modeling and simulation in polymers

Edited by
Purushottam D. Gujrati
and Arkadii I. Leonov
Modeling and Simulation
in Polymers
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Edited by Purushottam D. Gujrati and Arkadii I. Leonov
Modeling and Simulation in Polymers
The Editors
Dr. Purushottam D. Gujrati

The University of Akron
Department of Polymer Science
302 Buchtel Common
Akron, OH 44325-3909
USA
Dr. Arkady I. Leonov
The University of Akron
Department of Polymer Engineering
Polymer Engineering Academic Center
Akron, OH 44325-0301
USA
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Contents
Preface XV
List of Contributors XIX
1 Computational Viscoelastic Fluid Mechanics and Numerical Studies
of Turbulent Flows of Dilute Polymer Solutions 1
Antony N. Beris and Kostas D. Housiadas
1.1 Introduction and Historical Perspective 1
1.2 Governing Equations and Polymer Modeling 6
1.3 Numerical Methods for DNS 10
1.3.1 Spectral Methods: Influence Matrix Formulation 11
1.3.1.1 The Semi-Implicit/Explicit Scheme 11
1.3.1.2 The Fully Implicit Scheme 13
1.3.1.3 Typical Simulation Conditions 15
1.3.2 The Positive Definiteness of the Conformation Tensor 15

1.4 Effects of Flow, Rheological, and Numerical Parameters on DNS
of Turbulent Channel Flow of Dilute Polymer Solutions 17
1.4.1 Drag Reduction Evaluation 17
1.4.2 Effects of Flow and Rheological Parameters 19
1.4.3 Effects of Numerical Parameters 26
1.5 Conclusions and Thoughts on Future Work 29
References 31
2 Modeling of Polymer Matrix Nanocomposites 37
Hendrik Heinz, Soumya S. Patnaik, Ras B. Pandey, and Barry L. Farmer
2.1 Introduction 37
2.2 Polymer Clay Nanocomposites and Coarse-Grained Models 40
2.2.1 Coarse-Grained Components 42
2.2.2 Methods and Timescales 42
2.2.2.1 Off-Lattice (Continuum) Approach 43
2.2.2.2 Discrete Lattice Approach 43
2.2.2.3 Hybrid Approach 44
2.2.3 Coarse-Grained Sheet 44
V
2.2.3.1 Conformation and Dynamics of a Sheet 47
2.2.4 Coarse-Grained Studies of Nanocomposites 50
2.2.4.1 Probing Exfoliation and Dispersion 51
2.2.5 Platelets in Composite Matrix 52
2.2.5.1 Solvent Particles 52
2.2.5.2 Polymer Matrix 55
2.2.6 Conclusions and Outlook 60
2.3 All-Atom Models for Interfaces and Application to Clay Minerals 61
2.3.1 Force Fields for Inorganic Components 62
2.3.1.1 Atomic Charges 64
2.3.1.2 Lennard-Jones Parameters 65
2.3.1.3 Bonded Parameters 67

2.3.2 Self-Assembly of Alkylammonium Ions on Montmorillonite:
Structural and Surface Properties at the Molecular Level 68
2.3.3 Relationship Between Packing Density and Thermal Transitions
of Alkyl Chains on Layered Silicate and Metal Surfaces 78
2.4 Interfacial Thermal Properties of Cross-Linked Polymer–CNT
Nanocomposites 79
2.4.1 Model Building 81
2.4.2 Thermal Conductivity 83
2.5 Conclusion 86
References 86
3 Computational Studies of Polymer Kinetics 93
Galina Litvinenko
3.1 Introduction 93
3.2 Batch Polymerization 95
3.2.1 Ideal Living Polymerization 95
3.2.2 Effect of Chain Transfer Reactions 97
3.2.3 Chain Transfer to Solvent 97
3.2.4 Multifunctional Initiators 102
3.2.5 Chain Transfer to Polymer 105
3.2.6 Chain Transfer to Monomer 109
3.3 Continuous Polymerization 111
3.3.1 MWD of Living Polymers Formed in CSTR 113
3.3.2 Chain Transfer to Solvent 116
3.3.3 Chain Transfer to Monomer 118
3.3.4 Chain Transfer to Polymer 120
3.4 Conclusions 123
References 125
4 Computational Polymer Processing 127
Evan Mitsoulis
4.1 Introduction 127

