Nanostructured Soft Matter: Experiment, Theory, Simulation and Perspectives

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NanoScience and Technology

NanoScience and Technology Series Editors: P. Avouris B. Bhushan D. Bimberg K. von Klitzing H. Sakaki R. Wiesendanger The series NanoScience and Technology is focused on the fascinating nano-world, mesoscopic physics, analysis with atomic resolution, nano and quantum-effect devices, nanomechanics and atomic-scale processes. All the basic aspects and technology-oriented developments in this emerging discipline are covered by comprehensive and timely books. The series constitutes a survey of the relevant special topics, which are presented by leading experts in the field. These books will appeal to researchers, engineers, and advanced students.

Applied Scanning Probe Methods II Scanning Probe Microscopy Techniques Editors: B. Bhushan and H. Fuchs

Applied Scanning Probe Methods VI Characterization Editors: B. Bhushan and S. Kawata

Applied Scanning Probe Methods III Characterization Editors: B. Bhushan and H. Fuchs

Applied Scanning Probe Methods VII Biomimetics and Industrial Applications Editors: B. Bhushan and H. Fuchs

Applied Scanning Probe Methods IV Industrial Application Editors: B. Bhushan and H. Fuchs

Roadmap of Scanning Probe Microscopy Editors: S. Morita

Scanning Probe Microscopy Atomic Scale Engineering by Forces and Currents Editors: A. Foster and W. Hofer

Nanocatalysis Editors: U. Heiz and U. Landman

Single Molecule Chemistry and Physics An Introduction By C. Wang and C. Bai Atomic Force Microscopy, Scanning Nearﬁeld Optical Microscopy and Nanoscratching Application to Rough and Natural Surfaces By G. Kaupp Applied Scanning Probe Methods V Scanning Probe Microscopy Techniques Editors: B. Bhushan, H. Fuchs, and S. Kawata

Nanostructures Fabrication and Analysis Editor: H. Nejo Fundamentals of Friction and Wear on the Nanoscale Editors: E. Gnecco and E. Meyer Lateral Alignment of Epitaxial Quantum Dots Editor: O. Schmidt Nanostructured Soft Matter Experiment, Theory, Simulation and Perspectives Editor: A.V. Zvelindovsky

A.V. Zvelindovsky

(Ed.)

Nanostructured Soft Matter Experiment, Theory, Simulation and Perspectives

With 261 Figures

Dr. A.V. Zvelindovsky (Ed.) Centre for Materials Science Department of Physics, Astronomy and Mathematics University of Central Lancashire Preston Lancashire PR1 2HE United Kingdom

Series Editors: Professor Dr. Phaedon Avouris

Professor Dr., Dres. h.c. Klaus von Klitzing

IBM Research Division Nanometer Scale Science & Technology Thomas J. Watson Research Center P.O. Box 218 Yorktown Heights, NY 10598, USA

Max-Planck-Institut für Festkörperforschung Heisenbergstr. 1 70569 Stuttgart, Germany

Professor Dr. Bharat Bhushan

University of Tokyo Institute of Industrial Science 4-6-1 Komaba, Meguro-ku Tokyo 153-8505, Japan

Ohio State University Nanotribology Laboratory for Information Storage and MEMS/NEMS (NLIM) Suite 255, Ackerman Road 650 Columbus, Ohio 43210, USA

Professor Dr. Dieter Bimberg TU Berlin, Fakutät Mathematik/ Naturwissenschaften Institut für Festkörperphyisk Hardenbergstr. 36 10623 Berlin, Germany

Professor Hiroyuki Sakaki

Professor Dr. Roland Wiesendanger Institut für Angewandte Physik Universität Hamburg Jungiusstr. 11 20355 Hamburg, Germany

A C.I.P. Catalogue record for this book is available from the Library of Congress

ISSN 1434-4904 ISBN 978-1-4020-6329-9 (HB) ISBN 978-1-4020-6330-5 (e-book) Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands In association with Canopus Publishing Limited, 27 Queen Square, Bristol BS1 4ND, UK www.springer.com and www.canopusbooks.com All Rights Reserved © Canopus Publishing Limited 2007 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means,electronic,mechanical,photocopying,microﬁlming,recording or otherwise,without written permission from the Publisher, with the exception of any material supplied speciﬁcally for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

Preface

“The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living.” Henri Poincar´e (1854 - 1912) The ancient Greeks, quite ingeniously, realised that all materials and their (now known as macroscopic) properties, including life itself, are due to a limited number of tiny, constantly moving building blocks and the connections (now called interactions) between these blocks. Receiving both scientiﬁc and non-scientiﬁc opposition, the idea faded and, despite some renaissance of atomistic ideas in the 17-19th centuries, it still took more than two thousand years, until the time of Einstein, for the idea of microscopic building blocks to be fully accepted. These ideas, begun during the golden age of physics in the 20th century, have led to a comprehensive understanding of such states of matter as gases and solids, which in turn have completely revolutionised everyday life in the developed world by introducing technological wonders such as modern cars, air traﬃc, semiconductor chips for computers and nuclear power. Another state of matter, ﬂuids, appeared to be much more diﬃcult to tackle, even in the case of simple liquids like liquid argon, a research favourite in the ﬁeld. Legend tells that Lev D. Landau, Physics Nobel Laureate, was said to have commented that there could be no theoretical physics of liquids, as they have no small parameters. Nonetheless, as the 20th century advanced, it also became possible to treat even this most slippery of subjects due, in part, to the introduction of computers and the development of computer simulation methods like molecular dynamics. The 20th century brought yet another revolution: the industrial production of novel classes of materials, which simply did not exist before. For instance, almost every aspect of our everyday life would change immeasurably if plastics should disappear and life would turn “blind”, “deaf” and rather miserable without liquid crystals for computer screens or mobile phones. Such new materials were given the name complex ﬂuids, and their building blocks are not simply atoms or small molecules, but include block copolymers, surfactants, amphiphiles, colloids, liquid crystals, biomacromolecules, such as proteins and DNA, and various composites of the above. Complex ﬂuids possess features of both ﬂuids (for instance, they can ﬂow) and solids (they can have an internal structure often with various well

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Preface

resolved symmetry groups). These structures have a characteristic scale for their building blocks which is in the range of nanometers to microns, but the building blocks can be made (synthesised) with various degrees of complexity, so more than one size scale can be involved. Some structures can be formed spontaneously from a homogeneous mixture of the building blocks, a process referred to as self-assembly, which can be hierarchical and occur on various time scales depending on the complexity of the building blocks. Self-assembly is related to self-organization, which makes complex ﬂuids similar to living matter, so they can serve as model systems for biological systems and bioinspired materials. In the last decades of the 20th century the term complex ﬂuids started to be substituted by a more general one that is better suited to the overall concept of condensed matter: soft matter. The transition between millennia was marked by a burst of soft matter research, due, in part, to the fact that computers had then reached a level of power allowing the simulation of experimental size systems, thus enabling the very ﬁrst “virtual experiments” of such complex systems to be performed. This development made the links between theory and experiment truly symbiotic. Nanostructured soft materials, even apart from future technological perspectives beyond our imagination, are fascinating and beautiful. This research ﬁeld is growing so fast that there has been no single book that provided an overview of the many diﬀerent perspectives on both fundamental concepts and recent advances in the ﬁeld. A group of very enthusiastic contributors has now ﬁlled this gap; and the present book is the ﬁrst comprehensive monograph on nanostructured soft matter. It covers materials ranging in size from short amphiphilic molecules to block copolymers to proteins and also discusses colloids, hybrids, microemulsions and bio-inspired materials such as vesicles. Each chapter is written by active world-class researchers in the ﬁeld who oﬀer the reader an interdisciplinary view from diﬀering perspectives. They combine the experimental approaches of Chemistry and Physics, e.g. scattering techniques, electron and Atomic Force microscopy, with various Theoretical Physics, Mathematics and advanced computer modelling methods. We hope the book will be useful for both active and starting researchers as well as for undergraduate students; or, citing one of the anonymous referees of the original proposal for this book: “There is something for everyone in this book and it would represent a very useful text for those both operating at the forefront of nano-science and those entering the ﬁeld . . . ” I wish to thank the publishers at Canopus for assistance in the production of this book. I also thank Drs. R. McCabe, S. V. Kuzmin and N. Kiriushcheva. My editorial eﬀort is dedicated to Prof. A. V. Zatovsky (1942-2006), who ﬁrst introduced me to the wonders of Soft Matter.

Preston, Lancashire, January 2007

AVZ

Contents

Preface A. V. Zvelindovsky (ed.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V

Part I Experimental Advances Microemulsion Templating W. F. C. Sager . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Nanofabrication of Block Copolymer Bulk and Thin Films: Microdomain Structures as Templates Takeji Hashimoto and Kenji Fukunaga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Characterization of Surfactant Water Systems by X-Ray Scattering and 2 H NMR Michael C. Holmes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Polyelectrolyte Diblock Copolymer Micelles: Small Angle Scattering Estimates of the Charge Ordering in the Coronal Layer Johan R. C. van der Maarel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Structure and Shear-Induced Order in Blends of a Diblock Copolymer with the Corresponding Homopolymers I. W. Hamley, V. Castelletto and Z. Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Electric Field Alignment of Diblock Copolymer Thin Films T. Xu, J. Wang and T. P. Russell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Control of Block Copolymer Microdomain Orientation from Solution using Electric Fields: Governing Parameters and Mechanisms Alexander B¨ oker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

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Structure and Dynamics of Cylinder Forming Block Copolymers in Thin Films Larisa Tsarkova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

Part II Mathematical and Theoretical Approaches

Mathematical Description of Nanostructures with Minkowski Functionals G.J.A. Sevink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Scaling Theory of Polyelectrolyte and Polyampholyte Micelles Nadezhda P. Shusharina and Michael Rubinstein . . . . . . . . . . . . . . . . . . . . . 301 The Latest Development of the Weak Segregation Theory of Microphase Separation In Block Copolymers I. Ya. Erukhimovich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Coarse-Grained Modeling of Mesophase Dynamics in Block Copolymers Zhi-Feng Huang and Jorge Vi˜ nals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Eﬀective Interactions in Soft Materials Alan R. Denton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

Part III Computer Simulations

Ab-initio Coarse-Graining of Entangled Polymer Systems J.T. Padding and W.J. Briels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 Computer Simulations of Nano-Scale Phenomena Based on the Dynamic Density Functional Theories: Applications of SUSHI in the OCTA System Takashi Honda and Toshihiro Kawakatsu . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 Monte Carlo Simulations of Nano-Conﬁned Block Copolymers Qiang Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 Understanding Vesicles and Bio-Inspired Systems with Dissipative Particle Dynamics Julian C. Shillcock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 Theoretical Study of Nanostructured Biopolymers Using Molecular Dynamics Simulations: A Practical Introduction Danilo Roccatano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555

Contents

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Understanding Liquid/Colloids Composites with Mesoscopic Simulations Ignacio Pagonabarraga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

List of Contributors

Alexander B¨ oker (Germany) Physikalische Chemie II, Universit¨ at Bayreuth. Wim J Briels (The Netherlands) Computational Biophysics, Faculty of Science and Technology, University of Twente, Enschede. Valeria Castelletto (United Kingdom) Department of Chemistry, University of Reading. Alan R Denton (USA) Department of Physics, North Dakota State University, Fargo. Igor Erukhimovich (Russia) Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, Moscow, and Physics Department, Moscow State University. Kenji Fukunaga (Japan) Polymer Laboratory, Ube Industries, Ltd., Chiba. Ian W Hamley (United Kingdom) Department of Chemistry, University of Reading.

Takeji Hashimoto (Japan) Advanced Science Research Centre, Japan Atomic Energy Agency, Ibaraki. Michael C Holmes (United Kingdom) Centre for Materials Science, Department of Physics, Astronomy and Mathematics, University of Central Lancashire, Preston. Takashi Honda (Japan) Japan Chemical Innovation Institute, and Department of Organic and Polymeric Materials, Tokyo Institute of Technology. Zhi-Feng Huang (USA, Canada) Department of Physics and Astronomy, Wayne State University, Detroit and McGill Institute for Advanced Materials and Department of Physics, McGill University, Montreal. Toshihiro Kawakatsu (Japan) Department of Physics, Tohoku University, Sendai.

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Johan R C van der Maarel (Singapore) Department of Physics, National University of Singapore.

Nadezhda P Shusharina (USA) Department of Chemistry, University of North Carolina at Chapel Hill.

Johan T Padding (The Netherlands) Computational Biophysics, Faculty of Science and Technology, University of Twente, Enschede.

Larisa A Tsarkova (Germany) Physikalische Chemie II, Universit¨ at Bayreuth.

Ignacio Pagonabarraga (Spain) Departament de F´isica Fondamental, Universitat de Barcelona.

Jorge Vi˜ nals (Canada) McGill Institute for Advanced Materials and Department of Physics, McGill University, Montreal.

Danilo Roccatano (Germany) School of Engineering and Science, Jacobs University Bremen. Michael Rubinshtein (USA) Department of Chemistry, University of North Carolina at Chapel Hill. Thomas P Russell (USA) Polymer Science and Engineering Department, University of Massachusetts, Amherst. Wiebke F C Sager (Germany) IFF-Soft Matter, Forschungszentrum J¨ ulich. G J Agur Sevink (The Netherlands) Leiden Institute of Chemistry, Leiden University. Julian C Shillcock (Germany) Theory Department, Max Plank Institute of Colloids and Interfaces, Potsdam.

Jiayu Wang (USA) Polymer Science and Engineering Department, University of Massachusetts, Amherst. Qiang Wang (USA) Department of Chemical and Biological Engineering, Colorado State University, Fort Collins. Ting Xu (USA) Department of Materials Science and Engineering, University of California, Berkeley. Zhou Yang (United Kingdom) School of Materials, The University of Manchester. Andrei V Zvelindovsky (United Kingdom) Centre for Materials Science, Department of Physics, Astronomy and Mathematics, University of Central Lancashire, Preston.

Microemulsion Templating W.F.C. Sager Institute for Solid State Research (IFF)-Soft Matter, Forschungszentrum J¨ ulich, 52425 J¨ ulich, Germany

1 Introduction Surfactant molecules have for a long time been employed in stabilising ﬁnely dispersed matter since they have, due to their amphiphilic character, the tendency to adsorb at hydrophilic/hydrophobic interfaces such as water/air, water/oil or water/(hydrophobic) solid and oil/(hydrophilic) solid. In many inorganic or organic (colloidal) particle preparation routes, diﬀerent types of surfactants have been added to stabilise the formed particles sterically or electrostatically against (irreversible) aggregation (and, if applicable, coalescence) processes. Apart from their stabilising properties, surfactants have progressively gained more attention because they may self-assemble under certain conditions in binary and ternary systems of surfactant(s), water and/or oil into thermodynamically stable nano-heterogeneous systems, with a variety of diﬀerent morphologies. These self-assembled systems can in principle be used as a kind of micro- or better nanoreactor to separate and control nucleation and growth processes and as a template to direct growth and to control the morphology of the forming solid phase. Over the last two to three decades our knowledge of self-assembled surfactant systems has increased dramatically and a number of these systems have been thoroughly investigated for advanced materials synthesis [1–5]. Figure 1 shows in the central part a schematic phase diagram that displays the existence regions of a variety of morphologically diﬀerent self-assembled surfactant phases together with their multi-phase equilibria that can principally form in a ternary system of water, oil and a medium- or long-chain surfactant around its balanced state [6]. The structural polymorphism ranges, at low surfactant concentrations, from spherical and cylindrical micellar surfactant aggregates via swollen micelles and oil-in-water (o/w) droplet microemulsions on the water rich side to water-in-oil (w/o) droplet microemulsions and inverted micelles on the oil rich side, whereby the polar head group of the surfactant is turned inside. At intermediate water/oil ratios bicontinuous microemulsions are stable, consisting of an interwoven network of

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Fig. 1. Schematic Gibbs triangle (central part) displaying self-assembling structure formation in binary and ternary surfactant systems, stabilised by a strong amphiphile close to its balanced state [6]. The existence regions of the diﬀerent phases together with all corresponding two- and three-phase equilibria are presented. The compositions of the coexisting phases are indicated by the end points of the tie-lines in the two-phase regions, and by the corners of the three-phase triangles. More details about the phase diagram are given in Sect. 2.2. The outer part shows examples for solid phases of diﬀerent morphologies that have been prepared from the indicated self-assembled surfactant phases and will be described in more detail in Sects. 3–5. Central part reproduced with permission from [6].

water and oil channels. At higher surfactant concentrations hexagonal or cubic phases are formed, consisting of micellar and reverse micellar aggregates packed into ordered crystalline arrays, as well as lamellar liquid crystalline phases that are built up by a stack of planar surfactant bilayer sheets separated by water and/or oil. The phase diagram and the general phase evolution in binary and ternary self-assembled surfactant systems will be discussed in detail in Sect. 2.2 and 2.3. The outer part of Fig. 1 shows a selection of solid phases with diﬀerent morphologies that have been prepared from the indicated self-assembled surfactant systems. Besides thermodynamically stable surfactant self-assemblies, kinetically stabilised systems such as ordinary emulsions and uni- or multi-lamellar vesicles have been utilised. As can already be seen from these examples, self-assembled surfactant systems have successfully been employed in the synthesis of diﬀerently shaped organic and

Microemulsion Templating

5

inorganic nanoparticles and, more recently, of nanostructured or nanoporous (composite) materials [7]. Reﬂecting the changing demands in advanced materials science, emphasis has shifted from ﬁrst preparing only (monodisperse) spherical particles to controlling the particle morphology and ﬁnally to manufacturing well-deﬁned nanostructured materials. Since it has been realised that new developments in materials science synthesis are possible only if research eﬀorts are not solely concerned with ﬁnding novel materials but also with the morphological tailoring of the desired materials, special attention has been paid toward the end of the last century to investigating new techniques for microstructural design. These eﬀorts include the preparation of well-deﬁned particles of low polydispersity and of increasingly small dimensions down to the nanometer scale, ordering of these particles into two- and three-dimensional arrays, directed growth, as well as hierarchical structuring and control from the nanoto the meso-scale. In most of these processes, solid phase preparation and precipitation in surfactant self-assemblies has played a substantial role. In the case of mono-sized nanoparticles, precipitation in reversed micelles or w/o microemulsions has often been the ﬁrst successful route to prepare such small particles for diﬀerent inorganic materials and has therefore been employed to study quantum size eﬀects [8]. The preparation of ordered mesoporous materials such as silica by precipitation in crystalline surfactant phases opened a fast developing ﬁeld in the area of hybrid inorganic–organic mesoporous solids and zeolites [2, 9]. The basic idea of using the internal (dispersed) phase of the (thermodynamically) stable systems as nanoreactors lies in the possibility of achieving a controlled compartmentalisation and well-determined structure of the reaction medium. Precipitation and thus formation of the solid phase can be better controlled since it takes place in a more or less isolated and (ultra-) small space as, e.g., the critical supersaturation of the dissolved solid phase precursor is reached. (Direct) templating by precipitation from the continuous medium onto the surface of preformed micellar structures can be obtained if the precursor species of the forming solid phase undergo interactions with the surfactant head groups. This allows (molecular) orientation and directed ordering. The principle of both the deﬁned isolation of space and the structural templating has been adapted from nature. Natural examples include the preparation of magnetic particles in bacteria and biomineralisation processes in algae [10]. Precipitation in self-assembling systems has therefore increasingly been studied for biomimetic approaches and the development of novel preparation routes which are based on them [1, 10]. However, one should keep in mind that surfactant self-assembled systems are highly dynamic systems and that the precipitation processes taking place in them can be quite complex due to the large number of possible interactions between the mould and the templated structure. Preparation processes are therefore not always straightforward. To obtain solid phases that are commensurate in size and shape with the self-assembled surfactant system from which they are formed, detailed

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knowledge of the templating system as well as the precipitation reaction leading to the desired solid phase formation is a prerequisite. Generally, the ﬁnal morphology of the solid phase depends on the reaction kinetics (material exchange and solid phase formation), the reorientation of the surfactant selfassembled system as well as the interactions between the surfactant and the precipitating solid. In the latter case surfactants may block oriented growth due to speciﬁc adsorption of the surfactant onto certain surfaces (crystal net plains) of a forming crystalline phase, or direct ordering and aggregation of nanosized particles into complex inorganic particle–surfactant assemblies that can reach mesoscopic length scales [2, 11, 12]. Since microemulsion templating has developed into a broad ﬁeld and oﬀers a rich playground for new materials synthesis routes, only a few examples (see Fig. 1) to explain the main strategies and principles have been selected for this contribution. The chapter is organised as follows: In Sect. 2 the microemulsion template will be introduced, wherein special attention is paid to explaining the conditions under which the diﬀerent self-assembled surfactant phases form, allowing for a targeted morphological tailoring of the templating mould. Basic features and trends in the phase evolution and the tuning of the diﬀerent morphologies will be explained within the concept of the curvature energy. Most emphasis is laid on the structural changes and (exchange) kinetics of droplet phases. Sect. 3 deals with the preparation of well-deﬁned spherical and rod-like inorganic nanoparticles from single and aggregated w/o droplet phases. In Sect. 4, o/w (micro)emulsion droplet phases and bicontinuous microemulsions are discussed in view of the preparation of latex particles and polymer networks. Sect. 5 describes the basic ideas of employing diﬀerent crystalline phases to prepare ordered mesoporous solids.

