. At the highest shear rate studied (0.075), there is a


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27

slight temperature rise (~ 0.13) in the system due to viscous heating. Previously, it was

shown that a temperature gradient can cause migration of chains in a nanochannel (in

addition to the wall hydrodynamic interaction).[99] To separate out these two effects,

isothermal (constant Tfluid) shear flow simulations were carried out on the system and the

results for the chain distribution are shown in Figure 2.5 for this case. These results are very

similar to those in Figure 2.2 and indicate that the different migration behavior exhibited by

the two chains in the same solution that is subjected to shear flow is mainly due to the wall

hydrodynamic interactions at these conditions.

0 1 2 3 4 5 6 7 8 9 10z

0

0.5

1

1.5

2

2.5

3


Equilibrium

Equilibrium

We = 48.3We = 1.0We = 72.4We = 1.6

Figure 2.5: Chain center of mass density profiles for chains of length 20 (lines with open symbols) and 4

(filled symbols only) as a function of distance from the wall. Simulation conditions are: Couette flow

(shear rates = 0.05 and 0.075),= 0.8, isothermal flow conditions with Tfluid= 0.9 and WCA interactions.

The channel Reynolds number values are: 4.8 for the shear rate of 0.05 and 7.2 for the shear rate of0.075. The ratios of channel height (H) to the bulk chain radius of gyration (Rg) are: 8.1 (N= 20) and 22.4

(N= 4).


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Effect of channel height

We have studied the cross stream chain migration behavior in four different channels

with heights 18.0, 21.0, 27.0 and 33.0. Figure 2.6 to Figure 2.9 show the concentration

profiles of the chain center of mass for these four channel heights. The concentration profiles

for all the channel heights studied are qualitatively similar. In all cases, a steric depletion

layer is observed next to the channel surface in the absence of shear flow. The concentration

profiles in presence of shear flow show that the chains migrate further away from the channel

surface, the amount of migration increasing with an increase in the shear rate. We have

quantified the degree of chain migration by tracking the thickness of the chain depletion layer

which is defined as the distance from the channel surface at which the concentration profile

assumes a value of unity (i.e. uniform concentration).

0 3 6 9z

0

0.5

1

1.5

2

2.5

3


Equilibrium

We = 48.3We = 72.4

Figure 2.6: Chain center of mass density profiles for chains of length 20 as a function of distance from the

wall. Simulation conditions are: Couette flow (shear rates = 0.05 and 0.075),= 0.8, Twall= 0.9 and WCAinteractions. Density profile is shown for a channel height of 18. The ratio of channel height (H) to the

bulk chain radius of gyration (Rg) for this case is 6.9.


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0 1 2 3 4 5 6 7 8 9 10z

0

0.5

1

1.5

2

2.5


Equilibrium

We = 48.3We = 72.4


wall. Simulation conditions are identical to those for Figure 2.6. Density profile is shown for a channel

height of 21. The ratio of channel height (H) to the bulk chain radius of gyration (Rg) for this case is 8.1.

0 1 2 3 4 5 6 7 8 9 10 11 12 13z

0

0.5

1

1.5

2

2.5


EquilibriumWe = 48.3We = 72.4


wall. Simulation conditions are identical to those for Figure 2.6. Density profile is shown for a channel



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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16z

0

0.5

1

1.5

2

2.5


Equilibrium

We = 48.3We = 72.4


wall. Simulation conditions are identical to those for figure 2.6. Density profile is shown for a channel


0.75

1.25

1.75

2.25

2.75

3.25

3.75

6 8 10 12 14

Equilibrium

We = 48.3

We = 72.4

Figure 2.10: Normalized depletion layer thickness plotted as a function of normalized channel height for

a chain of length 20. Simulation conditions are same as those for Figure 2.6. Depletion layer thickness

and channel height are normalized by the radius of gyration of the chain.

d/Rg

H/Rg


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A plot of the normalized depletion layer thickness as a function of channel height is

shown in Figure 2.10 for conditions of equilibrium and shear flow. We note that the

depletion layer thickness at equilibrium shows a very weak variation (if at all) with channel

height as is expected for a steric depletion layer. On the other hand, for chains subjected to

shear flow, the thickness of the depletion layer increases monotonically with an increase in

the channel height.

Effect of concentration

The hydrodynamic interactions are expected to be increasingly shielded as the

solution concentration increases and hence the cross-stream chain migration process is

expected to be strongly affected by the chain concentration in the solution. We investigated

this effect by studying the chain migration phenomenon in solutions in the concentration

range of 0.11 C* - 1.64 C*. Our results show that the solutions with a lower concentration of

chains show a qualitatively different chain migration behavior than the higher concentration

solutions. Specifically, at the lower concentration (0.11 C*, see Figure 2.11a), chains

migrate away from the surface when subjected to the shear flow.

The tendency for the migration of chains away from the surface in presence of shear

flow decreases with an increase in the solution concentration. At the higher concentrations

(results for solution concentration of 1.64 C* are shown in Figure 2.11b), there is a weak

tendency for chain migration towards the surface in the presence of shear flow.