4.1.1 Polymer Processing 127
VI Contents
4.1.2 Historical Notes on Computations 128
4.2 Mathematical Modeling 130
4.2.1 Governing Conservation Equations 130
4.2.2 Constitutive Equations 130
4.2.3 Dimensionless Groups 134
4.2.4 Boundary Conditions 138
4.3 Method of Solution 140
4.4 Polymer Processing Flows 143
4.4.1 Extrusion 143
4.4.1.1 Flow Inside the Extruder 143
4.4.1.2 Flow in an Extruder Die (Contraction Flow) 146
4.4.1.3 Flow Outside the Extruder – Extrudate Swell 149
4.4.1.4 Coextrusion Flows 150
4.4.1.5 Extrusion Die Design 153
4.4.2 Postextrusion Operations 154
4.4.2.1 Calendering 155
4.4.2.2 Roll Coating 157
4.4.2.3 Wire Coating 162
4.4.2.4 Fiber Spinning 163
4.4.2.5 Film Casting 169
4.4.2.6 Film Blowing 173
4.4.3 Unsteady-State Processes 176
4.4.3.1 Blow Molding 176
4.4.3.2 Thermoforming 178
4.4.3.3 Injection Molding 181
4.5 Conclusions 185
4.6 Current Trends and Future Challenges 187
References 188

5 Computational Approaches for Structure Formation
in Multicomponent Polymer Melts 197
Marcus Müller
5.1 Minimal, Coarse-Grained Models, and Universality 197
5.2 From Particle-Based Models for Computer Simulations
to Self-Consistent Field Theory: Hard-Core Models 201
5.2.1 Hubbard–Stratonovich Transformation: Field-Theoretic
Reformulation of the Particle-Based Partition Function 201
5.2.2 Mean Field Approximation 206
5.2.3 Role of Compressibility and Local Correlations
of the Fluid of Segments 210
5.3 From Field-Theoretic Hamiltonians to Particle-Based Models:
Soft-Core Models 211
5.3.1 Standard Model for Compressible Multicomponent Polymer
Melts and Self-Consistent Field Techniques 211
5.3.2 Mean Field Theory for Non-Gaussian Chain Architectures 213
Contents VII
5.3.2.1 Partial Enumeration Schemes 213
5.3.2.2 Monte Carlo Sampling of the Single-Chain Partition Function
and Self-Consistent Brownian Dynamics 214
5.3.3 Single-Chain-in-Mean-Field Simulations and Grid-Based Monte
Carlo Simulation of the Field-Theoretic Hamiltonian 217
5.3.3.1 Single-Chain-in-Mean-Field Simulations 217
5.3.3.2 Minimal, Particle-Based, Coarse-Grained Model: Discretization
of Space and Molecular Contour 219
5.3.3.3 Monte Carlo Simulations and Advantages of Soft
Coarse-Grained Models 220
5.3.3.4 Comparison Between Monte Carlo and SCMF Simulations:
Quasi-Instantaneous Field Approximation 221
5.3.4 Off-Lattice, Soft, Coarse-Grained Models 225