2 The Microemulsion Template Surfactant molecules can, due to their amphiphilic nature, stabilise a number of diﬀerent microscopically heterogeneous systems. In the binary water– surfactant and oil–surfactant systems, surfactants self-assemble into normal (hydrophobic part inside) and reverse (hydrophilic part inside) micellar aggregates, respectively, with well-deﬁned aggregation numbers. At high surfactant concentration diﬀerent liquid crystalline phases are formed. The morphology of the micellar aggregates depends on the geometry and concentration of the surfactant molecules and external variables such as temperature or concentration of added salt. In the ternary water–oil–surfactant system, transparent and thermodynamically stable microemulsions can form under certain conditions. These microemulsion phases consist of structurally well-deﬁned nanosized domains of water and oil, separated by a monomolecular surfactant ﬁlm. The morphology and characteristic length depend on the water-to-oil ratio and the elastic properties (e.g., spontaneous curvature and rigidity constants) of the surfactant ﬁlm. Generally, droplet phase microemulsions exist if either oil

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7

or water is only present in small amounts, whereas bicontinuous microemulsions are found at comparable amounts of water and oil. The spontaneous or natural curvature of the interfacial surfactant ﬁlm depends on the geometric properties of the hydrophilic and hydrophobic moieties of the surfactant molecule and these, in turn, on their interactions with water and oil, respectively. If the temperature and/or ionic strength of the aqueous phase is varied, a rich phase behaviour is generally revealed, whereby microemulsion phases can coexist with water and/or oil excess phases as well as liquid crystalline phases forming two- and three-phase equilibria. If two-phase systems, consisting of a microemulsion phase in coexistence with a water or oil excess phase formed at low surfactant concentrations are mechanically agitated, ordinary or macro-emulsions can be obtained with a milky appearance. In this case the excess phase has mechanically been dispersed into, generally, micrometer-sized droplets, that can be kinetically stabilised due to the adsorption of surfactants onto the newly formed interface. The interface is not saturated and the size of the droplets depends on the mechanical energy applied to prepare the emulsion and on the outcome of coalescence and coagulation processes as well as Ostwald ripening that ﬁnally lead to phase separation. In this section self-assembled as well as kinetically stabilised surfactant systems will be introduced in view of their application as precipitation media and as templates. Since microemulsion phases form at intermediate conditions between the (binary) micellar (nanometer scale) systems and macroemulsions (micrometer scale), they will be treated in this context. Sect. 2.1 starts with the binary systems, whereby surface activity, micelle formation and the factors that inﬂuence the morphology of the micellar phase formed are discussed. The latter will be performed within the framework of a geometrical model (packing parameter approach). For more details on the diﬀerent classes of surfactants, micellar systems and a description of the structural evolution within the curvature energy concept, the interested reader is referred to more advanced textbooks such as [13–15] and [16]. Microemulsion systems will be introduced in the second part of Sect. 2.1 and their phase behaviour and structural evolution are described in Sect. 2.2. It has been found that the properties of the microemulsion phases are dominantly determined by properties of the interfacial surfactant ﬁlm that separates the water and oil domains. Basic features of the microemulsion phases are described within the framework of the curvature free energy. This concept is based on the statistical mechanics of curved interfaces, whereby the main properties are described in terms of spontaneous curvature and the rigidity constants for mean and Gaussian curvature. Since a full theoretical treatment is not in the scope of this Chapter, emphasis is only placed on explaining the general trends in (i) the phase behaviour and (ii) the structural evolution of the microemulsion phases and (iii) the macroscopic interfacial tension between microemulsion phases and water and/or oil excess phases in the two- and three-phase equilibria observed. In this context, formulas are only derived for droplet phases. More information on the curvature free energy and its application to more complex (microemulsion) phases,

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e.g., isotropic and crystalline bicontinuous phases, can be found in [17, 18]. Further knowledge of microemulsions can be obtained from [15, 16] and [19]. To obtain a better understanding of the precipitation reactions performed in droplet phases, Sect. 2.3 deals with the stability and exchange kinetics of microemulsion droplet phases. A comparison with ordinary emulsion droplet systems is ﬁnally given in Sect. 2.4. 2.1 Surfactant Self-assembly, Micelles and Microemulsions

Fig. 2. Left: Schematic diagram of a typical adsorption isotherm for an aqueous surfactant solution: Surface tension σ plotted against the logarithm of the surfactant concentration cS . Reproduced from [20]. The sketched insets show the state of the surfactant molecules in the water phase (bulk) and at the water/air surface. Right: Surfactant parameter range for diﬀerent surfactant aggregates. Reproduced from [13].

The Binary Water–Surfactant System: Surface Activity and Micelle Formation Amphiphiles are surface active molecules. If a surfactant is added to an aqueous solution, it will be distributed between the surface (water/air surface) and the bulk water phase. The distribution coeﬃcient depends on the solubility of the surfactant molecules in water. Upon addition of surfactant, the surface tension decreases due to the presence of the surfactant molecules at the water/air surface. The surface pressure Π caused by the spreading of the surfactant molecules is given by Π = σ0 − σ where σ is the surface tension at a given surfactant concentration cS and σ0 the surface tension of the bare surface without added surfactant. Fig. 2 shows a schematic plot of the Gibbs adsorption isotherm, typically observed for surfactants in aqueous solution.

Microemulsion Templating

9

The surface excess Γ , excess concentration of surfactant in the interface, is given by dσ 1 (1) Γ =− RT d ln cS T where R is the gas constant and T the absolute temperature. At a certain concentration, the so-called critical micelle concentration (cmc), it becomes energetically more favourable for the surfactant molecules to aggregate into micelles. The area of the surfactant in the saturated macroscopic ﬂat interface can be calculated from the surface excess for surfactant concentrations close to the cmc [21]. Above the cmc the surface tension stays constant since all additionally added surfactant molecules form micelles in the bulk water phase and the surface excess concentration remains nearly constant. Plateau values for the surface tension (cS > ccmc ) lie for conventional surfactants typically in the range of 25–35 mN m−1 . The driving force for micellisation is the hydrophobic eﬀect, which is mainly of entropic origin. The contribution to the free energy that favours micelle formation stems from a gain in entropy of the surrounding water molecules (hydrophobic or Tanford eﬀect) [14,15]. The presence of the hydrocarbon tail in the molecular dispersed state (below the cmc) induces an increased ordering of the surrounding water molecules (water cluster formation), which is not the case if the hydrocarbon chains form the micellar core and are shielded by the polar head groups from the bulk water phase. Since the operating aggregation forces are opposed by the repulsion between the head groups, ﬁnite (self-assembled) micellar aggregates are formed with a well-deﬁned number of surfactant molecules Np per micelle (Np ∼ 50–80). The size and shape of the micellar aggregates depend on the geometry of the surfactants. This dependence can be well described by a geometric model that is based on the (critical) packing parameter v/(al), where l and v correspond to the length and volume of the aliphatic tail and a corresponds to the eﬀective head group area. Np can be determined by dividing the total volume of a given micellar aggregate by v or the total surface area by a. For spherical micelles, a packing parameter of v/(al) ≤ 1/3 is obtained if one assumes that the radius of the micelle Rmic cannot exceed l: 4 3 3 πRmic

2 v 1 4πRmic → ≤ . (2) v a al 3 Whereas for a saturated hydrocarbon chain, v and l can easily be calculated from the van der Waals radii, a depends not only on the size of the head group but also on pH, surfactant and electrolyte concentration for ionic surfactants [15]. Head group areas are usually determined from measured values of the size of the micelle and the number of surfactant molecules per micelle. The head group area is typically in the range of 55–70 ˚ A2 . For ionic surfactants, the eﬀective head group area decreases if the ionic strength of the aqueous

Np =

, Np =

10

W.F.C. Sager

phase increases by addition of a salt. Due to screening of the electric charge at higher ionic strength, repulsion between the head groups of neighbouring surfactant molecules decreases and consequently the eﬀective head group area is smaller. For commonly used nonionic surfactants that are composed of a hydrocarbon chain and a water-soluble ethyleneoxide chain (Ci Ej , where the subscripts denote the number of methyl Ci and oxyethylene Ej repeats), the head group area increases with j and decreases for given j with increasing temperature. Increasing temperature breaks the hydrogen bonds between the ethyleneoxide oxygen atoms and the surrounding water molecules and results in a smaller eﬀective head group area. The diﬀerent possible packing geometries of surfactant molecules are listed in the table within Fig. 2 together with the ranges of packing parameters and the structure of the aggregates they form. Generally, surfactants shaped in a manner where the eﬀective head group takes more space than the tail region (v/al < 1/3) form spherical micelles. If v/al approaches 1, ﬁrst cylindrical micelles and ﬁnally almost ﬂat ﬁlms form as they occur in vesicles (bilayers) and lamellar crystals. If v/al > 1, the surfactants are no longer soluble in water. When dissolved in an apolar solvent (oil) they form inverse micelles. In this case the head groups form the interior of the reverse micelle and are shielded by the apolar moiety from the bulk oil. Tuning from strongly curved micelles to ﬂat surfactant sheets and even to oppositely curved aggregates (inverse micelles) can generally be achieved by either increasing the bulkiness (volume) of the hydrophobic moiety (e.g., double tailed instead of single tailed surfactants) and/or by decreasing the head group area. The latter can be realised by addition of salt in the case of ionic surfactants, decreasing j and increasing temperature in the case of nonionic surfactants and/or addition of long chain alcohols in multicomponent mixed systems. In the binary watersurfactant system micelles are generally formed at concentrations above the cmc. At high surfactant concentrations, diﬀerent liquid crystalline phases may exist, including hexagonal, cubic and lamellar phases (see water-surfactant side of the Gibbs triangle in Fig. 1). With increasing surfactant concentration in water v/al may rise up to values of above 1. For a description of the diﬀerent morphologies of micellar and crystalline structures in binary systems within the curvature free energy concept, the reader is referred to [18, 22]. The Ternary Water–Oil–Surfactant System: Microemulsions and (Multi)Phase Equilibria If a small amount of a third component barely miscible with water, such as a medium- or long chain alkane (oil), is added to micelles in water (or if water is added to reverse micelles in oil), this third component can be solubilised in the interior of (reverse) micelles. In the case of regular micelles, ﬁrst swollen micelles and then o/w microemulsions are formed upon the addition of oil, while w/o microemulsions form from reverse micelles upon addition of water.

Microemulsion Templating

11

If a larger amount of the third component is added, a number of diﬀerent (coexisting) phases may form and the phase diagram can become rather complex, especially at higher surfactant concentrations. Microemulsions are structurally well-deﬁned self-organising mixtures of water, oil and surfactant(s) that can form a wide variety of thermodynamically stable (mainly transparent) phases. These comprise droplet phases that consist of more or less spherical nanosized (20–300 ˚ A) droplets of oil in water (o/w microemulsions or L1 -phase) and water in oil (w/o microemulsions or L2 -phase), as well as bicontinuous mono- and bilayer (L3 ) phases. The latter two consist of an interwoven network of water and oil channels, separated by a saturated (mono- or bilayer) surfactant ﬁlm (see Fig. 3). If the temperature and/or ionic strength of the aqueous phase is varied, a rich phase behaviour is generally revealed, whereby microemulsion phases can coexist with water and/or oil excess phases as well as liquid crystalline phases forming two- and three-phase equilibria. The diﬀerent phases and coexistence regions occurring in a three-component water, oil and surfactant system can be represented in a triangular phase diagram (Gibbs triangle). At each point in the diagram, the concentration of each species is determined by drawing a line parallel to the corresponding axis, lying along one edge of the triangle. The central part of Fig. 1 shows a Gibbs triangle displaying a variety of structurally diﬀerent phases, which can principally form in a ternary system of a strong amphiphile close to its balanced state, together with their two- and three phase coexistence regions. At high surfactant concentrations the diﬀerent liquid crystalline phases reach from the water-surfactant and oil-surfactant (inverse structures) sides into the triangle. These phases form two and three-phase equilibria at intermediate surfactant concentration with the microemulsion phases at lower surfactant concentrations (lower 1/3 of the diagram). At the balanced state of surfactant, the volumes of the hydrophilic and hydrophobic moieties are nearly equal (v/al ≈ 1) and the interfacial surfactant ﬁlm is on average not curved. Therefore, bicontinuous microemulsions (and lamellar and bicontinuous crystalline phases at high surfactant concentrations) occur in the central part of the Gibbs triangle. Isotropic bicontinuous microemulsions exist only over a small surfactant concentration range at comparable amounts of water and oil. At lower surfactant concentrations, the amount of surfactant present is not high enough to form a bicontinuous structure over the whole sample. In this case the sample splits into a bicontinuous middle phase and almost pure water and pure oil excess phases. The compositions of the three coexisting phases are given by the corners of the three-phase triangle. According to the Gibbs phase rule, a three-component system can separate into a maximum of three coexisting isotropic phases at given temperature and pressure. The general phase behaviour and the structural evolution in microemulsion systems is described in Sect. 2.3. Microemulsions possess a huge internal interface since the size of the water and/or oil domains ranges from about 20–500 ˚ A. If one considers 10 g of a microemulsion that consists of, e.g., water droplets in oil and contains 1

12

W.F.C. Sager

g surfactant and 1 g water, the interfacial area formed by the surfactant ﬁlm amounts to about 500 m2 . Such large internal interfaces can only be formed if the interfacial tension of the macroscopic water/oil interface has become very small. In these cases, the gain in entropy leads to a spontaneous microemulsion formation (spontaneous emulsiﬁcation). Water/oil interfaces that are saturated with a surfactant (plateau values in Fig. 2) reach values of σ = 10−1 –10−3 mN m−1 . At these low interfacial tensions curvature eﬀects start to play a dominant role. As will be shown in the following subsection (Sect. 2.2), this curvature energy is responsible for the rich phase behaviour displayed. Historically, the droplet phases are the ﬁrst and also most studied microemulsion structures. The reason for this lies probably in the fact that their existence regions are larger and thus easier to ﬁnd, if, e.g., the temperature or salt concentration is varied. The term “microemulsion” was originally coined by Hoar and Schulman [23] in 1943 and was applied to a transparent sample, prepared by emulsifying an oil in an aqueous surfactant solution wherein a fourth component, a so-called cosurfactant (generally an alcohol of intermediate chain length such as hexanol) has been added. Upon titrating the emulsion system with the alcohol, they observed a sudden decrease in turbidity and obtained ﬁnally a transparent o/w dispersion. They concluded that the size of the droplets must be smaller than the wavelength of visible light and thus microscopically small, without knowing that they are actually in the nanometer range. The type of the dispersion (o/w or w/o) follows in principle the Bancroft rule, which had been found earlier for ordinary emulsions [24]: The medium that forms the continuous phase of the dispersion is the one in which the surfactant is better soluble. The bicontinuous structure was ﬁrst postulated by Scriven in 1976 [25]. A typical micrograph of a bicontinuous structure obtained by freeze fracture transmission electron microscopy is shown in Fig. 3. Best evidence of bicontinuous microemulsions is provided by NMR-self diﬀusion measurements when samples with diﬀerently deuterated compounds are used. For droplet phases, the dispersed phase and the surfactant display the same diﬀusion coeﬃcient, that depends on the size of the droplets following Stokes’s law [15], while the diﬀusion coeﬃcient for molecules of the continuous phase is much faster. For the bicontinuous structures, the diﬀusion coeﬃcients for water and oil are of the same order while that of the surfactant is much lower, since it is located in the interface. 2.2 Phase Behaviour and Structural Evolution Curvature Energy Model and Phase Behaviour of Microemulsions Over the last two decades it has been realised that the structure and phase behaviour of microemulsion systems is, apart from the composition, determined by the elastic properties of the surfactant ﬁlm that separates the water and oil domains. These have successfully been described using the presently

Microemulsion Templating

13

Fig. 3. Freeze fracture transmission electron micrographs (FF-TEM) of a bicontinuous microemulsion, containing 7 wt% C12 E5 , 37 wt% octane and 56 wt% water. The sample was prepared by rapid freezing, fracturing and subsequent deposition of a thin tantalum-tungsten coating to obtain a replica. Contrast between the oil and water domains occurs because the coating forms a mottled pattern on the oil regions. Reproduced from [26]. Copyright (1988) American Chemical Society.

well-established approach based on the statistical mechanics of curved interfaces that was introduced by Helfrich in 1973 [27]. The Helfrich free energy FH describes the bending energy of an arbitrarily shaped surface in terms of the spontaneous curvature, H0 , the rigidity constant associated with bending, κ, and the rigidity constant associated with Gaussian curvature, κ ¯, 2 FH = dA f0 + 2κ (H − H0 ) + κ ¯K . (3) The above form for the free energy is an expansion up to second order in curvature and features an integral over the whole surface area, A, of the mean curvature, H = 1/2(1/R1 + 1/R2 ), and Gaussian curvature, K = (1/R1 R2 ) with R1 and R2 the principal radii of curvature at a certain point on the surface A. The sign of the principal radii of curvature is generally taken as positive when the ﬁlm curves toward oil and negative when it is curved toward water, see Fig. 4. In terms of R1 and R2 , FH can be written as FH =

2κ dA σ − R0

1 1 + R1 R2

κ + 2

1 1 + R1 R2

2

κ ¯ + , R1 R2

(4)

14

W.F.C. Sager

where R0 = 1/H0 is the radius of spontaneous curvature and σ is the surface tension of the planar surface. f0 is deﬁned such that σ comprises all terms that are independent of curvature: σ = f0 + 2κH02 . ¯ cannot be obtained from a At present the four parameters σ, R0 , κ and κ molecular model allowing the direct input of the properties of the surfactant molecules and the interactions of the hydrophilic and hydrophobic moieties with water and oil, respectively. They can, however, be determined experimentally or simulated for simple model systems with well deﬁned interaction potentials (e.g., a “surfactant” with Lennard-Jones interactions [28]). A global insight into the general phase behaviour can be gained by minimising the Helfrich free energy for the diﬀerent phases. Phase boundaries or existence regions can then be obtained in terms of R0 and, e.g., κ ¯ /κ. It is therefore important to know how they are related to experimentally accessible variables, such as temperature or salt concentration. Since a complete treatment of microemulsion phases within the curvature energy concept is beyond the scope of this chapter, attention is only paid here to its manifestation in the general phase behaviour of microemulsion systems. In this context, formulas will only be derived for microemulsion droplet phases.

spontaneous curvature interfacial ﬁlm curved toward (packing parameter v/al microemulsion type multiphase equilibria

H0 > 0 oil

H0 ∼ =0 no preferred curvaturebalanced state

H0 < 0 water

<1

1

> 1)

o/w (L1 -phase)

bicontinuous

w/o (L2 -phase)

Winsor I (2Φ)

Winsor III (3Φ)

Winsor II (¯ 2Φ)

Fig. 4. Curved surfactant monolayers at the water/oil interface, with the corresponding values for the spontaneous curvature H0 and the type of microemulsion formed in the single phase and in multiphase equilibria.

The spontaneous curvature, H0 , is deﬁned as the curvature H a surfactant ﬁlm will adopt when the ﬁlm is totally unconstrained. In this case the ﬁlm is in its lowest free energy state and thus ∂FH /∂H = 0 for H = H0 . H0

Microemulsion Templating

15

corresponds to the energetically most favourable packing conﬁguration of the surfactant molecules in the interface. It can principally be compared with the packing parameters for micelles and depends on the molecular geometry of the surfactant and the solubilisation of the hydrophilic part in the aqueous phase and of the hydrophobic part in the oil. The latter depends in turn on formulation parameters such as temperature, (pressure), salt concentration in the aqueous phase, kind of oil used and type and concentration of cosurfactants (e.g., alcohol). Fig. 4 sketches the possible conformations of surfactant monolayers at a water/oil interface and gives the corresponding values for H0 and the type of microemulsion formed in the single phase and in multiphase equilibria. Positive and negative values for H0 favour the formation of droplets with spherical or cylindrical geometry. When H0 is zero (balanced state), the ﬁlm will preferably form ﬂat planar bilayers or bicontinuous structures with saddle shaped curvature for which R1 = −R2 . For a given surfactant, the spontaneous curvature can be tuned and adjusted by the aforementioned formulation variables. For nonionic surfactants of the Ci Ej type temperature has the strongest eﬀect on H0 . Upon increasing the temperature the hydrogen bonds between the hydrating water molecules and the ether groups of the ethyleneoxide moiety Ej break, whereby the surfactant becomes increasingly less soluble in the aqueous phase. The interfacial surfactant ﬁlm turns from being curved toward oil (H0 > 0) via its balanced state (H0 ∼ = 0) to become curved toward water (H0 < 0). The actual transition temperatures depend on the length of the ethylene Ci and ethylene oxide Ej blocks of the surfactant and the oil used. Smaller i, longer j and long chain aliphatic oils favour values of H0 > 0. For ionic surfactants the strongest inﬂuence is the salt concentration of the aqueous phase. If the salt concentration is increased the eﬀective charge of the head group will be screened and the surfactant heads repel each other less which can lead to a change from H0 > 0 → H0 < 0. The temperature dependence of H0 is for an ionic surfactant ﬁlm with H0 < 0 → H0 > 0 opposite to the salt dependence, since the Debye screening length is given by 1/κD = (εRT /2F 2 I)1/2 , where ε is the permittivity, F the Faraday constant, and I the ionic strength of the aqueous phase. The temperature dependence for nonionic and ionic surfactants is therefore opposite. For both kinds of surfactant, the addition of alcohols with more than four C-atoms stabilises structures which are curved toward water. Ionic surfactants with a strong head group (e.g., −SO− 3 or ), such as sodium dodecylsulfate (SDS), can only be brought into their −SO− 4 balanced state (or even curved toward water) if salt and a long chain alcohol are added, while ﬁlms of a double tailed ionic surfactant, such sodium di(2-ethylhexyl)sulfosuccinate (AOT), are close to their balanced state. The (bending) rigidity constant or bending elastic modulus, κ, is, in principle, a measure of how easily an interfacial surfactant ﬁlm can be bent and inﬂuences both the static and dynamic properties of microemulsions. κ is expressed in units of energy with values typically between 1-20 kB T , where kB is the Boltzmann constant. Experimentally, κ can be obtained from ellipsometric

16

W.F.C. Sager

measurements of ﬂuctuating ﬂat interfaces [29] or from electro-optical birefringence measurements of the form ﬂuctuations of microemulsion droplets [30]. The persistence length of the interfacial ﬁlm ξk describes the length over which the ﬁlm is locally ﬂat. High values of ξk indicate ﬂat surfaces, whereas low values are found with highly curved surfaces. ξk is given by 2πκ ξk = l exp (5) kB T where l is a molecular length that corresponds to the length of the surfactant. If the bending elastic constant κ is of the order of kB T , ξk is microscopic and the interfacial ﬁlm is ﬂexible. Over a large volume fraction range (see Figs. 1 and 5), a bicontinuous structure will exist if the spontaneous curvature of the surfactant ﬁlm is adjusted to be close to zero (H0 ∼ = 0) by, e.g., variation of T , salt and/or alcohol concentration. If κ is much larger than kB T , ξk is macroscopic and the ﬁlm is rigid. In this case, the droplet structure exists up to very high volume fractions and phase inversion takes place preferably via a lamellar crystalline phase. Flexibility is increased when i) short chain hydrocarbon chains or cosurfactants are added, ii) double chain surfactants with unequal chain lengths are used and iii) short chain oils are employed, that can easily penetrate into the surfactant ﬁlm, or iv) the salt concentration of the aqueous phase is increased. For the rigidity constant associated with Gaussian curvature or the saddle splay modulus, κ ¯ , few reliable measurements are available. The integral of the Gaussian curvature depends solely on the topology of the surface (Gauss–Bonnet theorem). Deformations of the ﬁlm leave the integral over K thus unchanged. This value only changes if the ﬁlm breaks due to the formation of holes or if two separated domains join. The typical trend is that κ ¯ is negative. The magnitude of it is usually smaller than that of κ for the same system. Experiments have shown that κ and κ ¯ are approximately constant over the temperature range encountered, but depend for ionic surfactant layers strongly on the salt concentration of the aqueous phase [31]. Generally, the interfacial layer is more stiﬀ when no screening occurs with little or no salt added. Phase and Structural Evolution in Microemulsions Figure 5 shows a phase evolution typically observed for nonionic microemulsion systems with increasing temperature. The same trend can in principle be obtained for ionic microemulsion systems with increasing salt concentration in the aqueous phase, but focus will be placed only on the nonionic system. At low temperatures, the surfactant is preferably soluble in the aqueous phase and forms, at concentrations above the cmc, micelles that can take up only small amounts of oil. Within the central miscibility gap at low temperatures an o/w microemulsion is formed which is in equilibrium with an almost pure oil excess phase (all tie-lines decline toward the oil corner), which is classiﬁed as the Winsor I microemulsion system [32]. With increasing temperature,

Microemulsion Templating

17

Fig. 5. Phase evolution resulting in Winsor I-III-II microemulsion equilibria, as observed for microemulsions stabilised by a nonionic surfactant with increasing temperature and for microemulsions stabilised by an ionic surfactant with increasing salt concentration. The compositions ci of the coexisting phases are indicated by the end points of the tie-lines and the corners of the three phase triangle, whereby the subscripts l, m, or u stand for lower, middle or upper phase, respectively. The bars below and above the 2 in the two-phase systems indicate whether the microemulsion is formed in the bottom or top phase. These also indicate in which phase the surfactant is preferably soluble. The thick lines mark the critical tie-line out of which (at Te ) and into which (at Tμ ) the three-phase triangle develops and collapses, respectively. T0 is the temperature at which the three-phase triangle is isosceles.

the hydrophilic moiety becomes less soluble in water and the interfacial ﬁlm becomes less curved toward oil (H0 > 0 → H0 ∼ = 0). When H0 approaches zero, the lower microemulsion phase of the Winsor I systems separates into a surfactant-rich, bicontinuous middle phase and a water-rich phase (Winsor III, which is also depicted in Fig. 1). The three-phase triangle develops out of a, so-called, critical tie-line, which is indicated as a thick line. At T = T0 the three-phase triangle is isosceles in shape. The concentrations of the three coexisting phases ci are given by the corners of the triangle, see Fig. 5. For samples that are prepared at overall concentrations within the three phase triangle, only the volumina but not the concentrations of the three coexisting phases diﬀer. The degree of freedom, f , is given by the Gibbs phase rule: f = C − P at constant temperature and pressure, where C corresponds to the number of components and P to the number of phases [15]. For a system consisting of three components and three phases f = 0. Upon further increasing the temperature the surfactant moves via the middle phase into the oil phase, whereby the middle phase takes up an increasingly large amount of oil

18

W.F.C. Sager

and the upper, the originally almost pure oil phase, takes up more water and surfactant. cm moves clockwise around the central miscibility gap, while the bottom phase declines in oil and surfactant and c moves to the water corner. If the compositions of both upper phases become identical, they merge (H0 ∼ = 0 → H0 < 0) into a two phase system (Winsor II system), that is composed of a w/o microemulsion (upper phase) in equilibrium with an almost pure water phase (lower phase). In this case the tie-lines decline toward the water corner. In the Winsor I-III systems the curvature of the surfactant ﬁlm in the diﬀerent assemblies/structures formed is related to the spontaneous curvature H0 . At higher surfactant concentrations, enough surfactant is available to solubilise all excess water and/or oil in the corresponding microemulsion droplets or bicontinuous phases, respectively, to span the whole system and as a result single phase microemulsions are formed.

