Series in Soft Condensed Matter Vol.
UNDERSTANDING SOFT CONDENSED MATTER VIA MODELING AND COMPUTATION
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All living organisms consist of soft matter. For this reason alone, it is important to be able to understand and predict the structural and dynamical properties of soft materials such as polymers, surfactants, colloids, granular matter and liquids crystals. To achieve a better understanding of soft matter, three different approaches have to be integrated: experiment, theory and simulation. This book focuses on the third approach — but always in the context of the other two.
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UNDERSTANDING SOFT CONDENSED MATTER VIA MODELING AND COMPUTATION
Series in Soft Condensed Matter Vol.
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Wenbing Hu • An-Chang Shi
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UNDERSTANDING SOFT CONDENSED MATTER VIA MODELING AND COMPUTATION Editors
Wenbing Hu
Nanjing University, China
An-Chang Shi
McMaster University, Canada
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SERIES IN SOFT CONDENSED MATTER Founding Advisor: Pierre-Gilles de Gennes (1932–2007) Nobel Prize in Physics 1991 Collège de France Paris, France
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Polymer Thin Films edited by Ophelia K. C. Tsui and Thomas P. Russell
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Polymers, Liquids and Colloids in Electric Fields: Interfacial Instabilities, Orientation and Phase Transitions edited by Yoav Tsori and Ullrich Steiner
Vol. 3
Understanding Soft Condensed Matter via Modeling and Computation edited by Wenbing Hu and An-Chang Shi
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Foreword
The study of Soft Condensed Matter has stimulated fruitful interactions between physicists, chemists, and engineers, and is now reaching out to biologists. A broad interdisciplinary community involving all these areas of science has emerged over the last 30 years, and with it our knowledge of Soft Condensed Matter has grown considerably with the active investigations of polymers, supramolecular assemblies of designed organic molecules, liquid crystals, colloids, lyotropic systems, emulsions, biopolymers and biomembranes, among others. Taking into account that research in Soft Condensed Matter involves ideas coming from physics, chemistry, materials science as well as biology, this series may form a bridge between all these disciplines with the aim to provide a comprehensive and substantial understanding of a broad spectrum of phenomena relevant to Soft Condensed Matter. The present Book Series, initiated by the late Pierre-Gilles de Gennes, comprises independent book volumes that touch on a wide and diverse range of topics of current interest and importance, covering a large number of diverse aspects, both theoretical and experimental, in all areas of Soft Condensed Matter. These volumes will be edited books on advanced topics with contributions by various authors and monographs in a lighter style, written by experts in the corresponding areas. The Book Series mainly addresses graduate students and junior researchers as an introduction to new fields, but it should also be useful to experienced people who want to obtain a general idea on a certain topic or may consider a change of their field of research. This Book Series aims to provide a comprehensive and instructive overview of all Soft Condensed Matter phenomena. The present volume of this Book Series, edited by Wenbing Hu and An-Chang Shi, provides a thorough set of examples for the power of modeling and computer simulations of Soft Condensed Matter. Each chapter addresses a different technique or system, but as an ensemble, all contributions provide evidence on how simulation approaches can complement v
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theory and experiment, bridging the gap between atomistic models and coarse-grained ones. The diversity of the selected fields is representative to the large spectrum of Soft Condensed Matter. As special effort has been made to review important concepts and simulation techniques on an introductory level, we expect that this volume will serve as a useful guide to graduate students and young investigators who aim to get a broad overview on modeling approaches and simulation techniques used for Soft Condensed Matter. Within the next few years, our Series on Soft Condensed Matter will grow continuously and eventually cover the whole spectrum of phenomena in Soft Condensed Matter. We hope that many interested colleagues and scientists will participate and profit from these endeavours.
David Andelman and G¨ unter Reiter Series Editors
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Preface
Soft condensed matter physics is concerned with the study of soft materials. Examples of soft materials are polymers, bio-macromolecules, liquid crystals, surfactant solutions, colloidal suspensions, and lipids. The territory of these materials extends to plastics, rubber, textiles, pharmaceuticals, foodstuffs and biomaterials, which are important and useful materials for our daily life and high-tech industries. Furthermore, soft materials provide the ground for the birth of new physics based on the theme of softness, malleability and fragility. At the extreme, all living organisms can be regarded as soft matter. For these reasons, it is important to be able to understand and predict the structural and dynamical properties of soft materials. The present volume of the book series in soft condensed matter focuses on the modeling and simulation of soft matter, as an essential investigation often concerted with the efforts of experiment and theory. One of the advantages of computer simulations is that they allow us to vary controlling parameters of microscopic models and to visualize physical processes in detail. This makes the simulation approach particularly valuable in the study of complex physiochemical systems such as soft condensed matter. Although simulations of a given physiochemical system are ideally carried out on atomistic models, using this approach for macroscopic systems sets the demand for us to track a huge number of particles, of the order of 1023 or so, which is a formidable task even for current computational abilities. In order to overcome this difficulty, a variety of coarse-grained models have been developed. The tasks of modeling and simulation are two-fold: to develop model parameters in the coarse-grained models arising from atomistic models, and to understand structure and dynamics of soft matter from simulations of the coarse-grained models. Both tasks require the development of sophisticated simulation methodologies and algorithms. This book collects a set of mini-reviews on a number of topics in soft condensed matter physics, emphasizing the advantages of simulation vii
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approaches. Each mini-review demonstrates how a carefully chosen algorithm leads to the understanding of some key properties of soft matter. Because the book is aimed at non-specialists, topics in each chapter are placed in a broader context combining experimental and theoretical approaches while technical issues are discussed at an introductory level. The content of this book can be roughly divided into five parts, covering colloids, polymers, bio-macromolecules, liquid crystals and fluid droplets. Using these various soft matter systems as a platform, a number of simulation techniques are introduced, including molecular dynamics simulations, Monte Carlo simulations, self-consistent field theory, density functional theory, quantum chemistry modeling, and finite element method of fluid mechanics. The first part begins with a brief overview by Daan Frenkel on the role of computer simulations in the study of soft matter, in particular, of colloids. The rationale of this approach is based on the fact that the weakness of cohesive energies of soft matter allows the entropic forces to drive phase transitions. This intrinsic feature of soft matter can be captured by an extension of the classical density functional theory, which is introduced by Hartmut L¨ owen in the second chapter. The non-equilibrium problems of soft matter, like colloids, liquid crystals and polymer solutions can be tackled by the dynamic density functional theory derived from the Smoluchowski equation. The second part focuses on phase separation and crystallization in polymeric systems. Marcus M¨ uller first describes the modeling of liquid–liquid demixing in polymer blends and solutions, as well as simulations of interfacial properties, phase boundaries in confined geometries, and wetting. Then, Feng Qiu et al. present an introduction to the self-consistent-field theory (SCFT) for the study of the phase behavior of block copolymers, focusing on a new generic reciprocal-space method for the discovery of ordered block copolymer phases. Furthermore, Toshihiro Kawakatsu introduces the recent development of dynamic SCFT in the study of nonequilibrium behaviors of diblock copolymers with various geometries of microdomains. Turning to polymer crystallization, Takashi Yamamoto combines molecular dynamics and Monte Carlo simulations to study crystallization of helical polymers, using isotactic polypropylene as the model system to demonstrate the relationship between the molecular chirality and the symmetry of the lattice. Finally, Wenbing Hu presents a study of the interplay of liquid–liquid demixing and polymer crystallization in multi-component systems. Dynamic Monte Carlo simulations are performed
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according to thermodynamic designs of interplay based on a mean-field theory. The third part of the book focuses on modeling and simulation of biologically important macromolecules containing hydrophobic interactions, hydrogen bonding and electrostatics. Chwen-Yang Shew and Kenichi Yoshikawa make a comparison between experiments and modeling on collapse transition of a giant DNA. The effects of chain rigidity, charge density, inter- and intra-molecular segregation, and temperature have been extensively studied. Jing Ma et al. present techniques combining quantum and molecular mechanics for the modeling of hydrogen bonding in solvated long oligomers and cyclic peptides. Naji et al. discuss the unusual features of electrostatic interactions between macromolecules. In the fourth part, Rauzah Hashim presents a survey of computer simulations of liquid crystals. Five classes of modeling have been identified, followed with applications of variants and hybrid models derived from these classes to elucidate the nature of mesophases and their relationships with molecular geometries and intermolecular forces. In the last part, James J. Feng et al. introduce finite element computations on the basis of diffuse-interface theory, demonstrating unusual behavior of interfaces between complex fluids. Two examples are partial coalescence of a droplet on a planar interface, and the self-assembly of droplets suspended in a nematic liquid crystal. We expect that this book will be beneficial to those readers working in the interdisciplinary area of soft matter physics and biophysics with some background in physics, chemistry, biological sciences and materials sciences. We believe that the selected contributions on the whole reflect the diversity of recent research activities in soft condensed matter, and provide useful guidance to graduate students and young investigators who are starting their research career in modeling and simulation of soft matter.
Editors Wenbing Hu Nanjing, Jiangsu Province, China An-Chang Shi Hamilton, Ontario, Canada
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Contents
Foreword
v
Preface
vii
Introduction. Role of Modeling in Soft Matter Physics D. Frenkel 1.
Applications of Density Functional Theory in Soft Condensed Matter H. L¨ owen 1
2
3
4
Freezing of Spheres . . . . . . . . . . . . . . . . . . . . . . . 1.1 Phenomenological results . . . . . . . . . . . . . . . . 1.2 Independent treatment of the different phases . . . . 1.3 Unifying Microscopic theories . . . . . . . . . . . . . 1.4 Phase diagrams of simple potentials . . . . . . . . . . 1.5 Density Functional Theory (DFT) . . . . . . . . . . Brownian Dynamics . . . . . . . . . . . . . . . . . . . . . . 2.1 Brownian dynamics (BD) . . . . . . . . . . . . . . . 2.2 BD computer simulations . . . . . . . . . . . . . . . 2.3 Dynamical density functional theory (DDFT) . . . . 2.4 An example: Crystal growth at imposed nucleation clusters . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Hydrodynamic interactions . . . . . . . . . . . . . . Rod-Like Particlesx . . . . . . . . . . . . . . . . . . . . . . 3.1 Statistical mechanics of rod-like particles . . . . . . . 3.2 Simple models . . . . . . . . . . . . . . . . . . . . . . 3.3 Brownian dynamics of rod-like particles . . . . . . . 3.4 “Active” (self-propelled) Brownian particles . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1
9 11 11 12 13 14 17 23 23 27 27 29 31 33 33 38 40 42 42
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2.
Polymer Phase Separation M. M¨ uller
47
1 2
47 56 56
3
3.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Phase Behavior in the Bulk . . . . . . . . . . . . . . . . . . 2.1 Predictions of the mean-field theory . . . . . . . . . . 2.2 Estimating the Flory-Huggins parameter for simulation models . . . . . . . . . . . . . . . . . . . . 2.3 Simulation techniques for computing the bulk phase behavior . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Compressible mixtures . . . . . . . . . . . . . . . . . Outlook: Interfacial Properties, Phase Boundaries in Confined Geometry, and Wetting . . . . . . . . . . . . . . .
Self-Consistent Field Theory of Block Copolymers F. Qiu, A.-C. Shi and Y. Yang 1 2
3 4 5 4.
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Introduction . . . . . . . . . . . . . . . . . . . . . Self-Consistent Field Theory of Block Copolymers 2.1 Polymer model and partition function . . . 2.2 Chain propagators . . . . . . . . . . . . . . 2.3 Self-consistent mean-field theory . . . . . . Reciprocal-Space Formulation . . . . . . . . . . . . Applications of the Reciprocal-Space Method . . . Summary . . . . . . . . . . . . . . . . . . . . . . .
3
63 70 74 85
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. 85 . 90 . 90 . 92 . 93 . 95 . 100 . 101
Dynamic Self-Consistent Field Theories for Polymer Blends and Block Copolymers T. Kawakatsu 1 2
59
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Formalism for Dynamic SCF Theory . . . . . . . . . 2.1 Diffusion flux . . . . . . . . . . . . . . . . . . . . . . 2.2 Convection flux . . . . . . . . . . . . . . . . . . . . . 2.3 Flux induced by external fields . . . . . . . . . . . . Dynamic SCF Theories in Slow Diffusion Regime . . . . . . 3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . 3.2 Formation process of mesophases of block copolymer melt . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Dynamics and non-equilibrium domain structures in thin films and near solid surfaces . . . . . . . . . . .
105 105 107 108 110 111 112 112 113 115
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4
5 5.
. . 117 . . . . .
. . . . .
Molecular Dynamics in Crystallization of Helical Polymers: Crystal Ordering and Chirality Selection T. Yamamoto 1 2 3
4
5
6 6.
Structural phase transitions induced by external fields . . . . . . . . . . . . . . . . . . . . . . . . . Beyond Diffusion Dynamics — Hydrodynamics, Viscoelasticity and Hybrid Techniques . . . . . . . . . . 4.1 Hydrodynamic effects on domain formation . . . 4.2 Dynamic SCF with viscoelastic properties . . . . 4.3 Hybrid simulations with particles and fields . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Our Strategies for Simulating Crystallization in Helical Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular Models and Simulation Methods . . . . . . . . . 3.1 MD simulations . . . . . . . . . . . . . . . . . . . . . 3.2 MC simulation . . . . . . . . . . . . . . . . . . . . . Crystallization of the Bare Helix . . . . . . . . . . . . . . . 4.1 A primary nucleation of a single polymer in vacuo . 4.2 Crystallization of a single polymer on a growth front 4.3 Order-disorder transition and crystal chirality . . . . 4.4 Development of chiral crystal . . . . . . . . . . . . . Simulations for iPP, a Helical Polymer with Side Groups . . 5.1 Collapsing of a single iPP chain in vacuo . . . . . . . 5.2 Crystallization of a single chain with definite chiral recognition . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Crystallization and polymorph selection . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .
Interplay of Liquid-Liquid Demixing and Polymer Crystallization W. Hu 1 2 3 4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . Simulation Techniques . . . . . . . . . . . . . . . . . . . . . Results and Discussion . . . . . . . . . . . . . . . . . . . . . 4.1 Liquid-liquid demixing enhanced by crystallizability . 4.2 Crystal nucleation enhanced by prior L-L demixing .
121 121 124 127 129
133 134 136 138 138 140 142 142 147 150 156 163 164 166 170 175
179 179 182 184 186 186 188
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4.3
5 7.
Elucidation of Single Molecular Observation of a Giant DNA C.-Y. Shew and K. Yoshikawa
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1 2
208
3
4
5
6
7
8
8.
Crystal nucleation enhanced by prior L-L demixing in the single-chain systems . . . . . . . . . . . . . . . . 199 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Polymer Physics Aspects of DNA Conformation . . . . . . Manipulation and Measurement DNA Conformation In Vitro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Condensing agents . . . . . . . . . . . . . . . . . . . 2.2 Single molecular images . . . . . . . . . . . . . . . . 2.3 Limitation of traditional light scattering . . . . . . . All-or-none Conformation Transition of DNA . . . . . . . . 3.1 Discrete conformational transition of DNA . . . . . . 3.2 Chain stiffness and discrete conformational transition . . . . . . . . . . . . . . . . . . . . . . . . Dynamics of Conformational Relaxation . . . . . . . . . . . 4.1 Time dependent conformational behavior . . . . . . . 4.2 Folding and unfolding kinetics . . . . . . . . . . . . . Conformational Hysteresis . . . . . . . . . . . . . . . . . . . 5.1 Characterization of hysteresis . . . . . . . . . . . . . 5.2 Thermodynamics in conformational hysteresis . . . . 5.3 Hysteresis under mechanical forces . . . . . . . . . . Effect of Charge on DNA Conformation . . . . . . . . . . . 6.1 Conditions to induce intramolecular segregation . . . 6.2 Polyelectrolyte analogy . . . . . . . . . . . . . . . . . 6.3 Phase diagram of intermolecular and intramolecular segregation . . . . . . . . . . . . . . . . . . . . . . . Temperature Effect of DNA Conformation . . . . . . . . . . 7.1 Temperature induced conformational change . . . . . 7.2 Competition of smaller ions on compaction . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . .
Theoretical Modeling of Hydrogen Bonding in Macromolecular Solutions: The Combination of Quantum Mechanics and Molecular Mechanics J. Ma, N. Jiang and H. Li 1
209 209 210 212 212 212 213 214 215 216 218 218 219 221 224 224 224 225 228 228 230 230 233
237
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 238
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3
4 9.
Fragmentation-Based QM/MM Simulations . . . . . . . . . 2.1 Solvent models . . . . . . . . . . . . . . . . . . . . . 2.2 Energy-based fragmentation QM . . . . . . . . . . . 2.3 Fragmentation QM/MM: Basic idea and formalisms 2.4 Fragmentation QM/MM simulations on poly(ethylene oxide) polyethylene . . . . . . . . . . . Simulations of Solvated Peptides Using Polarizable Force Field Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Fragmentation-based polarization model . . . . . . . 3.2 Configurations of solvated α-conotoxin GI and its analogues . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . .
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239 239 240 241 245 248 249 251 255
Exotic Electrostatics: Unusual Features of Electrostatic Interactions between Macroions A. Naji, M. Kanduˇc, R. R. Netz and R. Podgornik
265
1 2
266 268
3 4
5
6
7
8
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Scenery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Colloids, polymers and membranes: The mesoscopic scale . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Charges: From industry to biology . . . . . . . . . . 2.3 Theoretical challenge and coarse-grained models . . . Length Scales in a Classical Charged System . . . . . . . . From Mean-Field to Strong Coupling Regime . . . . . . . . 4.1 Weak coupling or mean-field regime . . . . . . . . . . 4.2 Strong coupling regime . . . . . . . . . . . . . . . . . Interactions between Like-Charged Surfaces . . . . . . . . . 5.1 WC regime: Repulsion . . . . . . . . . . . . . . . . . 5.2 SC regime: Attraction . . . . . . . . . . . . . . . . . Counterions with Salt . . . . . . . . . . . . . . . . . . . . . 6.1 Functional integral formalism . . . . . . . . . . . . . 6.2 Dressed counterions . . . . . . . . . . . . . . . . . . . 6.3 WC dressed counterion theory . . . . . . . . . . . . . 6.4 SC dressed counterion theory . . . . . . . . . . . . . Counterions between Randomly Charged Surfaces . . . . . 7.1 General formalism: The replica method . . . . . . . . 7.2 Disorder effects in the WC regime . . . . . . . . . . . 7.3 Disorder effects in the SC regime . . . . . . . . . . . Lessons . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
268 269 271 272 273 273 274 276 277 278 279 279 281 282 282 286 287 288 288 291
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10. Computer Modeling of Liquid Crystals R. Hashim 1
2
3
297
Introduction . . . . . . . . . . . . . . . . . . . . . . 1.1 What is a liquid crystal? . . . . . . . . . . . . 1.2 Theoretical approach to understanding liquid crystals . . . . . . . . . . . . . . . . . . . . . Introduction to the Computer Simulation . . . . . . 2.1 Computer simulation techniques . . . . . . . . 2.2 Simulation limitations . . . . . . . . . . . . . Liquid Crystal Models . . . . . . . . . . . . . . . . . 3.1 The lattice class . . . . . . . . . . . . . . . . . 3.2 The Gay-Berne class . . . . . . . . . . . . . . 3.3 Full atomistic class . . . . . . . . . . . . . . . 3.4 Conclusion . . . . . . . . . . . . . . . . . . . .
. . . . 297 . . . . 298 . . . . . . . . .
. . . . . . . . .
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. . . . . . . . .
11. Drop Dynamics in Complex Fluids J. J. Feng, X. Chen, P. Yue and C. Zhou 1 2
3
4 Index
Introduction . . . . . . . . . . . . . . . . . . . . . Partial Coalescence in Polymer Solutions . . . . . 2.1 Experimental observations . . . . . . . . . . 2.2 Numerical simulations . . . . . . . . . . . . Droplet Self-Assembly in Nematic Liquid Crystals 3.1 Experimental observations . . . . . . . . . . 3.2 Numerical simulations . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . .
301 304 305 306 307 308 317 325 331 339
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340 343 343 346 352 352 355 360 365
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Introduction Role of Modeling in Soft Matter Physics
Daan Frenkel Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, UK
Writing an introduction to a book on Modeling Soft Matter is difficult because this is a field that arguably has links with more other disciplines than any other branch of physics or chemistry. In fact, the contents of this book illustrate this and the subjects covered range from polymer science and quantum mechanics to hydrodynamics and biophysics. There are, however, certain unifying features in soft-matter science. The first one is, not surprisingly, that soft matter is “soft”. In practice, this means that soft materials have elastic moduli that are a million to a billion times smaller than that of, say, steel. This softness can be understood on the basis of dimensional arguments. Elastic moduli have the dimension energy/length3 . In hard (covalently bonded) materials, the relevant energy scale is that of the chemical bond, which is typically expressed in electronvolts. The characteristic length scale is the interparticle spacing in a solid. This distance is of the order of a few ˚ A. Hence, typical elastic constants of such materials are of the order of 109 Pa. Now soft matter. Here the 1 relevant energy is the thermal energy, kB T, which is about 40 eV. The typical distances span a wide range: from several nanometers to microns. As a consequence, typical elastic constants should have values between 1 and 105 Pa. For modeling, the above observation is of crucial importance. First of all, the fact that these materials consist of building blocks that have linear dimensions that may be up to 10,000 times larger than an atom, means that fully atomistic modeling is almost always hopeless. A single colloid may contain some 1010 atoms. A colloidal suspension of only 100 colloids in a solvent will contain 1012 particles. Clearly, atomistic modeling is not an 1
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option. The only exception — and a partial one at that — is the behavior of particles or structures at the very lower end of this range of length scales: e.g. small mesogens or proteins. But even there it is rare to see simulations that contain more than a few proteins in water. The atomistic simulation of an entire living cell is, for the foreseeable future, utterly impossible. There is a second case where atomistic simulations of soft matter may work, namely in cases where the local properties of the macro-molecular units are important. For instance, one can study the structure of a polymer crystal with atomistic simulations, precisely because the molecules are so large: in the crystal there are few free end points and, as a consequence, the local structure does not “feel” the length of the building blocks, only their lateral extent. However, in the overwhelming majority of cases soft matter has to be modeled using a coarse-graining approach: polymers are modeled as chains of self-avoiding Kuhn segments and colloids are modeled as spheres (or rods, or platelets), to name but two examples. The energy-scale is equally important: it is the thermal energy and hence thermal fluctuations are always important in soft materials. This is where the science of soft matter starts to diverge from that of, say, granular materials. In the latter, the length scales are large (larger even than for soft materials) but thermal fluctuations play no role. Where is the boundary between colloids and granular materials? The behavior of colloids is governed by the laws of statistical mechanics. In equilibrium, colloidal suspensions occur in the phase with the lowest free energy, and the dynamics of colloids in equilibrium is due to thermal (“Brownian”) motion. In principle, this should also be true for grains of sand. But in practice, it is not. In order for grains to behave like colloids, they should be able to evolve due to Brownian motion. There are two reasons why grains of sand do not. First of all, on earth, all particles are subject to gravity. The probability to find a particle of mass m at a height h above the surface of the earth is given by the barometric height distribution P (h) = exp(−mgh/kB T ) ,
(1)
where m is the effective mass of the particle (in the case of a colloid: the mass of the particle minus the mass of the displaced solvent), T is the temperature and kB is Boltzmann’s constant. The average height of the particle above the bottom of the container is equal to < h >= kB T /(mg). For a 1 milligram grain of sand at room temperature, < h >= O(10−14 ) cm. This tells us something that we all know: the beach is not seething with thermal motion. One way to delimit the colloidal regime is to require that
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< h > is larger than the particle diameter. Suppose we have a spherical particle with diameter σ and (excess) mass density ρ, then our criterion implies π gρσ 4 = kB T . (2) 6 For a particle with an excess density of 1 g/cm3 , the above equality is satisfied for σ ≈ 1 µm. So, on earth anything larger than a few microns is dominated by gravity, not by thermal motion: grains of sand are not colloids. Naively one might think that in space, much larger particles could behave like colloids. Yet, even if we would launch a grain of sand in a spaceshuttle, that grain would still not behave like a colloid. The reason is the following: colloidal particles should be able to move due to diffusion (i.e. thermal motion). How long does it take for a particle to move a distance equal to its own diameter? In a time t, a particle typically diffuses a distance √ 2Dt. For a spherical particle, the diffusion constant is given by the StokesEinstein relation D = kT /(3πησ), where η is the viscosity of the solution. Hence, a particle diffuses a distance comparable to its own diameter in a time τ = O(ησ 3 /kT ) .
(3)
For a 1µm-colloid in water, this time is of the order of one second. For a 1 mm grain of sand, it is of the order of 30 years. Hence, even though grains of sand in zero-gravity may behave like colloids, they will not do so on a human time-scale. Clearly, what we define as a colloid, also depends on the observation time. Again, 1 micron comes out as a natural upper limit to the colloidal domain. Although soft materials all have length scales within a certain range, their properties may still be very different. This is particularly clear when we consider the role of entropy in the phase behavior of soft matter. As I will briefly discuss below, entropy is extremely important (often even dominant) in driving phase transitions in colloidal systems. However, in polymer mixtures, entropy is much less important. So much so, that a mixture of deuterated polymer and hydrogenated polymers can (under suitable conditions) demix spontaneously: the entropy of mixing is too small to compensate for the tiny difference in interaction energies of H and D polymers. 1 The reason is that the interaction energy of a polymer of type A in a melt of type B is proportional to the number of atoms in that polymer. Yet the mixing entropy NA polymers of type A in a melt of type B is proportional to NA , the number of polymers. Clearly, the number of polymers is much
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smaller than the number of atoms. In “hard matter” physics, the 3 He-4 He mixture is the only example of isotopes demixing — and that only at very low temperatures. Now let us consider the role of entropy in colloidal phase transitions. The second law of thermodynamics tells us that any spontaneous change in a closed system results in an increase of the entropy, S. In this sense, all spontaneous transformations of one phase into another are entropy driven. However, this is not what I mean with the term “entropy-driven phase transitions”. When discussing ordering transitions one usually considers a system that can exchange energy with its surroundings. In that case, the second law of thermodynamics implies that the system will tend to minimize its Helmholtz free energy F = E − T S, where E is the internal energy of the system and T the temperature. Clearly, a system at constant temperature can lower its free energy in two ways: either by increasing the entropy S, or by decreasing the internal energy E. In order to gain a better understanding of the factors that influence phase transitions, we must look at the statistical mechanical expressions for entropy. The simplest starting point is to use Boltzmann’s expression for the entropy of an isolated system of N particles in volume V at an energy E, S = kB ln Ω
(4)
where kB , the Boltzmann constant, is simply a constant of proportionality. Ω is the total number of (quantum) states that is accessible to the system. In the remainder of this paper, I shall choose my units such that kB =1. The usual interpretation of Eqn. 4 is that Ω, the number of accessible states of a system, is a measure for the “disorder” in that system. The larger the disorder, the larger the entropy. This interpretation of entropy suggests that a phase transition from a disordered to a more ordered phase can only take place if the loss in entropy is compensated by the decrease in internal energy. This statement is completely correct, provided that we use Eqn. 4 to define the amount of disorder in a system. However, we also have an intuitive idea of order and disorder: we consider crystalline solids “ordered”, and isotropic liquids “disordered”. This intuitive picture suggests that a spontaneous phase transition from the fluid to the crystalline state can only take place if the freezing lowers the internal energy of the system sufficiently to outweigh the loss in entropy: i.e. the ordering transition is “energy driven”. In many cases, this is precisely what happens. It would, however, be a mistake to assume that our intuitive definition of order always coincides with the one based on Eqn. 4.
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In order to observe “pure” entropic phase transitions, we should consider systems for which the internal energy is a function of the temperature, but not of the density. Using elementary statistical mechanics, it is easy to show that this condition is satisfied for a system of hard particles. Suspensions of (uncharged, refractive-index-matched) colloids behave effectively as systems of hard particles. Whenever these systems order at a fixed density and temperature, they can only do so by increasing their entropy (because, at constant temperature, their internal energy is fixed). Such systems are conveniently studied in computer simulations. But, increasingly, experimentalists — in particular, colloid scientists, have succeeded in making real systems that behave very nearly as ideal hard-core systems. Hence, the phase transitions discussed below can, and in many cases, do occur in nature. Below I list a few examples of entropic ordering in hard-core systems. But I stress that the list is far from complete. The earliest example of an entropy-driven ordering transition is described in a classic paper of Onsager,2 on the isotropic-nematic transition in a (three-dimensional) system of thin hard rods. Onsager showed that, on compression, a fluid of thin hard rods of length L and diameter D must undergo a transition from the isotropic fluid phase, where the molecules are translationally and orientationally disordered, to the nematic phase. In the latter phase, the molecules are translationally disordered, but their orientations are, on average, aligned. This transition takes place at a density such that (N/V )L2 D = O(1). Onsager considered the limit L/D → ∞. In this case, the phase transition of the hard-rod model can be found exactly (see e.g. Ref. 3). At first sight it may seem strange that the hard rod system can increase its entropy by going from a disordered fluid phase to an orientationally ordered phase. Indeed, due to the orientational ordering of the system, the orientational entropy of the system decreases. However, this loss in entropy is more than offset by the increase in translational entropy of the system: the available space for any one rod increases as the rods become more aligned. In fact, we shall see this mechanism returning time-and-again in ordering transitions of hard-core systems: the entropy decreases because the density is no longer uniform in orientation or position, but the entropy increases because the free-volume per particle is larger in the ordered than in the disordered phase. The most famous, and for a long time controversial, example of an entropy-driven ordering transition is the freezing transition in a system of hard spheres. This transition had been predicted by Kirkwood in the early fifties4 on basis of an approximate theoretical description of the hard-
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sphere model. As this prediction was quite counter-intuitive and not based on any rigorous theoretical results, it met with wide-spread skepticism until Alder and Wainwright5 and Wood and Jacobson6 performed numerical simulations of the hard-sphere system that showed direct evidence for this freezing transition. Even then, the acceptance of the idea that freezing could be an entropy driven transition, came only slowly.7 However, by now, the idea that hard spheres undergo a first-order freezing transition is generally accepted. And, although the hard-sphere model was originally devised as an idealized and highly unrealistic model of an atomic fluid, it is now realized that this model provides a good description of certain classes of colloidal systems. The next step in this sequence came in the mid-eighties when computer simulations8 showed that hard-core interactions alone could also explain the formation of more complex liquid crystals. In particular, it was found that a system of hard sphero-cylinders (i.e. cylinders with hemi-spherical caps) can form a smectic liquid crystal, in addition to the isotropic liquid, the nematic phase and the crystalline solid. In the smectic (A) phase, the molecules are orientationally ordered but, in addition, the translational symmetry is broken: the system exhibits a one-dimensional density-modulation. Subsequently, it was found that some hard-core models could also exhibit columnar ordering.9 In the latter case, the molecules assemble in liquidlike stacks, but these stacks order to form a two-dimensional crystal. In summary, hard-core interaction can induce orientational ordering and one-, two- and three-dimensional positional ordering. This is rather surprising because, in particular for the smectic and the columnar phase, it was generally believed that their formation required specific energetic interactions. A recent addition to the list of entropic phase transitions is, interestingly, the liquid-vapor transition. Since the seminal work of van der Waals, the standard picture of the liquid-vapor transition is that it is due to the competition between entropic repulsion (the gas molecules strive to maximize the volume that they can access) and energetic attraction (due to the so-called van der Waals dispersion forces). Hence, the condensation of a liquid seems to be the prototypical example of an energy-driven phase transition: the system “pays” for the loss in entropy by delivering the heat of condensation to the surroundings where it will increase the entropy. However, in soft matter things are, again, different. This is in particular true in the case of colloidal particles coated with very long polymers (e.g. DNA) that may form bridges between different particles. If the binding is very strong, there are no “dangling” polymers. For instance, in the case of hard
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colloids linked by DNA, the low density phase consists of colloidal “dimers” linked by a number of DNA strands.10 As there are no dangling bonds, the energy of this system is independent of density. Yet, beyond a certain density, the system undergoes a transition from a dilute vapor phase to a dense liquid phase. The reason is that there are many more ways of arranging the bonds in a dense phase than in the gas phase. Hence, the entropy of the network drives the condensation. I have mentioned these examples to illustrate that modeling soft materials is an exciting activity. Often, we observe/predict phenomena that have no counterpart in “hard matter”. These unusual properties of soft materials are not simply of academic interest. Increasingly, complex self-assembling materials are being produced with properties that rely on the “soft” nature of their building blocks. In fact, all living matter is an illustration of the amazing diversity of structures that can be formed by soft building blocks. In order to unravel the underlying design principles, simulation still has a long way to go. The chapters in this book discuss different examples of progress in this direction. The editors, Professors Wenbing Hu and An-Chang Shi have succeeded in collecting a number of very interesting aspects of soft-matter modeling. As the reader will see, in all its diversity, the book shows the underlying unity of the subject. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
F. S. Bates and P. Wiltzius, J. Chem. Phys. 91, 3258 (1989). L. Onsager, Proc. NY. Acad. Sci. 51, 627 (1949). R. F. Kayser and H. J. Raveche, Phys. Rev. A17, 2067 (1978). J. E. Kirkwood, in Phase Transformations in Solids, R. Smoluchowski, J. E. Mayer and W. A. Weyl (eds.), Wiley, New York (1951), p. 67. B. J. Alder and T. E. Wainwright, J. Chem. Phys. 27, 1208 (1957). W. W. Wood and J. D. Jacobson, J. Chem. Phys. 27, 1207 (1957). The Many-Body Problem, J. K. Percus, editor, Interscience, New York, 1963. D. Frenkel, H. N. W. Lekkerkerker and A. Stroobants, Nature 332, 822 (1988). J. A. C. Veerman and D. Frenkel, Phys. Rev. A45, 5633 (1992). B. Bozorgui and D. Frenkel, Phys. Rev. Lett. 101, 045701 (2008).