5.4 An Application: Calculating Free Energies of Self-Assembling
Systems 227
5.4.1 Crystallization in Hard Condensed Matter Versus Self-Assembly
of Soft Matter 227
5.4.2 Field-Theoretic Reference State: The Einstein Crystal of
Grid-Based Fields 228
5.4.3 Particle-Based Approach: Reversible Path in External
Ordering Field 229
5.4.3.1 How to Turn a Disordered Melt into a Microphase-Separated
Morphology Without Passing Through a First-Order Transition? 229
5.4.3.2 Thermodynamic Integration Versus Expanded Ensemble
and Replica-Exchange Monte Carlo Simulation 232
5.4.3.3 Selected Applications 235
5.4.4 Simultaneous Calculation of Pressure and Chemical Potential
in Soft, Off-Lattice Models 238
5.5 Outlook 239
References 242
6 Simulations and Theories of Single Polyelectrolyte Chains 247
Arindam Kundagrami, Rajeev Kumar, and Murugappan Muthukumar
6.1 Introduction 247
6.2 Simulation 251
6.2.1 Simulation Method 251
6.2.2 Degree of Ionization 253
6.2.3 Size and Shape of the Polyelectrolyte 255
6.2.4 Effect of Salt Concentration on Degree of Ionization 256
6.2.5 Radial Distribution Functions 259
6.2.6 Dependence of Degree of Ionization on Polymer Density 259
6.2.7 Size and Structure of the Polyelectrolyte 262
6.2.7.1 Theoretical Background 262
6.2.7.2 Dependence of Radius of Gyration on Salt with Monovalent

Counterions 264
VIII Contents
6.2.7.3 Bridging Effect by Divalent Counterions 265
6.3 The Variational Theory 266
6.3.1 Free Energy 269
6.3.2 Effect of Coulomb Strength on Degree of Ionization and Size 275
6.3.2.1 Salt-Free Solutions 275
6.3.2.2 Divalent Salt and Overcharging 278
6.3.3 Chain Contraction: Contrasting Effects of Mono-
and Divalent Salts 279
6.3.4 Competitive Adsorption of Divalent Salts 279
6.3.5 Effect of Dielectric Mismatch Parameter 282
6.3.6 Effect of Monomer Concentration and Chain Length 282
6.3.7 Free energy Profile 284
6.3.8 Diagram of Charged States: Divalent Salt 287
6.3.9 Effect of Ion-Pair Correlations 290
6.3.10 Collapse in a Poor Solvent 291
6.3.11 Bridging Effect: Divalent Salt 295
6.3.12 Role of Chain Stiffness: The Rodlike Chain Limit 299
6.4 The Self-Consistent Field Theory 301
6.4.1 Extension of Edwards Formulation 303
6.4.2 Transformation from Particles to Fields 309
6.4.2.1 Transformation Using Functional Integral Identities 309
6.4.2.2 Hubbard–Stratonovich Transformation 310
6.4.3 Sum Over Charge Distributions 312
6.4.4 Saddle-Point Approximation 312
6.4.5 Numerical Techniques 314
6.4.5.1 Finite Difference Methods 315
6.4.5.2 Spectral Method: Method of Basis Functions 316
6.4.5.3 Pseudospectral Method 318

6.4.6 Fluctuations Around the Saddle Point 320
6.5 Comparison of Theories: SCFT and Variational Formalism 322
6.5.1 Self-Consistent Field Theory for Single Chain 322
6.5.2 Variational Formalism 325
6.5.3 Numerical Techniques 327
6.5.4 Degree of Ionization 328
6.5.5 Term-by-Term Comparison of Free Energy: SCFT and
Variational Formalism 330
6.6 Conclusions 339
References 339
7 Multiscale Modeling and Coarse Graining of Polymer Dynamics:
Simulations Guided by Statistical Beyond-Equilibrium
Thermodynamics 343
Patrick Ilg, Vlasis Mavrantzas, and Hans Christian Öttinger
7.1 Polymer Dynamics and Flow Properties We Want
to Understand: Motivation and Goals 343
Contents IX
7.1.1 Challenges in Polymer Dynamics Under Flow 343
7.1.2 Modeling Polymer Dynamics Beyond Equilibrium 344
7.1.3 Challenges in Standard Simulations of Polymers
in Flow 346
7.2 Coarse-Grained Variables and Models 347
7.2.1 Beads and Superatoms 348
7.2.2 Uncrossable Chains of Blobs 350
7.2.3 Primitive Paths 351
7.2.4 Other Single-Chain Simulation Approaches to Polymer Melts:
Slip-Link and Dual Slip-Link Models 353
7.2.5 Entire Molecules 354
7.2.6 Conformation Tensor 355
7.2.7 Mesoscopic Fluid Volumes 357