Fig. 6. Experimentally averaged mean curvature < H > (left) and characteristic length ξ (middle) obtained from FF-TEM and SANS for the system C12 E5 -water(n-octane). Around T¯ = 32.6◦ C, < H > changes sign and ξ is maximal. Interfacial tension (right) of the interface between the water-rich and oil-rich phases, indicated by σab , in the Winsor I, III and II systems for the same microemulsion system, obtained by spinning drop measurements. The interfacial tension shows a minimum when ξ is maximal. Reproduced with permission from [33].

The main trends found in microemulsions prepared from the H2 O-octaneC12 E5 system are summarised in Fig. 6. The data was collected by Strey and co-workers [33,34] using small angle neutron scattering (SANS) and FF-TEM to determine the characteristic sizes ξ and main curvatures of the water and oil domains as they evolve with temperature. In addition, spinning drop measurements were performed to determine the interfacial tension σab between the water-rich and oil-rich phases of the Winsor I, III, II systems. In the microemulsion region, the mean curvature varies linearly with temperature (H ∼ T − T0 ), while the characteristic length ξ shows a maximum at T = T0 , when the mean curvature passes through zero. Experimentally, it has been found that T0 lies close to the mean temperature T¯ of the threephase temperature interval that can be obtained from T¯ = (Tu + T )/2. Tu

Microemulsion Templating

19

and T are the temperatures at which the isotropic three-phase triangle appears and vanishes, respectively, in the Gibbs triangle (see Fig. 5). At high and low temperatures, both H and ξ level oﬀ to their limiting values of the corresponding (reverse) micellar systems. The interfacial tension σab shows a minimum when ξ is at a maximum. The minimum is more pronounced with higher amphiphilicity (amphiphilic strength) of the surfactant.

2 g

Fig. 7. Schematic phase diagrams performed at the indicated cuts through the phase prism. Left: ﬁsh-cut; right: one-phase-channel. The sketched structures within the one-phase regions display the structural evolution observed by diﬀerent experiments, including small-angle scattering and FF-TEM. Reproduced with permission from [33].

The determination of all the (co)existence regions and phase boundaries within the Gibbs triangle is quite time consuming. If one is particularly interested in the temperature evolution, it is often suﬃcient to perform certain cuts through the phase prism constructed from the Gibbs triangle as base and temperature as ordinate. Two of the most instructive phase diagrams obtained in this way are displayed in Fig. 7. The left hand diagram shows schematically a phase diagram obtainable by measuring the phase boundaries occurring in samples with an equal amount of water and oil as a function of surfactant concentration γ and temperature. This is a cut through the prism that starts in the middle of the water–oil side of the triangle and heads torwards the surfactant corner (see inset in Fig. 7). In this way it cuts through the central miscibility gap and the three-phase triangle at temperatures between T and

20

W.F.C. Sager

Tu . At T < T Winsor I (2Φ) phase equilibria are formed within the central miscibility gap, while Winsor II phase (¯2Φ) equilibria form at T > Tu (see also Fig. 5). The three-phase region then shows up as a kind of “ﬁsh-head” shape, that is surrounded by the two-phase regions. At very low surfactant concentration γ, the surfactant is molecularly dispersed in the water or the oil phase, respectively, depending on in which medium it is preferably dissolved at a given temperature. At low temperatures (T < T¯), the surfactant is soluble in water. If micellar aggregates form at concentrations above the cmc, they contain a certain amount of oil that depends on the spontaneous curvature H0 at that temperature. Figure 7 also shows the cmc-lines that indicate from which surfactant concentration onward microemulsion droplets form in the water (cmcwater ) and oil phases (cmcoil ). Both lines decline toward the threephase region. The two-phase region in front of the “ﬁsh-head” corresponds to the two-phase region that surrounds the three-phase triangle in Fig. 5 along the water-oil side of the Gibbs triangle. At T = T¯ (balanced state of the surfactant) the surfactant molecules are most eﬃcient and show the lowest solubility in a water-oil system. Above the three-phase region, the surfactant is preferably dissolved in the oil, and Winsor II microemulsion systems are formed with diﬀerent droplet sizes, which depend on the temperature and thus H0 . If, at a given temperature more surfactant is added, more microemulsion droplets of the same size can be formed and the microemulsion phase takes up more of the excess phase. Considering, e.g., a Winsor I system at T < T , more oil becomes solubilised in the o/w microemulsion. If more surfactant is present more oil can be “encapsulated” in order to keep almost the same size. At a certain surfactant concentration all excess phase is taken up and a single o/w microemulsion is formed. When more oil is solubilised in the o/w microemulsion phase at temperatures closer to T¯, more oil is present at low surfactant concentrations and the 2Φ → 1Φ boundary is shifted to lower surfactant concentrations. The point at which the boundaries of the three-phase region and the one-phase region intersect γ˜ , corresponds to the corner of the isosceles three-phase triangle that indicates the composition of the middle phase cm (see Fig. 5). γ˜ marks the lowest surfactant concentration needed to prepare single phase microemulsions. In its balanced state the surfactant ﬁlm can solubilise the highest amount of water and oil, the domain size is maximized and the surface tension between the excess phases is at a minimum. The sketched structures within the one-phase, “ﬁshtail”-like region in Fig. 7 display the micro-structural evolution that is observed experimentally. Droplet phases form within the tail at high surfactant concentrations and T < T¯ (o/w) or T > T¯ (w/o). The transition from one droplet structure to the other can either occur continuously via cylindrical structures and bicontinuous phases or via a 1st order phase transition to the lamellar phase Lα that might intersect with the microemulsion phases at temperatures close to T¯ (see Fig. 1). The phase diagram on the right in Fig. 7 presents a cut through the phase prism at constant surfactant concentration, whereby the concentration is chosen such that at T¯ the bicontinuous one-phase microemulsion is just formed.

Microemulsion Templating

21

The surfactant concentration is thus chosen to be slightly higher than γ˜. In this way, a continuous one-phase microemulsion channel can be obtained that leads from the water to the oil side with a large bicontinuous region at intermediate water/oil ratios. The sketches of the structural evolution within the channel demonstrate nicely that the microemulsion morphology is governed by the composition and the properties of the interfacial surfactant layer. If the temperature is varied the latter is primarily determined by H0 . The Helfrich energy may be applied to derive expressions for the droplet radius and the interfacial tension in Winsor I and II droplet phases and to determine the 2Φ → 1Φ boundary, when only spherical droplets are considered. For spheres R1 = R2 = Rs and volume (Vs ) and surface area (As ) are given by Vs = (4/3)πRs3 and As = 4πRs2 . For a given sample of a droplet microemulsion phase in equilibrium with an oil (T < T¯) or water (T > T¯) excess phase, the surfactant concentration and thus the total surface area A can be assumed not to change with temperature, since almost all surfactant is situated in the droplet interfaces of the w/o or o/w microemulsion phases. The radius and the droplet volume increase in the direction of T¯. The total interfacial area A is given by A = Ns 4πRs2 , with Ns the number of spherical droplets. The Helfrich free energy for a spherical droplet phase, Fs , can be written as

2κ + κ ¯ 4κ + . (6) Fs = Ns 4πRs2 σ − R0 Rs Rs2 Minimisation of Fs can then be performed with respect to the droplet radius Rs and the number of droplets Ns under the condition of constant interfacial area and thus surfactant concentration. Minimisation with respect to Rs gives κ 2κ ∂Fs = 8πNs σRs − 16πNs =0→σ= . ∂Rs R0 R0 Rs

(7)

The interfacial tension between the droplet and the excess phases is in good approximation proportional to the square of the droplet radius. The expression for σ in (7) was ﬁrst derived by de Gennes [35]. Minimisation with respect to Ns gives

∂Fs 2κ + κ ¯ 2κ + κ ¯ 4κ 4κ = 4πRs2 σ − + − =0→σ= ∂Ns R0 Rs Rs2 R0 Rs Rs2

(8)

and thus Rs =

1+

¯ 1κ 2κ

R0 .

(9)

Since the bending constant in the range considered depends only weakly on the temperature, the droplet radius in the Winsor I and II systems is pro-

22

W.F.C. Sager

portional to the radius of spontaneous curvature, which is tuned by the temperature. On approaching T¯ from high and low temperatures |R0 | increases, leading to an increase in Rs and a decrease in σ. To obtain the 2Φ →1Φ boundary, one has to determine when the droplet radius in the Winsor I and II systems becomes equal to the droplet radius in the neighbouring one-phase regions. The radius of the droplets in the onephase region is determined by the following argument. If there is no excess phase present, then not only the total surface area A but also the total volume Vtot = Ns (4/3)πRs3 is determined and consequently Rs = 3Vtot /A = 3ω. The phase boundary, which is normally referred to as the solubilisation limit, occurs at ¯ 1κ ω 1 κ ¯ = Rs = 3ω = 1 + R0 → 2+ . (10) 2κ R0 6 κ So far, entropic eﬀects have been neglected. Including translational entropy and a radial polydispersity for the droplets allows the droplet polydispersity to be calculated; this shifts the phase boundaries only slightly [36]. To obtain the complete picture not only spherical droplets but also cylindrical domains and interconnected cylinders have to be considered. For more information about the phase behaviour of nonionic and ionic surfactant–water-oil systems the reader is referred to review articles by the group of Kahlweit [37], (nonionic) [38], (ionic) [39]. 2.3 Statics and Dynamics of Droplet Phases One-phase w/o or o/w droplet microemulsions have been thoroughly investigated over the past three to four decades. They have mainly been studied as model systems to apply liquid state theories [40], to study dynamic percolation phenomena [41] and to investigate their exchange kinetics in view of their employment as reaction media [42]. Recently, the Helfrich energy has been used to describe shape ﬂuctuations [43], droplet aggregation [44] and the formation of cylinders [36] and interconnected networks [45]. In this section, only results, most relevant for particle preparation, are brieﬂy described. Droplet Phase Existence Regions, Droplet Aggregation and Cylinder Formation Within the one-phase region, the total area and total volume are ﬁxed by the concentrations of surfactant and the dispersed phase. This imposes a constraint of area and volume conservation. Figure 8 shows schematically the existence region of a w/o microemulsion stabilised by an ionic surfactant, such as AOT. The phase diagram can be obtained by a cut through the phase prism as indicated in the inset. The cut is performed by varying the water/surfactant weight ratio, rw , at constant oil and thus droplet weight

Microemulsion Templating

23

Fig. 8. Schematic phase diagram showing the existence region of a w/o microemulsion stabilised by an ionic surfactant, such as AOT, as a function of the water/surfactant ratio and temperature. The cut through the phase prism is performed by variation of rw =(water/wt%)/(surfactant/wt%) at constant oil and thus droplet weight fraction.

fraction. Since the temperature dependence of ionic and nonionic surfactants is opposite, the lower phase boundary corresponds to the solubilisation limit boundary (see Fig. 7 for comparison). Close to the solubilisation limit, Rs /R0 equals 1 and spherical droplets are formed with a polydispersity of about 20%. Their radius depends linearly on the (molar) water/surfactant ratio, since Vtot = vw NA [water] and A = aNA [surfactant], with vw the molar volume of a water molecule, NA the Avogadro number and the concentrations given in mol L−1 . The smallest radii reached are those of the inverse micelles and the largest that of the corresponding radius at the solubilisation limit at a given temperature. If the temperature is increased, H0 → 0 and the interfacial ﬁlm tends to be less curved towards the water phase. In the corresponding Winsor II system, the water volume is not ﬁxed and fewer droplets with larger radii are formed. In the one-phase region, if Rs /R0 1 the system minimises its free energy by forming a certain amount of aggregated droplets or cylinders. Small angle x-ray or neutron scattering curves, measured for AOT stabilised w/o microemulsions at diﬀerent temperatures and interpreted in terms of the sticky

24

W.F.C. Sager

hard sphere model [46] for the structure factor, have generally shown that the droplets become more attractive with increasing temperature. The degree of attraction depends on the oil used; at constant temperature, w/o emulsions prepared with long chain alkanes, such as decane and dodecane, show more attraction then systems with less hydrophobic oils as the continuous phase. Recently, the droplet→cylinder transition has been studied in detail, both experimentally [47] and theoretically [36]. Within the one-phase region spheres and cylinders coexist. When the cylindrical droplets start to form, they are of a certain minimal length that corresponds to a few multiples of their diameter. If one compares the radius of the spherical droplets at the lowest temperature with that of the cylinders, Rc , at the highest temperature measured, the geometrical limit of Rc = 0.67Rs is reached. The geometric limit is the point at which all spheres transform into cylinders. At temperatures close to the upper phase boundaries, the cylinders start to interconnect and form (bicontinuous) networks. The existence of these networks can best be demonstrated by conductivity measurements. If the temperature is increased the conductivity may show a steep increase over two to three orders of magnitude, reaching values of the same order as that of the aqueous phase [41]. At the upper phase boundary, either two coexisting oil-continuous microemulsions (low rw -values) form or the microemulsion coexists at high rw -values with a lamellar phase (Lα ). The transition from a microemulsion consisting of spherical droplets only to a lamellar phase can easily be determined by setting the contributions that stem from the curvature energy in the Helfrich free energy equal for both phases: Fs,curv = −

4κ κ ¯ 2κ + κ ¯ ω 1 2 + . + = F = 0 → = L ,curv α R0 Rs Rs2 R0 12 κ

(11)

Since the lamellar phase is composed of ﬂat surfactant sheets their contribution to the curvature energy is equal to zero. In Fig. 8 both the solubilisation limit and the transition to the lamellar phase are given for phases consisting of inﬁnitely long cylinders, which can be obtained in the same way as described above for spherical droplets using 1/R1 = 1/Rc and 1/R2 = 0. Vtot and A are then given by Vtot = Nc πRc2 L and A = Nc 2πRc L, with Nc the number and L the length of the cylinders. A full description including the coexistence of spheres and cylinders taking the radial and length polydispersity into account is given in [36]. Exchange Kinetics in Droplet Phases Micellar systems and microemulsions are highly dynamic systems. Surfactant molecules that are dissolved in the continuous phase exchange rapidly with those located at the interface of the assembled structures. Fast relaxation measurements have shown that exchange of monomeric surfactants between micelles or microemulsion droplets takes place with relaxation times of

Microemulsion Templating

25

τ1 = 10−6 -10−3 s. Relaxation times associated with the much slower formation or desintegration of entire micelles lie typically in the range of τ2 = 10−3 1 s [48]. In droplet microemulsions exchange of the dispersed phase also occurs. If, for example, two w/o microemulsions are brought together containing the same amounts of surfactant, aqueous phase and oil, but diﬀerent types of solubilised species in the internal water pools, material exchange sets in immediately and leads to a redistribution of the two species over the entire dispersed droplet phase. Solubilisation of two reactants A and B that can undergo a chemical reaction to form C, can then be utilised to measure the exchange kinetics. It is now generally accepted that material exchange between droplets, necessary for particle formation, takes place via the formation of intermediate dimeric droplet aggregates. Dimers allowing free passage of the solubilised species were originally proposed by Fletcher et al. [49]. For AOT w/o microemulsions, exchange of solubilisate between the water pools of the microemulsion droplets occurs with a second order rate constant of kex = 106 –108 dm3 mol−1 s−1 ; thus two to four orders of magnitude slower than the droplet encounter rate as predicted from simple diﬀusion theory. This indicates that successful fusion events take place only in 0.01-1% of binary droplet collisions [49], [50]. For microemulsions stabilised by nonionic surfactants of the ethyleneoxide monoalkylether (Ci Ej ) type exchange rates of kex = 108 –109 dm3 mol−1 s−1 were measured [51]. Rate constants were found not to depend on the size of the solubilised species but on droplet size, temperature and factors inﬂuencing the rigidity of the interfacial surfactant ﬁlm, such as the kind of surfactants and oil used, presence of cosurfactants (alcohol) and salt concentration of the aqueous phase [49]. The extent to which the interdroplet exchange rate is inﬂuenced by the presence of droplet aggregates and cylindrical droplets has so far not been investigated. Generally it has been observed that the exchange rate is low in droplet phases with rigid interfaces and high if water channels start to form. 2.4 Comparison Between Micro- and Macroemulsions (Macro)emulsions generally form when water and oil are vigorously mixed in the presence of a surface active compound. They are normally of milky appearance and display droplet sizes ranging from 10 μm down to 0.1 μm. The ﬁnal size of the emulsion droplets depends on the mechanical energy applied in the preparation process and on their stability. The free energy ΔFemul to disperse a liquid of volume V with drops of radius R in another liquid is 3V . (12) ΔFemul = σ R If the freshly created droplet interfaces are not stabilised during the emulsiﬁcation process, coagulation and coalescence processes cause a quick increase in droplet size and lead to rapid macroscopic phase separation. Employing surfactants in the emulsiﬁcation process enables the surface tension and thus

26

W.F.C. Sager

the mechanical energy required to be lowered and to stabilise the formed droplets electrostatically or sterically. Since macroscopic emulsions are only kinetically stabilised, their preparation, stabilisation and destabilisation resembles those of solid colloidal particles. Figure 9 shows schematically the destabilisation processes that can principally occur in, e.g., o/w emulsions. Flocculation and coagulation lead to (ir)reversible droplet aggregation, while the density diﬀerence between the water and oil phases causes creaming of the oil droplets. Coalescence processes and Ostwald ripening cause an increase in droplet size and lead eventually to phase separation. Ostwald ripening occurs only if molecules of the dispersed phase are slightly soluble in the continuous phase. Since the Laplace pressure is higher for more curved interfaces, small droplets display a higher tendency to dissolve. Since oil molecules inside the droplets are in equilibrium with those dissolved in the continuous phase, material transport occurs via molecular diﬀusion from small to large droplets. In this way, the larger droplets increase in size, while the smaller ones disappear.

Fig. 9. Processes that generally lead to a destabilisation of (o/w) emulsions. Creaming occurs normally before phase separation. The broken lines around the droplets indicate that the interfaces are not fully saturated.

Since emulsions are stabilised by surfactants, they combine the properties associated with colloidal particles and surfactant self-assembled systems. Within a certain water/oil ratio, the type of surfactant used as emulsiﬁer determines the type of emulsion formed. Emulsions displaying long-term stabilities are normally prepared at surfactant concentrations higher than the cmc and formulation conditions adjusted in such a way that the surfactant is not in its balanced state (H0 = 0). At surfactant concentrations above the cmc the macroscopic interfacial tension remains constant (see Fig. 2) and micelles or microemulsion droplets, formed in the continuous phase, serve as a surfactant reservoir to stabilise the newly formed interface. The corresponding thermodynamic equilibrium phases for o/w and w/o macroemulsions are the Winsor I and II systems. To prepare the emulsions the water and oil phases do not necessarily have to be in equilibrium but they will ﬁnally phase-separate into the Winsor systems. If after the emulsiﬁcation, micelles or microemulsion droplets are still present in suﬃcient concentration, they may introduce

Microemulsion Templating

27

ﬂocculation by depletion [52]. The attractive depletion forces are caused by a diﬀerence in osmotic pressure. If two emulsion droplets approach each other by Brownian motion, the distance can become so small that no swollen micelles can come in between them. The osmotic pressure of the continuous phase is then much lower between the approaching droplets than in the surrounding phase, resulting in an attractive force that can lead to droplet aggregation. At the balanced state of the surfactant, the prepared emulsions are not stable and phase separate quickly. T¯ then corresponds to the phase inversion temperature PIT [53] of emulsions. Phase separation can thus also be induced by changing a formulation variable after the emulsion has been formed. Changes in temperature, salt or alcohol concentration that cause H0 → 0 destabilise the emulsion. For more information concerning the relatively broad ﬁeld of emulsion preparation, stabilisation and emulsion breaking, the reader is referred to [15] and [54]. In order to obtain a better overview of the diﬀerent emulsions systems employed in particle preparation routes, Table 1 lists important properties and characteristics of microemulsions when compared to ordinary macroemulsions.

3 Precipitation of (Inorganic) Nanoparticles in Droplet Phases Microemulsion droplet phases, e.g., water-in-oil microemulsions stabilised by ionic or nonionic surfactants or surfactant mixtures, have been extensively utilised over the past two decades as nano-reactors for the preparation of welldeﬁned small spherical particles for a variety of diﬀerent inorganic materials. These include metals, semiconductors, metal halides, as well as magnetic, ferroelectric, superconducting and ceramic precursor particles. Since the use of microemulsions has for the ﬁrst time enabled the preparation of monodisperse particles of a few nanometres in diameter for some materials, a large number of investigations could focus on studying the optical, electrical and magnetic properties of the particles synthesised and on the study of quantum size effects [1, 55–60]. The advantages w/o microemulsions oﬀer are directly linked to the easily performable tailoring of the compartmentalised reaction medium in terms of droplet size and concentration as well as the extremely small size of the aqueous cores. Compared to conventional methods using homogeneous reaction media, better control of the size and size distribution of the nanoparticles and an enhancement of the reaction rates have often been achieved. In recent years, interest has shifted from making monodisperse nanoparticles of mostly spherical shape of diﬀerent materials to the control of particle morphology (anisotropic and high-axial-ratio particles), introduction of crystallinity and oriented growth [61,62] as well as the ordering of (spherical) nanoparticles into 2D or 3D structured arrays and superlattices [63, 64]. Microemulsions are both highly dispersed and highly dynamic systems, which makes them a good candidate for the precipitation of inorganic nanopar-

28

W.F.C. Sager Table 1. Main characteristics of (macro)emulsions and microemulsions

Stability

(Macro)emulsion kinetically stabilised

Microemulsion thermodynamically stable

•

•

•

•

Properties of the interfacial ﬁlm

•

• •

Emulsion structure and domain size

• • •

• Surfactant (type and concentration)

• • • •

form only if (mechanical) energy is applied to the system. emulsion characteristics depend on the way the emulsion is formed and the amount of mechanical energy applied. occurrence of coagulation and coalescence processes as well as Ostwald ripening that lead to an increase in droplet size and ﬁnally to macroscopic phase separation.