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Chapter 1 Applications of Density Functional Theory in Soft Condensed Matter Hartmut L¨ owen Heinrich-Heine University D¨ usseldorf, Germany Applications of classical density functional theory (DFT) to soft matter systems like colloids, liquid crystals and polymer solutions are discussed with a focus on the freezing transition and on nonequilibrium Brownian dynamics. First, after a brief reminder of equilibrium density functional theory, DFT is applied to the freezing transition of liquids into crystalline lattices. In particular, spherical particles with radially symmetric pair potentials will be treated (like hard spheres, the classical one-component plasma or Gaussian-core particles). Second, the DFT will be generalized towards Brownian dynamics in order to tackle nonequilibrium problems. After a general introduction to Brownian dynamics using the complementary Smoluchowski and Langevin pictures appropriate for the dynamics of colloidal suspensions, the dynamical density functional theory (DDFT) will be derived from the Smoluchowski equation. This will be done first for spherical particles (e.g. hard spheres or Gaussian-cores) without hydrodynamic interactions. Then we show how to incorporate hydrodynamic interactions between the colloidal particles into the DDFT framework and compare to Brownian dynamics computer simulations. Third orientational degrees of freedom (rod-like particles) will be considered as well. In the latter case, the stability of intermediate liquid crystalline phases (isotropic, nematic, smectic-A, plastic crystals etc) can be predicted. Finally, the corresponding dynamical extension of density functional theory towards orientational degrees of freedom is proposed and the collective behaviour of “active” (self-propelled) Brownian particles is briefly discussed.
Introduction Apart from their fundamental importance, soft matter materials (as colloidal dispersions, liquid crystals and polymer coils) represent excellent 9
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realizations of strongly interacting classical many-body systems. In this sense, soft matter provides us with model systems which in turn means that a reasonable (coarse-grained) modelling of soft matter is very realistic and can lead to valuable physical insight. This is one of the main reasons why theory, computer simulations and experiments go hand-in-hand in recent soft matter research. A classical many-body system is typically characterized by “effective” pairwise interaction forces [1], derived from an (optimal) effective pair potential V (r) where r denotes the interparticle distance [2]. Lots of theoretical and simulational efforts have been spent in the past to derive or to compute the effective interaction. Subsequently, once the effective pair interaction is known, equilibrium properties like structural correlations and phase transitions can be obtained by theory and computer simulation. This route can also be inversed: If for a particular shape of V (r) novel behaviour is predicted by theory or computer simulation, various soft matter systems (in particular mixtures) can be exploited to find a convenient realization of the prescribed potential V (r). While the concept of effective interactions is valid for static equilibrium quantities, it typically breaks down for dynamical correlations [3] and nonequilibrium situations [4]. The dynamics of colloidal particles and polymers in solution is Brownian due to the separation of time scales between the mesoscopic particles and the solvent molecules. The overdamped particle motion can be described using the Smoluchowski equation or the stochastically equivalent Langevin picture [5, 6]. A wealth of nonequilibrium effects have been discovered in these Brownian systems [7, 8]. In this book chapter we highlight the role of density functional theory in order to calculate phase diagrams in equilibrium and Brownian dynamics in nonequilibrium. Density functional theory represents a microscopic approach to many-body effects where the pair interaction potential V (r) is the only input. Computer simulations, on the other hand, are necessary to provide “benchmark” data to test the theory. For the topics considered here, these are mainly Monte-Carlo simulations for the equilibrium phase diagrams and Brownian dynamics simulations for nonequilibrium dynamics. In this contribution, we focus in particular on the freezing transition of simple systems, governed by a radially-symmetric pair potential (e.g. hard or soft spheres) and on orientational degrees of freedom relevant for e.g. rod-like systems. For both cases, phase diagrams and effects of nonequilibrium Brownian dynamics are described. Some examples from recent research in this field are used to illustrate the capacity of dynamical
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density functional theory as compared to Brownian dynamics computer simulations [9]. 1. Freezing of Spheres 1.1. Phenomenological results Experiments show that liquids freeze into periodic crystalline structures at low temperatures or high densities. In these states, the translational symmetry of the system is broken, i.e. the one-particle density N ρ(r) = δ(r − ri ) (1) i=1
is inhomogeneous. Here . . . denotes a canonical average, and ri (i = 1, . . . , N ) are the particle positions. As freezing is ubiquitous and general, it is one of the most important phase transition in nature. The basic question is: when does it happen? Answering this question is one of the central tasks of statistical physics. A full microscopic theory is highly desirable which uses the interaction forces as an input and predicts the thermodynamic conditions for freezing as an output. Since freezing is a collective effect, this is a very demanding task. Before turning to such a microscopic approach, let us first collect some empirical facts for freezing, as for more details and references, see [10–15]. We shall also summarize known phase behaviour for simple model potentials gained by computer simulations. i) Lindemann-criterion of melting If a = ρ−1/3 denotes a typical interparticle spacing (with the number density ρ being the number of particles per volume), then one can examine the root mean-square displacement u of a particle around a given crystalline i , i denoting a lattice site index, which is defined as lattice position R i )2 ri − R (2) u = ( This quantity can also be viewed as the spread of the inhomogeneous oneparticle density ρ(r) around a lattice position, see Figure 1. The Lindemann parameter L = u/a measures the fluctuations around the lattice positions in terms of the lattice constant. The traditional Lindemann rule of melting states that a solid melts if L ≈ 0.1. Computer simulations have confirmed this phenomenolgical rule where the actual value
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ρ(r)
i
L
R1
R2
r
Fig. 1. One-dimensional sketch of the inhomogeneous one-particle density ρ( r ) in a 1 and R 2 . The spread of the density peak is crystalline solid with lattice points at R embodied in the Lindemann parameter L.
of L at melting varies between 0.129 for hard spheres and 0.18 for the onecomponent plasma. But it is always roughly one order of magnitude smaller than the lattice constant itself. ii) Hansen-Verlet rule of freezing Different to the Lindemann rule, the Hansen-Verlet rule starts from the liquid side of the freezing transition and states that the freezing occurs if the amplitude of the first peak in the liquid structure factor S(k) exceeds 2.85. Originally found for Lennard-Jones systems, this rule has been confirmed also for other interactions like hard-sphere, plasma and Yukawa pair potentials. Without any notable exception a value of 2.5-3.0 was found near equilibrium freezing. However, the peak can be much higher in a metastable glassy state. 1.2. Independent treatment of the different phases The simplest theoretical approach is to construct different theories for the different thermodynamic states (solid and liquid). In particular the internal energy of the solid phase can be accessed by a simple lattice sum of the given pair potentials. In particular, different candidate lattices can be assumed at fixed averaged density, and the one with minimal potential energy will be the stable one for temperature T = 0. Finite temperature corrections based on a harmonic phonon-picture can be added on top of that resulting in a (Helmholtz) free energy F of the solid state. Likewise the free energy of the liquid can be gained by using for instance liquid integral equation theories [16] where different closure schemes may be adopted. Combined with the free energy of the solid, a Mawell double-tangent construction for the isothermal free energy per particles
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F/N
ρe
ρs
ρ
Fig. 2. Sketch of the Maxwell double tangent construction to the free energy per particle in the liquid and solid phase resulting in the two coexistence densities ρ and ρs .
versus density ρ leads to the coexisting liquid and solid densities ρ and ρs , see Figure 2. The double tangent ensures the equality of the chemical potential and the pressure in the two phases. If this is repeated for various temperature, the full phase diagram emerges. In three spatial dimensions, freezing is typically a first order transition with a considerable density jump ∆ρ = ρs − ρ .
1.3. Unifying Microscopic theories Both from a fundamental and esthetic point of view, a unifying theory which treats both the liquid and the solid phase on the same footing is desirable. In the past decades, there have been considerable advances in this field. In three spatial dimensions, classical density functional theory (DFT) can be used to get a liquid-based description of the solid phase. Here the solid is viewed as a strongly inhomogeneous liquid with strong density peaks. Freezing in DFT is therefore a condensation of liquid density modes [14]. Conversely, in two spatial dimensions the Kosterlitz-Thouless approach is a solid-based approach which treats the liquid as a solid phase with an accumulation of defects. In the following we shall focus on the three-dimensional freezing and on density functional theory. We emphasize that a unifying treatment is mandatory for the description of solid-liquid interface and phenomena like crystal nucleation and growth out of an undercooled melt where indeed a single theory for both phases is needed.
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1.4. Phase diagrams of simple potentials Let us first summarize some familiar phase diagrams for various model pairwise interactions. These were obtained mainly by “exact” computer simulation of a many-body system [17] and therefore provide “benchmark” data for a microscopic theory. a) Hard spheres The simplest nontrivial interaction potential is that for hard spheres of diameter σ. The potential reads ∞, r ≤ σ V (r) = (3) 0, r>σ The internal energy is completely ideal U = 32 N kB T , i.e. the averaged potential energy is zero. Hence Helmholtz free energy F = U − T S scales with kB T alone (as kB T is the only energy scale for hard spheres). Therefore, for hard spheres, the entire thermodynamic behaviour is governed by entropy alone. This becomes different for other interactions which possess an explicit energy scale. This is the main reason why hard spheres are the most important models for freezing. From computer simulations, the hard sphere phase diagram is shown in Figure 3. The only parameter is the density which is conveniently scaled in terms of a volume or packing fraction η = πρσ3 /6. The quantity η measures the ratio of the volume occupied by all spheres to the total available volume V of the system. For η → 0 an ideal gas is recovered, while maximal packing fo hard spheres occurs for η = ηcp = 0.74 corresponding to stacked layers of triangular crystals. In fcc-crystal (ABC stacking)
fluid 0
ηfluid =0.494
fluid
ηsolid =0.545
ηCP =0.74
η
crystal
Fig. 3. Hard sphere freezing diagram versus packing fraction η. The intuitive picture of freezing is also shown: at high densities a fluid state involves blocked configurations and more configurations are achieved by a periodic packing.
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between, there is a first order freezing transition with coexisting packing fractions ηell = 0.494 and ηs = 0, 545. The stable crystalline crystal is facecentred-cubic (fcc) which has an ABC stacking sequence. Interestingly, the freezing transition is driven by entropy. For η > ηs the solid state has a higher entropy than the fluid state clearly showing that entropy has nothing to do with structural order. More intuitively, a disordered fluid state at high densities implies jammed configurations, and much more configuration (i.e. higher entropy) is gained by taking as a reference configuration a solid and generating more configurations from slightly displaced particles (configurational entropy). b) Plasma The one-component plasma (OCP) with neutralizing background is defined by the pairwise Coulomb potential V (r) = V0 /r. By scaling the classical partition function, one can show that only the coupling param√ 3
4πρ
V
eter Γ = kB3T 0 determines the structure and phase behaviour. There is isochoric freezing from the fluid into a body-centered-crystal (bcc) at Γ = 178. c) Soft spheres Inverse power law potentials where V (r) = V0 (σ/r)n interpolate between the plasma (n = 1) and the hard sphere potential, formally obtained as n → ∞. Depending on n either bcc or fcc crystals are stable. d) Yukawa-system The Yukawa potential V (r) = V0 exp(−κr)/r applies e.g. to chargestabilized colloidal suspensions. Again κ interpolates between the OCP (κ = 0) and the hard-sphere-limit κ → ∞. The phase diagram depends λ 1 only on λ = κa (a = ρ− 3 ) and T˜ = kVB0T eλ and involves a fluid, a bcc solid and an fcc solid with a triple point as sketched in Figure 4. e) Lennard-Jones-system The Lennard-Jones the traditional model for rare gases, is potential, σ 6 σ 12 where is the energy and σ is the length given by V (r) = 4 r − r scale. This potential has a long-ranged attractive tail. Correspondingly it exhibits also a critical point separating a gas from a liquid and a triple point with gas-liquid-fcc solid coexistence.
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T
triple point 0.2 fluid fluid
fcc bcc bcc
0 plasma
2
4
6
λ
Fig. 4. Sketch of the Yukawa phase diagram in the plane spanned by λ and T˜. The special case λ = 0 is the one-component plasma.
f) Sticky hard spheres Sticky hard sphere possess a square-well attraction and are reasonable models for proteins. Here ∞ r ≤ σ V (r) = − σ ≤ r ≤ σ(1 + δ) (4) 0 elsewhere with a finite attraction range δσ and an attraction depth of − . The scaled range δ must be larger than about 0.25 in order to get a liquid-gas separation. For small δ < 0.05 there is a novel isostructural solid-solid-transition with a critical point [18]. g) Ultrasoft interactions Soft (floppy) objects like polymer coils have effective interactions which are even softer than the plasma, therefore these interactions are called “ultrasoft” [19]. A log-Yukawa interaction has been proposed for star polymers. Here the interaction diverges at the origin only logarithmically with distance, i.e. V (r) ∝ kB T ln(r/σ). The phase behaviour [20] involves fluid, bcc and fcc solids as well as body-centered-orthogonal and diamond lattices and exhibits reentrance effects.
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h) Penetrable interactions One may even describe soft objects with pair potentials which are finite at the origin. Examples are Gaussian potentials which are a good model for linear polymer coils. The phase behaviour involves again the fluid, bcc-solid and fcc-solid phase with fluid reentrance [21]. Penetrable interactions with other shapes exhibit again also “exotic” solid phases and reentrance [22]. Finite potentials which have a negative Fourier transform exhibit cluster crystals [23] where a lattice points is occupied by more than one particle. To summarize: (1) Hard and “harsh” potentials freeze into fcc lattices. (2) Soft repulsive potentials with an at least 1r singularity for r 0 freeze into bcc lattices. (3) Ultrasoft v(r) ∼ −ln( σr ) and penetrable (V (r 0) = V (0) < ∞) potentials show besides fcc and bcc structures, more open “exotic” lattices and reentrance effects. (4) If the Fourier transform of V (r) has negative parts, a cluster crystal occurs. (5) Attractions lead to gas-liquid coexistence and isostructural solid-solid transition. In conclusion, various shapes of the pairwise interaction potential can lead to a rich phase behaviour and there is the theoretical challenge to construct a microscopic approach in order to predict and reproduce this complex phase behaviour. As will be discussed in the sequel, classical density functional theory for inhomogeneous fluids does provide such an approach. 1.5. Density Functional Theory (DFT) a) Basics The cornerstone of density functional theory (DFT) is an existence theorem combined with a basic variational principle [10]. In detail, there exists a unique grand-canonical free energy-density-functional Ω(T, µ, [ρ]), which gets minimal for the equilibrium density ρ0 (r) and then coincides with the real grandcanoncial free energy, i.e. δΩ(T, µ, [ρ]) = 0. (5) δρ(r) ρ( r )=ρ0 ( r)
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In particular DFT is also valid for systems which are inhomogeneous on a microscopic scale. In principle, all fluctuations are included in an external potential which breaks all symmetries. For interacting systems in 3d, however, Ω(T, µ, [ρ]) is not known. Fortunately, there are few exceptions where the density functional is known exactly. First, for low density, the ideal-gas-limit is reached and the density functional can be constructed analytically (see below). Next leading orders for finite densities can be incorporated via a virial expansion which is quadratic in the densities. Conversely, in the high-density-limit, the meanfield approximation (see below) becomes asymptocially exact for penetrable potentials. Indeed this approximation also works surprisingly well for finite densities beyond overlap. Furthermore, the density functional is exactly known (as so-called Percus-functional) in one spatial dimension for the Tonks gas (had rods on a line). However, the latter system does not exhibit freezing. Please note that the knowledge of a functional is much more than a bulk equation of state since it can be applied to any external potential Vext (r). In principle, the application of DFT to freezing works as follows: First one has to chose an approximation. Then the density field is parameterized with variational parameters. In the homogeneous gas and liquid bulk phase one takes ρ(r) = ρ¯
(6)
where ρ¯ is a variational parameter. On the other hand, for the solid, the Gaussian approximation of density peaks on the lattice positions is an excellent choice [24]. α −3/2 2 n exp −α r − R ρ(r) = (7) π n Here both the lattice structure and spacing as well as the width α are variational parameters. Finally, for a given chemical potential µ and temperature T , one has to minimize the functional Ω(T, µ, [ρ]) with respect to all variational parameters. As a result one obtains the phase diagram in the µT plane. The procedure itself is sketched close to the solid-liquid transition in Figures 5 and 6. A solid-liquid transition line in the µT plane is schematically shown in Figure 5 and we consider a path with fixed µ and increasing T crossing the solid-liquid transition at µ = µcoex and T = Tcoex . Coexistence implies that temperature T , chemical potential µ and pressure p are the
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same in both phases. Since in the bulk p = −Ω/V (V denoting the system volume) coexistence means that at given µ and T , Ω/V has two minima with equal depth. A contour plot of the density functional in the space of variational parameters is shown in Figure 6 for three different temperature on the path shown in Figure 5. The liquid minimum occurs at zero α while the solid is characterized by a minimum at finite α. The global minimum is the stable phase and at coexistence, the two minima have equal depth. μ solid μ coex liquid
Tcoex
T
Fig. 5. Solid-liquid coexistence line in the µT plane. The path along which three state points are discussed in Figure 6 is indicated.
b) Approximations for the density functional Let us first recall the exact functional for the ideal gas where V (r) = 0. It reads as (8) Fid (T, [ρ]) = kB T d3 r ρ(r) ln(ρ(r)Λ3 ) − 1 and minimization
δFid = kB T ln(ρ(r)Λ3 ) + Vext (r) − µ δρ(r) 0
leads to the generalized barometric law 1 Vext (r) − µ ρ0 (r) = 3 exp − Λ kB T
(9)
(10)
for the inhomogeneous density. In the interacting case, V (r) = 0, one can split F(T, [ρ]) =: Fid (T, [ρ]) + Fexc (T, [ρ])
(11)
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a)
T
solid
metastable liquid α
ρ
b)
at coexistence: μ=μ coex , T=Tcoex
αs solid
ρl
ρs
ρ
liquid (α=0) α
metastable solid
T>Tcoex
liquid
c)
ρ
Fig. 6. Contour plot of the grandcanonical free energy Ω(T, µ, ρ¯, α) for fixed T and µ as a function of two variational parameters α and the averaged density ρ¯. The latter is given by the lattice constant in the solid phase. a) with a stable solid phase, b) at solid-liquid coexistence, c) with a stable liquid phase.
which defines the excess free energy density functional Fexc (T, [ρ]). Approximations work on different levels. In the mean-field approximation, we set 1 d3 r d3 r V (|r − r |)ρ(r)ρ(r ) (12) Fexc (T, [ρ]) ≈ 2 In fact, the mean-field approximation (together with a correlational hole in the solid) yields freezing of the Gaussian potential [25] and is the correct starting point for cluster crystals for penetrable potentials [23]. The Ramakrishnan-Yussouff (RY) approximation is a perturbative treatment out of the bulk liquid which needs the bulk liquid direct correlation function c(2) (r, ρ¯, T ) as an input. A functional Taylor expansion
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around a homogeneous reference density up to second order yields kB T Fexc (T, [ρ]) ∼ d3 r d3 r c(2) (|r − r |, ρ¯, T )(ρ(r) − ρ¯)(ρ(r ) − ρ¯) =− 2 (13) The RY approximation leads to freezing for hard spheres and was historically the first demonstration that freezing can be described within DFT. The RY functional can readily be generalized to soft interactions [26] (as the OCP) and gives reasonable results for freezing there (though it is better to improve the functional by including triplet correlations). A non-perturbative functional is based on Rosenfeld’s fundamental measure theory (FMT). This works, however, only for hard spheres. In FMT we have Fexc [ρ] = d3 r Φ[{nα (r)}] (14) kB T with (15) nα (r) = d3 r w(α) (r − r )ρ(r ) where the six weight function are given explicitly as w(2) (r) πσ 2 w(2) (r) w(1) (r) = 2πσ σ w(2) (r) = δ −r 2σ (3) −r w (r) = Θ 2 (V2 ) w (r) w(V1 ) (r) = 2πσ r σ w(V2 ) (r) = δ −r r 2 with σ denoting the hard sphere diameter and w(0) (r) =
Φ = Φ1 + Φ 2 + Φ 3 Φ1 = −n0 ln(1 − n3 ) n1 n2 − nv1 · nv2 Φ2 = 1 − n3 1 3 n − n2 (nv2 · nv2 ) Φ3 = 3 2 8π(1 − n3 )2
(16) (17) (18) (19) (20) (21)
(22) (23) (24) (25)
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This FMT functional yields the Percus-Yevick solution of the direct correlation function as an output. It furthermore survives the dimensional crossover [27]: If the three-dimensional hard sphere system is confined within a one-dimensional tube, the exact Percus functional is recovered. Moreover, in a spherical cavity which holds one or no particle at all, the exact functional is recovered. This helps to understand that the constraint packing argument of freezing is geometrically included in the FMT. In fact (also with a tensor modification in Φ3 [28]), the FMT gives excellent data for hard-sphere freezing [27], see Table 1 and the recent review by Roth [9]. Table 1. Coexisting number densities and solid Lindemann parameter at coexistence for the hard sphere systems. “Exact” computer simulation data are shown as well as DFT data using the Ramakrishnan-Yussouff (RY) or Rosenfeld’s fundamental measure theory. ρl σ 3 ρs σ 3 L (: Lindemann) computer simulations 0.94 1.04 0.129 RY 0.97 1.15 0.06 0.94 1.03 0.101 Rosenfeld
Last but not least we mention perturbation theories which can be used for attractive tails. The total potential V (r) is then split into a purely repulsive short-ranged part Vrep (r) and a longer-ranged attractive part Vattr (r) such that V (r) = Vrep (r)+Vattr (r). The repulsive part is treated as an effective hard core with an effective (temperature-dependent) Barker-Henderson diameter σ(T ) =
∞
dr
1 − e−βVrep (r)
(26)
0
and the attractive part is treated within mean-field approximation. Accordingly, the total excess free energy functional reads as HS (T, [ρ])σ=σ(T ) Fexc (T, [ρ]) ∼ = Fexc 1 d3 r d3 r ρ(r)ρ(r )Vattr (|r − r |) + 2
(27)
This procedure yields good phase diagrams for both Lennard-Jones potentials and sticky-hard-sphere systems including the isostructural solid-solid transition [18].
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To summarize: (1) Rosenfeld’s FMT yields excellent data for hard sphere freezing. (2) The much less justified RY perturbative approach works in principle for softer repulsions. (3) The mean-field density functional approximation works for penetrable potentials. (4) Hard sphere pertubation theory yields stability of liquids and solid-solid isostructural transitions. 2. Brownian Dynamics 2.1. Brownian dynamics (BD) Colloidal particles are embedded in a molecular solvent and are therefore randomly kicked by the solvent molecules on timescales much smaller than the drift of the colloidal motion [5, 29]. Let us first discuss the Smoluchowski picture. Here the time-dependent density field is the central quantity. It should follow a simple deterministic diffusion equation. For noninteracting particles with an inhomogeneous time-dependent particle density ρ(r, t), Fick’s law states that the current density j(r, t) is r , t) j(r, t) = −D0 ∇ρ(
(28)
where D0 is a phenomenological diffusion coefficient. The continuity equation of particle conservation ∂ρ(r, t) + ∇ · j(r, t) = 0 ∂t then leads to the wellknown diffusion equation for ρ(r, t): ∂ρ(r, t) ≡ D0 ∆ρ(r, t) ∂t
(29)
(30)
ext (r) In the presence of an external potential Vext (r), the force F = −∇V acts on the particles and will induce a drift velocity vD resp. an additional current density jD = ρvD .
(31)
We now assume totally overdamped motion since inertia effects are small as the colloids are much bigger than the solvent molecules. This results in 1 F vD = = − ∇V r) (32) ext ( ξ ξ
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with ξ denoting a friction coefficient. For a sphere of radius R in a viscous solvent, ξ = 6πηs R, with ηs denoting the shear viscosity of the solvent (Stokes law). Now the total current density is r , t) − ρ(r, t) 1 ∇V ext (r) j = −D0 ∇ρ( (33) ξ In equilibrium, the one-particle density is a Boltzmann distribution ρ(r, t) ≡ ρ(1) (r) = ρ(0) (r) = A exp(−βVext (r))
(34)
Futhermore, in equilibrium, the total current has to vanish. Therefore, necessarily kB T (35) D0 = ξ + which is the so-called Stokes-Einstein relation. Hence j = − 1 (kB T ∇ρ ξ
ext ) and the continuity equation yields ρ∇V
∂ρ(r, t) 1 ext (r))) r , t)∇V = (kB T ∆ρ(r, t) + ∇(ρ( ∂t ξ
(36)
which is called Smoluchowski equation (for non-interacting particles). The same equation holds for the probability density w(r, t) to find a particle at position r for time t. For N non-interacting particles, 1 w(r, t) = ρ(r, t) , (37) N and the Smoluchowski equation reads: ∂w 1 ext )) = (kB T ∆w − ∇(w · ∇V (38) ∂t ξ Now we consider N interacting particles. Using a compact notation for the particle positions {xi } = {ri } = {x1 , x2 , x3 , x4 , x5 , x6 , · · · , x3N −2 , x3N −1 , x3N } r1
r2
(39)
rN
we assume a linear relation between acting forces on the particles and the resulting drift velocities. (The same compact notation is used for other multiple vectors.) The details of this relation embody the so-called hydrodynamic interations mediated between the colloidal particles by the solvent flow field induced by the moving colloidal particles. This linear relation is in general vi =
3N j=1
Lij ({xn })Fj
(40)
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∂ where Fj = − ∂x Utot where Utot involves both the internal and the interacj tion potential energy and v is the drift velocity. The underlying assumption in (40) is that the hydrodynamic interactions act quasi-instantaneously. This is justified by the fact that the timescale upon which a shear perturbation is traveling through the suspension within an interparticle distance is much smaller than that of Brownian motion. The coefficients Lij constitute the so-called 3N ×3N mobility matrix and can in principle be obtained by solving the Navier-Stokes equations of N spheres with appropriate stick boundary conditions of the solvent flow field on the particle’s surfaces. In general, Lij depends on rN , and we postulate:
symmetry Lij = Lji positivity
Fi Fj Lij > 0 for all Fi,j = 0
(41)
(42)
ij
With w({ri }, t) denoting the probability density for interacting particles, the suitable generalization of the continuity equation is 3N ∂ ∂w =− (vtot,n w) ∂t ∂x n n=1
(43)
with vtot,n =
3N m=1
Lmn
∂ (kB T ln w + Utot ) ∂xm
(44)
which leads to the generalized Smoluchowski equation fo interacting particles. ∂w ˆ = Ow (45) ∂t with the Smoluchwoski operator ˆ= O
3N
∂ ∂ ∂Utot Lnm (kB T + ) ∂x ∂x ∂xm n m n,m=1
(46)
In many applications, hydrodynamic interactions are neglected. This means that the mobility matrix is constant and a diagonal Lnm =
1 δnm ξ
(47)
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‘Brownian motion’
Fig. 7.
Typical trajectory of a randomly kicked Brownian particle.
This assumption, however, is only true for small volume fraction of the colloidal particles. Complementary to the Smoluchowski approach which considers diffusion in phase space, stochastic trajectories in real-space are the basic ingredients for the Langevin picture. A typical “cuspy” Brownian trajectories of a colloidal particle is shown in Figure 7. First, we consider only one particle in an external potential Vext (r) with random force f(t). The stochastic differential equation for a single particle is completely overdamped ext (r) + f(t) ξr˙ = −∇V
(48)
where f(t) mimicks the random kicks of the solvent and is a Gaussian random variable which fulfills f(t) = 0
(49)
fi (t)fj (t ) = 2ξkB T δij δ(t − t )
(50)
One can show that this is stochastically equivalent to the Smoluchowski equation of non-interacting particles. For interacting particles, the Smoluchowski equation is obtained from the following Langevin equations [5]: x˙ n (t) =
3N m=1
Lnm (−
3N ∂Utotal ∂Lnm + fm (t)) + kB T ∂xm ∂xm m=1
(51)
with Gaussian random variables fm which fulfill fm (t) = 0
fm (t)fm (t ) =
2L−1 k T δ(t mm B
(52)
− t ).