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0 1 2 3 4 5 6 7 8 9 10z

0

0.5

1

1.5

2


Equilibrium

We = 48.3We = 72.4

0 1 2 3 4 5 6 7 8 9 10

z

0

0.5

1

1.5

2

Chaincenterofmass

density

Equilibrium

We = 48.3We = 72.4

Figure 2.11: Chain center of mass density profiles for chains of length 20. Simulation conditions are:

Couette flow, = 0.8, channel height H = 21, Twall = 0.9 and WCA interactions. Figures show densityprofiles at equilibrium and at shear rates of 0.05 and 0.075. Density profiles are shown for systems with

concentrations of: (a) 0.11 C* and (b) 1.64 C*.

(a)

(b)


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0.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

2.1

0 0.5 1 1.5 2

Equilibrium

We = 48.3

We = 72.4

Figure 2.12: Normalized depletion layer thickness (normalized by chain radius of gyration) plotted as a

function of concentration for a chain of length 20 at equilibrium and at shear rates of 0.05 and 0.075.

Simulation conditions: Couette flow,= 0.8, channel heightH= 21, Twall= 0.9 and WCA interactions.

At this point, we note that the relaxation time of the chain is a function of solution

concentration. We have chosen to compare the chain migration behavior of different

solutions at the same shear rate rather than at the same Weissenberg number. The relaxation

time of chain increases with the solution concentration and thus, the systems with the higher

concentration will, in fact, have a larger Weissenberg number than those with the lower

concentration at the same shear rate. The degree of chain migration is quantified by plotting

the normalized depletion layer thickness as a function of the solution concentration (Figure

2.12). As seen, the depletion layer thickness in shear flow is larger than that at equilibrium

for the lower concentration solutions, whereas opposite behavior is observed in solutions

with concentrations comparable to or greater than the overlap concentration.

d/Rg

C/C*


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34

In addition to the chain density across the channel, the solution concentration is also

expected to affect the velocity profile in the system. Previous molecular simulations of shear

flow of polymer melts in nanochannels have shown the occurrence of velocity slip at the

channel surface.[103, 107] The effect of polymer solution concentration on the velocity

boundary condition at the channel surface is shown in Figure 2.13. The velocity profiles

show that the amount of velocity slip increases significantly with an increase in the solution

concentration from 0.11 C* to 1.64 C*.

0 3 6 9 12 15 18 21z

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Velocity

C = 0.11 C*C = 1.64 C*Wall velocity

Figure 2.13: Velocity profiles in the channel for solutions with concentrations 0.11 C* and 1.64 C*.

Simulation conditions are the same as those for figure 2.12. The velocities of the two walls are 0.525.

Molecular simulations provide the ability for developing a detailed molecular level

understanding of the chain conformational behavior in these flowing solutions. Firstly, the

radius of gyration (Rg) of the polymer chain was seen to show a very weak dependence on

the concentration: the chainRg in the solution with concentrationC= 1.64 C* was seen to be


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35

lower than the value at C= 0.11 C* by about 5% at equilibrium and by about 2% at the two

shear rates studied. In addition to the overall chain size, the differences in the chain

stretching behavior in the solutions of different concentrations were also determined by

monitoring the chain dimensions. This was achieved by representing the polymer chain in

terms of an equivalent ellipsoid with the same moment of inertia and then calculating the

semi-axis lengths of this ellipsoid.[103] As can be seen from Figure 2.14a, at equilibrium

(no-flow), the chain longest dimension shows little variation across the channel (with the

exception of a small peak near the two walls) for a solution of concentration 0.11C*.

On the other hand, at equilibrium, the chains near the channel surface in the higher

concentration solution (1.64 C*) are much longer than the ones away from the surface. In the

bulk like region in the middle of the channel, the chains in this higher concentration

solution are shorter than the ones in the dilute solution. As shear rate is increased (Figure

2.14b and 2.14c), the chain size difference between the solutions of different concentrations

gradually diminishes. At a shear rate of 0.075, the chain longest dimension profile for

solution with concentration 1.64 C* still shows a well defined peak near the channel wall,

however, away from the surface, the length of the chains is similar to that for the dilute

solution (0.11 C*). In summary, the overall chain size as determined byRgshowed a small

variation with concentration over the concentration range studied in this work. On the other

hand, significant differences in the shape of the chains were observed as the solution

concentration was varied.


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0 3 6 9 12 15 18 21z

4

5

6

Chainlongestsem

i-axislength

C = 0.11 C*C = 1.64 C*

0 3 6 9 12 15 18 21z

5

6

7

Chainlongestsemi-axislength

C = 0.11 C*C = 1.64 C*

0 3 6 9 12 15 18 21z

5

6

7

Cha

inlongestsemi-axislength

C = 0.11 C*C = 1.64 C*

Figure 2.14: Chain longest semi-axis length profile for chain of length 20 at (a) equilibrium and at shear

rates of (b) 0.05 and (c) 0.075. Simulation conditions are same as those for figure 2.12. Profiles are

plotted fo

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