• •

•

self-assembled systems that form spontaneously. properties do not depend on the preparation. domain structure and size are thermodynamically controlled; they depend on the composition and the elastic properties of the separating surfactant ﬁlm and thus on H0 , κ and κ ¯. domain size does not change with time.

stable emulsions are formed if the • macroscopic (bulk water/ oil) interfacial tension is reduced by 10– 40 mN m−1 . the emulsion droplet interface is • not fully saturated. large internal interface. •

Winsor phase equilibria form when the interfacial tension between the water- and oil-rich phase σ = 5– 10−3 mN m−1 . the interfacial ﬁlm is formed by a saturated surfactant mono- or bilayer. huge internal interface of up to 100 m2 g−1 .

•

the interfacial tension is comparable with κ. rich phase behaviour: droplet phases form if H0 = 0. For H0 ≈ 0 bicontinuous microemulsions form at low κ and Lα phases at high κ. characteristic length of 2–50 nm and low polydispersity. transparent.

interfacial energy is low but still larger than κ. the structure formed has the minimal surface/volume ratio. spherical droplets of oil (o/w) or water (w/o) with sizes of 0.1–10 μm and high polydispersity are formed milky appearance.

•

• •

water- (o/w) or oil-(w/o) soluble • surfactants. instable emulsions if the surfactant is at its balanced state. low surfactant concentrations. • stable emulsions form at surfac- • tant concentrations ≥ cmc: at concentrations a few times larger than the cmc ﬂocculation occurs.

water- (o/w) or oil-(w/o) soluble surfactants; at the balanced state of the surfactants either bicontinuous microemulsions or Lα phases form. high surfactant concentrations. surfactant concentration > cmc up to 20–30 wt%.

Microemulsion Templating

29

ticles. The surfactant-stabilised cavities provide a cage-like eﬀect that can control nucleation, growth and agglomeration. Particle nucleation and growth are governed by the occupation numbers of the reactants (solutes) in the aqueous droplet cores, the kinetics of the chemical reaction(s) leading to particle formation and the rate of solute exchange between the droplets. The ﬁnal particle morphology is inﬂuenced by the properties of the microemulsion system, the chemistry of the precipitation reaction and speciﬁc interactions between surfactants and the solid phase being formed. The particles prepared can be of amorphous, quasi-crystalline or crystalline nature and the set of independently variable and intrinsic parameters determining this is quite large. Besides the increasing use of microemulsion systems in preparation routes for nanoparticles, much less systematic work has, however, been performed on the factors that control the size, shape and stability of the particles formed. To prepare (monodisperse) particles of a given size and material, the proper precipitation conditions have to be determined experimentally and the chemical reaction conditions have to be matched with the stabilisation criteria of the microemulsion system. Metal ions can be introduced to the microemulsion via the aqueous core or in the case of anionic surfactants as surfactant counterions. The reactant agent can then be brought in via the dispersed phase of a second microemulsion of the same composition or added directly to the microemulsion if it is soluble in the continuous phase. In any case, reactants and concentrations should be chosen such that phase separation leading to uncontrolled precipitation conditions is avoided. If the reactions that lead to the formation of the solid phase are much faster than the solute exchange between the droplets, the latter becomes the rate limiting step. The exchange kinetics then dominates nucleation and growth allowing us to establish general trends that can be used for the tailoring of spherical nanoparticles. The observed dependence of the size and distribution of spherical nanoparticles on compositional parameters and on the exchange rate will be described in Sect. 3.1. The morphology of the solid phase formed is not in all cases commensurate with that of the microemulsions from which they are formed. If only a few nuclei are formed, particle growth due to solute exchange can lead to particle sizes that are a few times larger than the droplet cores. Furthermore, secondary growth processes such as (uncontrolled) aggregation or reorganisation and (re)crystallisation might lead to complex inorganic-surfactant assemblies that can reach mesoscopic length scales. Recently, w/o microemulsion systems have also been investigated to prepare non-spherical particles such as nanorods and nanoﬁbres. The underlying idea is to perform the precipitation in cylindrical aggregates or interconnected cylinder phases. However the extent to which the self-assembled surfactant systems can really serve as a mould for high-axial ratio particles has not yet been completely established. Conditions under which rod-like particles and ﬁbres have been obtained are described in Sect. 3.2.

30

W.F.C. Sager

3.1 Spherical Nanoparticles

Fig. 10. Schematic presentation of nucleation and growth processes occurring in the reduction of metal ions in w/o microemulsions. Metal ions M+ solubilised in the aqueous core become reduced when the reducing agent R enters the droplet interior. If the number of metal atoms is higher than the critical aggregation number n∗ , a metal cluster nucleus forms that grows due to the exchange of metal ions with other droplets.

The basic trends observed in the formation of nanoparticles in w/o microemulsions will be described in more detail, allowing a better insight into the nucleation and growth processes that are taking place. It has been found that under certain conditions the number density of microemulsion droplets and particles obtained can diﬀer by up to three to four orders of magnitude, indicating the importance of interdroplet solute exchange [8, 65]. Let us for the sake of simplicity consider the reduction of a metal ion under the condition that the reducing agent is brought in via the continuous phase and that all chemical reactions involved are fast compared to the interdroplet exchange (see Fig. 10). Metal particle formation then involves the following

Microemulsion Templating

31

steps: i) Diﬀusion of the reactant to the droplet interface, ii) instantaneous reduction of the metal ions, iii) metal particle nucleation and iv) growth via intramicellar attachment of metal atoms and intermicellar exchange of droplets. The ﬁnal particle size depends on the population balances and the interdroplet exchange rate constant kex . Since diﬀusion into the reversed micelles proceeds faster than the interdroplet exchange process, the initial distribution of the metal ions equals that of the metal atoms. It has been found by ﬂuorescence measurements that the ion distributions generally obey Poisson statistics. If the metal occupation number in the droplets is smaller than the critical nucleation number n∗ , nucleation can only occur after solute exchange between the droplets has taken place and n∗ has been reached. Nuclei formed grow via interdroplet exchange until all free metal ions are used up. The described situation allows the following trends to be predicted: In general, the more nuclei are formed then less free metal ions are left for particle growth. Low metal occupation numbers obtained at low metal salt concentrations and high droplet number densities lead to extended particle growth and low particle number densities. If kex is high, more nuclei can form due to faster exchange kinetics leading to smaller particles and higher particle number densities. The number of droplets Ns depends on the relative amounts of water and surfactant present, whereby Ns ∼[surfactant]3 /[water]2 . The fraction of particles having a particular size increases if n∗ is higher since more free metal atoms became available for growth. The polydispersity also becomes larger as n∗ is increased. This can be explained by the fact that for higher n∗ more free metal atoms are available for the formation of nuclei as well as for particle growth. The particles would therefore start gathering over wider ranges and larger sizes. Therefore, for systems in which n∗ is high, low and moderate salt concentrations are expected to give highly polydisperse particles. In Fig. 11 samples with silver particles that have been prepared in a waterC12 E5 -cyclohexane microemulsion system at diﬀerent water/surfactant weight ratios rw are displayed [66]. The silver particles were prepared by mixing an Ag+ ion containing microemulsion with a microemulsion of the same composition that contains NaBH4 as reducing agent. The microemulsions were either prepared with a 0.1 M AgNO3 or a 0.1 M NaBH4 solution. Reduction occurs by exchange of Ag+ and BH− 4 between the droplets. The total amount and the quantity of surfactant plus aqueous phase were kept constant, for the cut through the phase prism (see the inset in Fig. 11). As can be seen from the particle size distribution, the radius passes a maximum as rw is increased. In this series the overall concentration of silver ions increases with rw , whereas the number of microemulsion droplets decreases. Both eﬀects lead to higher occupation numbers, although the latter is much stronger than the former. Since nonionic and ionic surfactants show an opposing phase behaviour with respect to temperature, the upper boundary corresponds to the solubilisation limit and aggregation processes occur with decreasing temperature (H0 → 0) (see Fig. 8 for comparison). In this case the interdroplet exchange rate is expected to increase if the lower phase boundary is approached at high rw values.

32

W.F.C. Sager

Fig. 11. Solutions and TEM-micrographs obtained by reduction of silver ions in w/o emulsions by varying the water/surfactant weight fraction rw at constant oil weight fraction (cw = 0.2) performed at 23◦ C [66]. The colour changes from yellow to red (from left to right), because both absorption and scattering depends on the size of the particles (Mie-scattering). The phase diagram shown has been performed for the system water-C12 E5 -cyclohexane along the cut indicated in the Gibbs triangle as a function of rw and temperature at cw = 0.2. Since for nonionic surfactant systems, temperature has a much larger eﬀect on the phase behaviour than added salt, the phase boundaries change only slightly for the concentrations of AgNO3 and NaBH4 used for the precipitation. The arrow in the phase diagram marks the temperature at which the precipitation has been performed. In the Table the averaged radius and its standard deviation obtained from the TEM-micrographs are given in nanometres. Reprinted from [66]. Copyright (1992) with permission from Elsevier.

Thus the formation of more smaller particles is expected at high rw . If the number of droplets Ns present in, e.g., 3 g of microemulsion is estimated, then this number decreases from Ns ∼ 1020 at rw = 0.05 to Ns ∼ 1017 leading to average occupation numbers that are smaller than 10−2 at rw = 0.05 and about 25 at rw = 0.8. These values indicate that enormous growth takes place

Microemulsion Templating

33

for the very small droplets and explain the observed maximum in the radius if rw is increased. The polydispersity is increased at high occupation numbers.

Fig. 12. Particles of diﬀerent morphology obtained by reduction of copper [67] in water-Cu(AOT)2 -(iso-octane) microemulsions (left) and mineralisation of BaSO4 [69] in water-Ba0.019 Na0.0962 AOT-(iso-octane) microemulsions (right) at diﬀerent molar water/surfactant ratios w0 . Left: reproduced from [67]. Right: reproduced from [69]. Copyright (1997) American Chemical Society.

3.2 Nanorods and Filaments Figure 12 shows examples of nanorods and nanoﬁlaments and ﬁbres prepared from AOT w/o microemulsions. Pileni and co-workers have demonstrated that the system water-Cu(AOT)2 -(iso-octane) can be employed to prepare well deﬁned spherical copper nanoparticles and nanorods [5, 67, 68]. Cu(AOT)2 was prepared by exchanging the Na+ counter-ion for Cu2+ and the reduction was performed by addition of hydrazine under nitrogen atmosphere at room temperature. If water is added to spherical inversed Cu(AOT)2 micelles at low molar water/Cu(AOT)2 ratio w0 , they ﬁrst swell and thereafter elongate into cylinders that then form interconnected cylindrical networks. Mostly spherical Cu particles displaying polydispersities below 15% were obtained when spherical and cylindrical droplets were used as templates (w0 = 4). In this case the spherical Cu particles self-assemble into hexagonal arrays on the TEM-grid. As soon as the interconnected cylinders formed in the mould, mixtures of copper spheres and rods with up to 40% rods were obtained (w0 = 12). The rods were on average 18–25 nm long and 7–8 nm thick. Examples for both Cu nanospheres and nanorods are shown on the left side in Fig. 12. Further elongated rods could be obtained if NaCl or KCl were added. Both salts are known to inhibit growth along the [100] direction and favour that along the [110] direction.

34

W.F.C. Sager

The right hand picture of Fig. 12 shows barite ﬁbers and ﬁlaments that were obtained by addition of Na2 SO4 to a water-Ba0.019 Na0.962 AOT-(isooctane) microemulsion system at diﬀerent w0 -values. The crystallisation was performed within one week. At w0 = 10, a non-compact ﬁbre formed, consisting of 2–20 μm long multiple ﬁlaments, while for 12 ≤ w0 ≤ 29 single highly ordered crystalline barrite ﬁlaments 1–100 μm long and 20–200 nm wide were obtained. All ﬁlaments formed in this w0 -range were found to be single crystalline and elongated predominately along the [010] crystallographic axis displaying a uniform thickness and aspect ratio of 1000. These results indicate that the ﬁnal morphology is not determined by the microemulsion template. Extended oriented growth took place, which might have been induced by secondary growth processes inﬂuenced by the presence of surfactants in high concentrations [69, 70]. Studies of the time dependence in the formation of CaSO4 obtained from comparable systems showed that in most cases ﬁrstly spherical particles were formed that elongated into cylinders and ﬁnally into long ﬁbres [71].

4 (Micro)Emulsion Polymerisation 4.1 Classical Emulsion Polymerisation Compartmentalisation has become more relevant for the preparation of polymer particles (latexes) by radical polymerisation on an industrial scale than for the synthesis of inorganic materials. Over the last 50 years this has developed into a broad ﬁeld with an enormous variety of industrial and academic applications. In contrast to suspension polymerisation, where the monomer starts to nucleate in a suitable solvent (diluent) in which it is dissolved, classical emulsion polymerisation makes use of the low solubility of most common monomers in water. Monomers that contain (at least) one double bond are dispersed into micrometer-sized droplets in the water phase. The monomer molecules in the droplets are in equilibrium with monomer molecules dissolved in the water phase. When water-soluble initiators start the radical polymerisation, polymer chains grow until they become insoluble in the water-phase and form primary particles. In principle, polymer particles can form under homogeneous nucleation conditions and the formation of the critical nuclei can be described by homogeneous nucleation theory. For the nucleation process to be spontaneous, the energy ΔFspon required for creating the polymer-water interface has to be compensated. Energy is in this case gained by the aggregation of the hydrophobic parts of the growing chain, ΔFchain , so that the overall free energy of a growing chain is given by 4 (13) ΔF = ΔFspon + ΔFchain = 4πR2 σpw − πR3 fv, 3 where R is the radius of the incipient particle, σpw the interfacial tension between polymer and water and fv the free energy of condensation per unit

Microemulsion Templating

35

volume of the primary particle or nucleus. At very small sizes the ﬁrst term dominates and prevents the formation of particles. With increasing R, the released condensation energy overcomes the surface energy requirement, and ΔF passes through a maximum. Beyond the maximum, the critical nucleus can grow further. Particle growth then takes place by diﬀusion of monomer molecules from the emulsion droplets, which serve as a reservoir for monomers. (Ionic) surfactants have been widely used to reduce σpw and to stabilise the particles formed against aggregation. Employing a micellar phase as continuous medium ensures surfactant concentrations high enough for particle stabilisation and may lead to heterogeneous nucleation conditions and thus to particle formation which is not restricted by surface energy requirements. The largest application of water-born latex particles is probably in the paint industry, however increasingly greater applications are found in ﬂoor coatings (polishes), printing inks, adhesives, paper overprint varnishes, carpet backing and paper making. Water-based paints consist of a dispersion of inorganic pigment and latex particles made of a polymer that possesses a suﬃciently low glass transition temperature Tg . When the water evaporates from the applied paint, the particles coalesce to form a continuous paint ﬁlm. Using emulsion polymerisation submicrometer-sized latex particles made of, e.g., acrylonitrile, butadiene, styrene and polymethacrylate have been prepared. Advantages compared to suspension polymerisation are that i) high molecular weight polymers can be prepared at rapid reaction rates due to the fact that the free radicals grow in relative isolation and that ii) heat transfer can easily be controlled during the polymerisation due to the low viscosity of the medium even at high solid contents. In some cases, latexes with an extremely narrow size distribution and a homogeneous charge density can be obtained, that can be employed as ideal model systems to study order-disorder phenomena in colloidal suspensions. Since it is not possible to explain the different processes involved in emulsion polymerisation in more detail within the scope of this chapter by discussing the diﬀerent mechanisms proposed and the kinetic models applied, only a brief description of the diﬀerent stages observed during emulsion polymerisation in the presence of micelles in the aqueous phase is given next. This allows a better comparison to polymerisation in microemulsion systems. For more information the reader is referred to recent text books or text book contributions on emulsion polymerisation such as [72–74]. Emulsion Polymerisation in the Presence of Micelles Free radical polymerisation proceeds generally via initiation and propagation followed by termination and chain transfer. The reaction is started by the induced dissociation of an added initiator into two radicals that can attack the monomer and form new radicals with added monomer units. A constant overall concentration of propagating free radicals can be obtained if the initial radicals are formed under steady state conditions. In this case they are generated at

36

W.F.C. Sager

the same rate as they disappear, which is the case when the initiator decomposition is much slower than the subsequent reactions. In Fig. 13 three diﬀerent stages of emulsion polymerisation in the presence of micelles are sketched. Initially the continuous water phase may contain dissolved monomer, inactive monomer swollen micelles and the initiator I. Polymer latex particles are generated by two processes which may occur simultaneously. The ﬁrst is the entry of the free radicals R• into the micelles (micellar nucleation). Despite their large size, the monomer droplets are greatly outnumbered by the small micelles which have a much larger total surface area. The radicals enter the micelles rather than the large monomer droplets. The second process involves nuclei formation in the continuous phase, whereby oligomeric radicals (homogeneous nucleation) are formed. If they increase in size they may be solubilised in the micellar core or the surfactant might adsorb onto the nucleus when the chain becomes insoluble. In both cases polymer particles surrounded by surfactant are formed. In this ﬁrst interval particles grown are stabilised by adsorbing surfactants from surrounding micelles. By the end of interval I (at about 2–25% conversion), the micelles have either been nucleated to form new polymer particles or depleted to stabilise those particles formed. Most of the polymerisation occurs in interval II during which the nucleated polymer particles continue to grow due to diﬀusion of monomer molecules from the monomer droplets through the continuous medium and into the particles. The primary radicals which are formed in interval II enter the polymer particles. In particles which already have a growing polymer radical, termination occurs more or less rapidly depending on the particle size. After a short interval, another radical may enter the particle, and start a new chain. Thus, these single latex particles may at the end of the polymerisation contain a large number of polymer chains. It should be noted that termination between two growing polymer radicals, predominant in bulk, does not occur because the chains are isolated from one another in the heterogeneous solution. Finally, interval III is reached when all of the monomer in the monomer droplets is consumed and only the monomer in the still-swollen particles remains to be polymerised. 4.2 Polymerisation in Droplet and Bicontinuous Microemulsions Since 1980 microemulsions have been investigated in order to prepare latex particles within the nanosize range (< 50 nm), starting with o/w droplet phases. In this case, o/w microemulsions were prepared using a monomer, such as styrene or methyl methacrylate (MMA) as dispersed phase. Nucleation then takes place inside the droplets by diﬀusion of water soluble initiators into the droplet interior. The ﬁnal droplet size is determined by the oil/surfactant ratio and the droplet exchange processes taking place. At too high monomer concentrations phase separation occurs during the polymerisation. Stable latex dispersions are generally obtained at surfactant/monomer ratios ranging from 1–5, which are quite high in comparison to emulsion polymerisation in which the surfactant/monomer ratio is usually smaller than 0.03. Additionally,

Microemulsion Templating

37

Fig. 13. Schematic presentation of the diﬀerent intervals taking place in emulsion polymerisation in the presence of (monomer swollen) micelles M in the continuous aqueous phase. The micelles are below 100 ˚ A while the monomer swollen polymer particles PP reach sizes of about 500 ˚ A (I and II) to about 1000 ˚ A (III). The monomer supply originates from the emulsiﬁed monomer droplets ED of about 10 μm.

w/o microemulsions can be employed to polymerise water-soluble monomers, such as acrylic acid and 2-hydroxyethylmethacrylate (HEMA). Generally the monomer was ﬁrst dissolved in water and w/o microemulsions formed from the four components. Bicontinuous microemulsions were ﬁrst studied to improve the interdroplet exchange rates and later to obtain interconnected polymer networks [75]. In templating the morphology of microemulsions, it has often been observed that the solid phase that nucleates in the microemulsion cavities (e.g., droplets or channels) ﬁnally grows out of these cavities. The solid structures thus formed are not commensurate in size, and even shape, with the template, especially in polymerising bicontinuous microemulsions. When the monomeric oil starts to polymerise, surfactant molecules may adsorb on the forming polymer, so causing the surfactant concentration in the remaining microemulsion to change. If the monomer continues to solidify while morphological changes or even phase inversion or separation processes occur, the ﬁnal structure may even become homogeneous but will certainly diﬀer in size and shape [76]. Bicontinuous microemulsions have only successfully been solidiﬁed, yielding porous polymeric ﬁlms in the same size range, if the polymerisation reaction was suﬃciently fast and the surfactant itself took part in the reaction. In this case the surfactant needs to possess the same polymerisable group as the monomer in its hydrophobic moiety. The ﬁrst breakthrough in polymerising bicontinuous microemulsions was realised by Gan et al. [77, 78]. They obtained nanoporous polymeric materials by water-methylmethacrylate (MMA) microemulsions stabilised by a zwitterionic or a quaternary ammonium salt surfactant, possessing the same polymerisable group. Solidiﬁcation took place only in a certain composition range (water and surfactant). The channel width

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of the remaining water domains increased with decreasing surfactant or increasing water concentration. The resulting pore sizes could furthermore be decreased by the addition of a more water soluble monomer, which partly polymerised in the water phase and then precipitated onto the pore surfaces. In this way a continuous variation of the channel width from 4–60 nm could be obtained. A higher mechanical strength of the polymeric ﬁlm was attained if a crosslinker was added. The polymerisation reaction was either initiated using UV light or a redox initiator at 50◦ C. Meanwhile, polymerisable nonionic surfactants and microemulsions could also be prepared which allow for copolymerisation with vinylbenzenesulfonate [79].

Fig. 14. TEM micrographs of a sample prepared of 30 wt% water, 32 wt% MMA, 38 wt% (((acryloyloxy)-undecyl)dimethylammino) acetate (AUDMAA), a polymerisable zwitterionic surfactant, and 2 wt% cross-linking agent (ethylene glycol dimethacrylate) during polymerisation. The reaction was photoinitiated at 35◦ C and samples were taken after 3–8 min and after 1 h. The polymeric phase is presented by the lighter contrast. Reproduced from [77]. Copyright (1995) American Chemical Society.

Figure 14 shows the network formation as a function of time in a sample prepared of 30 wt% water, 32 wt% MMA, 38 wt% (((acryloyloxy)undecyl)dimethylammino) acetate (AUDMAA), a polymerisable zwitterionic surfactant, and 2 wt% cross-linking agent (ethylene glycol dimethacrylate) [77]. Numerous nanoparticles can be seen from the TEM micrograph after 3 min of polymerisation, when the sample is still very ﬂuid. A semigel formed

Microemulsion Templating

39

after 6 min with polymer particles in the form of interconnected clusters with interdispersed pores and sizes smaller than 250 nm. The sample was totally gelled after 8 min. The obtained nanostructured network resembles that of the original bicontinuous microemulsion. The dimensions of the channels were in the range of 50–70 nm in width and 100–200 nm in length. The structure did not alter much even after polymerisation for 1 h when a transparent solid polymer was obtained as the ﬁnal product.