7.3 Systematic and Thermodynamically Consistent Approach
to Coarse Graining: General Formulation 357
7.3.1 The Need for and Benefits of Consistent Coarse-Graining
Schemes 357
7.3.2 Different Levels of Description and the Choice
of Relevant Variables 358
7.3.3 GENERIC Framework of Coarse Graining 360
7.3.3.1 Mapping to Relevant Variables and Reversible Dynamics 360
7.3.3.2 Irreversibility and Dissipation Through Coarse Graining 360
7.4 Thermodynamically Guided Coarse-Grained Polymer
Simulations Beyond Equilibrium 363
7.4.1 GENERIC Coarse-Graining Applied to Unentangled Melts:
Foundations 363
7.4.2 Thermodynamically Guided Atomistic Monte Carlo
Methodology for Generating Realistic Shear Flows 365
7.4.3 Systematic Timescale Bridging Molecular Dynamics
for Flowing Polymer Melts 369
7.4.3.1 Systematic Timescale Bridging Algorithm 369
7.4.3.2 Fluctuations, Separating Timescale, and Friction Matrix 371
7.4.3.3 Results 371
7.5 Conclusions and Perspectives 372
References 374
8 Computational Mechanics of Rubber and Tires 385
Michael J. Poldneff and Martin W. Heinstein
8.1 Introduction 385
8.2 Nonlinear Finite Element Analysis 386
8.3 Incompressibility Conditions 389
8.4 Solution Strategy 393
8.5 Treatment of Contact Constraints 394
8.6 Tire Modeling 397

References 403
X Contents
9 Modeling the Hydrodynamics of Elastic Filaments
and its Application to a Biomimetic Flagellum 405
Holger Stark
9.1 Introduction 405
9.1.1 Lessons from Nature 405
9.1.2 A Historical Overview 406
9.1.3 A Biomimetic Flagellum 408
9.2 Elastohydrodynamics of a Filament 408
9.2.1 Theory of Elasticity of an Elastic Rod 408
9.2.2 Hydrodynamic Friction of a Filament: Resistive Force Theory 410
9.2.3 Hydrodynamic Friction of a Filament: Method of Hydrodynamic
Interaction 412
9.3 A Biomimetic Flagellum and Cilium 414
9.3.1 Details of the Modeling 414
9.3.2 Microscopic Artificial Swimmer 416
9.3.3 Fluid Transport 420
9.3.3.1 Two-Dimensional Stroke 421
9.3.3.2 Three-Dimensional Stroke 425
9.4 Conclusions 427
References 427
10 Energy Gap Model of Glass Formers: Lessons Learned
from Polymers 433
Puru D. Gujrati
10.1 Introduction 433
10.1.1 Equilibrium and Metastable States: Supercooled Liquids 433
10.1.2 Common Folklore 434
10.1.3 Systems Being Considered 435
10.1.4 Long-Time Stability 436

10.1.5 High Barriers, Confinement, and the Cell Model 437
10.1.5.1 Cell Model 437
10.1.5.2 Communal Entropy, Free Energy, and Lattice Models 440
10.1.6 Fundamental Postulate: Stationary Limit 441
10.1.7 Thermodynamics of Metastability 443
10.1.8 Scope of the Review 444
10.2 Modeling Glass Formers by an Energy Gap 446
10.2.1 Distinct SMSs 446
10.2.2 Entropy Extension in the Gap 446
10.2.3 Gibbs–Di Marzio Theory 447
10.3 Glass Transition: A Brief Survey 451
10.3.1 Experimentally Observed Glassy State 451
10.3.2 Glass Phenomenology 452
10.3.3 Fragility 453
10.3.4 Ideal Glass Transition as r!0 454
10.3.5 Kauzmann Paradox and Thermodynamics 456
Contents XI
10.3.6 Entropy Crisis and Ideal Glass Transition 457
10.4 Localization in Glassy Materials 459
10.4.1 Communal Entropy, Confinement, and Ideal Glass 459
10.4.2 Partitioning of Z
T
(T, V ) 463
10.5 Some Glass Transition Theories 464
10.5.1 Thermodynamic Theory of Adam and Gibbs 464
10.5.2 Free Volume Theory 465
10.5.3 Mode Coupling Theory 466
10.6 Progigine–Defay Ratio P and the Significance of Entropy 467
10.7 Equilibrium Formulation and Order Parameter 469
10.7.1 Canonical Partition Function 469