(53)
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2.2. BD computer simulations In Brownian Dynamic (BD) computer simulations we use the overdamped Langevin equation (48) with a single realization of the stochastic process. Applying this to a finite time step ∆t and integrating xn (t + ∆t) = xn (t) + ∆t ∆n =
Lnm (−
m=1
where
3N
t+∆t t
dt
∂Lnm ∂Utotal ) + ∆n + kB T ∆t ∂xm ∂xm m
Lnm fm (t )
is Gaussian distributed
m
and possesses the moments ∆n =
t+∆t t
dt
Lnm fm (t ) = 0
m
∆n ∆ = 2kB T ∆tLnn , n
which is the basic equation for BD computer simulations. For each step we need to generate a Gaussian random number and update the Langevin equation by calculating xn (t + ∆t). Although this algorithm is less efficient than MD/MC for equilibrium correlations, it is unavoidable for calculation of colloidal dynamics. 2.3. Dynamical density functional theory (DDFT) Here we derive a deterministic equation for the time-dependent one-particle density from the Smoluchowski equations [25]. We follow the idea of Archer and Evans [30]. First, we recall Smoluchowski equation for the N -particle density w(r1 , . . . , rN , t) ≡ w(rN , t) , rN = {r1 , . . . , rN } as N
∂w i+∇ i Utot (rN , t)]w i · [kB T ∇ ˆ =1 ∇ = Ow ∂t ξ i=1
(54)
with Utot (rN , t) =
N i=1
Vext (ri , t) +
N i,j=1 i<j
V (|ri − rj |)
(55)
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Here, hydrodynamic interactions have been neglected. Now the idea is to integrate out degrees of freedom. In fact, an integration yields 3 ρ(r1 , t) = N d r2 . . . d3 rN w(rN , t) (56) The 2-particle density is analogously obtained as ρ(2) (r1 , r2 , t) = N (N − 1) d3 r3 . . . d3 rN w(rN , t) Integrating the Smoluchowski equation with N ξ
(57)
d3 r2 . . . d3 rN yields
∂ 1 (ρ(r1 , t)∇ 1 Vext (r1 , t) ρ(r1 , t) = kB T ∆1 ρ(r1 , t) + ∇ ∂t 1 d3 r2 ρ(2) (r1 , r2 , t)∇ 1 V (|r1 − r2 |) +∇
(58)
r1 ,t) In equilibrium, necessarily ∂ρ( = 0 which implies ∂t r ) + ρ(r)∇V ext (r) + d3 r ρ(2) (r, r )∇V (|r − r |) kB T ∇ρ( 0=∇
(59) The constant must vanish for r → ∞ and is thus identical to zero. Therefore r ) + ρ(r)∇V ext (r) + d3 r ρ(2) (r, r )∇V (|r − r |) (60) 0 = kB T ∇ρ( This is also known as Yvon-Born-Green-hierarchy (YBG). In equilibrium, DFT implies: δF = µ − Vext (r) δρ(r) = kB T ln(Λ3 ρ(r)) +
(61) δFexc , since F = Fid + Fexc δρ(r )
(62)
We now apply the gradient which gives: ext (r) + kB T ∇ ln(Λ3 ρ(r)) + ∇ δFexc = 0 ∇V δρ(r) combined with YBG we obtain (|r − r |) = ρ(r)∇ · δFexc [ρ] d3 r ρ(2) (r, r )∇V δρ(r)
(63)
(64)
We postulate that this argument holds also in nonequilibrium. In doing so, non-equilibrium correlations are approximated by equilibrium ones at the same ρ(r, t) (via a suitable Vext (r) in equilibrium). Equivalently, one can say
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that it is postulated that pair correlations decay much faster to their equilibrium one than the one-body density. Therefore the basic approximation of DDFT is sometimes called adiabatic approximation. (For an alternate derivation, see Marconi and Tarazona [31, 32] or Espa˜ nol and L¨ owen [33].) Hence: ξ
∂ρ(r, t) B T ∇ρ( r , t) + ρ(r, t)∇V ext (r, t) = ∇(k ∂t δFexc ) + ρ(r, t)∇ δρ(r, t)
(65)
or equivalently ξ
∂ρ(r, t) r , t)∇ δΩ[ρ] = ∇ρ( ∂t δρ(r, t)
(66)
which constitutes the basic equation of dynamical density functional theory (DDFT). The applications of DDFT are numerous. The dynamics of a strongly inhomogeneous Brownian fluid has found to be a good agreement with BD computer simulations. [34, 35] 2.4. An example: Crystal growth at imposed nucleation clusters The dynamical density functional theory can also be used for the dynamics of the crystal. This is demonstrated by an example in the following. A two-dimensional system with the repulsive inverse power-law potential V (r) = V0 /r3 corresponds to dipoles in a plane whose dipole moments are perpendicular to the plane. Two-dimensional systems are best realized for superparamagnetic colloids pending at an air-water interface in an external magnetic field [36, 37]. The RY functional gives a reasonable description for the two-dimensional crystallization [26]. The dynamical version can be applied to a situation where a cluster of 19 particles from a prescribed cut-out of a triangular lattice is imposed and offered as a nucleation seed to an undercooled fluid. This cluster is first fixed and then instantaneously released. Depending on the lattice constant of the prescribed nucleation cluster, there is either subsequent crystal growth or relaxation back to the undercooled fluid. Two examples are shown in Figure 8. The one-particle density field is shown for fixed time as a contour plot. In both examples, the crystalline seed is compressed relative to the
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Fig. 8. Snapshots of the central region of the dimensionless density field ρ(r, t)/ρ of two colloidal clusters with strain parameters Aρ = 0.7 (left panel) and Aρ = 0.6 (right panel) at times t/τB = 0, 0.001, 0.1, 1.0 (from top to bottom; t/τB = 1.0 only for Aρ = 0.7). A is the area of the crystal unit cell. From [38].
stable bulk one. If the compression is significant but not too large (Figure 8, left panel), there is still crystal growth, but if the crystal is compressed further, the seed is too dissimilar to the bulk crystal as to initiate crystal growth (Figure 8, right panel). The same qualitative behaviour has been found in Brownian dynamics computer simulations [38]. This example shows that dynamical density functional theory represents a reliable microscopic approach to nonequilibrium phenomena like crystal growth in an external field [39]. A similar behaviour has been seen for
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growing crystal fronts [38, 40] and for vacancy dynamics in two-dimensional solids. 2.5. Hydrodynamic interactions Here we address the question how Lnm ({xj }) looks like explicitely. Solving the linearized Navier-Stokes equations with the appropiate stick boundary conditions on the particle surfaces, is a difficult problem. Furthermore it is problematic that i) Lnm ({xj }) is long-ranged in terms of distances between particles. ii) H.I. have many-body character, pair expansion only possible at low concentrations. iii) H.I. have quite different near-field behaviour. They are divergent lubrication terms. The linear relationship (40) can be rewritten as vn =
N
¯ F H nm m
(67)
m=1
¯ where each quantity H nm is a 3×3 matrix. In particular, we can discriminate the following cases: 1) No H.I. Hnm = ½ δnm ξ 2) Oseen-tensor ¯ =½ In the Oseen approximation, H nn ξ ¯ r − r ) for n = m ¯ H nm = H( n m
(68)
r
with the Oseen tensor ¯ r) = H(
1 1 , (½ + rˆ ⊗ rˆ) 8πηs r
rˆ =
r r
(69)
This is the leading far field term for two particles at large distance r. The symbol ⊗ denotes the dyadic product or tensor product. 3) Rotne-Prager-tensor In this approximation, the next leading correction is included. Hnn =
½ ξ
,
¯ (r − r ) Hnm = H RP n m
(70)
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with ¯ (r) = D0 H RP kB T
3 3 RH 1 RH [½ + rˆ ⊗ rˆ] + [1 − 3ˆ r ⊗ rˆ] 4 r 2 r3
(71)
Higher order expansions of higher order than r13 are possible. These include also terms of sphere rotation. Finally the triplet contribution can be estimated. The DDFT can be generalized to hydrodynamic interactions [41]. Again the starting point is the Smoluchowski equation which we now write in the form N Utot (rN , t) ∂w(rN , t) N ¯ = w(rN , t) (72) ∇i · Hij (r ) · ∇j + ∇j ∂t kB T i,j=1
We use the two particle approximation D 0 ¯ (rN ) ≈ ½δij + δij ω11 (ri − re ) + (1 − δij )ω12 (ri − re ) H ij kB T i=j
(73) on the level of the Rotne-Prager expression ω11 (r) = 0 3 σH 1 σH 3 σH 7 r ⊗ rˆ) + O(( ω12 (r) = (½ + rˆ ⊗ rˆ) + ( ) (1 − 3ˆ ) ) 8 r 16 r r where σH is the hydrodynamic diameter. Integrating Smoluchowski equation [25] then yields [41] kB T ∂ρ(r, t) δF [ρ] = ∇r · ρ(r, t)∇r D0 ∂t δρ(r, t) δF [ρ] + dr ρ(2) (r, r , t)ω11 (r − r ) · ∇r δρ(r, t) δF [ρ] + dr ρ(2) (r, r , t)ω12 (r − r ) · ∇r δρ(r, t)
(74) (75)
(76)
A possible closure is via the Ornstein-Zernike equation ρ(2) (r, r , t) = (1 + c(2) (r, r ))ρ(r, t)ρ(r , t) + ρ(r , t) dr ((ρ(2) (r, r , t) − ρ(r, t)ρ(r , t))c(2) (r , r )) (77)
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with c(2) (r, r ) = −β
δ 2 Fexc [ρ] δρ(r, t)δρ(r , t)
(78)
In an easier attempt, one can approximate ρ(2) (r, r , t) ≈ ρ(r, t)ρ(r , t)g(|r − r |, ρ¯)
(79)
where ρ¯ is a suitable averaged density and g(r, ρ¯) is a pair distribution function in the equilibrium bulk fluid. Finally we describe an example which involves hard sphere colloids of diameter σ moving in a time-dependent oscillating radial-symmetric trapping potential Vext (r, t). The potential reads as Vext (r, t) = V1 r4 + V2 cos(ωt)r4
(80)
and corresponds to a shape which switches between stable and unstable situations at the origin r = 0 but is globally stable. The amplitudes are V1 = 10kB T /4096σ 4, V2 = kB T /σ 2 and the external switching frequency is ω = 4πD0 /σ2 . As a result, the density profile is picking up the external frequency ω and exhibits a breathing mode, i.e. it is periodically expanding and compressed again by the external potential (80). Time-dependent density profiles ρ(r, t) are presented in Figure 9 in the steady breathing state. Figures 9(a) and (b) show results for hydrodynamic interactions included on the Rotne-Prager level. DDFT data are in very good agreement with Brownian dynamics computer simulations which include hydrodynamic interactions on the same Rotne-Prager level. On the other hand, in Figure 9(c) and (d), hydrodynamic interactions are ignored. The density profiles are qualitatively different to that shown in Figures 9(a) and (b) but again DDFT data are in agreement with Brownian dynamics computer simulations. This demonstrates that DDFT is a reliable microscopic theory both if hydrodynamic interactions are included or ignored. 3. Rod-Like Particles 3.1. Statistical mechanics of rod-like particles Density functional theory can readily be extended to rod-like particles which possess an additional orientational degree of freedom described by a unit vector u ˆ. A configuration of N particles is now fully specified by the
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Fig. 9. Steady-state DDFT (solid curves) and BD (noisy curves) results for the timedependent density profile ρ(r, t). In Fig. (a) and (b) hydrodynamic interactions are taken into account while in (c) and (d) they are neglected. (a) and (c) correspond to the expanding half period and (b) and (d) to the compressing half period, respectively. The profiles correspond to the following time sequence: t0 = 2.5τB , t1 = 2.6τB , t2 = 2.7τB , t3 = 2.75τB in (a) and (c), and t3 = 2.75τB , t4 = 2.85τB , t5 = 2.9τB and t6 = 3.0τB , τB = σ 2 /D0 . From Ref. [41].
set of positions of the center of masses and the corresponding orientations i, u {R ˆi , i = 1, . . . , N }, see Figure 10. Example for ansiotropic particles include (1) (2) (3) (4)
molecular dipolar fluids (e.g. H2 O molecule) rod-like colloids (e.g. tobacco-mosaic viruses) molecular fluids without dipole moment (apolar), (e.g. H2 molecule) plate-like objects (clays) The canonical partition function for rod-like particles now reads [42] 1 d3 R1 ... d3 RN d3 p1 ... d3 pN Z = 6N h N! 3 V V R3 R × d2 u1 ... d2 uN d3 L1 ... d3 LN e−βH (81) S2
S2
R3
R3
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orientation vector
Ri center-of-mass coordinate
platelet i ûi
Ri
i and the orientational unit vector u Fig. 10. Sketch of the center-of-mass position R ˆi for the ith particle both for a rod-like and plate-like particle.
with the Hamilton function H=
N 2 pi 1 ¯ −1 Li + L i (Θ) 2m 2 i=1
+
N
i, u Vext (R ˆi )
+
N 1 j, u v(Ri − R ˆi , u ˆj ) 2 i,j=1
(82)
i=1
which comprises the kinetic energy, the pair interaction energy and the ¯ is the inertia tensor and S the unit-sphere in 3d. external energy. Here Θ 2 (1) Again the central quantity is the one-particle density ρ0 (r, u ˆ) which is defined as N (1) ˆ) := δ(r − Ri )δ(ˆ u−u ˆi ) (83) ρ (r, u 0
i=1
Integrating the orientations over the unit sphere S2 results in the density of the center-of-masses 1 (1) d2 u ρ0 (r, u ˆ) (84) ρ0 (r) = 4π S2
On the other hand, the globally averaged orientational order is gained by integrating over the center-of-mass corrdinates and given by 1 (1) d3 r ρ0 (r, u ˆ) (85) f (ˆ u) = V V
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û1
1
2
r
û2
R2
R1
Fig. 11. Sketch of two interacting rods. The interaction potential depends on r = 1 − R 2 and u R ˆ1 , u ˆ2 .
In analogy to the isotropic case, one can define a pair correlation function N ! 1 − R i )δ(R 2 − R j )δ(ˆ δ(R u1 − u ˆj )δ(ˆ u2 − u ˆj ) 1, R 2, u g(R ˆ1 , u ˆ2 ) :=
i,j=1 i=j
1, u 2 , uˆ2 ) ρ0 (R ˆ1 )ρ0 (R (1)
(1)
(86)
Now different phases are conceivable which can be classified and distinguished by their one-particle density field. 1) Fluid (disordered) phase, isotropic phase Here the center-of-mass-positions and orientations are disordered: (1)
ˆ) = ρ0 = const ρ0 (r, u
(87)
2) Nematic phase Here, positions are disordered and orientations are ordered, i.e. (1)
ˆ) = ρf (ˆ u) ρ0 (r, u
,
u ˆ0 : nematic director
(88)
Typically the orientation is distributed around a nematic director uˆ0 . In order to quantify orientational order, it is convenient to introduce a nematic order parameter. In fact, this is defined via the second rank tensor N 1 3 1 ¯= ˆi − ½ (89) Q u ˆi ⊗ u N i=1 2 2 where the dyadic product is
uix uix uix uiy uix uiz ˆi = uiy uix uiy uiy uiy uiz u ˆi ⊗ u uiz uix uiz uiy uiz uiz
(90)
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¯ is trace-less One can easily show that the tensor Q 1 Tr(3ˆ ui ⊗ u ˆi − ½) 2 1 = 3 · 1 − 3 = 0 2
¯= TrQ
(91) (92)
¯ is clearly symmetric and hence diagonalizable with three Furthermore Q eigenvalues λ1 ≥ λ2 ≥ λ3 where λ3 must be −λ2 −λ1 . The largest eigenvalue λ1 is called nematic order parameter S. The corresponding eigenvector is called nematic director. For perfect alignment along u0 we have ui ≡ u ˆ0 for all i. Then, S = 1. If the two lower eigenvalues are identical, λ2 = λ3 , the call it a uniaxial nematic phase. If λ2 = λ3 , the orientation is called biaxial. In the isotropic phase: S = 0. Orientational distributions are accessible experimentally, by e.g. birefringence. 3) Smectic-A phase The so-called smectic-A phase is positionally ordered along an orienation axis u ˆ0 . The associated one-particle density is periodic in the z-direction along uˆ0 : (1)
ˆ) = ρ(z, u ˆ) z-periodic ρ0 (r, u
(93)
4) Smectic-B phase The smectic-B phase is similar to the smectic-A phase but exhibits an inplane triangular lattice. 5) Columnar phase The columnar exhibits crystalline order perpendicular to the director uˆ0 but is disordered positionally along u0 . The one-particle density field thus reads as (1)
ˆ) ρ0 (r, uˆ) = ρ(x, y, u
(94)
6) Plastic crystal In a plastic crystal, the positions are ordered in all three spatial directions but the orientations are disordered. Therefore: (1)
ρ0 (r, uˆ) = f (r)
(95)
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7) Full crystalline phases Finally, in the full crystalline phase, positions and orientations are both ordered. The list (1)-(7) of liquid crystalline phases is by far not exhaustive. There are more “exotic” phases such as an AAA stacked phase, a smecticC with tilted rods, or cholesteric phases which possesses a helical twist with a pitch length lp . 3.2. Simple models Let us now discuss simple models for interactions between anisotropic rods. The phase behaviour of hard objects, as shown in Figure 12, is dominated by the shape. Temperature scales out in this case such that packing fraction alone (apart from the particle shape) is the only parameter. Hard spherocylinders have been studied, as well as hard platelets (“hard coins”) or thin needles which arise from spherocylinders in the limit of infinite aspect ratio L D → ∞. limit: L D h
L
D hard spherocylinders Fig. 12.
∞
hard coins
hard needles
Sketch of differently shaped hard bodies.
A) Analytical results by Onsager L In the limit p = D → ∞, a virial expansion up to second order is getting asymptotically exact. There is a isotropic-nematic transition which can be calculated analytically [43]. It is first order with a density jump. The scaled coexistence densities of the coexisting isotropic and nematic phases are
ρiso L2 D = 4.189 . . .
(96)
ρnem L2 D = 5.376 . . .
(97)
and At coexistence, the nematic order parameter in the nematic phase is S = 0.784. However, in practice, one needs pretty large aspect ratios (about p 200) in order to get reasonably into this Onsager limit.
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B) Computer simulations Full phase diagrams can be obtained by Monte Carlo computer simuations [44]. Hard spherocylinders show for various densities ρ and aspect ratios p the following stable phases: isotropic, plastic, ABC stacking, AAA stacking, smectic-A, and nematic phases. All those transitions are purely driven by entropy. Another anisotropic system are hard ellipsoids which are characterized by two different axial lengths a and b. The phase diagram involves isotropic, nematic fully crystalline and plastic crystalline phases [45]. There is a remarkable symmetry in x → x1 for the topology of the phase diagram where x = ab . x < 1 correspond to oblate and x > 1 to prolate objects. C) Density functional theory Again density functional theory tells us that there exists a unique grandcanonical free energy functional Ω(T, µ, [ρ(1) ]) (functional of the one-particle (1) density) which becomes minimal for the equilibrium density ρ0 (r, u ˆ) and then equals the real grand canonical free energy, i.e. δΩ(T, µ, [ρ(1) ]) =0 (98) δρ(1) (r, u ˆ) ρ(1) =ρ(1) (r,ˆu) 0
Here, the functional can be decomposed as follows 3 (1) Ω(T, µ, [ρ ]) = kB T d r d2 u ρ(1) (r, u ˆ)[ln(Λ3 ρ(1) (r, u ˆ)) − 1] + d3 rd2 u (Vext (r, u ˆ) − µ)ρ(1) (r, u ˆ) + Fexc (T, [ρ(1) ]) (99) The first term on the right hand side of equation (99) is the functional Fid [ρ(1) ] for ideal rotators. The excess part Fexc(T, [ρ(1) ]) is in general unknown and requires approximative treatments. For hard spherocylinders, Fexc (T, [ρ(1) ]) can be approximated by a smoothed density approximation (SMA) [46] yielding several stable liquidcrystalline phases, namely: isotropic, nematic, smectic-A and ABC crystalline. A modified weighted density approximation (MWDA) was subsequently proposed [47] which improves upon the SMA by exhibiting stable plastic crystalline and AAA crystals as well. An important recent progress was archieved by generalizing Rosenfeld’s fundamental measure theory from hard spheres to hard objects with any
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shape [48]. For spherocylinders the functional was worked out explicitly. This functional could be exploited also for attractions by employing a perturbation theory for the attractive parts in the potential. Finally, a meanfield density functional for rods with soft segments was proposed and studied [49]. 3.3. Brownian dynamics of rod-like particles In order to derive a dynamical density functional theory (DDFT) for rodlike particles one can start from the Smoluchowski equation for the full probability density distribution w(r1 , · · · , rN ; u1 , · · · , uN , t) of N rods with their corresponding center-of-mass positions rN = (r1 , · · · , rN ) and orienu1 , · · · , u ˆN ) which reads [6] tations uˆN = (ˆ ∂w ˆS w =O ∂t where the Smoluchowski operator is now given by N 1 N N ¯ u )· ∇ r · D(ˆ ˆS = ∇ O ∇ + U ( r , u ˆ , t) i ri r i kB T i i=1 ˆi · R ˆ i U (rN , u ˆi + 1 R ˆN , t) +Dr R kB T
(100)
(101)
ˆN , t) is the total potential energy. Here the rotation operator where U (rN , u ˆ ˆ i = u uˆi and the anisotropic translational diffusion i is defined as R ˆi × ∇ R tensor is given by ¯ u ) = D u ˆi ⊗ u ˆi + D⊥ (1 − u ˆi ⊗ u ˆi ) D(ˆ i
(102)
The two diffusion constants D and D⊥ , parallel and perpendicular to the orientations reflect the fact that the translational diffusion is anisotropic. For hard spherocylinders there are valid approximations for D and D⊥ [50]. Following the idea of Archer and Evans the can integrate 2Smolu [30] one chowski equation by now applying N d3 r2 · · · d3 rN d2 u1 · · · d un on both sides of Eqn (100). This results in [49]: (r, u F ˆ, t) 1 ∂ρ(r, u ˆ, t) ¯ u) · ∇ r · D(ˆ r Vext (r, u r ρ(r, u =∇ ρ(r, u ˆ, t) · ∇ ˆ, t) + ˆ, t) − ∂t kB T kB T 1 ˆ · Rρ( ˆ r, u r Vext (r, u + Dr R ˆ, t) − k 1 T T (r, u ˆ, t) + ˆ , t) ρ(r, u ˆ, t)∇ B kB T
(103)
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with an average force 3 r v2 (r − r , uˆ, u ˆ, u ˆ , t)∇ ˆ ) F (r, u ˆ, t) = − d r d2 u ρ(2) (r, r , u
(104)
and average torque 3 ˆ 2 (r − r , u T (r, u ˆ, t) = − d r d2 u ρ(2) (r, r , u ˆ, u ˆ , t)Rv ˆ, u ˆ )
(105)
The two-particle density which is in general unknown can be approximated in equilibrium by using r δFexc (T, [ρ0 ]) ˆ)∇ F (r, u ˆ, t) = ρ0 (r, u δρ0 (r, u ˆ)
(106)
ˆ δFexc [ρ] T (r, u ˆ, t) = ρ0 (r, u ˆ)R δρ0 (r, u ˆ)
(107)
respectively
Similar as in the isotropic (spherical) case we now employ the “adiabatic” approximation. We assume that the pair correlations in nonequilibrium are the same as those for an equilibrium system with the same one-body density profile (established by a suitable Vext (r, u ˆ, t)). The resulting dynamical equation for the time-dependent one particle density ρ(r, u ˆ, t) is then given by Ref. [49]: ∂ρ(r, u ˆ, t) ˆ, t)] ¯ u) · ρ(r, u r · D(ˆ r δF [ρ(r, u =∇ ˆ, t)∇ kB T ∂t δρ(r, u ˆ, t) δF [ρ( r , u ˆ , t)] ˆ ˆ + Dr R ρ(r, u ˆ, t)R (108) δρ(r, u ˆ, t) with the equilibrium Helmholtz free energy density functional u ρ(r, u ˆ) ln(Λ3 ρ(r, u ˆ)) − 1 F[ρ] = kB T d3 r dˆ 3 + Fexc (T, [ρ]) + d r dˆ u ρ(r, u ˆ)Vext (r, u ˆ, t)
(109)
This sets the frame for dynamical density functional theory (DDFT) for rods. As for a special application of DDFT to dynamics in the confined isotropic phase we refer to [49] where the mean-field approximation for the functional was employed. More recent work has used the Rosenfeld functional for hard spherocylinders [48] for driven nematic phases [51, 52].
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3.4. “Active” (self-propelled) Brownian particles “Active” particles are self-propelled by their own intrinsic motor. In fact, apart from swimming bacteria, there are artificial microswimmers made by colloidal particles [53, 54]. Ignoring hydrodynamic interactions, these swimmers can simplest be modelled by rod-like particles which are driven by a constant force along their orientations; the force corresponds to an effective drift velocity and mimicks the actual propulsion mechanism. On top of the intrinsic propulsion, the particles feel Brownian noise of the solvent. The corresponding motion is intrinsically a nonequilibrium one and even the dynamics of a single Brownian swimmer was solved only recently [55, 56]. Starting from the Smoluchowski equation with an appropriate intrinsic drift term, a dynamical density functional theory can be derived using the same adiabatic approximation (64) as in the “passive” case. The resulting equation of motion for the one-particle density then [57] has an extra term on the right-hand side of Eqn. (108), namely ¯ u)ρ(r, u D(ˆ ˆ, t)ˆ u −F0 ∇
(110)
where F0 denotes the internal driving force. This represents a microscopic theory for concentrated “active” matter. For swimmers in a two-dimensional channel, the time-dependent density profiles were found to be in agreement with Brownian dynamics computer simulations [57] even if a crude Onsager-like density functional approximation [43] was used. Qualitatively, the transient formation of hedgehog-like clusters ear the channel boundaries was reproduced by the dynamical density functional theory. 4. Conclusions In conclusion, there is a variety of topologies for equilibrium phase diagrams, even for relatively simple radially-symmetric pair potentials V (r). Examples include freezing into crystalline lattices with unusual open structure and reentrant melting effects if the interaction is soft and isostructural solid-to-solid transition for system with short-ranged attractions. The effects are predicted by Monte Carlo computer simulations and can in principle be confirmed by density functional theory of freezing. Colloidal particles can be used as model systems to realize these interactions. Rod-like systems with anisotropic interactions, on the other hand, exhibit liquid crystalline phase with different degrees of orientational and
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positional ordering. Again density functional theory of freezing can be formulated and be applied to predict the topology of the phase diagram. Rodlike colloidal particles represent ideal model systems to explore and test the phase behaviour experimentally. Effects of colloidal dynamics in a solvent both in equilibrium and nonequilibrium are conveniently simulated by Brownian dynamics computer simulations. The density functional theory can be extended towards dynamics, so-called dynamical density functional theory to tackle various nonequilibrium phenomena. This was demonstrated for crystal growth at imposed nucleation seeds and for the collective behaviour of “active” Brownian particles. In general, dynamical density functional theory is in good agreement with the simulations. This may be different for undamped Newtonian dynamics which occurs for example in molecular liquid crystals or in a complex plasma [58]. Acknowledgement I thank R. Evans, M. Rex, H. H. Wensink, S. van Teeffelen, U. Zimmermann, T. Glanz, M. Kohl and A. H¨ artel for many helpful suggestions. This work was supported by the DFG (SPP1296 and SFB TR6 (project D3)). References [1] A. F. Th¨ unemann, M. M¨ uller, H. Dautzenberg, J.-F. Joanny, H. L¨ owen, Adv. Polym. Sci. 166, pages 113-171 (2004), Springer Series. [2] H. L¨ owen, G. Kramposthuber, Europhys. Letters 23, 637-678 (1993). [3] G. A. Vliegenthart, P. van der Schoot, Europhys. Lett. 62, 600-606 (2003). [4] J. Dzubiella, H. L¨ owen, C. N. Likos, Phys. Rev. Letters 91, 248301 (2003). [5] M. Doi, S. F. Edwards, The Theory of Polymer Dynamics, Oxford Science Publications, Clarendon Press Oxford (1986). [6] J. K. G. Dhont, An Introduction to Dynamics of Colloids, Elsevier, Amsterdam, 1996. [7] H. L¨ owen, J. Phys.: Condensed Matter 13, R415-R432 (2001). [8] H. L¨ owen, Particle-resolved instabilities in colloidal dispersions, Soft Matter, 6, 3133-3142 (2010). [9] R. Roth, J. Phys.: Condensed Matter 22, 063102 (2010). [10] R. Evans, Advances in Physics 28, 143 (1979). [11] D. W. Oxtoby, in: Liquids, Freezing and Glass Transition, edited by J. P. Hansen, D. Levesque, J. Zinn-Justin, North Holland, Amsterdam, pages 145189, 1991. [12] H. L¨ owen, Physics Reports 237, 249 (1994).