5 Templating of Crystalline Phases The synthesis of inorganic mesoporous materials using ionic surfactant templates, ﬁrst reported in 1992 [80, 81], led to the discovery of a novel family of molecular sieves called M41S (see Fig. 16) and opened up a new ﬁeld of organised matter chemistry. The synthesis was performed by combining appropriate amounts of silica (and alumina) precursors, an alkyltrimethylammonium halide surfactant (e.g., cetyltrimethylammonium bromide, CTAB), a base and water. The inorganic phase is formed by hydrolysis of an alkoxide (e.g., tetramethylorthosilicate) that reacts with water followed by condensation. The mixture is then aged at elevated temperatures for 24–144 h, which results in a solid precipitate. The organic-inorganic mesostructured product is calcined A and a at about 500◦ C after washing. With well-deﬁned pore sizes of 15–100˚ hexagonal symmetry, these ﬁrst mesoporous (alumino)silicate materials break past the pore-size constraint (< 15 ˚ A) set by microporous zeolites. Surfactant– mediated synthesis has since been intensively investigated and employed to form a variety of mesoporous materials. The extremely high surface areas (> 1000 m2 g−1 ) and the precise tuning of pore sizes has made these materials highly attractive for applications in catalysis, membrane and separation technology, and molecular engineering. The M41S materials also ushered in a new approach in materials synthesis where, instead of the use of single molecules as templating agents, as in the synthesis of zeolites, self-assembled (molecular) surfactant aggregates or supramolecular assemblies are employed as the structure directing agents [82]. Recently, considerable attention has been focussed on tailoring the chemical composition of ordered mesoporous silicas for novel applications. The introduction of inorganic heteroatoms such as Al, Ti, or V into the silica framework, the synthesis of non-silica analogues (oxides of Al, Ti, V, W, Sb, Pb), and attachment of metallocene derivatives via pendant Si-O-H groups (to form Si-Θ metal linkages) have been explored as routes to the preparation of catalytically active, ordered mesophases. Furthermore, cocondensation of siloxane and organosiloxane precursors has been investigated to produce inorganic-organic networks. These organically functionalised, uniformly porous, ordered silica materials oﬀer a high potential in applications in organometallic chemistry, catalysis and host-guest systems [83]. There is still some debate on the actual mechanism that leads to the formation of the mesoporous materials. In principle, two diﬀerent paths are pos-

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sible, which are depicted in Fig. 15, for preparation of mesoporous materials with hexagonally ordered pores. In the original preparation route from 1992, the synthesis was performed at very low surfactant concentrations of about 1 wt%. This concentration is above the cmc but much too low to allow the formation of liquid crystalline arrays such as the hexagonal phase H1 that is built up of hexagonally ordered rodlike micelles. In this case, cooperative processes must be at work which require speciﬁc interactions between the soluble silicate species and the surfactant head group. The processes involved include speciﬁc binding of the silica precursor to the interfacial surfactant ﬁlm, polymerisation of the silica at the interface and a kind of interfacial charge density matching that leads to the mechanism sketched in Fig. 15, depicting an ordering of the co-adapted micelles into a hexagonal array. Pattern replication then proceeds by further condensation and polymerisation of silica within the interstitial spaces of the coassembled template. Post synthesis removal of the organic phase aﬀords an ordered mesoporous inorganic material with channel dimensions commensurate with the width of the rodlike surfactant micelles (for more insight into the mechanisms, see also [84]). Besides preparation via cooperative assembling, mesoporous silica has also been prepared by direct templating of a liquid crystalline phase, leading to a monolithic solid material that spans the whole sample.

Fig. 15. Two principally diﬀerent paths to prepare mesoporous silica (MCM-41). The cooperative approach has been performed by adding a silica precursor to a surfactant solution at low surfactant concentration. Speciﬁc interactions between the silicate precursors and the surfactant molecules are necessary to obtain coassembly into a hexagonal array. The monolithic approach has been performed at high surfactant concentrations and employs direct templating of a hexagonal liquid crystal. Reprinted with permission from [82].

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Fig. 16. Mesoporous M41S silica obtained by direct templating of an H1 phase of C12 E8 (left) and of the Lα -phase of C16 E8 (middle) [85] together with schematic illustrations of the mesoporpous M41S materials of hexagonal (MCM-41, space group: p6m), cubic (MCM-48, space group: Ia¯ 3d) and lamellar (MCM-50, space group p2) structure [82]. Silica is presented by the lighter contrast. Left and middle: reproduced from [85]. Right: reproduced from [82].

Figure 16 shows mesostructured materials that have been synthesised by direct templating of the corresponding liquid crystalline phases (monolithic approach) [85]. The mesoporous silica with hexagonal symmetry was obtained from the H1 phase of C12 E8 . The silica was prepared by adding tetramethylorthosilicate (TMOS) in quantities of 0.25 mole equivalent with respect to water to an H1 phase of C12 E5 with a surfactant/water ratio of 50 wt% at pH = 2. Methanol released from the hydrolysis of TMOS was removed under mild vacuum, since it destroys the H1 phase, before the sample was left at room temperature for 18 h to allow the monolith to form. The TEM-micrograph on the left in Fig. 16 shows a hexagonal array of regularly sized holes of 28 ˚ A diameter separated by ∼ 12 ˚ A-thick silica walls. X-ray diﬀraction shows that the value for the lattice parameter decreased only slightly upon solidifying the H1 -phase. The monolithic approach could also be applied in templating cubic and lamellar liquid crystals. The central portion of Fig. 16 shows mesoporous silica with a lamellar structure obtained from the Lα -phase of C16 E8 . When the same nonionic surfactants used in the monolithic approach were employed in the cooperative approach at much lower concentrations, no mesostructured precipitate was formed, indicating that direct templating was responsible for the formation of the monolith rather than speciﬁc interactions. Acknowledgement. This chapter is part of the Lecture notes of the 33rd Spring School of the Institute of Solid State Research (IFF) on “Soft Matter: Complex Materials on Mesoscopic Scales” (Foschungszentrum J¨ ulich, 2002, ISBN 3-89336297-5. The author would like to thank S. Uhorczuk for her assistance with ﬁgure drawings.

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References 1. Fendler J. H. (1987) Chem. Rev. 87:877 2. Ozin G. A. (1992) Adv. Mater. 4:612 3. Mann S., Burkett S. L., Davis S. A., Fowler C. E., Mendelson N. H., Sims S. D., Walsh D., Whilton N. T. (1997) Chem. Mat. 9:2300 4. Capek I. (2004) Adv. Colloid Interface Sci. 110:49 5. Pileni M. P. (2003) Nat. Mat. 2:145 6. Davis H. T., Bodet J. F., Scriven L. E., Miller W. G. (1987). In: Meunier J., Langevin D., Boccara N. (eds) Physics of Amphiphilic Layers. Springer-Verlag, Berlin, p. p. 310-327 7. Sager W. F. C. (1998) Curr. Opin. Colloid Interface Sci 3:276 8. Pileni M. P. (1993) J. Phys. Chem. 97:6961 9. Ying J. Y., Mehnert C. P., Wong M. S. (1999) Angew. Chem. Int. Ed. 38:56 10. Wong K. K. W., Mann S. (1998) Curr. Opin. Colloid Interface Sci. 3:63 11. Stein A., Melde B. J., Schroden R. C. (2000) Adv. Mater. 12:1403 12. Soten I., Ozin G. A. (1999) Curr. Opin. Colloid Interface Sci. 4:325 13. Hamley I. W. (2000) Introduction to Soft Matter: Polymers, Colloids, Amphiphiles and Liquid Crystals. Wiley, New York 14. Israelachvili J. (1992) Intermolecular & Surface Forces, 2nd Ed. Academic Press, London, p. p. 341-381 15. Evans D. F., Wennerstr¨ om H. (1999) The Colloidal Domain: Where Physics, Chemistry, Biology and Technology Meet, 2nd Ed. Wiley-VCH, New York 16. Gelbart W. M., Ben-Shaul A., Roux D. (1994) Micelles, Membranes, Microemulsions and Monolayers. Springer-Verlag, Berlin 17. Safran S. A. (1994) Statistical Thermodynamics of Surfaces, Interfaces and Membranes. Addison-Wesley, Reading, MA 18. Hyde S., Andersson S., Larsson K., Blum Z., Landh T., Lidin S., Ninham B. W. (1997) The Language of Shape-The Role of Curvature in Condensed Matter: Physics, Chemistry and Biology. Elsevier Science B. V., Amsterdam 19. Degiorgio V., Corti M. (1985) Physics of Amphiphiles: Micelles, Vesicles and Microemulsions. North-Holland, Amsterdam 20. Overbeek J. Th. G. (1986) K. Ned. Akad. Wet. Ser. B 89:61 21. Hunter R. J. (1993) Introduction to Modern Colloid Science. Oxford University Press, Oxford 22. Seddon J. M. (1996) Ber. Bunsen. Phys. Chem. 100:380 23. Hoar T. P., Schulman J. H. (1943) Nature 152:102 24. Bancroft W. D., Tucker C. W. (1927) J. Phys. Chem. 31:1681 25. Scriven L. E. (1976) Nature 263:123 26. Jahn W., Strey R. (1988) J. Phys. Chem. 92:2294 27. Helfrich W. (1973) Z. Naturforsch. C 28:693 28. Goetz R., Gompper G., Lipowsky R. (1999) Phys. Rev. Let. 82:221 29. Kellay H., Binks B. P., Hendrikx Y., Lee L. T., Meunier J. (1994) Adv. Colloid Interface Sci. 49:85 30. Van der Linden E., Bedeaux D., Borkovec M. (1989) Physica A 162:99 31. Mitchell D. J., Ninham B. W. (1989) Langmuir 4:1121 32. Winsor P. A. (1954) Solvent Properties of Amphiphilic Compounds. Butterworths, London 33. Strey R. (1994) Colloid Polym. Sci. 272:1005

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Strey R. (1996) Curr. Opin. Colloid Interface Sci. 1:402 De Gennes P. G., Taupin C. (1982) J. Phys. Chem. 86:2294 Blokhuis E. M., Sager W. F. C. (2001) J. Chem. Phys. 115:1073 Kahlweit M., Strey R., Busse G. (1990) J. Phys. Chem. 94:3881 Kahlweit M., Strey R., Firman P., Haase D., Jen J., Schom¨ acker R. (1988) Langmuir 4:499 Kahlweit M., Strey R., Schom¨ acker R., Haase D. (1989) Langmuir 5:305 Koper G. J. M., Sager W. F. C., Smeets J., Bedeaux D. (1995) J. Phys. Chem. 99:13291 Kahlweit M., Strey R., Haase D., Kunieda H., Schmeling T., Faulhaber B., Borkovec M., Eicke H. F., Busse G., Eggers F., Funck T., Richmann H., Magid L., S¨ oderman O., Stilbs P., Winkler J., Dittrich A., Jahn W. (1987) J. Colloid. Interface Sci. 118:436 Fletcher P. D. I., Holzwarth J. F. (1991) J. Phys. Chem. 95:2550 Borkovec M., Eicke H. F. (1988) Chem. Phys. Lett. 147:195 Blokhuis E. M., Sager W. F. C. (1999) J. Chem. Phys. 111:7062 Tlusty T., Safran S. A. (2000) J. Phys.-Condens. Mat. 12:A253 Baxter R. J. (1968) J. Chem. Phys. 49:2270 Svergun D. I., Konarev P. V., Volkov V. V., Koch M. H. J., Sager W. F. C., Smeets J., Blokhuis E. M. (2000) J. Chem. Phys. 113:1651 Stokes R. J., Evans D. F. (1997) Fundamentals of Interfacial Engineering. WileyVCH, New York Fletcher P. D. I., Howe A. M., Robinson B. H. (1987) J. Chem. Soc. Faraday Trans. I. 83:985 Jain T. K., Casin G., Badiali J. P., Pileni M. P. (1996) Langmuir 12:2408 Fletcher P. D. I., Horsup D. I. (1992) J. Chem. Soc. Faraday Trans. 88:855 Binks B. P., Fletcher P. D. I., Horsup D. I. (1991) Colloids Surfaces 61:291 Shinoda K. (1967) J. Colloid. Interface Sci. 24:4 Becher P. (ed) (1983) Encyclopedia of Emulsion Technology, Vol. 1, Basic Theory. Marcel Dekker, New York Hou M. J., Shah D. O. (1988). In: Attia Y. A., Moudgil B. M., Chander S. (eds) Interfacial Phenomena in Biology and Materials Processing. Elsevier, Amsterdam, p. p. 443-458 Pileni M. P. (1993) J. Phys. Chem. 97:6961 Boutonnet M., Kizling J., Stenius P. (1982) Colloids Surfaces 5:209 Cushing B. L., Kolesnichenko V. L., O’Connor J. (2004) Chem. Rev. 104:3893 Burda C., Chen X., Narayanan R., El-Sayed M. A. (2005) Chem. Rev. 105:1025 Uskokovic V., Drofenik M. (2005) Surf. Rev. Lett. 12:239 Walsh D., Mann S. (1995) Nature 377:320 M¨ oller M., Spatz J. P. (1997) Curr. Opinion Colloid Interface Sci. 2:177 Fendler J. H. (1997) Curr. Opinion Solid State Materials Sci. 2:365 Pileni M. P., Motte L., Billoudet F., Mahrt J., Willig F. (1997) Mater. Lett. 31:255 Natarajan U., Handique K., Mehra A., Bellare J. R., Khilar K. (1996) Langmiur 12:2670 Barnickel P., Wokaun A., Sager W., Eicke H. F. (1992) J. Colloid. Interface Sci. 148:80 Tanori J., Pileni M. P. (1997) Langmuir 13:639 Pileni M. P. (2001) Langmuir 17:7476

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69. Hopwood J. D., Mann S. (1997) Chem. Mat. 9:1819 70. Li M., Mann S. (2000) Langmuir 16:7088 71. Rees G. D., Evans-Gowing R., Hammond S. J., Robinson B. H. (1999) Langmuir 15:1993 72. Lovell P. A., El-Aasser M. S. (1997) Emulsion Polymerisation and Emulsion Polymers. John Wiley, Chichester 73. Gilbert R. G. (1995) Emulsion Polymerisation. Academic Press, London 74. Fitch R. M. (1997) Polymer Colloids: A Comprehensive Introduction. Academic Press, San Diego 75. Chow P. Y., Gan L. M. (2005) Adv. Polym. Sci. 175:257 76. Antonietti M., Hentze H. P. (1996) Colloid Polym. Sci. 274:696 77. Gan L. M., Li T. D., Chew C. H., Teo W. K., Gan L. H. (1995) Langmuir 11:3316 78. Gan L. M., Chieng T. H., Chew C. H., Ng S. C. (1994) Langmuir 10:4022 79. Liu J., Teo W. K., Chew C. H., Gan L. M. (2000) J. Appl. Polym. Sci. 77:2785 80. Kresge C. T., Leonowicz M. E., Roth W. J., Vartuli J. C., Beck J. S. (1992) Nature 359:710 81. Beck J. S., Vartuli J. C., Roth W. J., Leonowicz M. E., Kresge C. T. , Schmitt K. D., Chu C. T. W., Olson D. H., Sheppard E. W., McCullen S. B., Higgins J. B., Schlenker J. L. (1992) J. Am. Chem. Soc. 114:10834 82. Ying J. Y., Mehnert C. P., Wong M. S. (1999) Angew. Chem. Int. Ed. 38:56 83. Mann S., Burkett S. L., Davis S. A., Fowler C. E., Mendelson N. H., Sims S. D., Walsh D., Whilton N. T. (1997) Chem. Mat. 9:2300 84. Flodstrom K., Wennerst¨ om H., Teixeira C. V., Amenitsch H., Linden M., Alfredsson V. (2004) Langmuir 20: 10311 85. Attard G. S., Glyde J. C., G¨ oltner C. G. (1995) Nature 378:366

Nanofabrication of Block Copolymer Bulk and Thin Films Microdomain Structures as Templates Takeji Hashimoto1 and Kenji Fukunaga2 1

2

Advanced Science Research Center, Japan Atomic Energy Agency, Tokai-mura, Ibaraki 319-1195, Japan Polymer Laboratory, Ube Industries, Ltd., Ichihara, Chiba 290-0045, Japan

1 Introduction and Background In this chapter we shall discuss applications of block copolymers (bcps) to nanotechnologies and nanosciences. Our objectives here are to explore the methods and principles concerning fabrications of ordered structures of bcps having various symmetries with nano-sized periodicity to create new materials with interesting structures and properties. We deﬁne this kind of fabrication as nano-fabrication. In other words we aim to control or manipulate selforganized microdomain structures of bcps, in both nonequilibrium and equilibrium states, and utilize them as templates for further nano-fabrication toward advanced devices and materials, such as tunable photonic crystals [1–4], quantum dots and nanowires [5–9], nanohybrids with inorganic materials and nanometal particles [10–12], photovoltaics and photoluminescence [13–16], etc. We shall present the bcp templates in both bulk (Sect. 2) and thin ﬁlms (Sect. 3). 1.1 Bulk Before going into detailed discussion, it would be useful to review the basic physical factors which control the morphology of bcp microdomains as templates. Let us discuss the simplest bcp of A-b-B diblock copolymers (dibcps) in which one end of polymer A is covalently bonded with one end of polymer B. A-b-B dibcps in bulk are interesting two-component polymer systems. Without this covalent bond connecting A and B polymers, the two component system is a binary polymer mixture which can undergo a macrophase separation into two macrophases with a single interface at thermal equilibrium, as shown in part (a) of Fig. 1. Here important parameters are the short-range interactions between A-A, B-B, and A-B as also indicated in the legend of Fig. 1(a). On the other hand, with this covalent bond, the two component

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Takeji Hashimoto and Kenji Fukunaga

system becomes A-b-B dibcp which can undergo microphase separation and which can have multi-interfaces in bulk, as shown in Fig. 1(b). In this case, the bulk bcp is nothing other than an assembly of interfaces on which the chemical junctions between A and B exist. Here in addition to the short-range interactions, the long-range interactions play an important role as shown in the legend in Fig. 1(b). The long-range interactions arise from connectivity between two polymers A and B and their packing in the respective domains with a demand of incompressibility. Interplay of the short-range interactions and the long-range interactions is a key physical factor in bcp systems [17].

Fig. 1. Comparison of a self-assembled structure for a binary mixture of polymer A and B and that for a diblock copolymer A-b-B. Reprinted from [54]. Copyright (2005) The Chemical Society of Japan.

The long-range interactions involve conformational entropy of polymer chains (and hence elastic energy of polymer chains) as well as translational entropy of polymer chains (or junctions of A and B) along the interface. Thus the interplay of the short-range and long-range interactions is considered to be a competition between energetics and entropies. In the case when the short range van der Waals interactions between A and A and between B and B are stronger than those between A and B, A and B are subjected to net repulsive interactions, under which macro- and microphase separation can occur in the two component system. It is well known that various microdomain structures, including BCCSphere (sphere in body-centered-cubic lattice), Hex-cylinder (hexagonallypacked cylinders), a bicontinuous microdomain structure of the so-called double gyroid network structure with Ia¯3d space group symmetry, and alternating lamellae [18] can be tailored by changing the relative length of the two blocks as shown in Fig. 2. This is a tailoring based on chemistry. This is one of the consequences of the interplay of those two physical factors as described above.

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Fig. 2. Changes in the tailored microdomain morphology by increasing a chain length of A block chain relative to that of B block chain. Only the cases where A block chains are shorter than or equal to B block chains is shown.

We should note also the fact that bulk bcp usually have grains, as also shown in Fig. 2 for the lamella morphology for example, within which orientation of the ordered domain is consistent. The control of the grain size further requires a control of the ordering process of bcp, i.e., a tailoring based on physics. 1.2 Thin Films Let us now consider the case of thin bcp ﬁlms. In this case, in addition to (i) the short range interactions of their own in bulk, there are (ii) short-range interactions of A and B with air and substrate surfaces as indicated in Fig. 3. The two kinds of the short-range interactions (i) and (ii) further interplay with each other. The long range interactions of their own in bulk interplay also with those arising from the conﬁned space of the bcp in between the air surface and the substrate surface. The two kinds of the long-range interactions further interplay with each other. The interplay in the bcp thin ﬁlm is therefore much richer than that in bulk, simply because there are competitions or interplays within and between each of these interactions. This enrichment further enriches manipulations of ordered microdomain structures as templates. We know that lamella-forming bcp thin ﬁlms have parallel lamellae, in the case when their substrates and/or air surfaces have diﬀerent interfacial ten-

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Takeji Hashimoto and Kenji Fukunaga

sions against A and B, as shown schematically in Fig. 3(a). If the substrate and air surfaces are neutral against A and B, the thin ﬁlms have perpendicular lamellar orientation [19], as shown also in Fig. 3(b). The perpendicular lamellar systems may be more interesting than the parallel ones for use of them as a template for further fabrication.

Fig. 3. Parallel (a) and perpendicular orientation (b) of bcp microdomains in thin bcp ﬁlms as a consequence of short-range interactions of block chains A and B with surface and substrate surface and of long-range interactions of block chains A and B in a conﬁned space inherent in the thin ﬁlm thickness. γAir (K) and γsubstrate (K) denote interfacial tension between air and block chain K (K = A or B) and that between substrate and block chain K, respectively.

In this chapter we shall deal with microdomain structures developed in both bulk ﬁlms (Sect. 2) and thin ﬁlms (Sect. 3). In the case of the thin ﬁlms we shall focus on kinetic pathway along which the orientation and microdomain structure in thin ﬁlms change towards equilibrium. The clariﬁcation of the pathway is crucial for the manipulation of the template structures in thin ﬁlms for nanofabrication. In the case of bulk ﬁlms, one of the microdomains in bulk bcps can be transformed into empty space (Sect. 2.1). When the microdomains are co-continuous in 3d space, such a fabrication as described above creates continuous phase of empty space. We shall also discuss a controlled alignment of metal nanopraticles into microdomain structures as templates in Sect. 2.2.