10.7.2 Free Energy Branches 470
10.7.3 Order Parameter and Classification of Microstates 470
10.8 Restricted Ensemble 471
10.8.1 Required Extension in the Energy Gap 471
10.8.2 Restricted and Extended Restricted PFs 471
10.8.3 Metastability Prescription 472
10.9 Three Useful Theorems 472
10.10 1D Polymer Model: Exact Calculation 475
10.10.1 Polymer Model and Classification of Configurations 475
10.10.2 Exact Calculation 477
10.11 Glass Transition in a Binary Mixture 480
10.12 Ideal Glass Singularity and the Order Parameter 484
10.12.1 Singular Free Energy 484
10.12.2 Order Parameter 485
10.12.3 Relevance for Experiments 486
10.13 Conclusions 488
Appendix 10.A: Classical Statistical Mechanics 490
Appendix 10.B: Negative Entropy 491
References 492
11 Liquid Crystalline Polymers: Theories, Experiments,
and Nematodynamic Simulations of Shearing Flows 497
Hongyan Chen and Arkady I. Leonov
11.1 Introduction and Review 497
11.1.1 Low Molecular Weight and Polymeric Liquid Crystals 497
11.1.2 Molecular and Continuum Theories of LCP 498
11.1.3 Soft Deformation Modes in LCP 500
11.1.4 Specific Problems in LCP Theories 502
11.1.5 Experimental Effects in Flows of LCP 503
11.2 General Equations and Simulation Procedures 504
11.3 LCP and their Parameters Established in Simulations 508

11.4 Results of Simulations 511
11.4.1 Simulations of Steady Shearing Flows 511
11.4.2 Simulations of Transient Start-Up Shear Flows 514
XII Contents
11.4.3 Simulations of Relaxation after Cessation of Steady Flow 518
11.4.4 On the Time-Temperature Superposition in Weakly Viscoelastic
Nematodynamics 521
11.5 Conclusions and Discussions 522
References 524
Index 527
Contents XIII

Preface
Polymers are now one of the major materials used in many industrial applications.
The prediction of their behavior depends on our understanding of these complex
systems. As happens in all experimental sciences, our understanding of complex
physical phenomena requires modeling the system by focusing on only those
aspects that are supposedly relevant to the observed behavior. Once a suitable
model has been identified, we have to validate it by solving it and comparing its
predictions with experiments. Solving the model usually requires some approx-
imation or ap proximations. The resulting solutions are known as theories under
those approximations. Indeed, the interpretation and a nalyses of experimental
data depend in a crucial way on the way a system is modeled and the availability of
corresponding reliable theories. This is also true for polymers, wh ose behavior
can be very c omplex. After more than 70 years of research and development,
polymer science and engineering has reached a high level of sophistication.
Application of modern computational tools and methods, and modeling and
theories of observed behavior, made it possible to solve, t hereby understand,
many basic and applied problems, and along with that helpe d to resolve some
fundamental problems in polymer physics, physicoche mistry, and polymer