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[13] J.-L. Barrat, J.-P. Hansen, Basic Concepts for Simple and Complex Liquids, Cambridge University Press, 2003. [14] H. L¨ owen, J. Phys.: Condensed Matter 14, 11897 (2002). [15] P. Tarazona, J. A. Cuesta, Y. Martinez-Raton, Density functional Theories of Hard Particle Systems, Lect. Notes Phys. 753, Springer, Berlin, pages 247-341 (2008). [16] J. P. Hansen, I. McDonald, Theory of Simple Liquids. 3rd Edition, Elsevier, Amsterdam, Academic Press 2005. [17] M. P. Allen, D. J. Tildesley, Computer Simulation of Liquids, Oxford Science Publications, Clarendon Press, Oxford, 1987. [18] C. N. Likos, Z. T. N´emeth, H. L¨ owen, J. Phys.: Condensed Matter 6, 10965 (1994). [19] C. N. Likos, H. L¨ owen, M. Watzlawek, B. Abbas, O. Jucknischke, J. Allgaier, D. Richter, Phys. Rev. Letters 80, 4450 (1998). [20] M. Watzlawek, C. N. Likos, H. L¨ owen, Phys. Rev. Letters. 82, 5289 (1999). [21] A. Lang, C. N. Likos, M. Watzlawek, H. L¨ owen, J. Phys.: Condensed Matter 12, 5087 (2000). [22] D. Gottwald, C. N. Likos, G. Kahl, H. L¨ owen, Phys. Rev. Letters 92, 068301 (2004). [23] C. N. Likos, A. Lang, M. Watzlawek, H. L¨owen, Phys. Rev. E 63, 031206 (2001). [24] R. Ohnesorge, H. L¨ owen, H. Wagner, Europhys. Letters 22, 245 (1993). [25] A. J. Archer, Phys. Rev. E 72, 051501 (2005). [26] S. van Teeffelen, N. Hoffmann, C. N. Likos, H. L¨owen, Europhysics Letters 75, 583 (2006). [27] Y. Rosenfeld, M. Schmidt, H. L¨ owen, P. Tarazona, Phys. Rev. E 55, 4245 (1997). [28] P. Tarazona. Phys. Rev. Letters 84, 694 (2000). [29] P. N. Pusey, in: Liquids, Freezing and Glass Transition, edited by J. P. Hansen, D. Levesque, J. Zinn-Justin, North Holland, Amsterdam, pages 145189, 1991. [30] A. J. Archer, R. Evans, J. Chem. Phys. 121, 4246 (2004). [31] U. M. B. Marconi, P. Tarazona, J. Chem. Phys. 110, 8032 (1999). [32] U. M. B. Marconi, P. Tarazona, J. Phys.: Condensed Matter 12, A413 (2000). [33] P. Espa˜ nol, H. L¨ owen, J. Chem. Phys. 131, 244101 (2009). [34] J. E. Hug, F. van Swol, C. F. Zukoski, Langmuir 11, 111 (1995). [35] R. J. Hunter, Foundations of Colloid Science. 2nd ed. Oxford University Press, Oxford, 1989. [36] N. Hoffmann, F. Ebert, C. N. Likos, H. L¨ owen, G. Maret, Phys. Rev. Letters 97, 078301 (2006). [37] L. Assoud, F. Ebert, P. Keim, R. Messina, G. Maret, H. L¨ owen, Phys. Rev. Letters 102, 238301 (2009). [38] S. van Teeffelen, C. N. Likos, H. L¨ owen, Phys. Rev. Letters 100, 108302 (2008). [39] G. Kahl, H. L¨ owen, J. Phys.: Condensed Matter 21, 464101 (2009).
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[40] S. van Teeffelen, R. Backofen, H. L¨ owen, A. Voigt, Phys. Rev. E 79, 051404 (2009). [41] M. Rex, H. L¨ owen, Phys. Rev. Letters 101, 148302 (2008). [42] D. Frenkel, in: Liquids, Freezing and Glass Transition, edited by J. P. Hansen, D. Levesque, J. Zinn-Justin, North Holland, Amsterdam, pages 689756, 1991. [43] L. Onsager, Proc. New York Acad. Sci. 51, 627 (1949). [44] P. Bolhuis, D. Frenkel, J. Chem. Phys 106, 666 (1997). [45] D. Frenkel, B. M. Mulder, J. P. McTague, Phys. Rev. Letters 52, 287 (1984). [46] A. Poniewierski, R. Holyst, Phys. Rev. Letters 61, 2461 (1988). [47] H. Graf, H. L¨ owen, J. Phys.: Condensed Matter, 11, 1435 (1999). [48] H. Hansen-Goos, K. Mecke, Phys. Rev. Letters 102, 018302 (2009). [49] M. Rex, H. H. Wensink, H. L¨ owen, Phys. Rev. E 76, 021403 (2007). [50] H. L¨ owen, Phys. Rev. E 50, 1232 (1994). [51] A. H¨ artel, H. L¨ owen, J. Phys.: Condensed Matter 22, 104112 (2010). [52] A. H¨ artel, R. Blaak, H. L¨ owen, Towing, breathing, splitting and overtaking in driven liquid crystals, Phys. Rev. E 81, 051703(1-5) (2010). [53] R. Dreyfus, J. Baudry, M. L. Roper, M. Fermigier, H. A. Stone, J. Bibette, Nature 437, 862-865 (2005). [54] A. Erbe, M. Zientara, L. Baraban, C. Kreidler, P. Leiderer, J. Phys.: Condensed Matter 20, 404215 (2008). [55] S. van Teeffelen, H. L¨ owen, Phys. Rev. E 78, 020101 (2008). [56] B. ten Hagen, S. van Teeffelen, H. L¨ owen, Condensed Matter Physics 12, 725-738 (2009). [57] H. H. Wensink, H. L¨ owen, Phys. Rev. E 78, 031409 (2008). [58] G. E. Morfill, A. V. Ivlev, Rev. Mod. Phys. 81, 1353-1404 (2009).
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Chapter 2 Polymer Phase Separation
Marcus M¨ uller Institut f¨ ur Theoretische Physik, Georg-August Universit¨ at, Friedrich-Hund-Platz 1, 37077 G¨ ottingen, Germany E-mail:
[email protected]
Computer simulations and numerical techniques for studying phase separation and interfacial properties of polymer blends and polymer-solvent mixtures are discussed and illustrated by a simulation study of coarsegrained models. The results of the simulation are quantitatively compared to the prediction of mean-field theory to identify the Flory-Huggins parameter. Different computational techniques for locating phase boundaries and free energies of interfaces between immiscible phases are described, and the relation between the bulk phase behavior, wetting and phase coexistence in confined geometry is explored.
1. Introduction Multicomponent mixtures comprising polymers find widespread use in industrial applications,1,2 e.g., purification, processing, or melt blending. The phase separation dictates the compositions of the coexisting phases, and the presence of interfaces between phases often dominate the material properties. Typically, additives of low molecular weight (e.g., solvents like hexadecane, toluene or carbon dioxide) can be dissolved in a polymer to some extent. These solvents are utilized to lower the glass transition temperature (i.e. plasticizer) and aid processability. If one increases the concentration further or changes the thermodynamic control variables — pressure or temperature — domains will form that are enriched in the polymer or preferentially contain solvent. For instance, this effect is utilized to fabricate 47
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polymer foams. A detailed knowledge of the phase behavior is crucial for understanding these processes. Blending polymers with different properties is a common strategy for producing materials3 with tailored properties that are difficult to achieve in a one-component material. A prototypical and industrially important example is the mixture of polystyrene (PS) with polybutadiene (PB). PS is a stiff and rather brittle material, which makes it inappropriate for many practical applications. A blend of it with the rubbery PB, however, meets the criterion of a tough material with a large stiffness. This rubber-toughened blend — high-impact PS — is fabricated in large amounts and used in a variety of applications (e.g., packaging materials). Unlike the situation in metallic alloys, however, different polymers of high molecular weight are typically immiscible but rather form domains, in which one or the other component is locally enriched. The size of the domains is kinetically determined and, typically, on the order of micrometers. Much effort has been targeted to tailor the property of the interfaces between the polymers, which, in turn, determine the properties of the material on macroscopic length scales. For instance, the lower the interfacial tension is, the smaller are the domains formed in shear flow (e.g., extruding the mixture).4 If the width of the interface is large enough for entanglements to form between different polymer species, the mechanical strength is greatly improved.5 Thus, the characterization, understanding and control of these properties on a microscopic level is of vital importance. Surface effects can dramatically alter the phase behavior compared to the bulk. The shift of the binodals, which characterize the composition of the two coexisting phases, upon confinement are illustrated in Fig. 1.7–9 At a solid substrate or the free surface of a polymer film to the vapor phase (vacuum), one component may enrich and form a wetting layer. The interplay between these surface effects and the interfaces in a phase-separated blend give rise to a complex phase behavior. These properties become increasingly important as the size of the confinement decreases and the sizes of the enrichment layers is comparable with the linear dimension of the confinement. For instance, in nano-fluidic devices the linear dimensions of channels may be on the same order than the size of a large macromolecule, or in a blend, which is loaded with a large volume fraction of nano-particles, the distance between the surfaces of nano-particles may shrink down to the molecules’ extensions. In this contribution, we will discuss simulation techniques to address the properties of phase-separating mixtures containing polymers. These com-
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49
80
kBT/ε
60
bulk L=2.8Re sym L=2.8Re anti
40
20
0 0.0
0.2
0.4
0.6
0.8
Twet
1.0
φ Fig. 1. Phase diagrams of a confined, symmetric, binary polymer blend. The bulk phase behavior, binodals for symmetric (capillary condensation) and strictly anti-symmetric (interface localization-delocalization transition) preference of the confining surfaces are shown. The arrow marks the wetting transition temperature, Twet , of the semi-infinite system. The film thickness, L, in units of the molecule’s end-to-end distance, Re , is given in the key. From M¨ uller and Binder.6
plex multi-component materials posses interesting properties on widely different time, length and energy scales and, consequently, different numerical techniques have been employed to investigate them. Predicting the thermodynamics of a mixture of two polymers requires information about the structure on the level of atomistic constituents and their mutual interactions. Typically, in the molten state, chemical repeat units attract each other, which makes the polymers condense into a dense liquid with vanishingly low vapor pressure. The attraction per chemical repeat unit is on the order of the thermal energy scale, kB T , where T denotes the temperature and kB is Boltzmann’s constant. This cohesive energy can be obtained by convoluting the interaction potential and the pair correlation function, which measures the probability of finding two repeat units a distance r apart. Even for the pure component, it is a rather challenging task to determine both quantities with sufficient accuracy in order to predict, for instance, the melt density.
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The free energy of mixing in a phase-separating mixture is on the order of kB T per molecule and thus, for a macromolecule of high molecular weight, this quantity only amounts to 10−3 − 10−5 kB T per chemical repeat unit. This tiny energy difference stems from the fact that like repeat units attract each other slightly more than unlike do. Moreover, minuscule changes in the packing, which are reflected in the pair correlation function, or changes of the molecular conformations upon mixing contribute to the thermodynamics of the blend. For most components of practical interest, neither the interaction potentials are known with sufficient accuracy nor it is possible to measure the free energy change upon mixing in simulations of atomistic resolution with the required precision. Fortunately, many properties of the phase behavior and the structure of interfaces between the coexisting phases in a polymer blend can be understood without resorting to an atomistic description. The large molecular weight imparts a large degree of universality on the structure and thermodynamics of phase-separating polymer systems, which makes them particularly amenable to a modeling on a coarse-grained level. Conceptually, these coarse-grained models are constructed by lumping a small number of chemical repeat units into an effective interaction center — denoted as segment in the following — and by calculating the effective interactions between these effective segments. The reduction of the number of degrees of freedom and the softness of the effective interactions on the coarse-grained level allow to investigate larger time and length scales by computer simulations. While this “systematic coarse-graining” procedure has successfully been applied to one-component systems,10,11 it meets significant challenges when applied to polymer blends: Besides the difficulties due to the lack of sufficiently accurate interaction potentials on the atomistic scale12 and the precision of the systematic coarse-graining procedure, the effective interactions on the coarse-grained scale are free energies rather than interaction energies. Generally, they are comprised of multi-body interactions instead of the pair-wise interactions between the constituents on the atomistic level, and they depend on the thermodynamic state (characterized by temperature and pressure), at which they have been derived.13,14 Given these difficulties of “deriving” the blend’s properties from the characteristics of the atomistic constituents, one often uses minimal, coarsegrained models15–17 that only incorporate the relevant interactions: connectivity of the repeat units along the macromolecule, repulsion between the unlike species, and near-incompressibility of the dense polymer liquid. The strength of each one of those interactions is characterized by a coarse-
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grained parameter or “invariant”.18 The justification for utilizing these coarse-grained models is twofold: On the one hand, many polymer polymer blends exhibit common phase behavior and structure in experiments, although their interactions on the atomistic level significantly differ. On the other hand, the structure of long macromolecules can be described by renormalization group theory,19 which exploits the scale-invariant structure of the polymer coil on the length scale ranging from a few chemical repeat units up to the size of the entire molecule. de Gennes demonstrated20,21 that the behavior of long, flexible macromolecules is formally similar to the universal behavior at phase transitions of second order, and this analogy identifies the relevant interaction that a minimal, coarse-grained model has to incorporate to reproduce the universal features of polymer melts.a In a dense melt, linearly connecting chemical repeat units into a flexible molecule gives rise to random walk-like conformations on large length 2 increases linearly with scales, i.e., the mean-square end-to-end distance R e e, the number of repeat units. The distribution of the end-to-end vector, R 2 2 is Gaussian and characterized by zero mean and variance Re ≡ Re . The Gaussian chain conformations on large length scales are completely described by the coarse-grained parameter, Re . This invariant, coarse-grained parameter sets the length scale and, for instance, dictates the length of composition fluctuations in a blend or the thickness of enrichment layers at a surface. In a dense polymer liquid, density fluctuations are small and the system is characterized by a uniform number density, ρo , of segments. This density cannot be used as an invariant, coarse-grained parameter because it depends on the definition of an effective segment. One rather uses the ¯ = (ρo R3 /N )2 , where N is the number of segments per molecule. quantity N e ρo /N is the polymer number density, which is independent from how many effective segments are distributed along the molecule’s contour. In a dense ¯ ∝ N is often referred to as the invariant demelt, Re2 ∝ N and, thus, N gree of polymerization. It measures the number of neighboring molecules, which are contained inside the volume Re3 of a polymer coil. In a typical ¯ = 103 − 106 adopts large values, which indicates experimental system, N a In this chapter, we mainly focus on polymer blends and only briefly discuss polymersolvent systems. Both systems share many qualitative characteristics and many simulation techniques for polymer blends can straightforwardly be applied to polymer-solvent mixtures. Polymer-solvent mixtures, however, typically do not exhibit the same degree of universality as polymer blends, because the interface between polymer and solvent is narrow and there is no separation of scales between solvent and atomistic structure.
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¯ is, the less that one molecule interacts with many neighbors. The larger N important become the long-ranged fluctuations. The strength of the repulsion between unlike molecules in a blend is characterized by χN , where χ denotes the Flory-Huggins parameter. This invariant, coarse-grained parameter sets the energy scale. It is defined by the free energy change upon mixing of a homogeneous system, which takes the form22,23 V ∆Fmix ¯ φ ln φ + (1 − φ) ln(1 − φ) + χN φ(1 − φ) (1) = N kB T Re3 where V is the volume of the system, and A and B denote the two species of homopolymers of the symmetric binary blend. 0 ≤ φA = φ ≤ 1 and φB = 1 − φA are the volume fractions (composition) of A- and B-segments, respectively, and both species are comprised of the same number of effective segments, N . If the constituents of the blend are structurally asymmetric, e.g., they differ in their chain extensions or segmental volumes, additional coarse-grained parameters are introduced to capture these asymmetries. The basic idea of minimal, coarse-grained models is that all systems (polymer blends in a laboratory or models utilized for computer simulations) with an identical set of invariant, coarse-grained parameters, Re , ¯ , and χN share the same equilibrium behavior. The information about N the detailed structure and interaction of the chemical repeat units on the atomistic levels is encoded in the small number of invariant, coarse-grained parameters and properties on larger length scales depend on the properties ¯ , and χN . Thus, minimal, coarseof chemical repeat units only via Re , N grained models are well suited to study the universal aspects of phase behavior, interface structure and wetting phenomena and are able to provide quantitative predictions about this class of materials on the length scale ranging from a few nanometers to microns. ¯ , and In addition to these invariant, coarse-grained parameters, Re , N χN , that describe the equilibrium behavior of a symmetric blend in the bulk, the model can be augmented by additional interactions. For example, if a polymer melt is in contact with a solid substrate, the density profile at the surface will rise to the bulk density of the melt over a short distance (e.g., smaller than nanometer). On this length scale, interactions on the atomistic scale between the polymer and the constituents of the solid substrate dictate the behavior and quantum chemical methods have to be employed to accurately predict them. A minimal, coarse-grained model, where each effective segment corresponds to a small number of chemical re-
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peat units, is not capable of capturing the fine structural re-arrangements on the sub-nanometer scale. In order to describe the formation of enrichment layers or wetting phenomena — collective phenomena which occur on a length scale set by Re — the dimensionless surface tension, γRe2 /kB T , serves to encode the atomistic structure at the surface. Once the invariant, coarse-grained parameters have been identified, computer simulations of minimal, coarse-grained models can predict the structure and thermodynamics of multi-component systems containing polymers in the bulk and under confinement. A particular advantage of computer simulations is the fact that the parameters can be independently and systematically varied and a multitude of structural, thermodynamic and kinetic properties on different length scales is simultaneously accessible, e.g., the molecular conformation at the interface between coexisting phases in a blend and the excess free energy of the interface. Therefore, computer simulations can provide insights of properties on the nanoscale that are difficult to directly observe in experiment, and they can correlate them with macroscopic thermodynamical behavior and structure. This information obtained by computer simulation allows for a stringent test of theories and the comparison of different models that give rise to the same invariant, coarse-grained parameters can investigate, which properties are universal, i.e., independent from the microscopic interactions of the specific model. In order to exploit this potential, computational models and simulation techniques shall address some of the following challenges: • The invariant, coarse-grained parameters shall be accurately determined by computer simulations. While the extension, Re , of the macromolecules ¯ , can be straightforwardly measured in exand the polymer density, N periments or computed in simulations, the Flory-Huggins parameter, χ, is commonly defined by comparing the results of experiments or simulations to the prediction of the mean-field theory of a binary polymer blend (cf. Eq. (1)). If this comparison is performed in a regime of the phase diagram and for quantities, where the approximations of the meanfield theory are justified, consistent results are obtained. We shall discuss various possibilities of identifying the Flory-Huggins parameter in Sec. 2.2. • The invariant, coarse-grained parameters shall adopt values that are characteristic for experimental systems. If the minimal, coarse-grained model incorporates harsh, excluded volume interactions between the ef-
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fective segments (e.g., the bond fluctuation model24,25 or a LennardJones bead-spring model26,27 ), the maximal density will be on the order of ρo b3 ∼ O(1), where b2 = Re2 /N denotes the statistical segment length. For flexible polymer models, b is comparable to the segment size. For larger densities, the liquid of segments will form a crystal or vitrify into a glass. In order to describe a polymer blend with an invariant degree of ¯ = 104 , which is quite typical for an experimental syspolymerization, N tem, the chain contour has to be comprised of many effective segments, N = N¯ /(ρo b3 ) ∼ O(104 ). This poses formidable computational problems: The relaxation time of chains with such a large value of N is enormous, τ ∼ Re2 /D ∼ N 3 , and a simulation box with√modest, linear dimension of ¯ (L/Re )3 N ∼ O(6 · 107 ), L = 4Re contains a huge number, ρo L3 = N of effective segments. Even on state-of-the-art supercomputers such calculations are not feasible. Thus, computer simulations of models with harsh, excluded volume interactions typically correspond to values of ¯ in the range 10 − 500, which is significantly smaller than experiN mental values. Therefore, long-ranged fluctuations are very important in these computational models; they are less important for experiments; and the mean-field theory, which ignores all fluctuations, corresponds ¯ → ∞. A low value of N ¯ is useful in order to highlight to the limit N the role of fluctuation effects and explore the limitations of mean-field theories. In order to achieve large values of N¯ in computational models, one can increase the polymer density ρo /N . A system of size L = 4Re and invari¯ = 104 , contains 6 400 polymer chains. If ant degree of polymerization, N one places N = 32 effective segments along the chain contour, which is just sufficient to approximate the random walk-like statistics of the chain conformations, then the system contains only 204 800 effective segments 3 ¯ but the segment density is ρo b = N /N ≈ 18 very large. Such values can only be achieved by replacing the excluded volume interaction of the segments by a soft repulsion without hard core. These soft potentials between effective segments on the coarse-grained scale arise naturally in the systematic coarse-graining procedure, where the degrees of freedom “inside an effective segment” are integrated out. Such soft potentials are typical for DPD models28 or models applied in conjunction with SingleChain-in-Mean-Field simulations.29–31 • Thermodynamic properties shall be accurately computed for a coarsegrained model. Since these thermodynamic characteristics involve the free energy, they are not simple functions of the segments’ coordinates,
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and special simulation techniques have been devised to locate phase coexistence or compute interface free energies. Some of those techniques are described in Sec. 2.3. • On accessible, human time scales, equilibrium on macroscopic length scales often cannot be attained. For instance, the coarsening of the domain structure in a binary blend arrests, once the domains grow to a size on the order of micrometers, and one experimentally observes that the macroscopic structure depends on processing history. Unfortunately, the kinetics of phase separation exhibits less degree of universality than the equilibrium phase behavior. While the gross features have been classified by Hohenberg and Halperin,32 the application of minimal, coarse-grained models for investigating the kinetics of phase separation is still in its infancy. On the one hand, it is not obvious that different dynamic process can be described by a single rescaling factor of time, which relates the time scale of the minimal, coarse-grained model to the experiment. The largest single-chain relaxation time, τ = Re2 /D (with D being the self-diffusion coefficient of the molecules) is an appropriate invariant for symmetric blends but generalizations to blends with a strong dynamical asymmetry (e.g., where one component vitrifies in the course of phase separation) are a challenge. On the other hand, the softer interactions on the coarse-grained scale between the effective segments may alter the qualitative characteristics of the dynamics. For instance, the combination of finite bond length and harsh, excluded volume interactions on the atomistic scale ensures that chain molecules cannot cross through each other in the course of their motion. This incrossability condition leads to the reptation dynamics of long macromolecules in a melt,33 where the dynamics of fluctuations perpendicular to the chain contour is limited by the topological constraints to a tube around the instantaneous conformation. Long, flexible molecules slither like snakes along their contours. Fortunately, the√tube diameter, dT , does not introduce a ¯ with the constant of proportionality new length scale but dT ∝ Re / N being universal, i.e., independent from the details of the chemical repeat units. In coarse-grained models, which are able to address large values of N¯ by replacing harsh, excluded volume interactions between effective segments by soft repulsion, non-crossability of bonds between effective segments is not “automatically” guaranteed. It has to be enforced by explicitly checking that bonds do not cross in the course of the simulation,34 or they have to be incorporated in an effective manner by slip-links.35
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2. Phase Behavior in the Bulk 2.1. Predictions of the mean-field theory The excess free energy of mixing, Eq. (1), of a molecule in a symmetric polymer blend is comprised of two terms. The first one, stems from the combinatorial entropy of the mixture due to the indistinguishability of different A-molecules and different B-molecules. It is on the order of kB T per molecule. The second contribution describes the interactions between the different polymer species and the Flory-Huggins parameter, χ, quantifies the repulsion between segments of different type. Generically, this repulsion of coarse-grained segments is comprised of enthalpic and entropic contributions. This simple, analytical mean-field prediction lays at the the basis of self-consistent field (SCF) calculations16–18,36–39 for spatially inhomogeneous polymer systems in the framework of the Gaussian chain model. Its predictions have been abidingly employed to analyze experiments on the phase behavior and interfacial properties. Some of the results are compiled below: • The critical point of a symmetric blend is located at: N ∼ χMF c N = 2 Tc
and
φMF = c
1 , 2
(2)
This result quantifies that already minuscule repulsions between segments of different species, which are only on the order of 1/N , give rise to phase separation, and it explains why most pairs of polymers do phase separate. • In a symmetric blend, the binodals are given by the implicit form χN =
1 φ ln 2φ − 1 1 − φ
(3)
Due to the mean-field approximation, the binodal in the vicinity of the critical point are parabolic, while much flatter binodals are observed in simulations (cf. Fig. 1) and experiments, which are a hallmark of the Ising universality class. • Since it is difficult to measure the free energy of mixing in computer simulations, one often considers the relation between the exchange potential ∆µ ≡ µA − µB and the composition of the mixture, where µA and µB denote the chemical potentials per molecule. 1 ∆µ ∂∆F √ = = ln φ − ln(1 − φ) + χN (1 − 2φ) ¯ V /R3 ∂φ kB T kB T N e
(4)
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composition fluctuation loop
(b)
(a)
Fig. 2. (a) Snapshot of an interface between two coexisting phases in a binary polymer blend in the bond fluctuation model.24 Within this coarse-grained lattice model, effective segments are represented by cubes on a lattice. Each segment blocks the 8 corners from double occupancy and segments along the chain are connected by one of 108 different bond vectors. In the symmetric binary blend, segments interact via a square-well potential that is extended over the nearest 54 lattice sites. Like segments attract each other with strength , while unlike species repel each other with . (invariant polymerization ¯ = 91, incompatibility χN ≈ 17, linear box dimension L ≈ 7.5Re , or numindex N ber of effective segments N = 32, interaction /kB T = 0.1, monomer number density ρ0 = 1/16.0). (b) Cartoon of the molecular configuration illustrating loops of a chain into the domain of opposite type, fluctuations of the local interface position (capillary waves) and composition fluctuations in the bulk and the shrinking of the chains in the minority phase. From M¨ uller.17
• The experimentally accessible, collective structure factor, S(k), of composition fluctuations is given by: N SRPA (k)
=
1 1 + − 2χN φgA (k) (1 − φ)gB (k)
(5)
in the homogeneous phase, χN < 2. gA (k) = SA /N denotes the normalized single-chain structure factor of A-polymers and k the wave-vector of composition fluctuations. If the chains obey Gaussian statistics, it is given by the Debye-function: 12 −(kRe )2 /6 2 − 1 + (kR ) /6 (6) e g(k) = e (kRe )4 • The properties of the interface are dominated by excursions of Apolymers into the B-rich phase, and vice versa. This is illustrated by
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a snapshot of a simulation in Fig. 2. The energy cost of each loop is comparable to the thermal energy scale, kB T . Each monomer of a loop contributes to this energy an amount of the order χkB T , and, consequentially, the number of monomers per loop scales like 1/χ. The spatial extension of the loops determines the interfacial width. In the Gaussian chain model the conformational distribution is Gaussian on all length scales, √ and therefore the spatial extension of loops is of the order Re / χN . In the limit of strong segregation (SSL), this quantity sets the width of the interface between coexisting phases: wSSL 1 = √ Re 6χN
(7)
• Each monomer in the interfacial region contributes to the interface ten√ sion an amount χ, and the interface tension scales as χρo Re / χN . The result of the SCF theory (including prefactor of order unity) is:40 σ R2 χN √SSL e = (8) ¯ 6 N kB T ¯ , the FloryUnlike the other coarse-grained parameters, Re and N Huggins parameter is defined by comparing experimental or simulation data to the prediction of the mean-field theory. We emphasize, however, that these predictions are derived invoking a mean-field approximation (or even stronger assumptions) in order to obtain a tractable, analytical description. Of course, one should only use these mean-field predictions to identify invariant, coarse-grained parameters in the regime of validity of the mean-field approximation, i.e., the regime where long-ranged fluctuations are of negligible importance. In the hypothetical limit of large invariant de¯ → ∞, any of the above quantities can be utilized gree of polymerization, N to identify the Flory-Huggins parameter, χ. Unfortunately, large degrees of polymerization are often not accessible in computer simulations, fluctuations are important, and the extrapolation of simulation results to N¯ → ∞ can be difficult because the convergence towards the asymptotic behavior may set in only for large chain lengths, N . Moreover, the predictions above are affected to a different extent by the mean-field approximation. Thus, identifying χ via the comparison between simulation results and mean-field ¯ → ∞ or paying special attention to predictions without extrapolating to N the role of fluctuation effects might result in significant errors and different quantities might yield different, mutually inconsistent estimates.
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3
MC (L=96)
6
2 0
0
3 0
0.5
1
φA
-1
ε=0.004kBT ε=0.007kBT 2.44(2φA-1)
-2 -3 0.0
0.2
0.4
φA
(a)
0.6
0.8
2
8
4
0
6 5
0
4
1
8
3 2
FH
1 0
1.0
~ χN
4
~ χN
1
FH 3D Ising (FSS fit)
4
N/S(k=0)
~ χkBT/ε
(kBTln[φA/φB]-∆µ)/2Nε
2
59
ln[φA/(1-φA)]/(2φA-1)
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0 0
0
0.2 0.4φ 0.6 0.8 A
MC
1
1 ~ χN=2Nzcε/kT with zc=2.44
2
(b)
Fig. 3. Estimation of the Flory-Huggins parameter for a symmetric polymer blend with chain lengths NA = NB = 64 obtained by semi-grandcanonical simulations of the bond fluctuation model. (a) First derivative of the Flory-Huggins free energy. The slope of the curve yields the effective coordination number zc = 2.44. The inset presents the value of the coordination number zc = χT /2 extracted from the collective structure factor. The horizontal line marks the estimate from the main panel. (b) Inverse maximum of the collective structure factor of composition fluctuations, N/S(k → 0), as a function of the incompatibility, χN . Symbols correspond to MC simulations for system size L = 96, the dashed curve presents the results of a finite size scaling analysis of the simulation data in the vicinity of the critical point, and the straight, solid line indicates the prediction of the Flory-Huggins theory. The critical incompatibility predicted by the Flory-Huggins ¯ ≈ 240) are theory and obtained from MC simulations of the bond fluctuation model (N indicated by arrows. The left inset compares the phase diagram obtained from simulations with the prediction of the Flory-Huggins theory. The right inset replots the composition of the coexistence compositions such that the mean-field theory predicts them to fall onto a straight line.
2.2. Estimating the Flory-Huggins parameter for simulation models A choice of possible identifications of the Flory-Huggins parameter is given below: (1) The relation between exchange potential ∆µ and composition φ¯A of the blend, cf. Eq. (4), or composition fluctuations/scattering, cf. Eq. (5), in the one-phase region far away from the critical point can be used to determine χ. (2) Measuring the composition of the two coexisting phases at large incompatibility and comparing the result to the prediction of the FloryHuggins theory, Eq. (3), one can estimate the χ-parameter. (3) The mean-field theory also relates the location of the critical point of demixing (cf. Eq. (2)) and scattering intensity of composition fluctuations to the Flory-Huggins parameter.