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2 Bulk Block Copolymers 2.1 Nanofabrication of Microdomains via Selective Chemical Degradation by Ozonolysis: Double Gyroid Networks Background We shall discuss here a nano-fabrication, which involves transformation of one of the microdomains A or B in A-b-B dibcp into vacant space by using a selective chemical degradation. Such fabrication may be applied to hexagonal cylinders, double gyroid network structure with Ia¯3d space group symmetry, and alternating lamellae of polystyrene-block -polyisoprene (PS-b-PI) and poly(2-vinyl pyridine)-block -polyisoprene (P2VP-b-PI). Ozonolysis selectively degrades only polyisoprene microdomains but does not essentially aﬀect PS or P2VP microdomains in these bcps. The degraded compounds can be leached out from the domains by soaking the ﬁlm in ethanol, which transforms the PI microdomains into empty space. We shall discuss an application of such a nano-fabrication as discussed above to the double gyroid network structure, a periodic co-continuous microdomain structure, which is formed in a very narrow composition range between those of cylinders and the lamellae (ca.33-36 vol % of network forming component in the case of PS-b-PI, [20, 21]). We can selectively degrade one of the “microphases” of double gyroid network structure. When the network phase is degraded and transformed into empty space, a set of two vacant network channels designated as “voided double-gyroid channel” is formed in the solid matrix of the major component (eg., PS or P2VP). On the other hand, when the matrix is selectively degraded, the free standing (doublegyroid-network) texture of PS or P2VP in vacancy is formed. The free standing texture may be stabilized by the existence of a grain boundary structure, which may be easily visualized in a simpler case of the free standing lamellae. Fig. 4 shows a grain boundary structure called Scherk’s ﬁrst surface (part a) [22–24], which is made in the boundary of two orthogonal sets of lamellae (part b). The dark and bright lamellae in one grain are interconnected respectively to the dark and bright lamellae in another grain through the grain-boundary interface as shown in part (a). Even when one kind of the lamellar microdomains is transformed into empty space, the rest of lamellar system remains intact through this grain boundary structure. Degradation of Network Domains: Voided Double-Gyroid Channel A double gyroid network structure was prepared by casting a mixture of PSb-PI block copolymer and PS homopolymer (designated as SI in this section) with an overall volume fraction of PI component of 0.34 from toluene solution having total polymer concentration of ca. 5 wt% into petri dishes or by casting a mixture of P2VP-b-PI and PI homopolymer (designated as VI in

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Fig. 4. Scherk’s ﬁrst surface (a) formed in the grain boundary between two orthogonal sets of lamellae (b). l1 and l2 designate unit vectors along the directions normal to the lamellar interface.

this section) with an overall volume fraction of PI of 0.4 from chloroform solution having total polymer concentration of ca. 5 wt%. The details of the bcps and homopolymers used can be found elsewhere [25, 26]. It is important to note that the molecular weight of the PS or PI homopolymer is so small compared with the molecular weight of the corresponding block that the homopolymer is always solubilized into the corresponding domain of the bcp. Film specimens of about 100-300μm thick were prepared by slowly evaporating the solvent over a week. The ﬁlms thus obtained were further dried until a constant weight was attained. The obtained VI ﬁlms were further treated with 1,4-diiodobutane (DIB) in order to crosslink the matrix phase of P2VP according to the method as detailed elsewhere [27, 28]. Fig. 5 represents a transmission electron microscopic (TEM) image for an ultrathin section of SI stained with OsO4 (part a) and that of VI crosslinked and stained with DIB (part b). In part (a) the PI microdomains (the gyroid networks) stained with OsO4 appear dark and gray, while the PS matrix unstained with OsO4 appears bright. On the other hand, in part (b) the PI microdomains (the gyroid networks) unstained with DIB appear bright and gray, while the P2VP matrix stained with DIB appears dark: consequently the contrast of the two images is reversed. Nevertheless the two images show a common characteristic of a periodic arrangement of a “double wave pattern” comprising a set of a small amplitude wave and a large amplitude wave. In part (a), the bright double wave pattern exists in the gray background, while in part (b), the dark double wave pattern exists in the gray background, as highlighted by the insets. The double wave appears to reﬂect the PS matrix in

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part (a) and the P2VP matrix in part (b), while the gray background appears to reﬂect the PI microdomains in both cases.

Fig. 5. TEM images for SI stained with OsO4 (part a) and for VI stained with diiodobutane (part b), both showing the periodic double wave pattern. From [26].

One can further observe a series of dark circles in part (a) or bright circles in part (b) along the double wave pattern. These circles are expected to be the PI microdomains oriented more or less normal to the ultrathin section. We shall discuss the morphological interpretations of this double wave pattern later in this section in conjunction with Fig. 7. It should be noted that the image in part (a) is somewhat distorted, probably due to deformation of the microdomain structure in the ultrathin-sectioning process. However the distortion is less noticeable in the image in part b, probably because hardening of the P2VP matrix by the DIB crosslinking suppressed the distortion. The same ﬁlm specimens used for TEM were subjected to ozonolysis for 24 h at room temperature, followed by rinsing with methanol [25,26]. The treated specimens were then freeze-fractured in a liquid nitrogen bath, sputter-coated with platinum, and observed under a scanning electron microscope with a ﬁeld-emission gun (FE-SEM). Fig. 6 presents a typical FE-SEM image for the SI sample. The image uniformly shows the ordered structure comprising the double wave pattern with the dark circles along the waves. In the SEM image, the bright region indicates undegraded PS matrix, while the dark and gray regions are the degraded vacant microphase corresponding to the PI microdomains before the ozone degradation. The dark circles indicate the vacant channel oriented nearly perpendicular to the surface of the freezefractured specimens. Similar FE-SEM patterns were also observed for the

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freeze-fractured surface of degraded VI specimens [26]. This SEM image is quite similar to the particular TEM images shown in Fig. 5 observed for the specimens before the degradation.

Fig. 6. FE-SEM image of freeze-fractured surface of ozone-degraded SI block copolymer having voided double-gyroid channel in the matrix of PS. The image exhibits facets between the two cleavage surfaces parallel to (211) plane with a step height diﬀerence. From [26].

Fig. 7 shows 3d computer graphics (CG) of one repeat unit of the double gyroid network structure with Ia¯3d space group symmetry. In part (a), only the double network phase shown in Fig. 2 is made empty in the solid matrix [25]. Therefore Fig. 7 (a) presents a CG for the voided double-gyroidchannel in the solid matrix, while Fig. 7(b) shows the 3d cross sectional view on a particular plane cut parallel to the (211) plane of the double gyroid network structure. The CG of 3d cross sectional image (part b) shows a double wave pattern with holes along the waves. Here the bright and gray regions certainly correspond to the matrix phase and the voided double-gyroid-channel, respectively. The CG pattern is exactly identical to that shown in the FE-SEM image in Fig. 6; This pattern should also explain the TEM images shown in Fig. 5(a) and (b), revealing that the TEM images reﬂect those obtained on ultrathin sections which happened to be cut parallel to the (211) plane of the double gyroid network. These results obtained in Figs. 5 to 7 illuminate that the double gyroid network of PI microdomains is successfully transformed into the vacant nanochannel by the nano-fabrication technique described here.

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Fig. 7. Computer graphics of one repeat unit of the double gyroid network structure with Ia¯ 3d space group symmetry. (a) The double network phase made empty in the solid matrix phase, and (b) 3d CG image obtained in a cross section of the structure in part (a) cut parallel to (211) plane.

The TEM images showed various kinds of patterns other than the double wave pattern, depending on the cutting direction of the ultrathin section with respect to the double gyroid network lattice, for example, the “wagon wheel” pattern observed when an ultrathin section is cut parallel to the (111) plane. Contrary to the TEM images, the FE-SEM images observed on the freeze-fractured surface after the ozonolysis exclusively show the double wave pattern, except for the “hexagonal doughnut” pattern as will be shown later in Fig. 8. Why do we observe only these two special patterns? The detailed analyses of the double wave pattern and the hexagonal doughnut pattern elucidated that the patterns reﬂect cleavages plane parallel to (211) plane and (110) plane, respectively [26]. The cleavage planes correspond to the planes which have the minimum area fraction of the matrix phase, i.e., 0.51 and 0.56 for the cleavage planes parallel to (211) and (110), respectively. Since the cross sectional area of the former is smaller than that of the latter, the double-wave type cleavage pattern appears more often than the hexagonal-doughnut type cleavage pattern. The hexagonal doughnut pattern found for the cleavage plane parallel to (110) plane is shown in Fig. 8 where part (a) shows the CG of the 3d cross sectional image cut parallel to (110) plane and part (b) shows a typical FE-SEM image. It should be noted that the two patterns (a) and (b) have common characteristics along the lines a, b, c, and d drawn as insets. The image (b) shows a facet between the two (110) cleavage planes with a step height diﬀerence. Across the facet, characteristic features of the pattern along the lines a to c are shifted to the lines a to c . The existence of the two special regular patterns, double wave pattern and hexagonal doughnut pattern, and evidence

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Fig. 8. Cross-sectional image showing hexagonal doughnut pattern. (a) 3d CG of cross-section cut at the particular planes parallel to (110) plane of the double gyroid network structure and (b) an FE-SEM image on the freeze- fractured surface of the SI ﬁlms subjected to ozonolysis.

of these particular patterns reﬂecting the patterns on the cleavage surfaces unequivocally indicate existence of the double gyroid network structure with Ia¯ 3d space group symmetry in the real system. The special patterns of the facets shown in Fig. 6 also reinforce the above assessment, as will be discussed in detail elsewhere [26]. Degradation of Matrix Domains: “Free-Standing” Double-Gyroid Texture Films having the double gyroid network structure comprising P2VP block chains in the matrix of PI block chains were prepared by solution casting of mixtures of P2VP-b-PI (number average molecular weight Mn = 6.7×104 and weight fraction of PI block wP I = 0.39) and homopolyisoprene (Mn = 4×103 ) which contain a total PI volume fraction of 0.66. The mixtures were dissolved into ∼ 5 wt% total polymer concentration with benzene. Thin ﬁlms about 1 mm thick were prepared from the solution in a petri dish by slowly evaporating solvent over approximately a month. The as-cast ﬁlms were further dried under vacuum for about a week until constant weight was attained. The ﬁlms thus prepared were exposed to 1.4-diiodobutane (DIB) vapor at 80o C for 72 h for selective crosslinking of P2VP double-gyroid-network phase. The DIB crosslinked ﬁlms were soaked in heptane and the ozone was bubbled in heptane for 48 h at room temperature. The cleaved compounds were leached out from the ﬁlms by soaking them in ethanol at room temperature for 24 h. Detailed analysis of the eﬃciency of the DIB crosslinking and ozonolysis

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can be referred to elsewhere [27]. Morphology of the as-cast ﬁlm untreated with DIB was investigated under TEM with ultrathin sections stained with OsO4 . The observed images such as the wagon wheel pattern and the double wave pattern image conﬁrmed that the ﬁlms are deﬁnitely composed of large grains (of order of 10μms) of the double gyroid network structure, though the images are not shown here.

Fig. 9. Frequently observed FE-SEM images on freeze-fractured surfaces of ﬁlms obtained after the nanofabrication described in the text. Based on [27].

The ﬁlms subjected to the DIB crosslinking and the ozonolysis were freezefractured, and the fractured surfaces sputter-coated with platinum were observed under FE-SEM. Fig. 9 (part a and b) shows frequently observed EFSEM images [27]. Fig. 10(a) and (b) represent the numerically constructed computer graphics of 3d images of the texture cross-sectioned with the plane parallel to (100) and (110) plane, respectively [27]. The images illuminate topographical features of the texture around the cross-sectional planes. The 3d CG-SEM images shown in Fig. 10 and the FE-SEM images shown in Fig. 9 are quite similar, elucidating that the free-standing double gyroid texture is successfully obtained. With improved image quality the FE-SEM image would allow us to characterize topological features of the texture more precisely, which merits future studies. 2.2 Nanohybrids of Metal Nanoparticles and Polymers In this section we shall present various ways of creating hybrids of metal nanoparticles and polymers (hybrid #1 to #5). Hybrid Formation by Mixing BCP and Pd-BCP (Hybrid #1) Let us consider a self-assembly of mixtures of a small amount of Pd-(P2VPb-PI)’s, which are Pd nanoparticles coordinated and stabilized by P2VP-b-PI

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Fig. 10. 3d computer graphics of the free-standing double-gyroid texture crosssectioned with a plane parallel to (100) (a) and (110) plane (b). From [27].

bcps, and a large amount of P2VP-b-PI bcps (designated hereafter Tp-P2VPb-PI), which are free from Pd nanoparticles but which are a major component forming a microdomain as a template for an incorporation of Pd-(P2VPb-PI)’s. The amount of Pd-(P2VP-b-PI) has to be small in order to avoid macrophase separation between Pd-(P2VP-b-PI) and Tp-(P2VP-b-PI). It is important to note that the mixture involves generally the macrophase separation as well as the microphase separation [29]. We like to induce only the microphase separation so that one of the microdomains selectively incorporates Pd nanoparticles as a consequence of Pd-(P2VP-b-PI)’s being solubilized in the template microdomain formed by Tp-(P2VP-b-PI). We are concerned with a spatial distribution of Pd nanoparticles in the template microdomains. The template bcps Tp-(P2VP-b-PI) used in this section form by themselves alternating lamellae of P2VP block chains and PI block chains. The spatial distribution of Pd nanoparticles is intimately related to conﬁgurations of P2VP-b-PI incorporating Pd nanoparticles, i.e. Pd-(P2VP-b-PI). Pd nanoparticles obviously like P2VP more than PI as will be elucidated below. If Pd particles are heavily adsorbed by many P2VP block chains as illustrated in part a of Fig. 11, so that P2VP block chains are coating the particles and PI block chains emanate from the particles as brushes, then the Pd nanoparticles should be incorporated into PI lamellae rather than P2VP lamellae. In our experience this is not the case, at least for this system. Part (b) of Fig. 11 represents the case where the Pd nanoparticles adsorb only a small number of the bcp chains, e.g., only one chain here. The P2VP block segments are extensively adsorbed to the Pd nanoparticles so that the subchains of P2VP free from the particles are short and hence a PI block chain has a conﬁguration such that one end of the PI chain is ﬁxed to the Pd nanoparticle but the rest of it is free from the particle, because P2VP and PI chains repel each other. If this were the case, the particles would always

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align preferentially near the interface of P2VP lamellar nanodomains, simply because the PI block chains of Pd-(P2VP-b-PI) favors the PI lamellae and Pd coated by P2VP block chains may favor the P2VP lamellae. This is not the case found in our experiments either. If the Pd nanoparticles adsorb (i) only a small number of P2VP chains, (ii) only a small fraction of segments per single P2VP chains as shown in part (c), and (iii) furthermore uniformly any parts of the P2VP chains, at the free end, middle, and near the junction point as shown in part (d), then we anticipate that the Pd particles are arranged more or less uniformly in the P2VP domains.

Fig. 11. Possible conﬁguration of P2VP-b-PI block copolymer chains adsorbed to Pd nanoparicles. Based on [30]. Copyright (2004) Materials Research Society of Japan.

Fig. 12 shows a typical TEM image of solution-cast ﬁlm specimens of a mixture of Pd-(P2VP-b-PI) and Tp-(P2VP-b-PI), both having the same lamella-forming P2VP-b-PI. In the TEM image obtained on the unstained ultrathin sections (part (a)), the dark and bright lamellae are P2VP and PI lamellae, respectively, and the dark dots are Pd nanoparticles as sketched in part (b). Interestingly enough we found that Pd nanoparticles are more or less uniformly distributed across the lamella interface. This piece of evidence should infer that the models (c) and (d) in Fig. 11 are plausible models for conﬁgurations of P2VP block chain incorporating Pd particles. The model is also consistent with the AFM studies of visualization of isolated single Pd(P2VP-b-PI), as will be described below. Fig. 13 shows an AFM image and corresponding sketches for Pd-(P2VP-bPI), representing two chains with contour length of 380 and 400 nm adsorbed to a single Pd particle [31]. The bright object has a height of 6.1 nm and diameter of 21 nm. The height is very close to the average diameter (7 nm) of the Pd nanoparticles as observed under TEM [31, 32], suggesting that the object is a single Pd particle. The width is much larger than 6.1 nm, which is

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Fig. 12. Nanohybrids prepared through the self-assembly of a Pd-(P2VP-b-PI) / Tp-(P2VP-b-PI) mixture.

well-known to be due to a smearing eﬀect induced by the ﬁnite size of the AFM probe tip. The string like object observed around the particle is believed to be the adsorbed block chain(s). The height analysis of the string shows a height value of 0.2 nm, consistent with the single block chain. The counter length of 500 nm measured for the block chain adsorbed to the particles is consistent with that of 799 nm calculated from the molecular weight of P2VP-b-PI.

Fig. 13. Tapping-mode AFM image for Pd-(P2VP-b-PI) (left) and its sketch (right). The two bcp chains are adsorbed by the Pd particle. Reprinted from [31]. Copyright (2003) American Chemical Society.

Here it is worth noting that the mixture of Pd-P2VP (the Pd nanoparticles coordinated by P2VP homopolymer) and Tp-(P2VP-b-PI) was found to give selective incorporation of Pd nanoparticles in the middle of the P2VP lamellar

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microdomains formed by Tp- (P2VP-b-PI). Those requiring further details are referred to [32]. In order to control physically (i.e., to tailor) the location of the particles across the interface, we can use the physics of bcp mixtures. Part (a) in Fig. 14 exaggerates the chain conformation of a small amount of larger-molecularweight bcp mixed in the matrix of a large amount of smaller-molecular-weight bcps, while part (b) exaggerates the chain conformation of a small amount of smaller-molecular-weight bcps mixed in the matrix of a large amount of largermolecular-weight bcps. In both cases, we assume that chemical junctions of the short and long block chains share the common interfaces and the block chains are microphase-separated in their respective domains, hence the short and long bcps satisfying the so-called cosurfactant criterion [17, 29, 33–37]. If a part of the minority chains shown by the thick lines are coordinating Pd nanoparticles with equal probability along the chains as shown in Fig. 11(d), then the Pd particles will be localized preferentially in the middle or near the interface in the case (a) and (b) in Fig. 14, respectively.

Fig. 14. Lamellar microdomain structures formed by mixtures of short and long symmetric bcps. The long and short bcps are minority components in parts (a) and (b), respectively. Reprinted from [30]. Copyright (2004) Materials Research Society of Japan.

Fig. 15 shows a histogram P (z) for the location of Pd nanoparticles across the interface for the three systems as indicated in the ﬁgure, where z indicates the location of Pd nanoparticles (deﬁned by d) across the interface in the P2VP lamellae normalized by thicknesses of the P2VP lamellae (dP 2V P ), z ≡ d/dP 2V P . In the case when the P2VP blocks in Pd-(P2VP-b-PI) have a molecular weight much smaller than the matrix-forming P2VP blocks in Tp-(P2VP-b-PI), the Pd nanoparticles were found to be localized near the interface (part a). If the two P2VP blocks have the same molecular weights, the Pd nanoparticles were found to be more or less uniformly distributed

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(part b). More rigorous analyses with element spectroscopic imaging tend to show a weak bimodal distribution of Pd particles with small peaks at 0.15 and 0.85 [38]. If the P2VP blocks in Pd-(P2VP-b-PI) have a molecular weight much larger than the P2VP blocks in Tp-(P2VP-b-PI), the Pd nanoparticles were found to be localized in the middle of the P2VP lamellae (part c). Thus the models (c) and (d) in Fig. 11 are supported by these independent experiments. Moreover, the results shown in Fig. 15 are consistent with the AFM result shown in Fig. 13. Further details can be found in [39].

Fig. 15. Histograms showing position of Pd nanoparticles in P2VP lamellae in the solution-cast mixtures of Pd-(P2VP-b-PI) and Tp-(P2VP-b-PI). Pd-Xk-b-Yk / Zk-b-Wk in the ﬁgure legend designates weight-average molecular weights of P2VP and PI block chains in Pd-(P2VP-b-PI) being X and Y in units of thousands, respectively, and those of the P2VP and PI block chains in Tp-(P2VP-b-PI) being Z and W in units of thousands, respectively. Relative lengths of P2VP blocks coordinating Pd particles and those acting as microdomain templates are shown, respectively, by the relative length of the lines with and without the dark dot in the top section of the ﬁgure, respectively. The dark dots on the lines are designated Pd nanoparticles coordinated by the P2VP block chains. Reprinted from [30]. Copyright (2004) Materials Research Society of Japan.

In-situ Formation of Metal Nanoparticles in BCP Microdomains as Templates In this section we shall discuss hybrid formation via reduction of Pd ions, Pd(II), into Pd atoms, Pd(0), in the presence of bcp microdomains as templates. The reduction involves a selective incorporation of Pd nanoparticles in one of the domains (hybrids #2, #3 and #5 below) or on the interface between the two domains (hybrids #4 below).

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Self-assembly of Microdomain Templates and Pd Nanoparticles from bcp Solutions (Hybrid #2). We shall consider here a way to create nanohybrids by a single-step process as schematically illustrated in Fig. 16. In this case, P2VP-b-PI bcp (M n = 3.1 × 104 , volume fraction of P2VP block chain, φP 2V P = 0.4, Mw /Mn = 1.04 with Mw being weight average molecular weight) and palladium acetylacetonate, Pd(acac)2 , were dissolved into a dilute solution of benzyl alcohol and chloroform as a mixed solvent as shown in part (a). The solution is homogeneous and the bcps are in the disordered state. Chloroform was ﬁrst evaporated at room temperature to obtain a concentrated solution of the bcp in benzyl alcohol in which the lamellar microdomain structures swollen with benzyl alcohol were already self-organized but the Pd ions were not yet reduced and uniformly distributed in both PI and P2VP lamellae as shown in part (b). Although Pd(II) appear as if they are localized in one of the lamellae in the ﬁgure, this is actually not true. They are conﬁrmed equally partitioned in both P2VP and PI lamellae swollen with benzyl alcohol by means of small-angle X-ray scattering (SAXS) studies [40]. This is probably because Pd(II)’s are solvated, interacting more strongly with the solvent than the P2VP and PI blocks.

Fig. 16. A single step hybridization process of metal nanoparticles in bcp templates (hybrid #2). Based on [30]. Copyright (2004) Materials Research Society of Japan.

Then the temperature of the system was raised to 140˚C where Pd (II)’s were reduced into Pd(0)’s, the Pd(0)’s were aggregated into nanoparticles, and benzyl alcohol was simultaneously evaporated. In this way ﬁlms in which Pd nanoparticles were selectively incorporated in P2VP nanodomain structures were obtained, because of speciﬁc interactions of Pd nanoparticles with P2VP. Fig. 17 shows a typical TEM image of the microphase-separated structure

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observed for an unstained ultrathin section of obtained ﬁlms (part a) and its corresponding model (part b) conﬁrmed by the TEM image obtained with the ultrathin sections stained with osmium tetraoxide [40], though the image is not shown here. The dark dots in part (a) are Pd nanoparticles. They are aligned in a row with a spacing of about 40 nm perpendicular to the row. The image obtained with the OsO4 stained specimen indicated dark lamellae composed of selectively stained PI phase and bright lamellae composed of unstained P2VP phase. The TEM image shows that Pd nanoparticles were selectively localized in the middle part of bright P2VP lamellae [40], as schematically shown in the inset at upper left corner of Fig. 17(b). The inset enlarges the part of the domain enclosed by the square in Fig. 17(b).

Fig. 17. Microphase-separated structure containing Pd clusters obtained by the single-step hybridization process as schematically shown in Fig. 16. Based on [30]. Copyright (2004) Materials Research Society of Japan.