mechanics. This book illustrates the p rogress achieved in using computational
methods to understand the behavior and reliability of various models in polymer
science and engineering. The editors wanted a well-balanced presentation from
scientists and engineers. Accordingly, their attempt was to seek contributions
from universities, indust ries, and natio nal laboratories so that the book cou ld
represent a wide array of topics of interest in the field. The main desire was to have
a book that will not only function as a resource for people active in the field but will
also help educate them and graduate students.
One of the tough jobs the editors had was of limiting the number of contributions
to keep the size of the book manageable. Their job was made easy though by
the contributors, who are well recognized in their respective fields. The editors
are pleased to present their reviews for the benefit of the reader. The selection of the
topics covered in this book in no way reflects their bias; rather, it reflects the
strengths of the contributors. The topics cover a range of problems in polymers,
XV
including liquid crystals and biopolymers. Since in many cases the science and
engineering are not well distinguishable, the editors decided to use a ‘‘ mixed’’
approach in presenting the contributions in the book in alphabetic order of the
authors.
The chapter by Beris and Housiadas (‘‘ Computational Viscoelastic Fluid
Mechanics and Numerical Studies of Turbulent Flows of Dilute Polymer Solutions’’ )
aims at resolving the famous long-standing problems of turbulent drag reduction.
This contribution describes recent efforts and achievements in direct computations
of near-wall turbulent flows of dilute polymer solutions and comparisons with
experimental data.
In spite of many efforts, attempts to model complicated properties of liquid
crystalline polymers (LCPs) are far from being complete. The constitutive equations
of continuum type for thermotropic LCPs were proposed only last year. Multipara-
metric character of these equations is the challenging problem for LCP simulations.
The chapter by Chen and Leonov (‘‘ Liquid Crystalline Polymers: Theories, Experi-

ments, and Nematodynamic Simulations of Shearing Flows’’ ) reviews the major
findings in this field, describes new continuum theory valid for thermotropic LCPs,
and illustrates simulations of their shearing flows.
Glasses form an important part not only of industry but also of our daily life. A
satisfactory understanding of glasses is not within our reach at present, mainly
because it represents a nonequilibrium state of matter, which seems to not follow
Nernsts postulate or the third law of thermodynamics. Gujrati in his chapter
(‘‘ Energy Gap Model of Glass Formers’’ ) addresses the issue by elevating to a
fundamental level the observed fact that the energy of a glass is much higher than
that of the corresponding crystal (an energy gap) even at absolute zero. The chapter
therefore deals with glass formation in a supercooled liquid for which a more
thermodynamically stable state exists in the form of a crystal. The resulting metast-
ability is studied thermodynamically by treating it as constrained equilibrium. The
theoretical model is also supported by performing simple exact calculations.
Polymer nanocomposites are prospective new materials. Nevertheless, their prop-
erties are still not well understood. The chapter by Heinz, Patnaik, Pandey, and
Farmer (‘‘ Modeling of Polymer Matrix Nanocomposites’’ ) demonstrates the applica-
tion of modern computational methods for investigating dispersion of various
nanofillers in polymer matrices under the action of microscopic forces. In addition,
an attempt is made to calculate the thermal conductivity in a model system of
nanotubes in polymer matrix.
Predicting flow properties of polymers such as interfacial slip is of paramount
importance in industries and poses a major challenge at present. It truly requires a
multiscale attack. Ilg, Mavrantzas, and Öttinger provide in their contribution (‘‘ Mul-
tiscale Modeling and Coarse Graining of Polymer Dynamics: Simulations Guided by
Statistical Beyond-Equilibrium Thermodynamics’’ ) a comprehensive treatment of
polymer flow dynamics by borrowing ideas from nonequilibrium thermodynamics.
They develop a multiscale modeling approach, which successfully bridges micro-
scopic and macroscopic scales. By using GENERIC formalism, they attempt to avoid
XVI Preface

