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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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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
Simulation Method.......................................................................................................... 47
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
Summary and Discussion................................................................................................ 69
IV. MOLECULAR HYDRODYNAMICS IN NANOPARTICLE SUSPENSIONS:
FRICTION FORCE AND TORQUE ON A ROUGH NANOPARTICLE ................. 72
Introduction..................................................................................................................... 72
Simulation Method.......................................................................................................... 74
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
Summary and Discussion................................................................................................ 93
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
Figure 2.15: Chain center of mass density profiles for chains of length 10 (lines with open
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
Figure 2.16: Chain center of mass density profiles for chains of length 10 (lines with open
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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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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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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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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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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)(.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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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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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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( 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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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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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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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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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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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
Chaincenterofmassdensity
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
Chaincenterofmassdensity
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
Chaincenterofmassdensity
Equilibrium
We = 48.3We = 72.4
Figure 2.7: Chain center of mass density profiles for chains of length 20 as a function of distance from the
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
Chaincenterofmassdensity
EquilibriumWe = 48.3We = 72.4
Figure 2.8: Chain center of mass density profiles for chains of length 20 as a function of distance from the
wall. Simulation conditions are identical to those for Figure 2.6. Density profile is shown for a channel
height of 27. The ratio of channel height (H) to the bulk chain radius of gyration (Rg) for this case is 10.4.
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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
Chaincenterofmassdensity
Equilibrium
We = 48.3We = 72.4
Figure 2.9: Chain center of mass density profiles for chains of length 20 as a function of distance 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 chain radius of gyration (Rg) for this case is 12.7.
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
Chaincenterofmassdensity
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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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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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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