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(4) The SCF theory makes explicit predictions for the width of the interface between coexisting phases (cf. Eq. (7)). Comparing experimental measurements or simulation results to those predictions also yields an estimate for χ. All estimates basically treat the Flory-Huggins parameter, χ, as a phenomenological quantity, determined by matching the properties of the blend to the predictions of the SCF theory and thereby absorbing all unknown information concerning blend miscibility.41 This procedure is well adopted to the concept of coarse-grained models, where the structure on the microscopic length scale determines the behavior on the large length scale only via a small number of parameters. The latter two approaches are, however, affected by long-wavelength fluctuations, namely composition fluctuations near critical points and capillary waves at interfaces (to be illustrated in Sec. 3). Measuring the dependence of the composition on the exchange potential is straightforward in the semi-grandcanonical ensemble, where the exchange potential ∆µ, is a control parameter and one observes the composition and its fluctuations in the simulation.42,43 The dependence φA (∆µ) (cf. Eq. (4)) and the composition fluctuations φ2A − φA 2 (cf. Eq. (5)) correspond to the first and second derivative of the Flory-Huggins free energy of mixing with respect to the composition, and therefore constitute a direct test of the Flory-Huggins free energy.22,23 Simulation data for the bond fluctuation model are presented in Fig. 3(a), where we plot the difference between the exchange chemical potential and the ideal gas contribution, (∆µ − kB T ln[φ¯A /φ¯B ])/2N , vs the composition φ¯A , where measures the repulsion between unlike segment species in this coarse-grained lattice model. In accord with the FloryHuggins estimate, Eq. (4), we observe a linear dependence on composition for all temperatures far above the demixing transition. This suggests that χ ∼ /kB T in our model at least for small values of , and we can identify the Flory-Huggins parameter, χ = 2zc /kB T with an effective coordination number zc = 2.44. In the inset of Fig. 3(a) we present the estimate for the effective coordination number via the inverse of the collective structure factor, Eq. (5). Within the statistical uncertainties, we again find that χ ∼ /kB T with a slight upwards parabolic correction for asymmetric compositions. This parabolic composition-dependence becomes stronger as we approach the critical point.44,45 Far above the critical temperature, however, the ratio
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adopts the value χkB T / = 2zc ≈ 4.88 consistent with the results of the main panel. Even though composition fluctuations are directly related to the experimental procedure, we prefer to extract the Flory-Huggins parameter from the excess of the exchange potential, Eq. (4), in the simulations, because the translational contribution diverges only logarithmically for φA → 0 or 1. Also in experiments, strongly asymmetric compositions might induce inaccuracies46 when one utilizes the structure factor, Eq. (5). Moreover, the composition is a self-averaging quantity that can be obtained with high precision in the simulations while the error of composition fluctuations, S(k = 0), does not decrease with system size. In Fig. 3(b) we use this estimate of the Flory-Huggins parameter, χ = 2zc /kB T , to plot the inverse structure factor as a function of χN . Far above the critical temperature, χN χc N = 2, the data are consistently describable by the mean-field theory, i.e., for φA = 1/2 we find the simple relation, N/S(k = 0) = 4 − 2χN . This plot constitutes a common experimental procedure to measure the Flory-Huggins parameter. In the ultimate vicinity of the critical point, however, deviations from the SCF theory become notable. The inverse structure factor extracted from the MC simulations does not vanish at χSCF N = 2, and the mean-field theory c overestimates the true critical temperature by a factor 1.12 for the chain length, N = 64 utilized in the simulation. Note, that this value corresponds ¯ = 240. Similar to simulations44 to an invariant polymerization index, N 47–50 and experiments, we observe in Fig. 3(b) the crossover from the meanfield dependence to the Ising behavior: If we approach the critical point at the critical composition, φA = 1/2, we will not observe a linear dependence of the inverse scattering factor with /kB T up to the ultimate vicinity of the critical point, but we rather find 1/S(k → 0) ∼ ( /kB T − /kB Tc )γ , where γ ≈ 1.24 denotes the critical exponent of the order parameter fluctuations in the three dimensional Ising model. By virtue of the universality at second order phase transitions, a polymer mixture exhibits the same critical behavior as other mixtures characterized by a single scalar order parameter. The SCF theory, being a mean-field theory and assuming random mixing, cannot describe these critical, long-ranged composition fluctuations. The SCF theory predicts a parabolic shape of the binodal at the critB ical point, while simulations and experiments observe |φA coex − φcoex | ∼ β ( /kB Tc − /kB T ) , where the critical exponent of the order parameter takes the value of the three-dimensional Ising universality class, β ≈ 0.32.
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The binodals are presented in the insets of Fig. 3(b) for N = 64 (and in Fig. 1 for N = 32). Note that the deviations are even more pronounced in thin films (see Fig. 1), where the binodals are characterized by the exponent β = 1/8 characteristic of the universality class of the two-dimensional Ising model. While the SCF theory fails in the ultimate vicinity of the critical point, at larger χN the binodals are well described by the SCF theory. We expect the region where long-wavelength composition fluctuations lead to a break-down of the SCF theory to become even smaller for larger invariant polymerization index, which is characteristic for experimental systems. ¯A )] φ In the left inset of Fig. 3(b) we plot the binodals in the form ln[φA2φ/(1− A −1 vs χN . Within the mean-field theory, all data fall onto a line with unit slope. This constitutes an alternative possibility to extract the Flory-Huggins parameter for a phase-separated blend. At low incompatibilities, in the ultimate vicinity of the critical point, we find deviations due to long-ranged fluctuations as discussed above. At very large incompatibilities, the FloryHuggins parameter becomes of the order unity for the short chain length considered in the simulations and, then, the repulsion between unlike segments is strong enough to change the structure and packing in the liquid of segments. To summarize, fitting the results of simulations to the predictions of the SCF theory far away from the critical point yields consistent results, i.e., all observations can be described by a Flory-Huggins parameter, which is proportional to the repulsion between unlike segments, /kB T for small values of . In the ultimate vicinity of the critical point, however, neglect of longranged composition fluctuations by the SCF theory affects the estimate of the Flory-Huggins parameter. Estimating the Flory-Huggins parameter from properties in the ultimate vicinity of the critical point leads to spurious effects due to fitting a mean-field theory to data which exhibit critical fluctuations according to the universality class of the three-dimensional Ising model. The estimate of the Flory-Huggins parameter does not only account for short-ranged correlations but also incorporates some effects of long-ranged critical fluctuations. If we defined the Flory-Huggins parameter via the critical properties, the Flory-Huggins parameter would not give an appropriate description further away from the critical point or for larger invariant degrees of polymerization because the effect of long-ranged composition fluctuations would become negligible but short-range correlations would remain important.
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2.3. Simulation techniques for computing the bulk phase behavior The study of phase coexistence by computer simulation has played a central role in the study of condensed-matter systems. In order to obtain a very rough estimate of the properties of the coexisting phases without applying any special simulation technique, one can consider the coexistence between two domains in a simulation cell. By exchanging particles across the interface and adjusting the volume of the coexisting domains, one establishes chemical and mechanical equilibrium. The size of the simulation cell has to be chosen large enough for the volume fraction of the interfaces, which separate the two domains, be negligible and the properties in the interior of the domains be not influenced by the presence of the interfaces. Rather large system sizes are necessary in order to obtain accurate results, and the equilibration across the interface via the diffusive motion of the polymers is an extremely slow process. Even within the framework of computationally efficient, coarse-grained models, this approach is only a viable simulation strategy for small oligomers. In the late 80’s, the Gibbs ensemble simulation technique51 has been devised, which permits the direct calculation of phase coexistence properties. Instead of separating the two coexisting domains via interfaces, one places them in two, separate simulation cells. In each simulation cell, periodic boundary conditions are applied to mimic the behavior of an infinitely extended, bulk system. If the segregation between the phases is sufficiently strong, each cell will contain one phase but no interface. Since one studies the bulk properties of a phase, significantly smaller system size already yield accurate results. Chemical and mechanical equilibrium between the two simulation cells are established by exchanging particles and volume between the two cells. If the overall composition of the system, i.e., the volume-weighted average of the two simulation cells, is located somewhere inside the miscibility gap, one will observe phase coexistence and the two coexisting phases will form in the corresponding simulation cells. Thus, the knowledge of the coexistence pressure or chemical potential is not required, which makes this a rather robust method. A powerful and popular alternative are advanced thermodynamic integration methods that relate the two coexisting phases via a reversible path. In the context of a binary polymer blend, simulations in the semigrandcanonical ensemble in conjunction with reweighting techniques have proven particularly useful. In the semi-grandcanonical ensemble, which has
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been introduced by Sariban and Binder in the late 80’s,44,52 the temperature, T , the total number of polymers, nA + nB , and the volume, V , are A fluctuates and it is fixed. The composition of the mixture, φ ≡ nAn+n B controlled by the exchange chemical potential, ∆µ (see Eq. (4)). While we illustrate this technique for the specific example of a symmetric, binary polymer blend, the simulation strategy carries over to other systems by replacing the composition φ by the order parameter that distinguishes the two coexisting phase (e.g., the total number of particles in case of a liquid-vapor transition) and ∆µ by the thermodynamically conjugated variable (e.g., the chemical potential in case of a liquid-vapor transition). In order to realize such an ensemble in the framework of Monte Carlo simulations, two types of Monte Carlo moves have to be utilized — canonical moves, which alter the chain confirmations, and semi-grandcanonical moves, which relax the order parameter. In the case of a symmetric polymer blend, the latter moves merely consists in switching the labels A B at fixed conformation of the molecules. For structurally asymmetric mixtures generalizations can be devised for special cases.42,43,53,54 One advantage of the semi-grandcanonical ensemble consists in the fast relaxation of fluctuations of the order parameter. Rather than decaying via the slow diffusion of long macromolecules in a dense melt, composition fluctuations relax via relabeling. While those Monte Carlo moves do not mimic the kinetics of composition fluctuations in an experimental system, where the order parameter is locally conserved, they do allow for an extremely efficient sampling of configuration space. In the simulations, one monitors the probability distribution, P (φ), of the order parameter. In the vicinity of phase coexistence, this distribution is bimodal (see Fig. 4). Each peak corresponds to one of the coexisting phases. If one increases the exchange chemical potential, ∆µ, the weight of the peak, which corresponds to the A-rich phase increases. Two phases coexist, if at constant temperature and (exchange) chemical potential, they will result in the same (osmotic) pressure. The latter quantity is related to the (semi-)grandcanonical energy, G(∆µ, V, T ), via G(∆µ, V, T ) πV =− = ln Z dφ P (φ) kB T kB T A-rich peak
(9)
where Z(∆µ, V, T ) denotes the partition function of the (semi-) grandcanonical ensemble. The condition of equal pressure between coexisting phases leads to the equal weight rule: Two phases will coexist, if the (exchange) chemical potential is adjusted such that the weights of the two
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probability P(φ)
10
65
0
2
exp(2Lσ/kT)
10
-10
10
-20
0.0
0.2
0.4 0.6 composition φ
0.8
1.0
Fig. 4. Probability distribution of the composition at = 0.02kB T (i.e., T /Tcbulk = 0.7215) in a system of geometry 3.8Re ×3.8Re ×7.5Re with periodic boundary conditions in all three directions. The two peaks correspond to the A-rich and the B-rich phase as indicated by the snapshots. In the vicinity of φ = 1/2 the typical configuration consists in two domains separated by two AB interfaces of size L2 . The slab configuration — an A-rich domain that spans the simulation cell in the two shorter directions and separated by two interfaces from a B-rich domain — is stable over a larger range of composition and for symmetric composition, φ = 1/2, the two AB-interfaces are farther apart than in a cubic cell, thereby, reducing the interactions between the two interfaces. From M¨ uller and Binder.55
peaks in the probability distribution of the order parameter are equal, i.e. (10) dφ P (φ; ∆µcoex , V, T ) = dφ P (φ; ∆µcoex , V, T ) B-rich peak A-rich peak It has been demonstrated that away from the critical point, where the probability distribution exhibits a pronounced two-peak structure, this estimate for coexistence value of the (exchange) chemical potential, ∆µcoex , has only exponentially small finite size effects.56,57 This general strategy has become very popular because of two advanced simulation and analysis techniques: (i) weighted histogram analysis and (ii) multicanonical simulations. The first method, introduced by Ferrenberg and Swendsen in the late 80’s,58,59 provides a simple means for extrapolating data generated at one set of control parameters to nearby points in thermodynamic space. Specifically, for the example discussed above, there is no need to perform many different simulations for different values of the
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exchange potential, ∆µ, in order to accurately locate phase coexistence according to the equal weight criterion (10), but if the simulation have been performed at value ∆µ and the corresponding probability distribution, P (φ; ∆µ, V, T ), has been sampled, the probability distribution at a slightly different value ∆µ is given by: A P (φ; ∆µ, V, T ) exp (∆µ k−∆µ)n BT (11) P (φ; ∆µ , V, T ) = 1 (∆µ −∆µ)nA dφ P (φ; ∆µ, V, T ) exp kB T 0 √ ¯ V /Re3 is the number of A-polymers in the simulation cell. where nA = φ N The same idea can be utilized to combine simulation data generated at different values of the control parameter, ∆µ. Consider a series of r semi-grandcanonical simulations at different ∆µi with i = 1, · · · , r. The histograms of the number of A-polymers are sorted and stored in the form, Hi (nA ) = H(nA , ∆µi , V, T ). Let #i = nA Hi (nA ) denote the number th of entries in the i histogram. Then, the results of the different simulations can be combined to yield an accurate estimate of the unnormalized probability distribution r ∆µnA H (n ) exp i=1 i A kB T P˜ (φ, ∆µ, V, T ) = (12) r ∆µi nA −Gi i=1 #i exp kB T where the value of the free energy, Gi , of the system at exchange chemical potential, ∆µi , has to be determined self-consistently by iterating the above equation and the normalization condition
√¯ NV −Gi = dφ P˜ (φ, ∆µi , V, T ) exp (13) kB T Re3 The second advance — multicanonical simulation technique — addresses the issue of efficiently sampling the bimodal probability distribution of the order parameter: In order to establish the equal weight criterion, the system must often tunnel between the two peaks of P (φ) in the course of the simulation. As can be seen in Fig. 4, the probability to find the system between the two peaks is excessively low. Sampling the states with the Boltzmann weight, P (φ), one will not encounter configurations with a φA
+φB coex
composition around coex 2 one of the two phases.
but the system will rather remain stuck in
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In order to mitigate this problem, one can sample configurations {ri }nA with a weight
−E[{ri }nA ] + ∆µnA (14) Pmuca[{ri }nA ] ∼ exp × e−w(nA ) kB T The first factor characterizes the weight of a configuration comprised of nA polymers of type A and energy E in the semi-grandcanonical ensemble. The second, “re-weighting” factor modifies the Boltzmann weight and it only depends on the number, nA , but not on the detailed conformations of the polymers. Thus, if one averages any property over configurations with a fixed number of A-polymers, one will obtain a result that corresponds to the canonical ensemble. The probability, √ Pmuca (φ) of finding the system ¯ V /R3 , of A-polymers — i.e., a in a state with a fixed number, nA = φ N e canonical slice of the configuration space — will differ from the probability, P (φ), in the semi-grandcanonical ensemble. However, both quantities are related by the simple formula Pmuca (φ) ∼ P (φ)e−w(φ)
(15)
In order to sample the peaks of P (φ) and the region between them efficiently in a multicanonical simulation, Pmuca (φ) must be approximately flat, i.e., independent from φ.b Thus, we should choose the re-weighting factors such that P (φ) + const ≈ ln P (φ) (16) w(φ) = ln Pmuca (φ) Formally, these equations are valid for an arbitrary choice of the reweighting factors but an inappropriate, “bad” choice does not allow for a sufficient sampling of the interesting interval of nA -values. The problem of sampling the probability distribution, P (φ), is therefore shifted to that of obtaining appropriate re-weighting factors, w. An optimal choice of w(φ), however, requires a working estimate of the probability distribution, P . Several strategies have been devised to overcome this dilemma: (i) multicanonical recursion,61 (ii) Wang-Landau sampling,62 (iii) transition matrix Monte Carlo63 or (iv) successive umbrella sampling.64 We refer the reader to the literature65 for a thorough discussion of the advantages of the the different methods in terms of simplicity of implementation, efficient use of information, detailed-balance issues and convergence rate. In the following, we briefly describe the successive umbrella sampling technique, which b Alternatively, one can require that the number of transitions from the A-rich to the B-rich phase and vice versa be maximal.60
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is very easy to implement. The guiding idea of umbrella sampling66 is to divide the pertinent range of the order parameter, φ, into smaller windows and to investigate one small window after the other. If the windows are sufficiently narrow, the probability P (φ) will not substantially vary within a window, and a crude estimate of the re-weighting factors, e.g., w(φ) =constant within a window, is often sufficient to ensure sampling of all values of the order parameter within a window. In the limiting case, the windows just consist of two neighboring values of the order parameter, nA and nA + 1. A histogram Hi (nA ) monitors how often each state is visited in the + ith window, n− i ≤ nA ≤ ni . Care must be exercised at the boundaries of a window to fulfill detailed balance:67,68 If a semi-grandcanonical identity switch attempts to leave the window, it is rejected and H(window edge) is incremented by unity. Another question, which may arise from the discussion of the boundary is the optimum amount of overlap to minimize the uncertainty of the overall ratio. Here we choose the minimal overlap − of one state at the interval boundaries, i.e., n+ i = ni+1 . This is simple to implement and sufficient to match the probability distributions at their boundaries. A larger overlap may reduce the uncertainty but requires a higher computational effort. + th Let Hi− ≡ Hi (n− i ) and Hi+ ≡ Hi (ni ) denote the values of the i histogram at its left and right boundary, and Ri ≡ Hi+ /Hi− characterize their ratio. After a predetermined number of Monte Carlo steps per window, the (unnormalized) probability distribution can recursively be estimated according to: Hi (nA ) Hi (nA ) P (nA ) H0+ H1+ · ··· = Πi−1 k=1 Rk · − = H0− H1− Hi− Hi− P (n0 )
with Ri ≡
Hi+ , Hi− (17)
+ when nA ∈ [n− i : ni ]. The ratios in Eq. (17) correspond to the Boltzmann factor associated with the free energy difference across the order parameter interval i. We now consider how the overall error depends on the choice of the window size:64 For clarity, we assume that no sampling difficulties are encountered in our system, i.e., the order parameter in which we reweight (i.e., number of A-polymers) is suitable to flatten and overcome all barriers in the (multidimensional) free energy landscape and the restriction of fluctuations of the average order parameter by the window size does not impart sam-
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pling errors onto the simulation.c Under these circumstances, it has been suggested69,70 that small windows reduce computational effort by a factor of r, where r denotes the total number of windows into which the sampling range is subdivided: The time τ to obtain a predetermined standard deviation δ of the ratio Ri in a single window is proportional to the square of − 2 the window size, ∆i = n+ i − ni . With τ ∼ ∆i , we get a total computation 2 time (for all windows) of tcpu ∼ r∆i , implying that the overall error of the simulation is also δ. This contrasts the behavior of a single large window; r = 1 and ∆i = r∆i yield tcpu = (r∆i )2 = rtcpu , which suggests that a window size as small as possible be chosen.69,70 Since we are interested in localizing phase coexistence, however, the pertinent error is related to the free energy difference of the two end points of the order parameter interval, i.e., the peaks of the probability distribution that correspond to the distinct phases. Therefore, we have to account for error propagation in Eq. (17) and − we obtain for the error, ∆, of the ratio P (n+ r )/P (n1 ) r √ ) P (n+ r = ∆≡δ δRi2 ∼ O(δ r). (18) − P (n1 ) i=1 Due to error propagation across the windows, the error of each individual √ subinterval has to be smaller than the total error, ∆, by a factor of r. Thus, the time that must be spent in each window to achieve an error √ of δ = ∆/ r is rτ and the total simulational effort is r2 τ ∼ (r∆)2 ∼ rtcpu , which is identical to the time required for a single large window. This argument demonstrates that the statistical error for a given computational effort is independent from the window size, i.e., the number of intervals the range of order parameter is divided into. A more detailed analysis including possible systematic errors due to (i) estimating probability ratios and (ii) correlations between successive windows can be found in Ref. 64. The computational advantage from subdividing the range of order parameter into windows does not stem from an increased statistical accuracy due to the small correlation times within a window, but from the fact that successive umbrella sampling does not require an independent and comc Restricting
the window size limits order parameter fluctuations to far less than those explored in a semi-grandcanonical simulation and each sub-simulation resembles more closely a simulation in the canonical ensemble than in the semi-grandcanonical ensemble. We emphasize, however, that local composition (order parameter) fluctuations are not restricted and that, ideally, configurations with a fixed order parameter have identical statistical weight in the canonical ensemble, in the ensemble used in our simulation and in the semi-grandcanonical ensemble.
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putationally costly generation of re-weighting factors. Making the window size small, we minimize the variation of the free energy across a single interval and achieve sufficiently uniform sampling without re-weighting factors. Therefore, successive umbrella sampling is as efficient as a multicanonical simulation with very good re-weighting factors, except that now w needs not be known beforehand. Moreover, the scheme is easy to implement on parallel computers and the range of order parameter can be easily enlarged by simply adding more windows. This computational scheme has successfully been applied to study phase equilibria and interface properties in polymersolvent mixtures,64,71 polymer-colloid mixtures72–75 and liquid crystals.76,77 2.4. Compressible mixtures The phase behavior of binary mixtures will become much more complex, if one relaxes the assumption of incompressibility. In this case, the free energy of mixing, ∆Fmix , depends on two order parameters — density and composition (ρ, φ) or the two densities of the species (ρA , ρB ). More importantly, the functional form of the free energy of mixing is much more complex, because both liquid-liquid demixing (as previously discussed for incompressible blends) and liquid-vapor phase separation are simultaneously possible. There is no single, coarse-grained parameter that characterizes the interaction between the molecules and dictates the phase behavior like the Flory-Huggins parameter in the incompressible limit. In fact, the interplay between liquid-liquid and liquid-vapor types of phase separation gives rise to six, qualitatively different types of phase behavior in the classification scheme of Konynenburg and Scott.78 Therefore, the phase behavior does depend on the detailed interactions of the computational model and a coarse-grained description has to be constructed with great care.71,79–83 Table 1. Critical points of the pure components from MC simulations and experiments. By comparing the critical temperatures from simulation and experiment, we identify CO2 = 4.201 · 10−21 J, HD = 5.787 · 10−21 J and CO2 /HD = 0.726. From the critical densities we derive σCO2 = 3.693 · 10−10 m, σHD = 4.523 · 10−10 m and σCO2 /σHD = 0.816. ρ refers to segment number densities. Tcrit ρcrit pcrit MC CO2 0.999 k 0.32 σ13 0.088 σ3 g EXP CO2 304 K 0.464 cm3 73.87 bar MC HD 1.725 k 0.27 σ13 0.022 σ3 g EXP HD 723 K 0.219 cm3 13.98 bar
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To give a specific example, we shall discuss the mixture of hexadecane (HD), a short alkane, and carbon dioxide (CO2 ). Carbon dioxide has attracted much recent interest because it is environmentally benign (i.e., nonpoisonous) and small pressure changes give rise to pronounced changes of the physical properties in the supercritical region. The polymer is represented by a flexible, Lennard-Jones bead spring model. We utilize a chain length of N = 5 effective segments, which roughly mimics the geometrical shape of a hexadecane molecule. Comparing the two parameters of the Lennard-Jones potential, and σ, with experimental results for the critical temperature and pressure, we identify energy and length scales.71,81 Carbon dioxide is represented by a single Lennard-Jones interaction center; the interactions between CO2 -beads are adjusted to reproduce the experimental critical point. A modified Lorentz-Berthelot mixing rule is used to describe the interactions between hexadecane and CO2 beads:84,85 σHD + σCO2 √ σHD−CO2 = (19) and HD−CO2 = ξ HD CO2 2 where ξ describes the deviations from the standard mixing rule. The semi-grandcanonical simulation techniques can be straightforwardly transferred to compressible mixtures, where one controls the two chemical potentials of the individual species and the number of polymer molecules and carbon dioxide molecules fluctuates. In order to insert (or remove) a macromolecule into a dense liquid, the configurational bias algorithm86 is employed in order to generate trial configurations that do not strongly overlap with the other particles of the liquid. In the simulation, one monitors the joint probability distribution of the two order parameters, ρHD and ρCO2 , and adjusts via weighted histogram analysis the chemical potentials in order for the peaks in the ρHD -ρCO2 plane to have equal weight (cf. Eq. (10)). This procedure is illustrated in Fig. 5, where this two-dimensional probability distribution is depicted. The data are close to the critical point and the two peaks in the finite-sized system indicate the compositions of the two phase, which will coexist in the vicinity of the critical point. At low temperatures (top/left panel), the two phases are comprised of a dense carbon dioxide liquid and a liquid that contains both, polymer and CO2 , i.e., liquid-liquid phase separation occurs into a liquid solvent and a dense polymer-solvent mixture. Upon increasing the temperature, the character of the phase coexistence gradually changes and, at the highest temperature (bottom/right panel), the two coexisting phases correspond to a liquid and a vapor of polymer.
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Fig. 5. Joint probability distribution of solvent and polymer density along the critical line in a finite-sized simulation cell for ξ = 0.886. The distributions correspond to temperatures T = 314K, 356K, 398K, 486K, 650K and 713K from up/left to down/bottom. For temperatures T = 314K and 356K the box size is L = 6.74σP while L = 9σP was employed for other temperatures. Reprinted with permission from Virnau et al.81 Copyright 2004 American Institute of Physics.
Fig. 6 shows projections of the phase diagram in the plane of temperature, T , and pressure, p, for two choices of ξ. The value ξ = 1 corresponds to the standard Berthelot rule, which describes the mixing of van-der-Waals interactions, while the value ξ = 0.886 matches the experimental behavior87,88 most closely. Solid lines represent the liquid-vapor phase equilibria of the two pure components that end in critical points marked by arrows. When a small amount of solvent is added to the pure polymer, the liquidvapor coexistence shifts and so does the critical point. The loci of critical points for the binary system form a critical line that is shown by the dashed line with squares for ξ = 1 and triangles for ξ = 0.886. In the former case — phase behavior of type I — the critical line connects the critical points of the two pure components and the two coexisting phases gradually change from vapor and solvent-rich liquid to vapor and polymer-rich liquid. This is not observed in experiments, where a phase diagram of type III is encountered. In a type-III diagram, the critical line that emerges from the liquid-vapor critical point of the polymer is not connected to the critical point of the pure solvent, but it gradually changes its character from a liquid-vapor co-
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sim: lv coexist. [=1.000 [=0.886
300 pressure p [bar]
73
200
100 critical point CO2
0 200
300
critical point C16H34
400 500 600 temperature T [K]
700
800
Fig. 6. Projection of the global phase diagram for a compressible mixture of hexadecane, C16 H34 , and carbon dioxide, CO2 , into the temperature-pressure plane for two values of the mixing parameter, ξ = 1 (square) and ξ = 0.886 (triangle). Simulation results for the liquid-vapor coexistence of the pure components are shown by solid lines and end in critical points that are indicated by arrows. The line of critical points that emerges from the critical point of the less volatile polymer component is indicated by symbols. Reprinted with permission from Virnau et al.81 Copyright 2004 American Institute of Physics.
existence to a liquid-liquid coexistence between a dense solvent-liquid and a polymer-liquid. A minuscule change of the interaction between the different components from ξ = 1 to ξ = 0.886 qualitatively alters the type of the phase diagram. The reduction of the attractions between CO2 and hexadecane can be traced back to quadrupolar interactions between CO2 molecules.83,89 The well depth CO2 of the pure solvent parameterized the attraction between carbon dioxide molecules that is caused by the joined effect of both, van-der-Waals and quadrupole interactions. The Berthelot mixing rule implicitly stipulates that all attractions result from van-derWaals interactions and utilizes the strength CO2 , which only partially stems from dipolar interactions. If one used a coarse-grained representation of the carbon dioxide that takes due account of the quadrupolar interactions, one could predict the phase behavior without introducing a modification, ξ, of the mixing rule. This has been confirmed by recent simulations and greatly improves the predictive power of the coarse-grained modeling.83
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3. Outlook: Interfacial Properties, Phase Boundaries in Confined Geometry, and Wetting As illustrated in Fig. 4, the transition from the A-rich phase to the B-rich phase in the course of a semi-grandcanonical simulation of a symmetric, binary polymer blend occurs via configurations that are comprised of two domains — an A-rich and a B-rich domain — that are separated by two interfaces in a system with periodic boundary conditions. Thus, the simulation technique does not only accurately locate the bulk phase behavior but additionally yields information about the interfacial properties. The configurational snapshots in Fig. 4 suggest that the free energy barrier for tunneling between the two homogeneous phases is associated with the excess free energy of the two interfaces separating the A-rich and B-rich domains. This is further corroborated by the dependence of the free energy profile, F (φ) = −kB T ln P (φ), on the composition. Around the φA
+φB
composition of the rectilinear diameter, coex 2 coex = 0.5, the free energy profile is flat, i.e., one can change the composition and thereby the distance between the two AB-interfaces at no free energy costs. Therefore, the two interfaces do not interact, and the excess free energy is simply the sum of two, independent contributions.90 2 × L2 σAB is the interfacial tension and σAB
B P (φA coex )+P (φcoex ) kB T kB T ln L 2 = ln + O B φA 2L2 L2 coex +φcoex P 2
(20)
where L2 denotes the (smallest) cross-section of the simulation cell. In Fig. 4, we have utilized a cuboidal shape of the simulation cell of geometry L × L × 2L. This helps to increase the distance between the interfaces and thereby reduces their mutual interactions without increasing the free energy barrier between the two coexisting phases, which has to be compensated by the re-weighting technique. The results of such a free energy calculation are presented in Fig. 7. They illustrate the comparison with the prediction of Eq.(8), analytical predictions for corrections to the behavior in the strong segregation limit,40 and SCF calculations. The agreement between the mean-field prediction and the simulation data is quantitative without any adjustable parameter. Therefore, measuring the interfacial free energy in a phase-separated blend far below the critical point of demixing is a suitable method (see Sec. 2) for measuring the Flory-Huggins parameter, χ.
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1.0 0.8
VI/VSSL
0.6 0.4 0.2 0.0 0.0
N=16 N=32 N=64 N=32 (capillary waves) SCF 1-4ln2/FN 1-4ln2/FN+1.1842/(FN) 0.1
1/FN
3/2
0.2
0.3
√ ¯ χN/6. The symbols correFig. 7. Interface tension in units of σSSL R2e /kB T = N spond to MC simulations for three chain lengths as indicated in the key. The interfacial tension has been extracted from the probability distribution of the composition in the semi-grandcanonical ensemble, and the spectrum of capillary waves (for N = 32). Data are compared to SCF results and the exact asymptotic corrections in the limit χN → ∞ uller and de Pablo.65 by Semenov.40 From M¨
While excess properties of interfaces (like the interfacial free energy or the total interfacial excess of chain end segments, solvent particles, etc) can be straightforwardly obtained from such simulations, special care has to be exerted when comparing profiles across interfaces. While the meanfield theory makes rather detailed predictions about the spatial variation of properties (e.g., segment densities, orientations, etc) across the interface, the interface is assumed to be perfectly flat and long-ranged fluctuations of the local interface position, i.e., capillary waves,91–94 are ignored. If one averaged profiles across the total interfacial area, then these capillary waves would give rise to a system-size dependent broadening of the profiles. Therefore, the interfacial width, cf. Eq. (7), does not provide a reliable means of identifying the Flory-Huggins parameter. Fig. 8(a) presents a snapshot of the local interface position, u, in a binary polymer blends, where those capillary waves are clearly visible. Panel (b) depicts the variation of the local position for different incompatibilities. The larger the incompatibility between the two species is, the larger is the interfacial tension, σAB , and the stronger are capillary waves suppressed.