thermodynamic inconsistency that is present in most coarse-grained models. They
achieve this by carefully separating timescales.
Studying charged polymers in aqueous solutions provides another example of a
major challenge in polymer technology, and is considered by Kundagrami, Kumar,
and Muthukumar (‘‘ Simulations and Theories of Single Polyelectrolyte Chains’’ ).
Only single chains are considered. Chain connectivity and topological considera-
tions are the sources of complication in understanding the interactions between the
solute and the solvent. They consider two different kinds of theoretical methods,
variational and self-consistent, and employ Langevine dynamics for their simula-
tions.
Studies of polymerization kinetics have a long history. Nevertheless, many pro-
blems in this field remain unresolved. Using computational methods, several of
these problems are clarified in the chapter by Litvinenko (‘‘ Computational Studies of
Polymer Kinetics’’ ). Special attention is paid to the effect of chain transfer reactions
on polymer molecular weight and applications to different types of polymerization
methods.
Modeling of polymer processing has more than a 60-year history. Computational
methods gave many possibilities for optimization of processing operations. The
chapter by Mitsoulis (‘‘ Computational Polymer Processing’’ ) provides a comprehen-
sive review of simulations and computational efforts for a majority of polymer
processing operations and also forecasts the future development in this important
part of polymer industry.
Multicomponent melts that are commonly found in such varied situations as
material fabrification, reinforcement, blending, and so on are discussed in the
chapter by Müller (‘‘ Computational Approaches for Structure Formation in Multi-
component Polymer Melts’’ ). Only equilibrium properties are discussed along with
computational approaches for coarse-grained models in the mean field approxima-
tion. Both hard-core and soft-core models are used to cover a multitude of scales of
length, time, and energy. Attention is also paid to methods that go beyond the mean
field approximation.

The chapter by Poldneff and Heinstein (‘‘ Computational Mechanics of Rubber
and Tires’’ ) reviews the latest achievements in using finite element analysis for
solving highly nonlinear problems of rubber and tire mechanics, with several
illustrative examples of industrial importance.
Application of ideas from polymers has begun to revolutionize bio-related dis-
ciplines. Therefore, this review book will not be complete without a chapter detailing
such an application. The last chapter (‘‘ Modeling the Hydrodynamics of Elastic
Filaments and its Application to a Biomimetic Flagellum’’ ) by Stark attempts to
model Natures successful strategies for propulsion such as of sperm cells and fluid
transport such as of mucus. The artificial cilium is based on a superparamagnetic
filament, actuated by an external magnetic field; the latter allows one to explore the
filaments capacity to transport fluid.
The editors hope that the collection of reviews will be beneficial to graduate
students, scientists, and engineers, whether practicing or just eager to familiarize
Preface XVII
themselves with new models and computational tools. The editors also take full
responsibility for any shortcoming of the book.
Finally, the editors offer their sincere thanks to Manfred Kohl for inviting us to
take on this project and to Stefanie Volk and Claudia Nussbeck, all at Wiley-VCH, for
their patience and help to ensure the completion of this project.
August 2009 Puru Gujrati and Arkady Leonov
Akron, OH, USA
XVIII Preface
List of Contributors
XIX
Antony N. Beris
University of Delaware
Department of Chemical Engineering
Newark, DE 19716
USA

Hongyan Chen
The University of Akron
Department of Polymer Engineering
Akron, OH 44325-0301
USA
Barry L. Farmer
Wright-Patterson Air Force Base
Air Force Research Laboratory
Materials and Manufacturing
Directorate
Dayton, OH 45433
USA
Puru D. Gujrati
The University of Akron
The Departments of Physics and
Polymer Science
Akron, OH 44325
USA
Martin W. Heinstein
Sandia National Laboratory
Computational Solid Mechanics and
Structural Dynamics
P.O. Box 5800 MS 0380
Albuquerque, NM 87185-0380
USA
Hendrik Heinz
The University of Akron
Department of Polymer Engineering
Akron, OH 44325
USA

Kostas D. Housiadas
University of the Aegean
Department of Mathematics
Karlovassi
Samos
Greece
Patrick Ilg
Polymer Physics, ETH Zürich
Department of Materials
CH-8093 Zürich
Switzerland
Rajeev Kumar
University of Massachusetts
Polymer Science and Engineering
Department
Amherst, MA
USA
and
University of California
Materials Research Laboratory
Santa Barbara, CA
USA
Arindam Kundagrami
University of Massachusetts
Polymer Science and Engineering
Department
Amherst, MA
USA
Arkady I. Leonov
The University of Akron