In this way, we can create hybrids of metal nanoparticles and bcp template where nanoparticles are aligned selectively in a row in one of the lamellae with a controlled spacing between the rows. The metal nanoparticles are driven towards the middle of the nanodomain due to the (rubber-like) elastic force of the P2VP chains emanating from the lamellar interface. We can create denser nanoparticles with increasing concentration of Pd(II)’s [40]. Incorporation of Pd Nanoparticles in Free-Standing Double-Gyroid Texture (Hybrid #3). Earlier in Sect. 2.1 we presented the nanofabrication method for preparation of the free-standing double-gyroid texture made out of crosslinked P2VP chains. We can incorporate Pd nanoparticles into this texture which can be regarded as a porous template having pore volume of 67%. The porous membrane containing Pd nanoparticles may be useful for applications such as membrane reactors [43–46]. The double-gyroid texture prepared as described previously (p. 54) was soaked in a warm bath (85◦ C) made from a 50ml mixture of

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1-proponal and toluene (25:75). This mixed solvent contained 0.3 grams of Pd(acac)2 which has been shown to be reduced to Pd atoms by 1-propanol [32]. The amount of Pd(acac)2 added to the solvents was intended to give an initial ion concentration in excess of the total number of pyridine units in the system which are thought to have attractive interactions with Pd atoms and particles. It is expected that this reduction process will occur not only within the P2VP texture but also within the bulk solution itself. The immersion time was controlled and varied from 1 h to 4 days as part of the experimental strategy. Additionally the toluene was replaced with other solvents; benzene,14 dioxane, and acetone in order to investigate eﬀects of the solvent used upon Pd nanoparticle growth. The microdomain structures of the samples prepared in this way were then examined by TEM. Ultrathin sections for TEM were obtained using a cryoultramicrotome. The size distribution of the Pd nanoparticles observed in the resulting micrographs was then analyzed by use of standard imaging software. This approach was endorsed by independent manual measurements on selected micrographs. Typically 300-400 particles were counted and measured within a 0.16μm2 area of the micrograph. Since the sampled area of the micrographs was constant, the number of particles found represents the particle number density, N. Fig. 18 (a) to (c) show representative TEM of the sample after reduction in a 1-propanol/toluene/Pd(acac)2 bath for a range of times [28]. The Pd particles, approximately spherical in shape, are visibly growing in size with the reduction time. Moreover in the ﬁnal stages of the time scale shown in Fig. 18(c) the particle number density, N, is decreasing. It also seems that the Pd particles are to be found in the P2VP phase (stained dark by DIB) and not in the void areas. The fact that this is not immediately apparent is a testament to the three dimensional nature of the double gyroid texture. Pd particles that may seem (from the micrographs) to be within the void phase (bright phase) of the micrographs are more likely to be found at a lower or upper overlapping domain of the P2VP gyroid network in the ultrathin section used for the TEM observation. However it is unequivocally proved that the particles reside only in the P2VP microdomain by conducting similar experiments within a lamellar microdomain morphology. Fig. 18(d) shows just such a micrograph, where the processing described above has been applied to the pure PI-b-P2VP polymer forming a lamellar morphology and this degraded structure together with the double gyroid network texture, has been immersed in the same Pd(acac)2 bath for 48 h in order to ensure the two structures were being subjected to the same reduction condition. It should be noted that the P2VP lamellae (stained dark by DIB) and the degraded PI lamellae (existing as a void space and hence appearing bright) are oriented with their interfaces parallel to the electron beam within the ultrathin sections prepared for the TEM observation so that two sets of lamellae are not overlapping. TEM images of the double-gyroid texture samples, subjected to the reducing reaction in the Pd(acac)2 bath, were collected in order to obtain a

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Fig. 18. TEM images of Pd-nanoparticle loaded in the free-standing gyroid texture in air matrix after reduction in 1-propanol/toluene/Pd(acac)2 for diﬀerent times at 85◦ C. (a) 13 h; (b) 24 h; (c) 48 h. The bright regions refer to the degraded PI domain (void space). The dark regions refer to the DIB stained P2VP phase. The darkest spots are the Pd nanoparticles. Part (d) represents TEM image of Pd-nanoparticle loaded into the free-standing P2VP lamellar microdomains in air. Pd nanoparticles are demonstrated to exist only in the dark P2VP lamellae. Reproduced from [28]. Copyright (2006) American Chemical Society.

quantitative measure of the nature of the particle growth mechanism. Particle analysis software allowed us to determine the population and size distribution as a function of time. Fitting the distributions with a normal Gaussian ¯ with time model, we can extract the growth of the mean particle diameter, D ¯ which goes as t1/6 (Fig. 19) [28]. There is a clear power low dependence for D (part a). At the same time we see that the number density, N , of particles appears to ﬁrst rise quickly as a function of time before decaying at a slower rate (part b). It is worthwhile considering the various possible processes that

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are at work during immersion of the polymer within the reducing bath. Both inside and outside of the swollen double-gyroid texture, Pd(II)’s are being reduced to Pd atoms, and these atoms are subsequently aggregating into larger particles.

¯ vs time for the Pd reduction in 1Fig. 19. Experimental D propanol/toluene/Pd(acac)2 at 85◦ C extracted from particle analysis (ﬁlled circles). Also shown are the power law predictions from the diﬀusion and coalescence model (without hydrodynamic interactions); full line; and the LSW theory; dashed line. (b) The number of particles in a 0.16 μm2 area of the TEM micrograph as a function of time. Reproduced from [28]. Copyright (2006) American Chemical Society.

The number of these particles that make their way into the swollen P2VP polymer network will be a complex function of several parameters that depend on the relative interactions between the Pd(II), the Pd atom, the solvent, and the P2VP texture. The picture is slightly simpler if we consider only Pd(II)s that have diﬀused and adsorbed into the P2VP texture. This process is shown schematically in Fig. 20. The adsorbed ions will be reduced to Pd atoms in the presence of 1-propanol in the swollen cross-linked texture. These atoms quickly aggregate to small Pd particles (process 1). There may be an exchange of ions between the inside and the outside of the network (process 2). The number of Pd particles will increase with time as a consequence of the reduction and aggregation. These particles will then coalesce within the texture into larger particles (process 3a & 3b). As a consequence the particle number tends to decrease with time. However the precise nature of this latter stage is subject to speculation. There are two feasible dominant modes of particle growth. The ﬁrst, often known as Ostwald ripening predicted by Lifschitz–Slyozov–Wagner (LSW) [47, 48], might occur by an evaporation/condensation mechanism of small Pd

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particles existing in the neighborhood of large Pd particles whereby Pd atoms from a smaller particle aggregate evaporate into the matrix, diﬀuse toward the large aggregate driven by the Gibbs–Thomson eﬀect, and condense to the large particle (process 3a). The alternative model would be that of diﬀusion and coalescences of particles into a larger aggregate [49, 50](process 3b). ¯ as tn . Both models are characterized by a power law time dependence for D, The LSW model predicts n = 1/3. The diﬀusion and coalescence model has n = 1/6 and 1/3, respectively, in the absence and presence of hydrodynamic interactions [51]. These predictions are plotted in Fig. 19(a), and it is clear that the diﬀusion and coalescence model without hydrodynamic interactions has a very good agreement.

Fig. 20. A schematic of the various processes that will occur during the immersion of the gyroid network into the Pd reducing bath. The three processes highlighted are discussed in the text. Reproduced from [28]. Copyright (2006) American Chemical Society.

The coarsening process of metal particles via either of the two models mentioned were originally proposed to occur in hot melts of binary metal alloys, e.g., Oki et al. [52]. Whether or not such coarsening of Pd nanoparticles occurs in the swollen polymer network at low temperatures is not self-evident. We should note that the state of Pd atoms changes from gaseous state to liquid state and eventually to solid state. The coarsening process described above should occur when the particles are still in liquid state. The coarsening of particles via absorption of gaseous atoms would occur as long as their surfaces are still in the liquid state. There are other caveats to the use of coarsening models, since the exponents are based on theory where the total number of Pd atoms is conserved

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during the coarsening process. Since there is a bath of Pd(II) surrounding the texture and the atoms can constantly be fed into the system via diﬀusion and reduction of Pd(II), this will not necessarily be the case. We can argue that there is a given concentration of Pd(II) inside and outside of the P2VP texture immediately before the reduction of Pd(II) starts to occur. There may be an exchange or interdiﬀusion of ions between the inside and outside of the texture during the reduction, driven by osmotic pressure of ions, but the total ion concentration subject to reduction inside and outside of the texture will not be altered by the exchange of ions. The reductions of Pd(II) inside and outside the texture are eﬀectively independent of each other, and the reduced atoms eﬀectively stay in the same region, i.e., either within the texture or outside of it. In this way a quasi-conserved system might exist. Such a notion is corroborated by the behavior of N as a function of time. The initially fast increase in particle number density, N , could refer to the reduction of Pd(II) to atoms and the aggregation of atoms into particles as shown by process 1 in Fig. 20. The subsequent slower decay of N suggests that the large number of small Pd particles thus created grow into a smaller number of larger Pd particles via process 3a or process 3b. It is prudent at this stage to only show that we can control the size and distribution of Pd particles dynamically. Such control will be critical in the potential application of such materials. There are other factors that can inﬂuence the nature of Pd particle growth in the matrix, such as the degree of cross-linking achieved with DIB, and the nature of co-solvent that is used with 1-propanol. For further details the reader is referred to [28]. Incorporation of Metal Nanoparticles on the Surface of the Voided Double-Gyroid Channel (Hybrid #4). In Sect. 2.1 we described creation of the voided double-gyroid-channel in the matrix of PS blocks. We can introduce metal plating of the surface of the PS matrix by using commercial reagents for the non-electrolytic plating [25]. The metal plating should be essentially equivalent to the incorporation of metal particles on the surface of the channel. First the specimens with the vacant nanochannels were soaked in a “sensitizer” (an aqueous solution of SnCl2 ) so that the surface adsorbs Sn2+ . The sensitized ﬁlms were then soaked in an “activator” (an aqueous solution of PdCl2 ) so that the surface adsorbs Pd0 through the reaction Sn2+ + Pd2+ → Sn4+ + Pd0 . Finally the activated ﬁlms were soaked in a 1:4 mixture of “nickel A” (an aqueous solution of NiCl2 ) and “nickel B” (an aqueous solution of reductant) for the nickel plating. Pd0 was used as the catalyst for the reduction of Ni2+ . All the commercial reagents were diluted with water to avoid rapid clogging of the vacant nanochannel with the reduced nickel metal. Since the matrix phase of PS is not swollen with the aqueous medium, we expect that the reduced metal particles are preferentially localized in the interface between the voided nanochannel and the PS matrix.

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Fig. 21(a) shows a typical TEM image of the ultrathin section of the nickel plated nanochannel specimen obtained without staining. The nickel metal plated on the nanochannel surface gives a dark gray contrast in the TEM image. The PS matrix free from the metal plating appears bright and the vacant nanochannel with its surface plated with metal appears gray and dark. The bright PS matrix phase has the double wave pattern (with a series of dark circles along the waves as highlighted with a visual guide in Fig. 21(b)), though slightly distorted. The dark circles correspond to a part of the nanochannel oriented perpendicular to the thin section. The special pattern reﬂects the metal-plated nanochannel having the double gyroid network structure. The inset to Fig. 21(a) or (b) shows an enlarged image of the dark circles. The image highlights a deposition of metal particles on the channel surface, leaving an empty space in the center of the channels.

Fig. 21. (a) TEM image of nickel-plated voided double-gyroid nanochannel and (b) its visual guide for the double wave pattern. Based on [30].

Incorporation of Pd Nanoparticles in Thin BCP Films (Hybrid #5). In Sect. 3.3 later, we shall discuss control of lamellar microdomain orientation in spin-cast poly(styrene)-block -poly(methyl methacrylate) (PS-bPMMA) thin ﬁlms. We found that the perpendicular or parallel lamellae can be obtained on substrates having reduced roughness parameter qs Rs larger or smaller than a critical value (qs Rs )c , respectively. Here qs and Rs are the characteristic wave number and root-mean squared amplitude of the surface roughness of the substrate, respectively. We can incorporate metal nanoparticles selectively into one of the lamellar nanodomains by reducing metal ions

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Fig. 22. Selective incorporation of Pd nanoparticles into thin ﬁlms of PS-b-PMMA having perpendicular lamellae by sublimation of Pd(acac)2 complex at 200◦ C into the ﬁlms and in-situ selective reduction in PS lamellae. Based on [54].

in the microdomain template. There are two methods of creating hybrids of metal nanoparticles in thin bcp ﬁlms. The methods involve thermal reduction of metal ions into metal nanoparticles in the thin ﬁlm templates. We designate here this type of hybrid as hybrid #5. One method involves spin-casting of PS-b-PMMA bcp solutions containing Pd(acac)2 into thin ﬁlms on rough or smooth substrates and then heating of the thin ﬁlms for reduction of Pd(acac)2 . The selective interactions of Pd(acac)2 to one of the domains in the as-spun ﬁlms and/or the diﬀerence in the reduction rate in the two types of domain determine the selectivity in the incorporation of Pd nanoparticles into one of the domains. Another method involves spin-casting of thin ﬁlms of PS-b-PMMA free from Pd(acac)2 and then sublimation of unreduced Pd(acac)2 complex into the spun-cast thin ﬁlms where selective reduction occurs in-situ [53], as schematically shown in Fig. 22(a). Figs. 22 (b) and (c) show typical hybrids prepared on rough substrates of indium tin oxide, as observed by plan (through) view TEM and tapping mode AFM, respectively. The TEM image (b) indicates the perpendicular lamellae of PMMA (bright regions) and of PS (dark regions). The dark dots

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representing Pd nanoparticles are selectively localized into PS lamellae, presumably because the reduction rate is much faster in PS lamellae than in PMMA lamellae. Origin of the dark and bright contrast on the PS and PMMA lamellae is due to a selective abrasion of PMMA lamellae by electron beams. The AFM image in part (c) shows the surface structure on the same domain structure as observed in TEM. It is particularly interesting to note that the bright dots in AFM correspond to Pd nanoparticles which locate on the top surface of the thin ﬁlms. The images (b) and (c) indicate that the particles exist on the surface of the thin ﬁlms as well as in their interior. The hybrids are believed to have important applications as patterned media with a large magnetic memory density.

3 Thin Film Block Copolymer 3.1 Surface-Induced Alignment and Thickness Quantization As mentioned in Sect. 1, various very intriguing nano-structures have been found in bulk bcps. As well as in the bulk bcps, there is a growing interest in thin ﬁlm bcps [55]. Thin ﬁlms typically have a thickness less than about ten times of the microdomain spacing of the constituting bcp. In such thin ﬁlms, as described in Sect. 1.2, the microphase-separated structures within the thin ﬁlms are much more enriched than those in the bulk. The bcp component preferentially attractive to one surface develops the microdomains on this surface [56, 57]. In this section we shall consider lamella-forming bcps in bulk. In the case of a lamella-forming dibcp, the boundary surface gives rise to a lamella of the one component preferred by the surface. Thus the non-neutral surface generally generates a parallel lamellae structure (lamellar microdomains with their lamellar interfaces parallel to the substrate surface) throughout the entire ﬁlm (called surface-induced alignment of microdomains) [58]. In this parallel lamellae structure, the ﬁlm thickness should be restricted to some quantized values of the lamella spacing. When the substrate surface and the air surface prefer diﬀerent components in the diblock copolymer (asymmetric boundary condition), the ﬁlm thickness becomes a half-integer (n + 1/2) multiple of the lamella spacing d. When the initial ﬁlm thickness is not commensurate with this quantization, lateral mass transport in the plane of the thin ﬁlm is induced so that the local ﬁlm thickness is adjusted to some quantized value (thickness quantization) and thereby the parallel lamellae are established in the entire ﬁlm [59]. The substrate generally leads the parallel lamella formation, which is well known to be due to short range interactions. Recent studies, however, revealed some new aspects on the eﬀects of the short-range interactions with the substrate: The short-range interactions of bcps with the substrates cause not only parallel lamella formation but also long range eﬀects on microdomain

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formation in thin ﬁlms. We shall highlight the long range eﬀects of the substrate on the non-equilibrium microdomain formation process. A rich variety of the long-range eﬀects arising from the substrate will be shown below. 3.2 Substrate Eﬀect on Parallel Lamellae Formation For the lamella-forming bcp, the substrate surface is believed to determine which lamella constitutes the ﬁrst layer on the substrate. However, there can be cases in which the morphology most preferred by the substrate surface is not the lamellar structure. An example was found on the substrate which has a very strong attractive interaction with one of the blocks in the lamella-forming bcp (case A). In this case, the conformations of the preferred block chains on the surface are considerably perturbed compared with those in the bulk, and thereby the substrate induces a perturbation of the lamella structure near the substrate [60]. Another example is the case of thin ﬁlms of a lamella-forming triblock terpolymer on the substrate which is preferential to the middle block of the terpolymer (case B). In this case, the parallel lamella alignment is not expected near the substrate surface [61, 62]. In the above examples, the substrate induces a special structure which is never observed in bulk bcp. The problem was found to be most highlighted in the non-equilibrium process of bcp thin ﬁlms. First, the non-equilibrium process will be presented in dibcp thin ﬁlms on a substrate which has strong interactions to one block component (case A). Later the non-equilibrium process observed for the triblock terpolymer thin ﬁlms on the substrate which preferentially adsorbs the middle block (case B) will be presented. Diblock Copolymer Thin Films on Substrate Which Strongly Adsorbs One Block Component As-Spun State Thin ﬁlms of polyisoprene-block -poly(2-vinylpyridine) dibcps (PI-b-P2VP) (Mn = 67.9 × 103 , Mw /Mn = 1.31, volume fraction of PI φP I = 0.49, and bulk showing lamellar morphology with the lamella spacing D = 44 nm) were prepared by spin-cast from a solution in chloroform on hydrogen-terminated silicon (SiH) [60]. The initial ﬁlm thickness z0 was about 25 nm, which is nearly equal to the half lamella thickness in the bulk. P2VP is preferentially adsorbed by the SiH substrate. On the other hand, PI has a lower surface free energy than P2VP and thereby tends to segregate toward the free surface. The thin ﬁlm having half-lamella thickness is anticipated to form a monolayer of bcp brushes comprising PI-half-lamella on the free surface and P2VP-halflamella on the substrate surface, respectively, at thermal equilibrium.

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The as-spun ﬁlm surface was explored with a Tapping Mode AFM [63]. The AFM height image (Fig. 23a) showed the objects slightly protruding (∼ 1 nm) from the surrounding area. They seem to form the close-packed hexagonal structure. In the AFM phase image of the same area (Fig. 23b), the protrusions appear as bright phase. Since the glassy (rubbery) phase appears bright (dark) in the phase image, the protrusion and the matrix are considered to be P2VP and PI microdomains, respectively. The glass transition temperature of P2VP is much higher than that of PI, and thereby the P2VP is vitriﬁed earlier than the PI as the solvent evaporates during the spincast. After the vitriﬁcation of P2VP domains, the PI matrix further shrinks as the residual solvent evaporates, and hence it is depressed relative to the P2VP.

Fig. 23. AFM height (a) and phase images (b) of the as-spun PI-b-P2VP thin ﬁlm of 25 nm thickness. The gray level of part a and b represents a height range of 8 nm and the phase shift range of 10 deg, respectively. Power spectral density of the phase image acquired in extended area (2μ × 2μ) is shown in the inset to part (b). The structure observed in the area indicated by the white circle in part a and b is schematically drawn in part c. In part c, the bright phase corresponds to P2VP domains protruded in part a and appearing harder in part b. Reproduced from [60]. Copyright (2006) American Chemical Society.

Formation of the laterally microphase-separated structure, rather than the parallel lamellae which are laterally uniform, may be a consequence of the strong interactions of P2VP block chains with the substrate. In order to clarify the inﬂuences of the substrate, it is very instructive to explore the structure

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of the ﬁrst layer on top of the substrate surface. The strong adsorption of the thin ﬁlms to the substrate makes it diﬃcult to remove the thin ﬁlms from the substrate for the preparation of TEM specimens. On the other hand, due to this strong adsorption, the PI-b-P2VP monomolecular layer ﬁlm on the substrate, which should resemble the ﬁrst layer structure developed in the half-lamella thickness ﬁlms, can be obtained by simply soaking the PI-b-P2VP thin ﬁlms on the SiH substrate as a whole in solvent [60]. The AFM height image of the obtained PI-b-P2VP monomolecular ﬁlm is shown in Fig. 24a. This ﬁlm surface shows small protrusions appearing bright, and the protrusions exhibit a short-range spatial order. The protrusions are considered to be condensed globules of PI bound to the monomolecular layer of P2VP preferentially adsorbed to the substrate surface. Part (b) shows the binarized AFM height image in order to highlight the spatial distribution of the PI globules. The binarization threshold was chosen so that the areas occupied by the PI globules appear white. The PI globules indicated by the round markers seem to form hexagons as speciﬁed by the solid white line (Fig. 24b). A schematic model of this monomolecular ﬁlm is shown in part c. Although the spatial distribution of the PI globules have not attained a well-deﬁned long-range order, they seem to show a locally ordered honeycomb-like pattern as schematically indicated by the broken line. The unit cell of this structure is indicated by the solid line in part (c) and is reproduced in part (d) together with the PI globules marked by the open circles. The length L0 of the unit cell deﬁned in part c and d is about 63nm as determined from the AFM images. This value is larger than the bulk lamellar spacing. Now let us recall the as-spun ﬁlm structure as already shown in Fig. 23; P2VP domains form the close-packed hexagonal structure. The hexagon resembles the grid of the PI globules on top of the P2VP monomolecular ﬁlm (Fig. 24). The spatial distribution of the PI globules is determined by the conﬁgurations of the P2VP block chains adsorbed to the SiH surface, as schematically shown in Fig. 25 (see the layer α). The conﬁgurational changes have a high kinetic barrier due to the strong interactions between the P2VP and the SiH and hence due to the stiﬀness of the P2VP layer. In the PI-b-P2VP half-lamella thickness ﬁlms too, the spatial distribution of the PI globules on top of the ﬁrst P2VP layer (layer α) is believed to be almost the same as that observed in the monomolecular ﬁlm. Therefore the following model (Fig. 25) may be considered for the structure of the as-spun thin ﬁlms. The condensed PI globules G form on top of the physisorbed P2VP layer (the layer α in Fig. 25). Their spatial distribution on top of this ﬁrst P2VP layer shows a short-range order with the unit cell as represented in Fig. 24d. In the second or higher molecular layer (the layer β in Fig. 25b), the P2VP chains are expelled from the positions occupied by the PI globules in the ﬁrst molecular layer in order to minimize the unfavorable contact between PI and P2VP. As a result, the P2VP chains segregate into the complementary space in the second or higher layer with respect to the PI globule (G) (see Fig. 25b).

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Fig. 24. AFM height image of the monomolecular ﬁlm surface (Part a). The image in part b is a zoom-in of the area indicated by dotted square in part a and is binarized such a way that the PI globules appear bright. Some of the PI globules are highlighted by the round markers in part b. Part c represents a schematic model for the structure of the monomolecular ﬁlm which is composed of PI globules on the top of the P2VP layer adsorbed to SiH. Part d shows the unit cell (solid line) for the locally ordered PI globules on the monomolecular ﬁlm surface. Open circles represent the positions of PI globules. Part e shows the unit cell of the as-spun thin ﬁlm surface. Filled circles represent the P2VP domains whose position vectors are shown by u1 and u2 . L1 and L0 are anticipated to be identical and are actually found to be almost identical. Reproduced from [60]. Copyright (2006) American Chemical Society.