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0.15
0.10 P(u)
u(x,y)
5 60
0
ε = 0.02 ε = 0.024 ε = 0.03 ε = 0.04 ε = 0.05 ε = 0.07 ε = 0.1 ε = 0.146
0.05
-5 -10
40
x 20
20
0.00 −16
40
60
(a)
−8
0 u
y
8
16
(b)
Fig. 8. (a) Fluctuations of the local interface position in a binary polymer blend at = 0.03T (χN ≈ 5.1). (b) Distribution of the local interface positions as a function of the incompatibility /T . From Werner et al.95
In fact, this effect can be utilized to measure the interface tension (see Fig. 7) which yields consistent results. The semi-grandcanonical simulation strategy can also be employed to locate phase coexistence in confinement or to measure the free energy difference of the two coexisting phases with respect to a surface. To this end, periodic boundary conditions are only utilized in two directions and, in the third direction, surfaces are introduced, which preferentially attract the A-component of the mixture with an energy, wall . Using the semigrandcanonical reweighting technique in conjunction with re-weighting, the system tunnels between an A-rich phase in contact with the surface and a B-rich phase in contact with the surface. If the chemical exchange potential, ∆µ, is adjusted as to fulfill the equal weight criterion (cf. Eq. (10)), one accurately locates the phase transition in a slit-like pore. The results of such a procedure are depicted in Fig. 1 for symmetric surfaces, which both attract the A-species, and anti-symmetric surfaces, where the lower surface prefers A with the same strength as the upper surface prefers B. If the semi-grandcanonical simulation is performed at the bulk coexistence value of the chemical potential, the simulation measures the free energy difference between an A-rich bulk phase in contact with the surface and a B-rich phase in contact with the surface. Since the two phases coexist, the semi-grandcanonical free energies of the bulk regions are identical and the difference solely stems from the surface free energy difference, i.e., one computes L2 ∆γ. Knowing the interface tension, σAB , and the difference of
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Fig. 9. Interface tension σAB and difference in surface tensions ∆γ = γW B − γW A as a function of inverse temperature /k B T obtained from simulations. Approximations for the interface tension σAB = bρo χ/6 in the strong segregation limit and ∆γ = 2Φdwall wall = wall /4 (dashed-dotted lines) are also shown. From Ref. 9. The inset shows the dependence of the contact angle on wall for the two temperatures, /kB T = 0.025 and 0.05. From M¨ uller and Binder.55
surface tensions, ∆γ, one can invoke Young’s equation96 σAB cos Θ = ∆γ
(21)
to accurately determine the contact angle, Θ, that an A-rich domain boundary makes with the surface. It turns out that utilizing Young’s equation is a very precise tool to determine the contact angle, which does not suffer from effects of finite domain size.7,9 The simulation techniques discussed provide a wealth of information about the phase behavior in the bulk and confinement and, simultaneously, yield excess properties of interfaces. In this contribution, we mainly focused on a simple system — a symmetric polymer blend — but the strategy carries over to other phase-separating systems, which simultaneously exhibit liquidliquid and liquid-vapor phase separation or liquid crystalline order. The two major advantages are that (i) the semi-grandcanonical scheme efficiently relaxes the order parameter of the transition, which would take a protracted time to relax in a polymer system if it were locally conserved and (ii) that a criterion of phase coexistence exists — the equal weight rule or sophisticated
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finite-size scaling techniques in the vicinity of the critical point — which allows for an accurate computation of the properties of the coexisting phases from small systems. Given the importance of phase separation for practical applications, the wealth of new materials created by blending and altering the macromolecular architecture, and the necessity to predict it or extrapolate it into parameter regions, which are experimentally difficult or dangerous to access, computer simulations of atomistic and coarse-grained models will continue making contributions to this field of material science. References 1. R. Cahn, P. Haasen, and E. Kramer, Materials Science and Technology: A Comprehensive Treatment. 12, VCH, Weinheim, (1993). 2. F. Garbassi, M. Morra, and E. Occhiello, Polymer Surface: From Physics to Technology. pp. Wiley, Chichester, (2000). 3. D. R. Paul and J. W. Barlow, Polymer blends (or alloys), Journal of Macromolecular Science-Reviews In Macromolecular Chemistry. C18, 109–168, (1980). 4. G. Taylor, The viscosity of a fluid containing small drops of another fluid, Proc. R. Soc. London A. 138, 41–48, (1932). 5. C. Creton, E. J. Kramer, and G. Hadziioannou, Critical molecular-weight for block copolymer reinforcement of interfaces in a 2-phase polymer blend, Macromolecules. 24, 1846–1853, (1991). 6. M. M¨ uller and K. Binder, Interface localization-delocalization transition in a symmetric polymer blend: a finite-size scaling Monte Carlo study, Phys. Rev. E. 63, 021602, (2001). 7. M. M¨ uller and K. Binder, The interplay between wetting and phase behaviour in binary polymer films and wedges: Monte Carlo simulations and mean field calculations, J. Phys.: Condens. Matter. 17, S333–S361, (2005). 8. M. M¨ uller, K. Binder, and E. V. Albano, Non-monotonous crossover between capillary condensation and interface localisation/delocalisation transition in binary polymer blends, Europhys. Lett. 50, 724–730, (2000). 9. M. M¨ uller and K. Binder, Wetting and capillary condensation in symmetric polymer blends: a comparison between Monte Carlo simulations and selfconsistent field calculations, Macromolecules. 31, 8323–8346, (1998). 10. J. Baschnagel, K. Binder, P. Doruker, A. A. Gusev, O. Hahn, K. Kremer, W. L. Mattice, F. M¨ uller-Plathe, M. Murat, W. Paul, S. Santos, U. W. Suter, and V. Tries, Bridging the gap between atomistic and coarse-grained models of polymers: Status and perspectives, Adv. Polym. Sci. 152, 41–156, (2000). 11. F. M¨ uller-Plathe, Coarse-graining in polymer simulation: From the atomistic to the mesoscopic scale and back, Chem. Phys. Chem. 3, 754–769, (2002).
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12. D. Heine, D. T. Wu, J. G. Curro, and G. S. Grest, Role of intramolecular energy on polyolefin miscibility: Isotactic polypropylene/polyethylene blends, J. Chem. Phys. 118, 914–924, (2003). 13. P. G. Bolhuis, A. A. Louis, and J. P. Hansen, Many-body interactions and correlations in coarse-grained descriptions of polymer solutions, Phys. Rev. E. 6402, 021801, (2001). 14. A. A. Louis, Beware of density dependent pair potentials, J. Phys.: Condens. Matter. 14, 9187–9206, (2002). 15. K. Kremer and K. Binder, Monte Carlo simulations of lattice models for macromolecules, Computer Physics Reports. 7, 259–310, (1988). 16. M. M¨ uller and F. Schmid, Incorporating fluctuations and dynamics in selfconsistent field theories for polymer blends, Adv. Polym. Sci. 185, 1–58, (2005). 17. M. M¨ uller, Comparison of self-consistent field theory and Monte Carlo simulations, Soft Condensed Matter, G. Gompper and M. Schick (eds), Wiley VCH 1, 179–281, (2006). 18. E. Helfand, Theory of inhomogeneous polymers: Fundamentals of the Gaussian random–walk model, J. Chem. Phys. 62, 999–1005, (1975). 19. L. Sch¨ afer, Excluded volume effects in polymer solutions, Springer, Berlin. (1999). 20. P. G. de Gennes, Exponents for excluded volume problem as derived by Wilson method, Physics Letters A. A, 339, (1972). 21. P. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca and London. (1979). 22. P. J. Flory, Thermodynamics of high polymer solutions, J. Chem. Phys. 9, 660, (1941). 23. M. L. Huggins, Solutions of long chain compounds, J. Chem. Phys. 9, 440, (1941). 24. I. Carmesin and K. Kremer, The bond fluctuation method: a new effective algorithm for the dynamics of polymers in all spatial dimensions., Macromolecules. 21, 819–2823, (1988). 25. M. M¨ uller, Miscibility behavior and single chain properties in polymer blends: a bond fluctuation model study, Macromol. Theory Simul. 8, 343–374, (1999). 26. K. Kremer and G. S. Grest, Dynamics of entangled linear polymer melts: a molecular-dynamics simulation, J. Chem. Phys. 92, 5057–5086, (1990). 27. G. S. Grest, M. D. Lacasse, K. Kremer, and A. M. Gupta, Efficient continuum model for simulating polymer blends and copolymers, J. Chem. Phys. 105, 10583–10594, (1996). 28. R. D. Groot and P. B. Warren, Dissipative particle dynamics: Bridging the gap between atomisticand mesoscopic simulation, J. Chem. Phys. 107, 4423– 4435, (1997). 29. M. M¨ uller and G. D. Smith, Phase separation in binary mixtures containing polymers: a quantitative comparison of single-chain-in-mean-field simulations and computer simulations of the corresponding multichain systems, J. Polym. Sci. B: Polymer Physics. 43, 934–958, (2005).
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30. K. C. Daoulas and M. M¨ uller, Single chain in mean field simulations: Quasiinstantaneous field approximation and quantitative comparison with Monte Carlo simulations, J. Chem. Phys. 125, 184904, (2006). 31. K. C. Daoulas, M. M¨ uller, J. J. de Pablo, P. F. Nealey, and G. D. Smith, Morphology of multi-component polymer systems: Single chain in mean field simulation studies, Soft Matter. 2, 573–583, (2006). 32. P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49, 435–479, (1977). 33. M. Doi and S. F. Edwards, The theory of polymer dynamics. Oxford University Press, (1986). 34. J. T. Padding and W. J. Briels, Time and length scales of polymer melts studied by coarse-grained molecular dynamics simulations, J. Chem. Phys. 117, 925–943, (2002). 35. A. E. Likhtman, Single-chain slip-link model of entangled polymers: Simultaneous description of neutron spin-echo, rheology, and diffusion, Macromolecules. 38, 6128–6139, (2005). 36. E. Helfand and Y. Tagami, Theory of interface between immiscible polymers, J. Chem. Phys. 57, 1812, (1972). 37. J. M. H. M. Scheutjens and G. J. Fleer, Statistical-theory of the adsorption of interacting chain molecules. 1. partition-function, segment density distribution, and adsorption-isotherms, J. Phys. Chem. 83, 1619–1635, (1979). 38. J. Noolandi and K. M. Hong, Interfacial properties of immiscible homopolymer blends in the presence of block copolymers, Macromolecules. 15, 482–492, (1982). 39. M. W. Matsen and M. Schick, Stable and unstable phases of a diblock copolymer melt, Phys. Rev. Lett. 72, 2660–2663, (1994). 40. A. N. Semenov, Theory of long-range interactions in polymer systems, J. Phys. (France) II. 6, 1759–1780, (1996). 41. J. Dudowicz, K. F. Freed, and J. F. Douglas, New patterns of polymer blend miscibility associated with monomer shape and size asymmetry, J. Chem. Phys. 116, 9983–9996, (2002). 42. M. M¨ uller and K. Binder, Computer-simulation of asymmetric polymer mixtures, Macromolecules. 28, 1825–1834, (1995). 43. M. M¨ uller, Effects of structural disparities in polymer blends – a Monte Carlo investigation, Macromolecules. 28, 6556–6564, (1995). 44. A. Sariban and K. Binder, Phase-separation of polymer mixtures in the presence of solvent, Macromolecules. 21, 711–726, (1988). 45. H. P. Deutsch and K. Binder, Critical-behavior and crossover scaling in symmetrical polymer mixtures – a Monte Carlo investigation, Macromolecules. 25, 6214–6230, (1992). 46. B. Crist, Thermodynamic interactions in isotope blends: Experiment and theory, Macromolecules. p. 5853, (1998). 47. F. S. Bates and G. H. Fredrickson, Block copolymer thermodynamics – theory and experiment, Ann. Rev. Phys. Chem. 41, 525–557, (1990).
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48. F. S. Bates, M. Schultz, J. Rosedale, and K. Almdal, Correlation of binary polyolefin phase behavior with statistical segment length asymmetry, Macromolecules. 25, 5547, (1992). 49. M. D. Gehlsen and F. S. Bates, Conformational asymmetry in poly(vinylcyc1ohexane) containing diblock copolymers, Macromolecules. 27, 3611, (1994). 50. D. Schwahn, G. Meier, K. Mortensen, and S. Janssen, On the n-scaling of the Ginzburg number and the critical amplitudes in various compatible polymer blends, J. Phys. II (France). 4, 837, (1994). 51. A. Z. Panagiotopoulos, Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble, Mol. Phys. 61, 813–826, (1987). 52. A. Sariban and K. Binder, Critical properties of the Flory-Huggins lattice model of polymer mixtures, J. Chem. Phys. 86, 5859–5873, (1987). 53. M. M¨ uller and K. Binder, An algorithm for the semi-grand-canonical simulation of asymmetric polymer mixtures, Comp. Phys. Comm. 84, 173–185, (1994). 54. M. M¨ uller and A. Werner, Interfaces between highly incompatible polymers of different stiffness: Monte Carlo simulations and self-consistent field calculations, J. Chem. Phys. 107, 10764–10776, (1997). 55. M. M¨ uller and K. Binder, The interplay between wetting and phase behaviour in binary polymer films and wedges: Monte Carlo simulations and mean field calculations, J. Phys. Condens. Matt. 17, S333–S361 (2005). 56. C. Borgs and R. Kotecky, J. Stat. Phys. 60, 79, (1990). 57. C. Borgs and S. Kappler, Phys. Lett. A. 171, 37, (1992). 58. A. M. Ferrenberg and R. H. Swendsen, New Monte Carlo technique for studying phase transitions, Phys. Rev. Lett. 61, 2635–2638, (1988). 59. A. M. Ferrenberg and R. H. Swendsen, Optimized Monte Carlo data analysis, Phys. Rev. Lett. 63, 1195–1198, (1989). 60. P. Dayal, S. Trebst, S. Wessel, D. Wurtz, M. Troyer, S. Sabhapandit, and S. N. Coppersmith, Performance limitations of flat-histogram methods, Phys. Rev. Lett. 92, 097201, (2004). 61. B. A. Berg, Multicanonical recursions, J. Stat. Phys. 82, 323, (1996). 62. F. Wang and D. P. Landau, Efficient multiple-range random walk algorithm to calculate the density of states, Phys. Rev. Lett. 86, 2050–2053 (2001). 63. G. R. Smith and A. D. Bruce, A study of the multi-canonical Monte Carlo method, J. Phys. A-Math. Gen. 28, 6623–6643, (1995). 64. P. Virnau and M. M¨ uller, Calculation of free energy through successive umbrella sampling, J. Chem. Phys. 120, 10925–10930, (2004). 65. M. M¨ uller and J. J. de Pablo, Simulation techniques for calculating free energies, Lec. Notes Phys. 703, 67–122, (2006). 66. J. P. Valleau, Thermodynamic-scaling methods in Monte Carlo and their application to phase equilibria, Adv. Chem. Phys. 105, 369–404, (1999). 67. B. J. Schulz, K. Binder, M. M¨ uller, and D. P. Landau, Avoiding boundary effects in Wang-Landau sampling, Phys. Rev. E. 67, 067102, (2003).
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68. N. B. Wilding and M. M¨ uller, Accurate measurements of the chemicalpotential of polymeric systems by Monte Carlo simulation, J. Chem. Phys. 101, 4324–4330, (1994). 69. D. Chandler, Introduction to Modern Statistical Mechanics. (Oxford University Press, New York, 1987). 70. D. Frenkel and B. Smit, Understanding Molecular Simulation. (Academic Press, London, 2002). 71. K. Binder, M. M¨ uller, P. Virnau, and L. G. MacDowell, Polymer plus solvent systems: Phase diagrams, interface free energies, and nucleation, Adv. Polym. Sci. 173, 1–104, (2005). 72. R. L. C. Vink and J. Horbach, Grand canonical Monte Carlo simulation of a model colloid-polymer mixture: Coexistence line, critical behavior, and interfacial tension, J. Chem. Phys. 121, 3253–3258, (2004). 73. R. L. C. Vink, J. Horbach, and K. Binder, Capillary waves in a colloidpolymer interface, J. Chem. Phys. 122, 134905, (2005). 74. R. L. C. Vink, J. Horbach, and K. Binder, Critical phenomena in colloidpolymer mixtures: Interfacial tension, order parameter, susceptibility, and coexistence diameter, Phys. Rev. E. 71, 011401, (2005). 75. R. L. C. Vink and M. Schmidt, Simulation and theory of fluid demixing and interfacial tension of mixtures of colloids and nonideal polymers, Phys. Rev. E. 71, 051406, (2005). 76. R. L. C. Vink and T. Schilling, Interfacial tension of the isotropic-nematic interface in suspensions of soft spherocylinders, Phys. Rev. E. 71, 051716, (2005). 77. R. L. C. Vink, S. Wolfsheimer, and T. Schilling, Isotropic-nematic interfacial tension of hard and soft rods: Application of advanced grand canonical biased-sampling techniques, J. Chem. Phys. 123, 074901, (2005). 78. P. H. V. Konynenburg and R. L. Scott, Critical lines and phase equilibria in binary van der waals mixtures, Phil. Trans. R. Soc. A. 298, 496–540, (1980). 79. P. Virnau, M. M¨ uller, L. G. MacDowell, and K. Binder, Phase diagrams of hexadecane – CO2 mixtures from histogram-reweighting Monte Carlo, Comp. Phys. Comm. 147, 378–381, (2002). 80. M. M¨ uller, L. G. MacDowell, P. Virnau, and K. Binder, Interface properties and bubble nucleation in compressible mixtures containing polymers, J. Chem. Phys. 117, 5480–5496, (2002). 81. P. Virnau, M. M¨ uller, L. G. MacDowell, and K. Binder, Phase behavior of n-alkanes in supercritical solution: a Monte Carlo study, J. Chem. Phys. 121, 2169–2179, (2004). 82. L. Yelash, M. M¨ uller, W. Paul, and K. Binder, Artificial multiple criticality and phase equilibria: An investigation of the PC-SAFT approach, Phys. Chem. Chem. Phys. 7, 3728–3732, (2005). 83. B. M. Mognetti, L. Yelash, P. Virnau, W. Paul, K. Binder, M. Muller, and L. G. MacDowell, Efficient prediction of thermodynamic properties of quadrupolar fluids from simulation of a coarse-grained model: The case of carbon dioxide, J. Chem. Phys. 128, 104501, (2008). 84. H. A. Lorentz, Annln Phys. 12, 127, (1881).
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85. D. C. Berthelot, r. hebd. Seanc. Acad Sci., Paris. 126, 1703, (1898). 86. J. I. Siepmann and D. Frenkel, Configurational-Bias Monte Carlo: a new sampling scheme for flexible chains, Mol. Phys. 75, 59–70, (1992). 87. G. Schneider, Z. Alwani, W. Heim, E. Horvath, and E. U. Franck, Phasengleichgewichte und kritische Erscheinungen in bin¨ aren Mischsystemen bis 1500 bar, Chem.-Ing.-Techn. 39, 649, (1967). 88. T. Charoensombut-Amon, R. J. Martin, and R. Kobayashi, Application of a generalized multiproperty apparatus to measure phase equilibrium and vapor phase densities of supercritical carbon dioxide in n-hexadecane systems up to 26 MPa, Fluid Phase Equilibria. 86, 89–104, (1986). 89. C. Mendui˜ na, C. McBride, and C. Vega, The second virial coefficient of quadrupolar two center Lennard–Jones models, Phys. Chem. Chem. Phys. 3, 1289–1296, (2001). 90. K. Binder, Monte Carlo calculation of the surface tension for two- and threedimensional lattice-gas models, Phys. Rev. A. 25, 1699–1709, (1982). 91. F. P. Buff, R. A. Lovett, and F. H. Stillinger, Interfacial density profile for fluids in critical region, Phys. Rev. Lett. 15, 621, (1965). 92. W. Helfrich, Elastic properties of lipid bilayers – theory and possible experiments, Zeitschrift Naturforschung C. 28, 693–703, (1973). 93. J. D. Weeks, Structure and thermodynamics of liquid-vapor interface, J. Chem. Phys. 67, 3106, (1977). 94. D. Bedeaux and J. D. Weeks, Correlation-functions in the capillary wave model of the liquid vapor interface, J. Chem. Phys. 82, 972, (1985). 95. A. Werner, F. Schmid, M. M¨ uller, and K. Binder, Intrinsic profiles and capillary waves at homopolymer interfaces: a Monte Carlo study, Phys. Rev. E. 59, 728–738, (1999). 96. T. Young, Philos. Trans. R. Soc. London. 5, 65, (1805).
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Chapter 3 Self-Consistent Field Theory of Block Copolymers
Feng Qiu,∗ An-Chang Shi† and Yuliang Yang‡ Department of Macromolecular Science, Fudan University, Shanghai, China Department of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada Self-consistent field theory (SCFT) is a powerful theoretical framework for the study of the phase behavior of polymeric systems, in particular, block copolymers. In this chapter the framework of SCFT is introduced, focusing on the reciprocal-space method to solve the SCFT equations for ordered phases of block copolymers. The approach utilizes Fourier representation of spatially varying functions. The SCFT equations are cast in terms of the Fourier coefficients. An advantage of the reciprocal-space approach is that the symmetry of the ordered phases can be easily and definitively deduced from the nonzero Fourier coefficients. The method has been used to successfully reproduced all the ordered phases observed in diblock copolymers and ABC linear triblock copolymers and predicted some previously unknown structures.
1. Introduction Block copolymers are macromolecules composed of two or more chemically distinct sub-chains (blocks) linked together by covalent bonds. Common examples of block copolymers are linear AB diblock copolymers, linear ABC triblock copolymers, and miktoarm (or star) ABC triblock copolymers. One intrigue property of block copolymers is that the competition of repulsion and connectivity between different blocks leads to the self-assemly of a rich array of ordered phases. The phase behavior of block copolymers has attracted tremendous attention in the past decades. In the content of ∗ Fudan
University University ‡ Fudan University † McMaster
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soft condensed matter physics, block copolymers have served as an ideal paradigm for the study of the self-assembly in soft materials. Due to a large number of theoretical and experimental studies over the last decades, remarkable progress on understanding the phase behavior of block copolymers has been achieved. It is well established that the self-assembly of block copolymers is governed by a delicate balance of the interaction energy and chain conformational entropy. The phase behavior of block copolymer melts is determined by several factors: the volume fraction and sequence of the chemically distinct blocks, the interaction strength between different components (usually quantified by the Flory-Huggins interaction parameters), the degree of polymerization of the macromolecules, an the topology of the block copolymers. For a given set of controlling parameters, block copolymers self-assemble to form a particular ordered phase so that the free energy of the system is minimized. Determination of the possible phases and transition sequences between these ordered phases has been the major focus of theoretical and experimental studies of block copolymers.1,2 For the simplest case of AB diblock copolymers, the equilibrium phases are largely governed by three parameters: the volume fraction of the A block, fA , the AB interaction parameter χ, and the degree of polymerization, N . Although the AB diblock copolymer is the simplest block copolymer, it exhibits a surprisingly large number of ordered phases: alternating lamellae (L), hexagonally packed cylinders (C), spheres packed in a bodycentered cubic or face-centered cubic lattice (S), double-gyroid network (G) and orthorhombic F ddd network (O 70 , which is a noncubic phase).3 It is important to point out that the determination of the complex networked phases is not a trivial task. The correct identifications of the two networked phases were only accomplished in recent years. For example, the first bicontinuous structure, the gyroid, was initially identified as a double diamond structure with the space group P3mm when it was first observed in experiment.4,5 The identification of this phase as the gyroid phase was made recently when high-resolution small angle X-ray scattering data become available.6 More complex block architectures, such as ABC triblock copolymers, offer opportunities to create a greater complexity of microphase morphologies. The richness of ordered structures in triblock copolymers stems from a greatly enlarged parameter space. At the mean-field level, there are at least five independent molecular parameters determining the phase behavior of ABC triblock copolymers: two independent compositions, i.e., volume fractions of A and B blocks (fA and fB ), and three integrated
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interaction parameters (χAB N , χBC N and χAC N ). Furthermore, chain topology, i.e., linear triblock copolymer versus non-linear triblock copolymers (star-shaped, comb- and branched-type), and block sequence in linear triblock copolymers, can also play important roles in determining the equilibrium structures. Consequently, much more complex ordered morphologies can be obtained in ABC triblock copolymers. At the present time, more than three-dozen ordered structures have been identified for ABC triblock copolymers.3 A centre theme emerged from previous studies on block copolymer phase behavior is that the availability of block copolymers with different architectures will lead to complex phase behavior and diverse ordered phases. For a given block copolymer architecture and block type, the major challenge is to discover the possible ordered phases from this particular block copolymers. This challenge can only be met by a combination of experimental and theoretical approaches. Extensive experimental efforts provide valuable contributions to the understanding of the complex morphologies of different block copolymers. On the other hand, because varying copolymer compositions and monomer-monomer interactions involve the synthesis of a large number of block copolymers, experimental exploration of the phase space is a very costly and time consuming task. From this perspective, theory and simulation provide an ideal tool for the investigation of the phase behavior of complex block copolymers. In particular, theoretical studies can be used to obtain a good understanding of the detailed relationship between the mesoscopic structures and molecular parameters (block type, composition, architecture, and interactions). Over the last decades the self-consistent field theory (SCFT) of polymers has emerged as a powerful theoretical frameworks for the study of the statistical thermodynamics of block copolymers.7–9 The SCFT of polymers can be traced back to the work of Edwards in 1960s.10 Helfand adapted this theoretical framework to treat block copolymers in 1975.11 Because of the complexity of finding exact solutions of the SCFT equations, several approximate methods were developed. These approximation methods can be roughly divided into two classes, the weak segregation theory and the strong segregation theory. The weak-segregation theory (WST) proposed by Leibler in 198012 assumes that under weak-segregation limit condition, i.e., near order-disorder transition, the composition profile of an ordered structure is a small sinusoidal fluctuation in the disordered or homogeneous phase. Therefore the free energy of the system can be expressed as a Landau expansion based on the random phase approximation (RPA)
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around the homogeneous phase. The weak segregation theory provides a good explanation for phase behaviors of block copolymers near ODT. On the other hand, the strong segregation theory (SST) proposed by Semenov in 198513 is based on the assumption that, at the strong segregation limit, the phase separation is almost complete. Therefore the free energy of an ordered structure can be separated into interfacial free energy and stretching free energy. Analytical expressions of these two contributions can be estimated for a given structure. The SST has been successful in constructing phase diagram of diblock copolymers, initially including equilibrium phases such as lamellae, sphere and cylinder. Complex phases, such as gyroid, perforated lamellae, double diamond, have been considered with the SST.14,15 The WST and SST approximations provide valuable insights into the self-assembly of block copolymers. On the other hand, most experimental systems correspond to intermediate degree of segregation, where the WST and SST become inaccurate. In this case numerical techniques become indispensible as they provide exact solutions to the SCFT equations. The initial attempt to obtain numerical solutions of SCFT equations was made by Helfand and coworkers.16 Subsequently Shull17 and Whitmore and coworkers18 constructed phase diagrams of block copolymer melts and solutions using approximate numerical techniques. The first exact threedimensional numerical solutions of the SCFT equations were obtained by Matsen and Schick using a spectral-method.19 The availability of the exact numerical solutions allowed precise calculations of free energies and phase diagrams for block copolymers. The main idea of the Matsen-Schick approach is to expand all spatially varying functions in terms of a finite set of basis functions determined from the symmetry of the ordered structure under consideration. With this powerful tool, an accurate phase diagram of diblock copolymer was successfully constructed.20 Subsequently, other systems of block copolymers, such as symmetric triblock copolymer melts21 and block copolymer blends,22 have also been investigated. Recently, Morse and coworkers have constructed phase diagrams of nonfrustrated linear ABC triblock copolymers by solving the SCFT equations using the spectral-method.23,24 Initially the reiprocal-space method utilizes basis functions which are determined from some assumed symmetries of the ordered phases. This restriction has been removed recently by Guo et al with a generic Fourier-space method for solving the SCFT equations.25 In this generic spectral-method, all spatially varying functions are expanded in terms of Fourier series. The Fourier series themselves are determined by discretizing the reciprocal space. This method can be viewed as a general-
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ization of the Matsen-Schick method. The new method has been validated by applying it to AB diblock copolymers. Furthermore, the capability of the method to predict new phases has been demonstrated via the application to linear and star ABC triblock copolymer melts. Parallel to the spectral-method, tremendous progresses have been made for the development of real-space methods to solve the SCFT equations. For the purpose of exploring new phases, Drolet and Fredrickson26 proposed a combinatorial screening method, which is based on a direct implementation of SCFT equations in real space with an adaptive arbitrary cell. This approach has been applied to a number of block copolymer systems, leading to the construction of two-dimensional phase maps of linear and star ABC triblock copolymers.27 In a similar approach, Bohbot-Raviv and Wang28 proposed an efficient strategy to minimize the free energy function in an arbitrary unit cell with respect to the composition profile and the dimensions of the unit cell. These real-space strategies to solve the SCFT equations are greatly aided by the development of new numerical techniques, such as the split-step method proposed by Rasmussen et al,29 for solving the modified diffusion equation in SCFT. The split-step method is numerically stable and efficient. It has become the method of choice for real-space solutions of SCFT equations used in combinatorial screening of novel stable and metastable phases of block copolymers. The computational advantage of he real-space method has been demonstrated in a recent study by Cochran em et al,30 in which accurate phase diagram of AB diblock copolymers has been constructed up to χN = 100. A comparison of the efficiency of the real-space and reciprocal-space method is given by Matsen recently.31 The combination of experimental, theoretical and simulation studies has led to a good understanding of the phases and phase transitions of diblock copolymers and non-frustrated linear triblock copolymers. Despite all these progresses, predicting the possible ordered phases from a given complex block copolymer still presents a challenging task. From a theoretical point of view, the challenge is to find the possible solutions of the SCFT equations for a given block copolymer system. Progresses in real-space and reciprocal space methods to solve the SCFT equations have made it possible to predict the ordered phases of block copolymers, as demonstrated in many examples from AB diblock and ABC triblock copolymers. An effective strategy to construct phase diagrams of block copolymers is to divide the task into two steps. The first step is to obtain possible candidate ordered phases by combining the discovery power of the real-space and generic reciprocalspace methods. The symmetry of the candidate phases can then be used
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as input to the more accurate Matsen-Schick method or the split-space real-space method to constructed the phase diagram. Because reviews of the real-space method9 and the Matsen-Schick method7,8 are available in the literature, we will focus on the generic reciprocal-space method in this chapter. 2. Self-Consistent Field Theory of Block Copolymers In this section, a brief description of self-consistent field theory for block copolymers is presented. It should be pointed out that several excellent reviews of self-consistent field theory have been published over the years.7–9,32,33 The readers are refereed to these sources for a systematic description of the self-consistent field theory framework. For simplicity, we will use AB diblock copolymer melts as an example first, followed by extensions to ABC star triblock copolymer melts. Generalization of the theory to other block copolymers or block copolymer blends is straightforward. 2.1. Polymer model and partition function The model system of an AB diblock copolymer melt consists of n block copolymer chains contained in a finite volume V . The copolymer chains are linear macromolecules formed from NA A-monomers connected with NB B-monomers (N = NA + NB is the degree of polymerization of the block copolymer). For simplicity, the A- and B-monomers are assumed to have the same volume v0 = 1/ρ0 , where the monomer density ρ0 , is defined as the number of monomers per unit volume. The conformation of the ith polymer chain is described by a space curve, Rα i (s), which specifies the spatial position of the s-th monomer at α-block (α = A or B for AB diblock copolymers). The monomer species can be described by a function γα (s), which is 1 when the s-th monomer belongs to the α-block, and 0 otherwise, i.e., γα (s) = δα,A for 0 < s < NA and γα (s) = δα,B for NA < s < N . It is P obvious that this function satisfies α γα (s) = 1 for all s. For a given chain configuration ({R(·)}), the α-monomer density at a spatial position r can be expressed as, n Z 1 X N ˆ φα (r) = dsδ(r − Ri (s))γα (s), (1) ρ0 i=1 0 where the hat on φα indicates that φα (r) is a functional of the chain conformation Ri (s). The probability distribution of a given chain conformation
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is denoted by p0 (R(·)). For the Gaussian chains considered here, the probability distribution has the standard form " 2 # Z N dR(s) 3 ds , (2) p0 (R(·)) = A exp − 2 2b 0 ds R where A is a normalization constant such that D(R(·))p0 (R(·)) = 1. For AB diblock copolymers, the probability of a chain configuration {R(·)} (= {R1 (·), R2 (·), · · · , Rn (·)}) can be written in the form, n Y B P0 ({R(·)}) = p0 (Ri (s)) δ RA , (3) i (fA ) − Ri (fA i=1
B where the delta function δ RA i (fA ) − Ri (fA ensures that the A-block is linked to the B-block at segment position NA = fA N such that the polymers are diblock copolymers. The partition function of the AB diblock copolymer melts can be given in term of the functional integral over all the chain conformations, Z h i wn ˆ , (4) ZC = c D {R(·)} P0 ({R(·)}) δ (1 − φA − φB ) exp −βV (φ}) n! where wc is the partition function of a single diblock copolymer chain ˆ due to kinetic energy, and β = 1/kB T is the Boltzmann factor. V ({φ}) is the monomer-monomer interaction functional, which is often given by the Flory-Huggins expression, Z ˆ = χρ0 drφA (r)φB (r), V ({φ}) (5)
where χ is the Flory-Huggins parameter. The delta-function in the partition function is defined by, Y δ(1 − φA − φB ) = {δ (1 − φA (r) − φB (r))} , (6) r
which is introduced to ensure that the incompressibility condition at every spatial position r is satisfied. The partition function ZC in its current form presents a formidable many-body problem in statistical mechanics. In order to proceed, it is useful to insert an identity in terms of the delta-functions into the expression of the partition function. The delta-functions are in turn converted to functional integrals in terms of the auxiliary fields, Z 1 = Dφα δ φα − φˆα =
Z
Dφα
Z
+i∞
−i∞
Dωα exp
Z
drωα (r)δ φα (r) − φˆα (r)
.