Department of Polymer Engineering
Akron, OH 44325-0301
USA
Galina Litvinenko
Karpov Institute of Physical Chemistry
Vorontsovo Pole 10
Moscow 105064
Russia
Vlasis Mavrantzas
University of Patras and FORTH-ICE/
HT
Department of Chemical Engineering
Patras GR 26504
Greece
Evan Mitsoulis
National Technical University of Athens
School of Mining Engineering &
Metallurgy
Zografou
157 80 Athens
Greece
Marcus Müller
Georg-August-Universität
Institut für Theoretische Physik
37077 Göttingen
Germany
Murugappan Muthukumar
University of Massachusetts
Polymer Science and Engineering
Department

Amherst, MA
USA
Hans Christian Öttinger
Polymer Physics, ETH Zürich
Department of Materials
CH-8093 Zürich
Switzerland
Ras B. Pandey
University of Southern Mississippi
Department of Physics and Astronomy
Hattiesburg, MS 39406
USA
Soumya S. Patnaik
Thermal and Electrochemical Branch
Energy Power Thermal Devision
Propulsion Directorate
Air Force Research Laboratory
Dayton, OH 45433
USA
XX List of Contributors
Michael J. Poldneff
External Science & Technology
Programs
The Goodyear Tire & Rubber Company
Innovation Center, P.O. Box 3531
D/410A
Akron, OH 44309-3531
USA
Holger Stark
Technische Universität Berlin

Institut für Theoretische Physik
Hardenbergstr. 36
10623 Berlin
Germany
List of Contributors XXI

1
Computational Viscoelastic Fluid Mechanics and Numerical
Studies of Turbulent Flows of Dilute Polymer Solutions
Antony N. Beris and Kostas D. Housiadas
1.1
Introduction and Historical Perspective
According to the late A.B. Metzner [1], the phenomenon of polymer-induced drag
reduction was independently discoveredby two researchers, K.J. Mysels in May 1945,
as reported by the discoverer at an AIChE symposium on drag reduction in 1970, and
B.A. Toms in the summer of 1945, as reported by the discoverer at the IUTAM
symposium on the structure of turbulence and drag reduction held in Washington,
DC in 1976. The original fluids studied in these two first experimental investigations
were micellar aluminum disoaps or rubber in gasoline (K.J. Mysels) and polymethyl
methacrylate in monochlorobenzene (B.A. Toms). However, due to the war, the first
records in an accessible publication were found later [2, 3] with journal contributions
even much later [4, 5]. Since that time, the field has literally exploded with 500 papers
until the seminal review by Virk [6] and 4900 papers by 1995 [1].
The first attribution of drag reduction to fluid viscoelasticity was by Dodge and
Metzner [7], whereas the first description of drag reduction as Toms effect was by
Fabula at the Fourth International Congress on Rheology, held in 1996 [8]. The first
measurements of viscoelasticity of drag-reducing fluids were performed by Metzner
and Park [9] and Hershey and Zakin [10]. The first articulations of a maximum drag
reduction asymptote for dilute polymer solutions were reported by Castro and
Squire [11], Giles and Pettit [12], and Virk et al. [13]. As far as the first proposed

mechanisms of drag reduction due to fluid mechanicaleffectsareconcerned, Lumley
attributed drag reduction to molecular stretching in the radial flow patterns in
turbulent flows [14]. Simultaneously, Seyer and Metzner [15] clarified it even further,
as due to high extensional deformation rates in radial flow patterns in turbulent
flows and high resistance to stretching of viscoelastic fluids. More recently, Lumley
and Blossey [16] elaborated further by arguing that polymer additives, by boosting the
extensional viscosity of the fluid, affect especially the structure of the turbulent
bursts; see also Ref. [17]. This same mechanism is also suspected to be operative
when other additives are employed, such as micellar surfactant solutions [2, 4, 18–20]
j
1