(7)
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This identity allows one to replace the density operator φˆα (r) in the interˆ by the density field φα (r). The range of integral of action potential V ({φ}) the auxiliary field ωα (r) is along the imaginary axis in the complex plane from. Inserting the above identity into the partition function and after some simple manipulation, the partition function can be rewritten in terms of a functional integral Z Y ZC = [Dφα Dωα ] δ (1 − φA − φB ) exp [−βFC ({φ}, {ω})] , (8) α
where the “Hamiltonian” functional FC is given by ( ρ0 Rg3 −V ln QC ({ω}) FC ({φ}, {ω}) = N +
Z
"
dr χN φA (r)φB (r) −
X α
ωα (r)φα (r)
#)
.
(9)
Here the functional QC ({ω}) is the partition function of a single diblock copolymer chain in the external field ωα (r), " # Z XZ N 1 QC ({ω}) = D {R(·)} P0 ({R(·)}) exp − dsωα (R(s)γα (s) . V 0 α
(10)
2.2. Chain propagators The above expression of the single-chain partition function QC ({ω}) is in the form of a path integral. Computationally it is convenient to cast the single-chain partition function in terms of the chain propagators G(r, t|r0 , t0 ), which is the conditional probability distribution of monomer t (t0 ) at spatial position r (r0 ), given that the monomer t0 (t) is located at r0 (r) under an external field ωα (r), Z 1 dr1 dr2 dr3 G(r3 , 1|r2 , s)G(r2 , s|r1 , 0). (11) QC ({ω}) = V It can be proven that the chain propagator satisfies the following differential equations, ) ( X 2 2 ∂ 0 0 G(r, t|r , t ) = γα (t) σα ∇ − ωα (r) G(r, t|r0 , t0 ), (12) ∂t α
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with the initial conditions, G(r, t|r0 , t) = δ(r − r0 ). Here σα2 = N b2α /6 and bα is the statistical segment length of the α-block. For convenience, it is useful to introduce the end-integrated chain propagators. There are two types of end-integrated propagators, the forward end-integrated propagator q(r, s) and the backward end-integrated propagator q † (r, s), because the two ends of a block copolymer chain are not equivalent. These end-integrated propagators are defined by, Z q(r, s) = dr0 G(r, s|r0 , 0), (13) †
q (r, s) =
Z
dr0 G(r0 , 1|r, s).
(14)
Obviously, these two end-integrated propagators also satisfy the same differential equations, ( ) X 2 2 ∂ q(r, s) = γα (t) σα ∇ − ωα (r) q(r, s), (15) ∂s α ( ) X 2 2 † ∂ † q (r, s) = − γα (t) σα ∇ − ωα (r) q (r, s), ∂s α
(16)
but with different initial conditions q(r, 0) = 1 and q † (r, 1) = 1. In terms of these two new definitions, the single-chain partition function can be rewritten as Z 1 QC ({ω}) = drq(r, s)q † (r, s). (17) V 2.3. Self-consistent mean-field theory The thermodynamics of the diblock copolymer melts is completely determined by the partition function ZC . However, exact evaluation of the partition function is in general a formidable task. Numerous methods have been proposed to evaluate this partition function, including approximate analytical and numerical methods as well as numerical simulations of the field theory. Among the many approximation strategies, the mean-field approximation has provided extremely useful information about the phases and phase transitions of block copolymers. Technically, the mean-field approximation assumes that a single field configuration dominates the functional integral defining the partition function. This particular configuration is the one corresponding to the maximization of the integrand exp [−βF({φ}, {ω})]. Consequently, this field
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configuration is obtained by minimization of the free-energy functional F({φ}, {ω}) with respect to φα (r) and ωα (r), leading to a set of coupled SCFT equations, ωα (r) = χN φβ (r) + η(r) Z fα 1 φα (r) = dsqα (r, s)q † (r, fα − s), QC 0
(18) (19)
where the chain propagator qα (r), s) satisfies the modified diffusion equation in the mean field ωα (r), ∂ qα (r, s) = σα2 ∇2 − ωα (r) qα (r, s), ∂s
(20)
and η(r) is a Lagrangian multipler, or a pressure field, which ensures the incompressibility condition, φA (r) + φB (r) = 1. Because both φα (r) and ωα (r) are determined self-consistently from the above self-consistent equations, the mean-field approximation is referred as the self-consistent mean-field theory (SCMFT). In the literature, the SCMFT is often referred simply as the self-consistent field theory (SCFT). Within the mean-field approximation, the free energy per chain of the system is obtained by inserting the mean field solution into the free energy expression. It is interesting to examine the structure of the SCMFT for a diblock copolymer melt. Within the mean-field approximation, the parameters entering the theory are the combination χN , the block volume fraction fα , and the effective Kuhn lengths σα . The parameter χN characterizes the degree of segregation and it is controlled by temperature and the molecular weight. On the other hand, the polymer structure is characterized by the block composition fα and the block Kuhn lengths σα . Within the SCMFT, the thermodynamic properties of a diblock copolymer melt is completely specified by χN , fα and σα . The method to solve the SCMFT equations is conceptually straightforward. The first step is to make an initial guess of the mean fields ωα (r), which bears the symmetry of the ordered phase under investigation. The modified diffusion equations with the appropriate initial and boundary conditions are then solved to obtain the propagators, qα (r, s). These propagators can then be used to compute the mean field concentrations, φα (r). The next step is to adjust the mean fields ωα (r) according to an iterative procedure so as to satisfy the self-consistent equations and the incompressibility condition. For a given set of controlling parameters such as {χN, fα , σα }
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for a diblock copolymer melt, there are many solutions to the mean field equations, corresponding to different morphologies. The phase diagram is constructed by finding the structures with the lowest free energy density. Because of the complexity of the theory even within the mean-field approximation, analytic solutions of the mean-field theory can only be obtained under special conditions, such as for homogeneous phase. In order to obtain analytic solutions of the SCMFT equations, approximation methods are needed. In particular, approximation methods have been developed at the weak segregation limit (WSL)12 and at the strong segregation limit (SSL).13 The connection of the SCMFT and the WSL and SSL theories has been reviewed by Matsen.7 To obtain an exact solution, numeric techniques are required. Since the formulation of the SCFT for block copolymers by Helfand in 1975,11 a great effort has been devoted to the solution of the SCFT equations. The first efficient and accurate method to solve the self-consistent meanfield equations is the reciprocal-space method developed of Matsen and Schick,19 which is based the expansion in terms of plane wave like basis functions. Recently, with the availability of increasing computing power and new numerical techniques, real-space methods have been developed to the level that they can be used to explore the possible phases for a given block copolymer architecture.9 Mathematically, the structure of a block copolymer system is described by a density profile which is a spatially varying function determined by the minimization of the free energy function. For an ordered phase the density profile is a periodic function of the space. In the real-space formulation, the density profile is described by a real function φ(r) in space. Numerical methods in real-space start with discretizing the real-space into a lattice and the density profile is specified by its values at the lattice point. In the reciprocal-space formulation, the density presented as a Fourier R profile isik·r integral over the wavevector k, φ(r) = dkφ(k)e . Numerical methods in reciprocal-space start with the discretization of the reciprocal-space into a discrete lattice of wave vectors. In what follows the reciprocal-space formulation is given in detail. 3. Reciprocal-Space Formulation In the last section the SCFT framework of block copolymers is developed in real-sapce. Mathematically, any spatially varying functions can be represented in the Fourier space or the reciprocal-space. Therefore the SCFT
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can be cast in the reciprocal space. The advantage of the reciprocal space approach is that an ordered phase is naturally described by a set of basis functions which are periodic functions in space. Specifically, the functions of interest within the SCFT, such as the mean-field concentrations, the mean-field potentials, and the end-integrated propagators, are expanded in terms of Fourier series. The SCFT is then cast in terms of the expansion coefficients. The reformulation starts with expanding the functions of interest using the plane waves, eiG·r , as the basis functions. The wave vectors {G} are determined by specific discretization of the reciprocal space. One useful method to discretize the reciprocal space is to use periodic boundary conditions applied to the computation box with specific size and shape. Once the reciprocal vectors are specified, any spatially varying functions of interest, ψα (r), can be expressed in terms of Fourier series as X ψα (r) = ψα,j eiGj ·r . (21) j
Without loss of generality, the following discussions utilize a rectangular computation box. In this case the wave vector {G} is specified by three indices, {G} = 2π (h/Dx , k/Dy , l/Dz ), where Dx , Dy , Dz are the sizes of the rectangular box, and h, k, l are integers. A reciprocal vector can be represented by a set of integers (h, k, l). With these notations, the chain propagators qα (r, s), segments density φα (r), and the mean-field ωα (r) are specified as, X qα (r, s) = qα,j (s)eiGj ·r , (22) j
φα (r) =
X
φα,j eiGj ·r ,
(23)
ωα,j eiGj ·r .
(24)
j
ωα (r) =
X j
One of the important characteristics of the basis functions (fj (r) = eiGj ·r ) is that they are are orthonormal eigenfunctions of the Laplace operator, ∇2 fj (r) = −λj fj (r) with the corresponding eigenvalues λj = 2 2 2 2 4π (h/Dx ) + (k/Dy ) + (l/Dz ) . It is convenient to order the basis functions such that their eigenvalues λj form a non-decreasing series, therefore the first wave vector is G1 = 0. These basis functions are othronormal, Z 1 drfi (r)fi (r) = δij . (25) V
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In terms of the Fourier coefficients, the SCFT equations become, ωα,j = χN (φβ,j − fβ δi,1 ) + ηi Z 1 X X fα † φα,i = dsqα,j (s)qα,k (fα − s)δGi ,Gj +Gk , QC j 0
(26) (27)
k
where the single-chain partition function QC = qα,1 (1) and ηi is introduced to ensure the incompressibility condition φA,i + φB,i = δi,1 . The diffusion equations of the end-integrated propagators in reciprocal space are rewritten as, X ∂qα,i (s) =− Hα (Gi , Gj )qα,j (s) ∂s j
† X ∂qα,i (s) † = Hα (Gi , Gj )qα,j (s), ∂s j
where the “Hamiltonian” of the chain propagators is specified by X Hα (Gi , Gj ) = σα2 G2i δGi ,Gj + ωα,k (s)δGi ,Gj +Gk .
(28)
(29)
(30)
k
The initial conditions of the propagators are qA,i (0) = δGi ,0 , qB,i (fA ) = † † qA,i (fA ), qB,i (1) = δGi ,0 and qB,i (fB ) = qA,i (fB ). In the reciprocal-space, the modified diffusion equation becomes a set of coupled linear differential equation, whose solution can be found in terms of the transfer matrix Tα (s) = e−Hα s , X qα,i (s) = Tα,ij (s)qα,j (0). (31) j
Th transfer matrix can be expressed in terms of the eigenvalues and eigenvectors of the propagator Hamiltonian Hα . Because the Hamiltonian is a Hermitian matrix, diagonalization of matrix Hα can always be performed by a unitary transformation, leading to Hα = Uα Dα Uα† , where Dα is a diagonal matrix with the real eigenvalues dα,i of Hα as its matrix elements Dα,ij = dα,i δij . The columns of the unitary matrix Uα are the normalized eigenvectors of Hα . In terms of the diagonalized matrix the transfer matrix is given by Tα (s) = Uα e−Dα s Uα† , whose matrix elements are, X † Tα,ij (s) = Uα,ik e−dα,k s Uα,kj . (32) k
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The Fourier coefficients of the end-integrated propagators can now be written in terms of the eigenvalues and eigenfunctions for s < fA , X † qα,i (s) = UA,ij e−dA,j s UA,j1 (33) j
† qα,i (s)
=
XX j
† UA,ik e−dA,k (fA −s) UA,kj qB,j (fA ).
(34)
k
Similar expressions are obtained for s > fA , X † † qα,i (s) = UB,ij e−dB,j (1−s) UB,j1
(35)
j
qα,i (s) =
XX j
† UB,ik e−dB,k (s−fA ) UB,kj qA,j (fA ).
(36)
k
Using these expressions for the propagators, the density profiles and the single chain partition function QC can be obtained in terms of these eigenvalues and eigenvectors. Once the above equations are solved, the free energy per chain of the system is evaluated by, X FC = − ln [qB,1 (1)] − χN φA,i φB,i . (37) nkB T i
For a given set of molecular parameters, ordered phases are obtained by solving the SCFT equations. The minimizing the free energy can be further carried out with respect to the computation box parameters Dx , Dy and Dz . Once a solution is obtained, the space group of the resulting ordered structures can be determined by the wave vectors Gi with nonzero coefficients φα,i . Although the above formulae are derived for diblock copolymer melts, extension of extension to other block copolymers with varying architectures is straightforward. The following is a brief description of the extension to the case of ABC star triblock copolymers. The numerical procedure for the triblocks is very similar to that of the diblocks, with different initial conditions for the modified diffusion equations of chain propagators. We denote the junction point in star ABC triblock copolymers as segment position s = 0, and the three end-monomers along the A, B and C arms are assigned as s = fA N, fB N and fC N , respectively. Here N is the degree of polymerization of star polymer, and fA , fB , and fC (fA + fB + fC = 1) are the volume fractions of the three star chains. Starting from the initial conditions of the coefficients of the backward end-integrated chain propagators, † qα,i (fα ) = δi,1 ,
(38)
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and following the same procedures for solving the differential equations in reciprocal-space as diblock copolymers, the coefficients for the backward end-integrated chain propagators expanded in terms of the eigenvalues and eigenfunctions are obtained for 0 ≤ s ≤ fα as, X † † qα,i (s) = Uα,ij e−dα,j (fα −s) Uα,j1 . (39) j
For topological constraint that three chemically distinct chains are connected into a common junction at s = 0 in an ABC star triblock copolymer, the initial conditions qα,i (0) for solving the forward end-integrated chain propagators equations must be satisfied, XX † qA,i (0) = qB,j (0)qC,k (0)δGi ,Gj +Gk , (40) j
qB,i (0) =
j
qC,i (0) =
k
XX
(41)
† qA,j (0)qB,k (0)δGi ,Gj +Gk .
(42)
k
XX j
† qC,j (0)qA,k (0)δGi ,Gj +Gk
k
With these initial conditions, the “forward” chain propagators are found for 0 ≤ s ≤ fα as, XX † qα,i (s) = Uα,ik e−dα,k s Uα,kj qα,j (0). (43) j
k
With the solution of the chain propagators, the density profiles are obtained via. Z 1 X fα † φα,i = dsqα,j (s)qα,k (s)δGi ,Gj +Gk . (44) QC 0 j,k
P † The single chain partition function is QC = i,j qA,i (0)qA,j (0)δGi ,−Gj . Again all these quantities can be expressed in terms of the eigenvalues and eigenvectors of the chain Hamiltonian. The mean-fields are determined selfconsistently from, ωA,i = χAB N φB,i + χAC N φC,i + ηi ,
(45)
ωB,i = χAB N φA,i + χBC N φC,i + ηi ,
(46)
ωC,i = χAC N φA,i + χBC N φB,i + ηi .
(47)
where the field ηi is introduced to ensure the incompressibility of the system φA,i + φB,i + φC,i = δi,1 .
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Once the SCFT equations are set up, their solutions are found by using an iterative procedure starting with the initial guess of the ω-fields. The chain propagators and density profiles are then computed for the given ω-field. These density profiles are used to constructed new ω-field. If the new ω-field agrees with the old one with a prescribed accuracy,9 the solution is considered to be converged. Otherwise a new ω-field is constructed by some mixing procedures and the new chain propagators and density are computed. This procedure continues till self-consistency is achieved. The accuracy of the numerical solutions depends on the number of reciprocal vectors used in the computation. Because many of the coefficients are zero when the wave vectors are from a generic set described above, the method is not designed to produce highly accurate solutions. Instead, it is a screening technique to obtain the symmetries of the possible ordered phases for given block copolymers. The knowledge of symmetry obtained from our method can be used as input to the more accurate and efficient method of Matsen and Schick. In this approach the basis functions are constructed choosing reciprocal lattice vectors according to the symmetry of ordered structures. In a general platform, the generic method and Matsen-Schick method can be used in tandem to produce accurate solutions for given block copolymers. 4. Applications of the Reciprocal-Space Method The reciprocal-space method of SCFT equations can be classified into two classes. The first class utilizes a restricted set of basis functions determined from a pre-assumed symmetry of the ordered phase, as initially proposed by Matsen and Schick.19 There are numerous applications of the Matsen-Schick method to a variety of block copolymer systems, starting with the seminal paper by Matsen and Schick19 on dliblock copolymers to the study of nonfrustrated ABC linear triblock copolymers by Morse and coworkers.23,24 The second class of the reciprocal-space method utilizes a generic set of plane waves as the basis functions as proposed by Guo et al.25 In this approach there is no restriction on the symmetry of the ordered phase. Instead, the symmetry of the ordered phase emerges from the solution of the SCFT equations. Specifically, the set of wave vectors corresponding to non-zero expansion coefficients are used to map out the symmetry of the solution. In this section a brief summary of the application of the generic reciprocal-space method is given.
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In order to test the validity of the generic reciprocal-space approach for discovering equilibrium morphologies of block copolymers, the method has been applied to linear AB diblock copolymers. The three classic phases (lamellae, hexagonally packed cylinders and body-centered-cubic spheres) and some complex phases including the gyroid and O 77 structures have been successfully reproduced.25 The relative stability of these ordered phases is in agreement with the well-established phase diagram of diblock copolymers. A non-trivial application of the generic reciprocal-space method is the study of linear ABC triblock copolymers.25 A phase diagram for a model of the frustrated ABC triblock copolymers has been constructed. A number of new phases are predicted for the triblock copolymers. Furthermore, the method is applied to a realistic model of SEBM triblock copolymer, in which the fascinating KP phase was predicted to occur at the parameters that mostly match the experiment conditions. Recently the generic reciprocal-space method has been used to study the phases and phase transitions of ABC star triblock copolymers.34 The distinct feature of strongly segregated ABC star triblock copolymers is that their junction points must be on space curves, leading to the formation of two-dimensional packing patterns. A number of ordered two-dimensional patterns have been predicted by solving the SCFT equations using the generic reciprocal-space method. Phase diagrams are constructed for two typical ABC star triblock copolymers. The predicted phase diagrams are in good agreement with available experimental and simulations results. 5. Summary In this chapter the self-consistent field theory for block copolymers is reviewed. Although a diblock copolymer melt is used as a model system, it should be emphasized that the theoretical methods of deriving the SCFT and techniques of solving the SCFT equations are very flexible and versatile. Extension of the theory to more complex block copolymers is straightforward. The current formulation allows a systematic analysis of the thermodynamic properties of block copolymer systems. The equilibrium phases and phase diagrams can be obtained by solving the SCFT equations. We have emphasized the newly developed generic reciprocal-space method to solve the SCFT equations for complex ordered structures of block copolymers. The approach is based on a generic Fourier-space method. The essential idea is to expand all the spatial varying functions into a generic set of basis functions. The solutions of the SCFT equations can then be
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obtained using any of the available numerical techniques, such as Matsen and Schick.19 The symmetry of the equilibrium phases emerges from the solutions. The space group of the ordered phases can be identified by mapping to scattering vectors. The essence of this reciptoal-space approach is to use the full power of the spectral method, while the symmetry of the ordered phases is not presumed. The generic reciprocal-space method has been applied to a number of block copolymers, including AB diblock copolymers and ABC linear and star triblock copolymers. The method is able to reproduce all known ordered phases of these block copolymers and prediction about a number of preciously unknown ordered phases has been made. Acknowledgments We would like to thank Dr. Guojie Zhang and Mr. Zhuojun Guo for their contributions in developing the generic reciprocal-space method. References 1. I. W. Hamley, The Physics of Block Copolymers. Oxford University Press c1998. 2. F. S. Bates and G. H. Fredrickson, Physics Today 52, 32 (1999). 3. A. J. Meuler, M. A. Hillmyer and F. S. Bates, Macromolecules 42, 7221 (2009). 4. E. L. Thomas, D. B. Alward, D. J. Kinning, D. C. Martin, D. L. Handin, L. J. Fetters, Macromolecules 19, 2197 (1987). 5. H. Hasegawa, H. Tanaka, K. Yamasaki, T. Hashimoto, Macromolecules 20, 1651 (1987). 6. D. A. Hajduk et al, Macromolecules 27, 4063 (1994). 7. M. W. Matsen, J. Phys.: Condens. Matter 14, R21 (2002). 8. Shi, A.-C. in Developments in Block Copolymer Science and technology, Edited by I. W. Hamley, Wiley & Sons 2004. 9. G. H. Fredrickson, The Equilibrium Theory of Inhomogeneous Polymers, Oxford University Press, New York, 2006. 10. S. Edwards, Proc. Phys. Soc. 85, 613 (1965). 11. E. Helfand, J. Chem. Phys. 62, 999 (1975); Macromolecules 8, 552 (1975). 12. L. Leibler, Macromolecules 13, 1602 (1980). 13. A. N. Semenov, Sov. Phys. JETP 61, 733 (1985). 14. A. E. Likhtman and A. N. Semenov, Macromolecules 27, 3101 1994. 15. P. D. Olmsted and S. T. Milner, Macromolecules 31, 4011 1998. 16. E. Helfand and Z. R. Wasserman, Macromolecules 9, 879 (1976). 17. K. R. Shull, Macromolecules 25, 2122 (1992).
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18. J. D. Vavasour and M. W. Whitmore, Macromolecules 25, 5477 (1992); M. Banaszak and M. W. Whitmore, Macromolecules 25, 3406 (1992). 19. M. W. Matsen and M. Schick, Phys. Rev. Lett. 72, 2660 (1994). 20. M. W. Matsen and F. S. Bates, Macromolecules 29, 1091 (1996). 21. M. W. Matsen, J. Chem. Phys., 108, 785 (1998). 22. M. W. Matsen, Macromolecules 28, 5765 (1995); 36, 9647 (2003). 23. C. A. Tyler and D. C. Morse, Phys. Rev. Lett. 94, 208302 (2005). 24. C. A. Tyler, J. Qin, F. S. Bates and D. C. Morse, Macromolecules 40, 4654 (2007). 25. Z. J. Guo, G. J. Zhang, F. Qiu, H. D. Zhang, Y. L. Yang and A.-C. Shi, Phys. Rev. Lett. 101, 028301 (2008). 26. F. Drolet and G.H. Fredrickson, Phys. Rev. Lett. 83, 4317 (1999). 27. P. Tang, F. Qiu, H. D. Zhang and Y. L. Yang, Phys. Rev. E 69 031803 (2004); J. Phys. Chem. B 108, 8434 (2004). 28. Y. Bohbot-Raviv and Z.-G. Wang, Phys. Rev. Lett. 85, 3428 (2000). 29. G. Tzeremes, K. O. Rasmussen, T. Lookman and A. Saxena, Phys. Rev. E 65, 041806 (2002); K. O. Rasmussen and G. Kalosakas, J. Polym. Sci.: Part B: Polym. Phys. 40, 1777 (2002). 30. E. W. Cochran, C. J. Garcia-Cervera and G. H. Fredrickson, Macromolecules 39, 2449 (2006); 39, 4264 (2006). 31. M. W. Matsen, Eur. Phys. J. E, in press (2009). 32. M. W. Whitmore and J. D. Vavasour, Acta. Polymerica 46, 341 (1990). 33. F. Schmid, J. Phys.: Condens. Matter 10, 8105 (1998). 34. G. J. Zhang, F. Qiu, H. D. Zhang, Y. L. Yang and A.-C. Shi, Macromolecules 43, 2981 (2009).
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Chapter 4 Dynamic Self-Consistent Field Theories for Polymer Blends and Block Copolymers Toshihiro Kawakatsu Department of Physics, Tohoku University, Sendai 980-8578, Japan E-mail:
[email protected] Understanding the behavior of the phase separated domain structures and rheological properties of multi-component polymeric systems require detailed information on the dynamics of domains and that of conformations of constituent polymer chains. Self-consistent field (SCF) theory is a useful tool to treat such a problem because the conformation entropy of polymer chains in inhomogeneous systems can be evaluated quantitatively using this theory. However, when we turn our attention to the dynamic properties in a non-equilibrium state, the basic assumption of the SCF theory, i.e. the assumption of equilibrium chain conformation, breaks down. In order to avoid such a difficulty, dynamic SCF theories were developed. In this chapter, we give a brief review of the recent developments of dynamic SCF theories, and discuss where the cutting-edge of this theory is.
1. Introduction One of the characteristic features of multi-phase polymeric materials, such as polymer blends or block copolymers, is the formation of mesophases where phase-separated domains are formed on the length scale of the gyration radius of the constituent polymer chains (i.e. so-called microphase separation). It is important to recognize that the physical properties of these polymeric materials very often depend on the structure and dynamics of the domains in mesophases.1–3 Typical examples are the self-assembling processes of polymer blends and block copolymers, where the final domain structure is often strongly affected by the kinetic pathway that the system traces from the initial state to the final state. Of course, the deformation of constituent chains due to the flow or to the deformation of the domains also contribute to this dynamics of mesophases. 105
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For the study of equilibrium domain structures and equilibrium chain conformations in mesophases, there is a well-established technique called (static) self-consistent field (SCF) theory1,4–8 Using this static SCF theory, one can predict the equilibrium domain structures in mesophases of polymeric systems not only qualitatively but also quantitatively. On the other hand, the techniques to study dynamic phenomena in polymeric systems on the length scale of the domains in the mesophases are still limited to Monte Carlo or molecular dynamics simulation techniques on molecular models, which require a large amount of computational cost. In order to improve the computational efficiency of the dynamic simulations, it is essential to construct a model that is based on the coarse-grained picture like the SCF theory. The dynamics of polymeric flows are roughly categorized into two regimes. One is the slow flow regime where only the phase separated domains are deformed but the chain conformations are still in equilibrium. In such a regime, we can still rely on the static SCF theory by assuming that the chain conformations are always in local equilibrium and the chains diffuse according to the gradient of the thermodynamic forces (Diffusion regime). The other extreme is the fast flow regime where the chain conformation are deformed considerably from its equilibrium one due to the external forces or by migration of the domains (Viscoelastic regime).5,9 There have been considerable works for the dynamic extensions of the SCF theory to the diffusion regime, and various simulation techniques and knowledge have been accumulated.3,5,10 For example, dynamics of microphase separation of diblock copolymer melts,10,11 phase behavior of complex polymeric systems,12,13 dynamics of block copolymers under external constraints and under external fields14–22 Contrary to the situation for these dynamic SCF theories in the diffusion regime, studies for the viscoelastic regime have not yet been well developed from the viewpoint of the SCF theory. Actually, there have been several trials to study such viscoelastic regime using dynamic SCF theories. Typical examples are studies on long-range hydrodynamic interactions,23–28 effects of long-range kinetic correlations in mesophases,29,30 chain dynamics in dilute polymer solutions (Rouse dynamics),31–33 the chain dynamics in dense polymer systems (reptation dynamics),34 and development of hybrid approaches between SCF theory and particles.27,35,36 Despite these investigations, the state of the art of the dynamic SCF theories for viscoelastic regime is still in a developing stage.
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In this short review, we try to figure out the current status of this developing dynamic SCF theories for both diffusion regime and for viscoelastic regime. This article is organized as follows. In the next section, we give a brief summary of the theoretical formulations of the dynamic SCF theories. In section 3, we review typical studies on the domain motion in the diffusion dynamics regime, where dynamics of polymer blends and block copolymer systems are discussed. Then, in section 4, several recent trials to extend the dynamic SCF to viscoelastic regime. Finally, in section 5, we give a conclusion. 2. Basic Formalism for Dynamic SCF Theory For a simple illustration of the dynamic SCF theories, let us consider multicomponent polymer blends (including block copolymers as components). The essence of SCF theory for polymers is the coarse-graining, where microscopic details of the polymer chains are averaged out and only several essential physical properties survive. In this coarse-grained level, the polymer chain conformations are well described by the Gaussian chain statistics, where the conformation entropy of the chains is evaluated using the path integrals.5 In the static SCF theories,1,4–8 the important coarse-grained variables that characterize the state of the system are the density distributions of individual type of segments. Let us denote the density distribution of Ktype segments as φK (r), where K is the index for the segment species. The calculation of the static SCF theory is performed so that the total free energy of the system is minimized with respect to these density distributions of segments. On the other hand, when we turn our attention to dynamic processes, we immediately recognize that the segment density distributions φK (r) are not the only relevant field variables to specify the state of the system. Extra variables we need are, the local flow velocities of each components and the stress field. In general, time evolution of the density field φK (r, t), which is now also a function of time, is described by the following continuity equation because it is a conserved quantity; ∂ φK (r, t) + ∇ · jK (r, t) = 0, ∂t where jK (r, t) is the flux for the K-type segments.
(1)
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The flux jK (r, t) is decomposed into three contributions, i.e. diffusion flux, convection flux and the flux induced by the external field such as an electric field, as conv ext jK (r, t) = jdiff K (r, t) + jK (r, t) + jK (r, t).
(2)
In the following, we will discuss these contributions to the dynamics of the segment density separately. 2.1. Diffusion flux The first term on the right-hand side of Eq. (2) accounts for the diffusion flux. The simplest and the most popular choice is the following Fick’s law for linear diffusion driven by the gradient of the local chemical potential µK (r, t): LKK ∇µK (r, t), (3) jdiff K (r, t) = − K
where the coefficients LKK are the Onsager kinetic coefficients, which is in general a function of the local density φK (r). The following approximation for this kinetic coefficient is often adopted; (0)
LKK = LK φK (r)δKK , (0)
(4)
where LK are constants and δKK is Kronecker’s delta symbol. This simple form of the kinetic coefficient is accepted because in most of the cases, the off-diagonal elements of the kinetic coefficient matrix, LKK (K = K ), are negligible compared to the diagonal elements LKK , which means that the effects of the dynamical cross coupling between different types of fluxes is small. In Eq. (4), ∇µK (r) corresponds to a thermodynamic force acting on the individual segment of K-type, which originates from the inhomogeneity in the segment density distributions. As the polymer flow is very viscous, each segment moves with a limiting velocity driven by this thermodynamic force (over damped case). Then, the diffusion flux is a product of the drift velocity of a single segment and the total numbers of segments per volume, which results in the expression described above. In the dynamic SCF theory, the system is in general non-equilibrium. Therefore, the chemical potential µK in Eq. (3) or Eq. (4) is also nonequilibrium. Then, how to evaluate this chemical potential is a big problem.
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If the motion of the segment density distributions is so slow that the chain conformations are always relaxed under the constraint of the instantaneous distributions of the segment densities φK (r, t) at time t, we can use local equilibrium assumption as in the static SCF theory and can evaluate the chemical potential µK (r, t). Only difference from the static SCF is that the segment density distributions φK (r) are not determined by the SCF calculation but they are regarded as constraints imposed to the SCF calculation. Then, these density distributions φK (r) are determined by integrating the equation of motion, Eq. (1), rather than to be determined in SCF calculation. In this case, the SCF scheme is modified so that the density distribution of K-type segments calculated in the SCF scheme at every time step, denoted as ϕK (r), should coincide with the given density field φK (r). This can be done by solving the following set of SCF equations:5 b2 ∂ QK (r, s) = K ∇2 QK (r, s) − βVK (r)QK (r, s) ∂s 6 ϕK (r) = CK dsQK (r, s)Q†K (r, NK − s) VK (r) = χKK ϕK (r) + λK (r),
(5)
K
where QK (r, s) and Q†K (r, s) are forward and backward path integrals (or sometimes called end point distributions) for the K-type polymer chains, bK is the size of the statistical segment, VK (r) is the self-consistent mean field potential, ϕK (r) is the segment density distribution calculated by the SCF scheme, CK is a normalization constant, χKK is the χ-parameter between K and K type segment pair, and λK (r) is the Lagrange multiplier that constrains the segment density distribution calculated by this SCF scheme, ϕK (r), to the given density distribution φK (r). Once the above SCF scheme is solved (usually an iteration procedure is used), the chemical potential µK (r) can be obtained by the following relation: µK (r) = −λK (r).
(6)
This relation is understood in the following way. As we discussed above, the SCF calculation generates an equilibrium state under the constraint ϕK (r) = φK (r) by using the Lagrange multiplier λK (r). If this Lagrange multiplier is absent, the density field φK (r) will move according to the thermodynamic driving force due to the chemical potential µK (r). Thus,
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the Lagrange multiplier λK (r, t) should cancel this thermodynamic driving force so that the system becomes static (local equilibrium). This balance between µK (r) and λK (r) gives Eq. (6). 2.2. Convection flux The convection flux in Eq. (2) is driven by the local convection flow field v(r, t), and is given by jconv K (r, t) = v(r, t)φK (r, t).
(7)
This flow velocity v(r) is determined by the following conservation equations for the mass and the momentum: ∇ · v(r, t) = 0,
(8)
and
∂ v(r, t) = ∇ · Π(r, t) + ρK(r, t), (9) ∂t where ρ is the total density of the polymeric fluid that is assumed to be constant due to incompressibility condition, Π(r, t) is the stress tensor and K(r, t) is the body force which will be defined below. The stress tensor Π(r) expresses the viscoelastic properties of the polymeric fluid. In general, it is described as: ρ
Π(r) = −p(r)1 + σ(r),
(10)
where p(r) is the local pressure that plays the role of Lagrange multiplier for the incompressibility condition for total density, 1 is the unit tensor, and σ(r) is the shear stress tensor that in general depends on the history of the flow. For polymeric flow, the inertia term ρvv that appears in the stress tensor for low molecular weight fluids is usually neglected because of the high viscosity. The shear stress σ(r) characterizes the viscoelastic properties of the fluid. The relation between the shear stress σ at time t and the history of the velocity gradient tensor κ ≡ (∇v)T is called the constitutive equation. The simplest one is the Newtonian fluid,24 for which the constitutive equation is given by σ(r) = η(r)κ(r) = η(r)(∇v)T ,
(11)
where η(r) is the viscosity that in general depends on the local composition. For more general case with viscoelastic properties, one has to assume a model constitutive equation, such as Maxwell model, Rouse model, or reptation model.5,9 These models will be discussed in section 4.2 later.
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The body force K(r) in Eq. (9) accounts for the effects from the inhomogeneity in the segment density distribution (interfaces), and is given by23,24 K(r) = −
φK (r)∇µK (r).
(12)
K
Contrary to the static SCF theories discussed in the previous chapters, dynamic SCF simulation is usually done in the real space because the main target system of dynamic SCF is deformed domain structures that have no crystalline symmetry, for which spectral method is no longer useful.5,37 2.3. Flux induced by external fields Imposing an external field to the polymeric system also induces a nontrivial polymeric flow. A typical example is the electric field, which is often used to make the phase separated domains in a thin film align along the direction of the electric field.38 Let us consider a situation where an electric field is imposed in the zdirection of an A/B binary polymer blend or an A-B block copolymer melt. Assuming slow spatial variation of the local compositions φK (r), we can derive following approximated expression for the flux due to this electric field39 jext K (r, t) = α {ez · ∇φK (r)} ez ,
(13)
where α is a constant that measures the effect of the electric field, and ez is a unit vector in z-direction. This expression means that the imposed electric field induces an anisotropic diffusion in its direction. The parameter α depends on both the electrostatic nature of the material and the strength of the electric field. It is given by α=
0 21 2 E vL,
¯ 0
(14)
where E0 is the amplitude of the electric field, v is the volume of a single polymer chain, L is the Onsager kinetic coefficient that appeared in Eq. (3) where we assumed LKK = LδKK for simplicity, 0 is the dielectric constants of vacuum, and 1 and ¯ are defined by the dependence of the local dielectric constant (r) on the local composition φA (r) as
(r) ≈ ¯ + 1 (φA (r) − φ¯A ), where φ¯A is the average composition of the A-component.
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3. Dynamic SCF Theories in Slow Diffusion Regime 3.1. Formulation When there is mesoscopic phase separated domains in the system, such as the microphase separation of block copolymers, these domains dominate the rheological properties of the total system in the low frequency regime or long time regime. In this case, we can adopt quasi-static approximation where the chain conformation is always equilibrium under the constraint imposed by the profile of the segment density distribution φK (r, t). If we further neglect the flow field, Eq. (2) is simplified to (0)
jK (r, t) = −LK φK (r)∇µK (r, t),
(15)
where we adopted the simplified form of the Onsager kinetic coefficients given in Eq. (4). In Eq. (15), the connectivity of polymer chain is taken into account into the SCF calculation through the chemical potential {µK }. This diffusion dynamics takes only the slowest Rouse mode of the constituent chains into account.40 If we apply this diffusion type model to polymer blends, the dominant contribution to the dynamics of whole system is the motion of the interfaces that separate the domains. However, for immiscible polymer blends, such domains continuously grow and result in a macrophase separation. In such a late stage of the macrophase separation, the gyration radius of constituent polymers are negligibly small compared to the domain size, and the effect of the conformation entropy of the polymer chains becomes less and less important. This is why SCF theory was not extensively applied to dynamics of macrophase separation of polymer blends. For such macrophase separations, the so-called Ginzburg-Landau (GL) type coarse-grained theory is often used.5,41 Situation changes drastically when we consider block copolymer melts or blends of homopolymers and block copolymers. In these systems, the domains do not grow to a macroscopic scale because of the connectivity of the subchains of the block copolymer or of the surfactant nature of the block copolymer. The former limits the domain size and the latter diminishes the driving force for the domain coarsening. As a result, the system shows a microphase separation on the length scale of the block copolymer chain. (The domain size is proportional to N 1/2 in the vicinity of the critical point and to N 2/3 well inside the coexistence region for a melt of a block copolymer with chain length N .5 ) Therefore, block copolymer systems are very suitable targets of the dynamic SCF theory, where the conformation entropy plays an important role.
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Another suitable target of dynamic SCF technique is a thin film of polymeric fluid. If the thickness of the film is of the same order as the gyration radius of the polymer chain, the polymers undergo considerable deformation due to the constraint imposed by the wall of the film. These systems can also be simulated with dynamic SCF theories. 3.2. Formation process of mesophases of block copolymer melt The first simulation based on the diffusion type dynamic SCF theory was reported by Fraaije on the formation process of lamellar phase of a symmetric diblock copolymer melt using a 2-dimensional system,10 and later they extended this simulation to 3-dimenional systems.11 Figure 1 is a series of snapshot pictures obtained by a 3-dimenisonal simulation on formation of lamellar phase of an A8 B8 symmetric diblock copolymer melt with χN = 16.0.11 The initial state is a uniform phase of the block copolymer quenched from a high-temperature disordered state. They simulated this process based on Eqs. (1) and (2), where jdiff is given by Eq. (15) and other fluxes are neglected. In this simulation, the authors assumed that diffusion dynamics, where the kinetic coefficients LK were (0) (0) given by LK = LK φK (r) with constant LK . (a)
(b)
(c)
(d)
Fig. 1. (a)–(c) Formation of a lamellar phase of an A8 B8 symmetric diblock copolymer melt from its initial uniform state with χN = 16.0. (d) An isosurface representation of the data is shown in (d). Reprinted with permission from Ref. 11.
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(a)
(b)
(c)
Fig. 2. (a) Dynamic SCF simulation on phase separation of a mixture of long and short A-B type block copolymers. Upper figures show the density distributions of long block copolymer, while the lower figures show the density distributions of A-segments of both long and short block copolymers. (b) Domain structure obtained by a 3-dimensional simulation on the same system. (c) A TEM image of the corresponding experimental system. Reprinted with permission from Ref. 13.
Although there exists GL type theory for block copolymer melts42 based on coarse-grained free energy model,43 dynamic SCF has several advantages. For example, the chain architecture can easily be introduced, and the simulation can be done even for larger value of χN , for which the GL free energy is no longer valid. One of the early simulations in this direction was performed on Pluronics, which are A-B-A type triblock copolymers made of two poly(propylene oxide) blocks and a poly(ethylene oxide) middle block.12 This triblock copolymer is a typical polymeric surfactant, and is widely used in chemical industries. Dynamic SCF simulations were performed for Pluronic/water solutions by changing the composition of the solution.12 It was shown that the simulations reproduce most of the phase diagram of the experimental system without assuming any crystal symmetries of the domain structures. Dynamic SCF simulations can also predict the domain structures and phase diagram of more complex systems. Figure 2 shows a phase separation dynamics of a mixture of a long and a short block copolymers of the same kind.13 This simulation was intended to reproduce the behavior of an experimentally investigated systems.44 Therefore the parameters such as the χ-parameters, chain lengths and the molecular structures of the block copolymers are all taken from the experimentally obtained data.44 Due to this reason, the system is a mixture of a slightly asymmetric A − B type diblock copolymers with different chain lengths. In Figs. 2(a) and (b), one
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can confirm that the lamellar domains are generated in a matrix of the disordered phase. Such a structure is similar to that observed in real experiment shown in Fig. 2(c). We also confirmed that the experimentally determined phase diagram is very well reproduced by the dynamic SCF simulations. It should be noted that the generated lamellar domains both in the experiment and in the dynamic SCF simulation are irregular ones. This means that the system is not yet reached the true equilibrium state. For such a non-equilibrium state without crystal symmetry, the dynamic SCF simulation in real space is a powerful tool. 3.3. Dynamics and non-equilibrium domain structures in thin films and near solid surfaces Another interesting target of dynamic SCF technique where the diffusion dynamics is dominant is the phase separation dynamics and irregular domain structures in thin films and near solid surfaces. Typical examples are surface-induced phase separation and selective adsorption of polymers to a solid or free surface. Application of dynamic SCF technique to these problems was done by Hasegawa and Doi.14 They investigated the adsorption of homopolymers to an attractive solid surface from a semidilute polymer solution using 1dimensional dynamic SCF technique. Using the path integrals calculated in the SCF simulation, one can extract information on the polymer chain conformations. As is shown in Fig. 3(a), they calculated the average lengths of tail, train, and loop parts in the adsorbed polymers. (The definitions of these terminologies are given in Fig. 3(b).) From these data, one can (a)
(b)
average length
10
tail
5 train
tail loop
loop 0
train
10-2 10-1 100 101 102 time
Fig. 3. (a) Time evolution of average lengths of tail, train and loop parts of adsorbed homopolymers. Reprinted with permission from Ref. 14. (b) Schematic explanation for the tail, train and loop parts.
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(a)
(b)
(c)
(d)
Fig. 4. Mesophases in thin block copolymer layer obtained by experiments ((a)-(c)) and dynamic SCF simulations ((d)). Reprinted with permission from Ref. 15.
understand that the adsorption process consists of two stages. In the first stage, segments that were close to the surface in the initial time are adsorbed rather independently. This is indicated by the emergence of long tails and loops. Then, using the adsorbed segment as a nucleus, connected segments to this adsorbed segment are successively adsorbed onto the solid surface. This leads to a formation of long trains, which replaces the initially generated tails and loops. Another example is the mesophase formation in thin films of block copolymers placed between two parallel solid plates.15 The data shown in Fig. 4 are comparison between experimental observation on SBS block copolymer film and corresponding dynamic SCF simulation of a thin film of block copolymer A3 B12 A3 and χN = 117, where the film thickness is gradually changed so that the various mesophases can be observed simultaneously. Not only traditional spherical and cylindrical phases, one can also observe perforated lamellar domains which is not stable in a 3-dimensional bulk system. Dynamic SCF technique is also applicable to a free surface between a polymer film and the air.16 In such a system, domains formed by the phase separation near the free surface induce deformation of this surface due to the difference in the interfacial tension. In Fig. 5, we show the domain structures and the surface roughening of an interface between A/B polymer blend film and the air obtained by dynamic SCF simulation. In
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(c)
(b)
Fig. 5. Formation of domains at the free surface of an A/B polymer blend film obtained by dynamic SCF simulations. (a) Segment density distributions and (b) deformation of free surface. (c) Phase diagram for the surface roughening. χAB and χBV are interaction parameter between A and B segments and that between A-segment and the void (air) segment. Reprinted with permission from Ref. 16.
this simulation, the air phase is modeled by an artificial “void” phase that has a very strong repulsive interaction with the polymer segments. Using a series of simulations, we can determine the phase diagram of the surface roughening as a function of the interaction parameters between A and B segments and that between A segment and the void (Fig. 5(c)). 3.4. Structural phase transitions induced by external fields When a mesophase of a polymeric system is subjected to a weak external field, such as an electric field or an external flow, in response to this external field, the phase separated domains are deformed. If the strength of the external field exceeds certain threshold value, a structural phase transition will take place. Here we discuss several examples of such a structural phase transition induced by external fields. The first example is concerned with the gyroid phase of diblock copolymers. Gyroid phase is composed of bicontinuous networks of two A-domains and two B-domains, each of which consists of 3-fold junctions.17 Figure 6 shows results of dynamic SCF simulations on the effect of an external shear flow on gyroid phase of an A7 −B13 diblock copolymer melt at χN = 20 done by Honda and Kawakatsu.17 The shear flow direction is in
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(a)
(b)
(d)
(e)
(c)
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Fig. 6. (a)–(f) Structural phase transition from a gyroid phase to a cylindrical phase induced by an external shear flow. Reprinted with permission from Ref. 17.
the [1 1 1] direction of the Gyroid structure (chosen as x-direction) and its velocity gradient in the [1 1 ¯2] direction (chosen as y-direction), where the shear rate γ˙ ≡ ∂vx /∂y is γ˙ = 0.001 (in unit of 1/L, where L is the kinetic coefficient which is assumed to be the same for A and B components). In this simulation, the effect of the flow is introduced by assuming v(r) in Eq. (7) as v(x, y, z) = (γy, ˙ 0, 0). (An example of the treatment of full hydrodynamics is discussed in section 4.1.) A coexistence of the gyroid phase and the cylindrical phase is observed in the late stage. This means that the transition proceeds as nucleation and growth process. For this coexistence, it is essential that the size of the simulation box is adjusted so that the free energy density is minimized with respect to the system box size at every time step. (We call this technique as “system size optimization”.) Otherwise the coexistence is smothered by the constraint of the constant system size.17 A similar strategy of adjusting the system size was used by Li et al. for 2-dimensional models of sheared micro phases of linear and star triblock copolymers.18 On the other hand, Fig. 7 shows the results of dynamic SCF simulations on a similar structural transition of gyroid phase but induced by an external static electric field imposed in the [1 1 1] direction of the gyroid structure.19
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Fig. 7. (a)–(f) Structural phase transition from gyroid phase to cylindrical phase induced by an external electric field. Reprinted with permission from Ref. 19.
The effect of the electric field is taken into account using Eq. (13). Differently from the case with shear flow (Fig. 6), we observe uniform transition from the initial gyroid structure to the final cylindrical structure. We confirmed that this uniform nature of the intermediate structure is not due to the effect of the system size optimization19 as in the case of the shear flow. The main difference between the shear flow case and this electric field case is the fact that the external field has two characteristic directions in the shear flow case, i.e. the flow direction and the direction of the velocity gradient, while the electric field has only one characteristic direction. This difference affects the kinetic pathways from the initial gyroid phase to the final cylindrical phase because in shear flow case, anisotropy in the domain structure is easily be generated due to the bi-directional nature of the shear flow. The effects of the external electric field is especially important when one try to align the micro domains in a thin film of block copolymer to a desired direction. It is usually not easy for an electric field to penetrate the bulk specimen of a polymeric material due to the screening effects by the dielectric property. However, by using a ultra thin film of block copolymers, such an alignment can be achieved.38 The effects of an external electric field on block copolymer thin film were extensively studied by Knoll et al.39 They studied the alignment of the cylinder domains in a thin film both by experiment and by dynamic
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(v) Fig. 8. (a) (i)–(v) Time evolution of the domain structures upon structural phase transition from initial perforated lamellar phase to cylindrical phase in a thin film of A-B diblock copolymer induced by an external electric field in the normal direction of the thin film. (b) Variation of the shapes of the final cylinders by changing the strength of the electric field. From (i) to (iv), the electric field becomes stronger. Reprinted with permission from Ref. 20.
SCF simulation for χN = 19.0 (weak segregation regime). They observed splitting of cylinders into small spheres due to the imposed electric field. Then, these spheres reconnect in a different way so that new cylinders that are parallel to the electric field are generated. This is the scenario of domain reorientation in weak segregation regime. If the system is in the strong segregation regime (larger χN ), rotation of the grains of ordered mesophase is expected instead of splitting and reconnection of domains. In this case, the domain rotation is achieved by migration of defects in the ordered domains. As we have discussed in section 3.3, block copolymers can show exotic domain structures, such as perforated lamellar domains, which are no longer stable in a 3-dimensional bulk system.15 Thus, it is interesting to study the stability of such exotic phase in thin film under the influence of an electric field. Ly et al. performed a series of dynamic SCF simulations on thin films of an A-B diblock copolymer20 or of an A-B-A triblock copolymer21 under an electric field. Figure 8 shows results of a dynamic SCF simulation on perforated lamellar phase in a thin film of A-B diblock copolymer under an electric field. The electric field is imposed in the normal direction of the film, in which direction the final cylinders are aligned.
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In this case, the kinetic pathway is rather complex. First, the narrow arms of the perforated lamellar domains shrink. Then, these arms are disconnected to form cylinder domains perpendicular to the film. The shape of the final cylindrical domains are determined by the competition between the effect of the electric field and the effect of the wall which prefers one of the two segment species. In this simulation, the interaction parameters between the polymer segments and the solid walls are chosen so that the cylinder-forming block dislikes the wall. Therefore the top and the bottom of the cylinders are shrinking when the electric field is weak (Figure 8(b)(i)(ii)). However, as we increase the strength of the electric field, the cylinder tend to be straight (Figure 8(b)(iii)-(vi)). 4. Beyond Diffusion Dynamics — Hydrodynamics, Viscoelasticity and Hybrid Techniques In the previous section, we have discussed the dynamic phenomena where the diffusion process and the deformation of domains are dominant. However, if the system is strongly and rapidly deformed by an external field or by a rapid migration of the interfaces, the characteristic time scale of the deformation becomes shorter than the characteristic relaxation time of the chain conformation. In such a case, the assumption of diffusion dynamics breaks down, and we have to introduce more microscopic dynamic processes into the model. In this section, we review the recent developments of the extensions of dynamic SCF theory to several interesting situations. 4.1. Hydrodynamic effects on domain formation Even though a polymer melt or a polymer blend is so viscous, hydrodynamic effects cannot be neglected in the long time behavior.45 This is because the hydrodynamic interaction is long ranged and does not have characteristic length scale (it decays as inverse distance). For example, in the case of the macrophase separation of a polymer blend, the hydrodynamic interaction may be negligible in the early stage of the phase separation. However, its importance gradually increases as the domains grow, and finally it dominates the dynamics of the domain coarsening. This property has been well verified experimentally46 and by simulations on time-dependent GL model.47,48 Even for the diffusion type dynamics, the dynamic SCF simulation requires a massy computation. The introduction of hydrodynamics requires an additional heavy computational cost, which is necessary to solve the
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Fig. 9. Time evolution of the free energy of an A8 B8 diblock copolymer melt that undergoes microphase separation from an initial uniform state. Two model systems, i.e. simple diffusion dynamics system (upper curve) and hydrodynamic system where Darcy’s law is used (lower curve), are compared. Reprinted with permission from Ref. 23.
time evolution equation for the Navier-Stokes equation for the velocity field, Eqs. (9)-(11): ρ
∂ v(r, t) = −∇p(r) + ∇ · [η(r)∇v(r)] + ρK(r, t), ∂t
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where the pressure p(r) is determined so that the incompressibility condition Eq. (8) is always satisfied. In order to avoid the heavy computational cost associated with the numerical solution of Eq. (16), Maurits, et al. used an assumption of Darcy’s law23 1 (17) ζv(r, t) = K(r, t), ρ where ζ = 6πη0 /ξ 2 is the friction constant of an object that has a characteristic size ξ. This assumption means that the segment is assumed to be a moving object in a static medium with viscosity η0 and is always moving with the termination velocity driven by the body force K(r). In Fig. 9, we show a comparison of the time evolution of the free energy between the two model systems, one is the simple diffusion system and the other is the hydrodynamic system where Darcy’s law is used.23 It is obvious that the hydrodynamic effects accelerates the phase separation. This acceleration originates from the alignment of domains induced by the long-range hydrodynamic interaction.
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Fig. 10. (a) Dynamic SCF simulation with hydrodynamic interaction is applied to polymeric flows in micro channels with various shapes. (b) Dynamic SCF with hydrodynamics is applied to polymeric flows with filler particles. Here the filler particles are treated as discrete molecules while the polymers are described by dynamic SCF theory. Reprinted with permission from (a) Ref. 25 and (b) Ref. 27, respectively.
On the other hand, dynamic SCF simulations based on the full hydrodynamic equation, Eq. (16), were performed recently.24–26 Honda and Kawakatsu investigated the effects of the hydrodynamic interaction on the formation process of mesophases of block copolymers.24 They simulated the formation processes of lamellar, cylinder, spherical and gyroid structure from initial disordered phase. Comparison between the two cases with and without the hydrodynamic interactions revealed that the well-ordered lamellar, cylindrical and spherical domains can only be formed for the cases with hydrodynamic interaction. This result corresponds to the observation by Maurits et al.23 However, even with the hydrodynamic interaction, well-ordered gyroid phase was not obtained. The main difference between the gyroid phase and the other phases is the connectivity of the domains. As the gyroid domains are interconnected, the flow field induced by the interfacial motion cannot propagate to distant points and cannot induce long-range dynamic/static correlations between domains. Hall et al. used the dynamic SCF with hydrodynamic interaction for the study of polymeric flows in microscopic channels with arbitrary shapes25 (Figure 10(a)) and those containing filler particles27 (Figure 10(b)). For the latter case of the simulation with fillers, they used a hybrid approach where the polymer flow is described by continuous field while the filler particles are described by discrete molecules,28,49–51 which will be discussed in section 4.3.
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As shown in Fig. 10, the dynamic SCF simulations give us an understanding of micro-fluidics and composite flows, and offers a way to design high-functional materials and micro machines. 4.2. Dynamic SCF with viscoelastic properties It is well known that polymeric materials show viscoelastic properties. The viscoelastic property of a fluid is described by the relation between the stress tensor σ(r, t) and the velocity gradient tensor κ(r, t) ≡ (∇v)T as5,9 σ(r, t) = S [{κ(r, t )}; t ≤ t}] ,
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where the right-hand side means that the stress at time t depends on the history of the deformation in the past. The relation Eq. (18) is called the constitutive equation. A simple example of the constitutive equation is the following one for a linear viscoelastic fluid: t G(t − t ) κ(t ) + κT (t ) dt , (19) σ(t) = −∞
where G(t) is called the stress relaxation function, and we assumed that the fluid is uniform and isotropic. The non-Markovian nature of the constitutive equation Eq. (18) or Eq. (19) means that the fluid has a memory. In the case of multi-component polymeric fluids, this memory is stored in the fluid in the form of (a) deformation of the domains, and (b) deformation of the polymer chains or polymer networks. The former (factor (a)) is already taken into account in the dynamic SCF theory using the diffusion dynamics. However, the latter (factor (b)) is not accounted for by the diffusion dynamics or by hydrodynamics discussed so far. There are two typical chain dynamics in polymer solutions and melts. One is the Rouse dynamics and the other is the reptation dynamics.5,9 The Rouse dynamics is valid for dilute polymer solutions without hydrodynamic interaction or for short time dynamics of dense polymer systems. In a dilute polymer solution, a chain undergoes a Brownian motion under the constraint of the chain connectivity. This constraint leads to the Rouse dynamics. On the other hand, reptation dynamics explains the dynamics for overall time regime of dense polymer melts and blends. In dense polymer systems, a polymer chain moves as if it is confined in a tube formed by the steric hindrance by the surrounding chains. This chain dynamics is called reptation dynamics.
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Because SCF theory can quantitatively evaluate a change in the polymer conformation, it should basically be possible to take the viscoelastic properties into SCF theory. However, both static and dynamic SCF theories assume that the chain conformation is always equilibrium, which contradicts with the chain conformation in deformed viscoelastic polymeric fluids. 4.2.1. Non-local kinetic coefficients with chain deformations Early works intended to overcome this difficulty in the framework of dynamic GL and SCF theories were done by introducing non-local kinetic coefficient.29,30,52 Instead of the time evolution equations with local kinetic coefficients, Eqs. (1)-(4), Kawasaki and Sekimoto proposed a time evolution equation with non-local kinetic coefficients for diffusion flux as ∂ φK (r, t) = dr LKK (r, r )∇ µK (r , t), (20) ∂t K
where LKK (r, r ) is the non-local kinetic coefficient which depends on the instantaneous deformed conformations of the chains.52 The non-locality comes from the dynamical correlations between segments at distant points r and r that are connected by polymer chains, which move in Rouse dynamics or in reptation dynamics.5,9 This non-local kinetic coefficient was applied to formation of lamellar phase of a block copolymer melt using dynamic GL and SCF simulations with reptation dynamics in Ref. 29 and with Rouse dynamics in Ref. 30, respectively. As is shown in Fig. 11, these simulations with non-local kinetic coefficient lead to an acceleration of the coarsening compared to that with local kinetic coefficient. This acceleration is a result of an apparent longrange transport of segments. 4.2.2. Full dynamic SCF theory with Rouse dynamics More direct way of introducing non-equilibrium chain conformation into dynamic SCF theory is to calculate the time evolution of the probability distribution of each bond vector connecting the segments along the chain. If we can calculate this probability distribution of bond vectors, we can obtain the instantaneous stress tensor σ(r, t) as a sum of the contributions from each bond vectors. Then, we can substitute the result into Eq. (10) and solve Eq. (9). Maurits et al. introduced Smoluchowski equation, which is the time evolution equation of the probability distribution of bond vector, into dynamic
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Fig. 11. Time evolutions of inverse of the average domain size of lamellar formation of a symmetric diblock copolymer. Results of two models with local (broken curve) and non-local kinetic coefficients (solid curve) based on reptation dynamics are compared. The ratio between the gyration radius of the chain Rg and the interfacial thickness ξ is chosen as (a) Rg /ξ = 2.0 and (b) Rg /ξ = 5.0, respectively. Reprinted with permission from Ref. 29.
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Fig. 12. A comparison of domain structures of diblock copolymer melt under a shear flow between dynamic SCF simulations with (a) simple diffusion dynamics and with (b) Rouse dynamics is shown. Reprinted with permission from Ref. 31.
SCF theory.31 When they calculated the shear stress under an external flow filed, they approximated the Rouse dynamics of a connected bonds along the chain by a set of independently moving dumbbells. By applying a shear flow on a diblock copolymer melt using both simple diffusion dynamics and Rouse dynamics, they observed that the response of the domains to the external flow is slowed down when they introduce the Rouse dynamics (Figure 12). Since these works, several groups extended dynamic SCF mainly with Rouse dynamics.32,33
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Fig. 13. Example of dynamic SCF simulation with reptation dynamics. (a) Two parallel plates with grafted polymer brushes. (b) Time evolution of the stress when a start-up flow is imposed. Inset figures are the conformations of a single chain before