These positions for the centers of the P2VP domains, together with the unit cell of the PI globules array in the ﬁrst molecular layer, are schematically indicated by ﬁlled circles in Fig. 24e. The centers of the P2VP domains are located in the vertexes of the unit cell of the PI globule array and thereby forms the close-packed hexagonal structure. In this model, the spacing L1 of the P2VP domains, which perforate the PI matrix and appear on the free surface, has to be almost identical to L0 (the unit cell size of the PI globule arrays, see Fig. 24c and d) on top of the ﬁrst layer α. Indeed, the observed

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Fig. 25. Schematic model of the as-spun thin ﬁlm structure (Part a). Part b displays a cross-sectional view of the model cut through the plane including the broken line and normal to the substrate shown in part a, which is parallel to vector u1 in Fig. 24e. In part b, α and β schematically designate the P2VP monomolecular layer and the second layer comprising PI and P2VP blocks on the top of the ﬁrst molecular layer, respectively. Although the PI globules (G) are not in this cross-sectional plane, their projection on the same plane is represented. L0 and L1 represent the characteristic length for the unit cell of the PI globules and that of the P2VP domains deﬁned in Fig. 24d and Fig. 24e, respectively. L1 ≈ L0 ≈ 63 nm was found experimentally in this system. Based on [60].

spacing L1 of the P2VP domains in the half-lamella thickness ﬁlm is the same as the observed L0 of the PI globules in the monomolecular ﬁlm. The P2VP domains observed in the as-spun ﬁlm of the half-lamella thickness may be considered as the P2VP cylinders standing up with respect to the special layer α in Fig. 25. Non-Equilibrium Pathway Toward Monomolecular Brush Layer Parallel to Substrate. Upon thermal treatment, the kinetically trapped non-equilibrium structure in the as-spun thin ﬁlms starts transforming into the equilibrium morphology. The as-spun ﬁlms of PI-b-P2VP in the half-lamella thickness were annealed at 210˚C in vacuum. After this heat treatment for one day, the P2VP protrusions on the free surface seen in Fig. 23 disappeared [60]. This indicates a transformation of the microphase-separated structure into a new structure. The thin ﬁlm structure was investigated by AFM after a stepwise removal of thin layers of polymer from the top of the thin ﬁlms, since the strongly

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adsorbing rigid substrate inhibited the microtoming of the thin ﬁlms necessary for the cross-sectional TEM observation [56, 57]. We concluded that the PI-b-P2VP surface can successfully be ablated by the UV-O3 treatment detailed elsewhere [60]. This method was applied to the samples after the heat treatment for one day and three days, respectively (Fig. 26a and b). After the UV-O3 treatment for various periods, the AFM height images were taken at the same lateral position of the sample. For the sample after the heat treatment for one day (part a), after removal of about 5nm of polymer from the sample surface (ex. image a1), the surface becomes non-uniform and shows protrusions with a short-range order in the AFM images. As the removed thickness, deﬁned by Δz, increases to more than 10 nm, the short-range ordered morphology becomes more signiﬁcant (see images a3 to a5). The surface, however, becomes again featureless when Δz exceeds 20 nm (ex. image a6).

Fig. 26. AFM height images of the PI-b-P2VP thin ﬁlm after the UV-O3 treatment for various etching thickness. For each of the AFM images, the residual thickness after the etching is indicated by the solid lines, respectively, in the schematic drawing of the thin ﬁlm shown in the center. The sample annealed at 210◦ C for one day and for three days prior to the UV-O3 etching procedure is shown in the left-hand part and in the right-hand part, respectively. The scale bar shown in the bottom left image applies to all the images. The shaded portion shown in the schematic drawing in part (c) represents the thickness range in which the short-range ordered morphology is observed on the surface after the UV-O3 treatment. Based on [60].

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In the case of heat treatment for three days (part b), the change in the surface topography along the progress of the UV-O3 etching shows a trend similar to what is shown in part (a): the initially featureless surface (image b1) turns to a corrugated surface with short-range order (image b2 and b3) and becomes featureless again (image b4 and b5). However, the thickness range in which short-range ordered structure is detected is reduced signiﬁcantly as the annealing time increases: 6 nm ≤ Δz ≤ 20 nm for the sample after heat treatment for one day; 8 nm ≤ Δz ≤ 14 nm for the sample after heat treatment for 3 days. The disappearance of the surface corrugation seems to indicate formation of a laterally homogeneous layer on top of the thin ﬁlm (see Fig. 27c). Since the surface energy of PI is lower than that of P2VP, the PI chains begin to cover the free surface of the thin ﬁlm via diﬀusion of junction points between P2VP and PI along the interface as shown in Fig. 27b. Eventually a thin layer of PI covers the entire top of the P2VP domains. However, the microdomain interfaces perpendicular to the ﬁlm surface still exist in the middle of the thin ﬁlm. When the etched surface intersects the microdomain interfaces perpendicular to the ﬁlm surface, the etched surface shows a laterally inhomogeneous topography. The reduction in the Δz range, in which the inhomogeneous surface topography is detected, indicates that the amplitude of the undulations of microdomain interfaces between PI and P2VP in the plane normal to the ﬁlm surface decreases with extension of the heat treatment. Noting that the half-lamella thickness ﬁlm of the PI-b-P2VP will most likely develop the parallel lamellae or bcp monolayer with the microdomain interface parallel to the substrate and air surface (PI-half-lamella on the free surface and P2VP-half-lamella on the substrate surface) in thermal equilibrium, the equilibration mechanism or process can be understood as shown schematically in Fig. 27. Through heat treatments for a prolonged time, the non-equilibrium microdomain structure as shown in Fig. 25 and in Fig. 27a is transformed to the parallel lamellae by way of decreasing the area of the microdomain interface perpendicular to the ﬁlm surface. The process involves translational diﬀusion of chemical junctions of the bcps along the interface oriented perpendicular to the substrate surface toward the interfaces parallel to the substrate surface as shown by the gray and white arrows in Fig. 27b. This diﬀusion creates new PI layers in the free surface side and new P2VP layers on the top of the ﬁrst layer α deﬁned in Fig. 25. Since the surface area parallel to the substrate does not change with the heat treatments, the transformation process tends to decrease total interfacial area and hence stretches the block chains normal to the interface, which is a natural trend for the equilibration process.

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Fig. 27. This schematic diagram visualizes the transformation mechanism of the PI-b-P2VP thin ﬁlm structure during heat treatment; (a) as-spun state, (b) intermediate state, and (c) the equilibrium state. The black dot on the microdomain interfaces schematically represents the junction point between the PI block and the P2VP block. After heat treatment, some of the junction points on the microdomain interfaces perpendicular to the substrate are moved on the interface parallel to and close to the substrate surface as indicated by the brighter arrows, and some of the junction points are moved on the interface parallel to and close to the free surface as indicated by the dark arrows. A depression of the parallel interface in the air side by the height d1 is compensated by an elevation of the parallel interface (in the substrate side) by the height d2 in such a way that volume conservation is satisﬁed. Eventually, the microdomain interfaces perpendicular to the ﬁlm surface that have developed in the as-spun ﬁlm are transformed into the interface parallel to the ﬁlm surface. Reproduced from [60]. Copyright (2006) American Chemical Society.

Triblock Terpolymer Thin Films on Substrate Preferentially Adsorbing Middle Block In the above section, the interactions of the substrate and the bcps were shown to force the bcps to reconstruct the microphase-separated structure under the particular boundary condition set by the interactions between polymers and substrates and thereby form some special structure on top of the substrate. The results were presented in the context of the lamella-forming dibcp thin ﬁlms. The substrate strongly aﬀects the conformation of the chains adsorbed to the substrate, and the special structure formed on the substrate has signiﬁcant inﬂuences on the non-equilibrium process in the thin ﬁlms. This substrate

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eﬀect appears to have a long-range eﬀect on the entire thin ﬁlm structure. This eﬀect is more clearly and explicitly highlighted in thin ﬁlms of ABC triblock terpolymers. In the case of ABC triblock terpolymers in bulk, the short-range interactions compete with each other among A, B, and C and so do the long-range interactions between A-B and B-C connectivities [17]. These enriched competitions give rise to an extremely rich variety of morphologies in bulk [17, 64]. Let us consider thin ﬁlms of ABC triblock terpolymers which form a three phase co-existing lamella morphology in bulk. In thin ﬁlms one should take into account short-range interactions with air and substrate surface and eﬀects of a conﬁned space on the long-range interactions as well. When the substrate preferentially adsorbs the middle block B, the lamellae expected for bulk have to be reconstructed to allow the B block chains to adhere on the substrate. As will be shown later, this constraint further enriches the microphase-separation in the thin ﬁlms, and clearly represents the longrange substrate eﬀect which has been partly shown in the previous section for the dibcp thin ﬁlms. As-Prepared State Thin ﬁlms of lamella-forming polystyrene-block -poly(2-vinylpyridine)-block poly(tert-butylmethacrylate) triblock copolymers, denoted as SVT terpolymer, where S, V, and T respectively denote polystyrene, poly(2-vinylpyridine), and poly(tert-butylmethacrylate), on SiOx or polyimide substrate were studied [62,65,66]. T block is the component having the lowest surface tension and is thereby expected to segregate to the free surface in the ﬁlm. Moreover this segregation gives rise to a parallel lamellae morphology with T lamella on the free surface. On the other hand, both the SiOx and the polyimide substrate preferentially adsorb the middle V block. The thin ﬁlms were prepared by using the dipping method with a solution of the SVT in THF as a common solvent. Fig. 28 a and b show the cross-sectional TEM image taken for the as-spun ﬁlm on the SiOx and polyimide substrate, respectively. In the TEM images, S, V, and T domains appear bright, dark, and gray, respectively (the samples were stained by OsO4 vapor). Although the preparation procedure is the same for both substrates, the ﬁlm thickness varies depending on the substrate (260 nm on SiOx , 200 nm on polyimide). This indicates the diﬀerence in the interactions between the substrate and the triblock terpolymer. The as-spun thin ﬁlm on the SiOx substrate (part a) shows “sponge-like” structures in phase-separated polymer blends [67] with a short-range spatial order. Three microphases of S (bright), V (dark), and T (gray) appear to be continuous in space. The characteristic spacing of the sponge-like domains (e.g. spacing of the bright domains) is about 71 nm. The free surface of this thin ﬁlm exhibits the undulation (not shown here) with a characteristic length equal to the aforementioned spacing of the sponge-like domains. On the other hand, the as-spun thin ﬁlm on the polyimide substrate (part b) shows a texture of dark domains within a gray matrix. Although three

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Fig. 28. Cross-sectional TEM image of the as-spun ﬁlm on SiOx (a) and polyimide (b). A zoom-in TEM image of the as-spun ﬁlm on SiOx is shown in the right-hand part of part (a). A schematic model of the microphase-separated structure of the as-spun ﬁlm on polyimide is shown in part (b). Based on [65, 66].

diﬀerent microphases are not obviously distinguished, this structure can be understood as a core-shell cylinder model as schematically shown in part (b). In this model, T block forms the gray matrix, and S and V blocks are segregated to the core and shell, respectively, which appear as dark domains. The TEM contrast expected for this model coincided well with the observed contrast across the dark domain [66]. The core-shell cylinders are separated by a characteristic spacing of about 70 nm. The characteristic spacing of the microphase-separated morphology developed in the “bulk” of the as-spun thin ﬁlms (middle portion of the as-spun thin ﬁlms) seems identical for the two diﬀerent substrates. However, the morphology itself seems somewhat diﬀerent, depending on the nature of the substrates, despite the same selectivity of interactions between these substrates and the block components.

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Self-Assembly into Parallel Lamellae In the case of the SVT triblock terpolymers, thermal annealing does not provide a successful route for the system to achieve thermodynamic equilibrium. This is partly due to the high viscosity of the terpolymer melts and the rather narrow temperature window between the highest glass transition and the lowest thermal degradation temperature of the respective blocks. This is also due to a large kinetic energy barrier for a complex metastable structure formed during sample preparation processes to overcome in order to achieve its equilibrium structure. Therefore, swelling the terpolymer thin ﬁlm with non-selective solvents, followed by drying of the thin ﬁlm, is a method applied to the sample as an eﬀective alternative route for “equilibration” [68–70]. Fig. 29 shows a series of cross-sectional TEM images of the SVT thin ﬁlms on the SiOx substrate (part a) and those on the polyimide substrate (part b) as a function of time of the solvent vapor treatment. Fig. 29a-1 to a-3 shows the surface-induced formation of the lamellar microdomains like the case of thermally equilibrated thin ﬁlms of symmetric dibcps [71]. After vapor exposure for 1 min (Fig. 29a-3), the lamellae are uniformly developed with their interfaces parallel to the surface of the thin ﬁlm (deﬁned as the “ﬁrststage” ordering process). After a further prolonged solvent-vapor treatment (Fig. 29a-4), a thickening of the lamellar microdomains was observed (deﬁned as the “second-stage” ordering process). This second-stage ordering process is accompanied by macroscopic dewetting of the SVT thin ﬁlms from the SiOx substrate (Fig. 29a-5). This dewetting process leaves the monomolecular ﬁlm strongly adsorbed to the SiOx substrate in the dewetted area indicated by B in Fig. 29a-5. On the other hand, the self-assembly process of the SVT lamellae on the polyimide substrate is very diﬀerent from what is observed on the SiOx substrate. The fragmented dark domains in the as-prepared ﬁlm (Fig. 29b-1) tend to connect to each other, leading to a more continuous structure after the short solvent-vapor treatment (for 5 s, not shown here). After the solvent treatment for 10 s, the dark domain completely covers the substrate surface (Fig. 29b-2). Another laterally continuous dark domain (indicated by L in Fig. 29b-2) has developed on the air surface side from the initially discontinuous domains. Detailed analysis of this structure showed that the structure shown in part (b-2) is a mesh-like structure (see Figure 3b in [66], where the dark cylinders (C) perforate the gray matrix, thereby bridging between the dark layer (L) and the dark layer on the substrate). The intermediate structure in part (b-2) eventually transformed into a lamellar structure oriented parallel to the ﬁlm surface after the solvent treatment for 1 min (Fig. 29b-3). After a further prolonged solvent-vapor treatment, the ordering of the lamellae is somewhat improved and all three phases become clearly distinguished in the TEM image (Fig. 29b-4). However, the spacing of the lamellae does not change, contrary to the case of the SiOx substrate. We therefore conclude that on the polyimide substrate the equilib-

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Fig. 29. (a) Cross-sectional TEM images showing the time-evolution of selfassembly in the thin SVT triblock terpolymer ﬁlm on SiOx : as-prepared (a-1), after the THF-vapor treatment for 5 s (a-2), 1 min (a-3), and 3 days (a-4). Part (a5) represents AFM height images of the ﬁlm after vapor treatment for 3 days. (b) Cross-sectional TEM images showing the time-evolution of self- assembly in the thin SVT triblock terpolymer ﬁlm on polyimide: as-prepared (b-1), after the THF-vapor treatment for 10 s (b-2), 1 min (b-3), and 1 day (b-4). Part (b-5) represents AFM height images of the ﬁlm after vapor treatment for 1 day. The black scale bar in (a-1) and (b-1) represents 100 nm and is common for all the TEM images. The triblock terpolymer ﬁlm are in the portion P. In the right part of (a-4) and (b-4), the lamellae structure is represented schematically. Part (a-5) and (b-5) shows the SVT that dewets on both substrates after the long solvent treatment. Based on [65, 66].

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rium lamellae are formed in a one-step ordering process (no lamella thickening process). This reveals a remarkable diﬀerence to the case of SiOx substrates, where the equilibrium lamellae formed in the two-step ordering process. First Layer on Substrate Both SiOx and polyimide preferentially adsorb V component in the SVT terpolymer. However, the strength of the interaction of V is diﬀerent in these two substrates. In the case of the SiOx substrate, soaking of the SVT thin ﬁlm samples in THF left a monomolecular ﬁlm of the SVT terpolymer, whereas, in the case of the polyimide substrate, the same procedure completely removed the thin ﬁlms. It indicates that the SiOx substrate has stronger attractive interactions with the V block in the SVT terpolymer than the polyimide substrate. This diﬀerence in the interaction strength should be responsible for the observed diﬀerence in the non-equilibrium process. Like the PI-b-P2VP thin ﬁlms presented in Sect. 3.2, the ultrathin ﬁlms of the SVT terpolymer were explored to understand the substrate eﬀect. The ultrathin ﬁlms on SiOx were obtained by soaking the SVT terpolymer thin ﬁlms on SiOx in the solvent, and its structure was unchanged after any heat treatment and/or solvent treatment. The ultrathin ﬁlms on polyimide were prepared by partly dissolving the 200 nm thick SVT terpolymer thin ﬁlms by THF [66]. The thickness of the remaining ﬁlms ranged between 10 and 30 nm. In the case of the polyimide substrate, the ﬁlms thus prepared will be referred to as ultrathin ﬁlms. As seen in Fig. 30a, the as-prepared ultrathin ﬁlm forms isolated V microdomains in a matrix of the other two blocks as schematically represented by black particles in the right-hand part. On the solvent-vapor treatment, the V blocks ﬁrst form a layer completely covering the polyimide surface (Fig. 30b), indicating that the V chains are selectively attracted by the polyimide. Subsequently, as schematically shown in the right-hand side of Fig. 30b, S and T blocks microphase-separate on top of the V layer. This ultrathin ﬁlm is supposed to be a monomolecular ﬁlm. Figure 31 shows the structure of the monomolecular SVT ﬁlm on the polyimide (a-1) after the vapor treatment and on the SiOx substrate (b-1), respectively. On the polyimide substrate (Fig. 31a-1), S and T blocks form periodic phase-separated domains across their common interfaces on top of the V layer as schematically shown in part (a-3). The height proﬁle shown in part (a-2) indicates areas of two diﬀerent height levels (A, and B). After the solvent-vapor treatment, S domains are vitriﬁed later than T domains and thereby shrink more, since the solvent is preferential to S [61]. Furthermore, the radius of gyration of the T block is larger than that of the S block. Hence, in the monomolecular ﬁlm, the T block is expected to be more protruded than the S block on top of the V layer, and the higher (A) and lower area (B) are therefore assumed to correspond to T and S microdomains, respectively. Part (a-3) displays schematically the chain conﬁguration of the SVT on the

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Fig. 30. Cross-sectional TEM images showing the time-evolution of self-assembly in an ultrathin SVT ﬁlm on the polyimide substrate: as-prepared (a), after THFvapor treatment for 1 min (b). The scale bar, common for all images, is shown in the bottom of the ﬁgure. As a visual-guide, the anticipated structures are schematically depicted on the right-hand side. In the schematic diagrams, the embedding resin, the substrate, the matrix comprising PS and PtBMA, and P2VP are represented by the dotted area, slashed area, gray area, and black area, respectively. In part (b), PS domains are explicitly depicted by the bright area in order to distinguish them from the PtBMA (gray) matrix [66].

polyimide substrate in order to highlight a comparison with that on the SiOx substrate (part b-3) to be discussed below. On the SiOx substrate, the SFM height image (part b-1) shows areas of three diﬀerent height levels (A, B, and C). As in the case of the polyimide substrate, V blocks form the ﬁrst layer on the SiOx substrate (part b-3). On the top of this V layer, the S and T blocks form microdomains, which do not seem to have common interfaces. Hence the height of the lowest level (C) in part (b-2) is expected to correspond to the top of the bare V layer. Noting that the gyration radius of the T block is larger than that of the S block, it is expected that the more protruded area A and the less protruded area B correspond to the T and S microdomains, respectively. The expected chain conﬁguration is schematically depicted in part (b-3). From Fig. 30b, the thickness of the V layer (dV ) is estimated to be around 6 nm for the case

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of the polyimide substrates. This value is considerably larger than the one observed on SiOx substrates (the height of the level C in Fig. 31b-2 being only 1.3 nm). This ﬁnding indicates that on the polyimide the V block is less collapsed toward the substrate and less expanded parallel to the substrate compared to the situation on SiOx . The lateral spacing L of the condensed domains comprising S and T end blocks is smaller on polyimide substrates than on SiOx substrates. This means that the area density of junction points on top of the V layer is higher on polyimide than on SiOx . Therefore, the S and T domains are expected to be more extended in the direction normal to the V layer on polyimide than on SiOx .

Fig. 31. AFM image showing topography of a monomolecular SVT ﬁlm on a polyimide substrate (part a-1) after the vapor treatment and on a SiOx substrate (part b-1). Part (a-2) and (b- 2) show height proﬁles along the white lines in part (a-1) and (b-1), respectively. Part (a-3) and (b-3) show schematically the chain conﬁgurations of the SVT on the polyimide substrate and on the SiOx substrate, respectively. Part (a-4) and (b-4) are blown-up TEM micrographs of the ﬁrst layer on the polyimide substrate (B in Fig. 29b-4) and that on the SiOx substrate (B in Fig. 29a-4), respectively. Here, the bright phase (supposed to be PS) is indicated by the white dotted circles. The spacing of the bright domains on the SiOx substrate (part b-4) is larger than that on the polyimide substrate (part a-4). Based on [66].

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Substrate Eﬀect on Self-Assembling Process On both the polyimide and the SiOx substrates, the non-equilibrium structure trapped in the as-prepared ﬁlm is eventually transformed into a parallel lamella structure in the ﬁrst-step of the ordering process as shown in Fig. 32a and b, respectively. However, the thicknesses of the lamellae formed in this step diﬀer signiﬁcantly on the SiOx and the polyimide substrates. The diﬀerence in the ﬁrst layer structure formed on the two substrates is responsible for this diﬀerence. On the polyimide substrate, the density of the junction points in the ﬁrst layer is rather high due to the weaker interactions between the V blocks and the substrate surface. The lamellae of equilibrium thickness are directly formed on the polyimide substrate in the one step ordering process (Fig. 29). However, on the SiOx substrate, thin lamellae are formed over several layers in this step. Here, due to extensive adsorption of V segments onto the polar substrate surface, the density of the junction points between S and V and between V and T in the ﬁrst layer is lower than that on the polyimide substrate as discussed before. The small number density of junction points may be inherited over several interfaces on top of the ﬁrst V layer, giving rise to a tendency toward formation of lamellae thinner than the equilibrium lamellae in the ﬁrst-step ordering process. Therefore, the substrate-polymer interactions exert a long-range eﬀect, inﬂuencing the lamellar spacing across the entire thin ﬁlm. This trend was found to be applicable to thin ﬁlms with thickness in the range from some 100 nm to some 1000 nm. The mobility of the V blocks on the SiOx surface is lower than that on the polyimide surface, which causes a slower equilibration of the junction density on the substrate and interfaces and hence of the lamellar thickness. 3.3 Eﬀect of Surface Roughness of Substrate on Lamella Orientation Perpendicular Lamellae One recent issue is how to produce perpendicular lamellae (lamellar microdomains with their lamellar interfaces perpendicular to the substrate surface) which is attractive for expanding applications of the thin ﬁlm structure. In the previous section, crucial eﬀects of the substrate on the nonequilibrium process of forming the parallel lamellae was presented. The difference in the strength of the short-range interactions between the substrate and the constituent block chains was found to have a long-range eﬀect on the self-assembled structure in the thin ﬁlms mediated by the particular chain conformation of the preferential component on the substrate. Non-neutral substrates having ﬂat surfaces generally formed parallel lamellae after thermal treatment and/or solvent treatment. So far, the ‘neutral’ surface [72], or the patterned surface with a laterally repeating chemical

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Fig. 32. Cross-sectional TEM images of three-phase coexisting SVT lamellae formed after THF-vapor treatment for 1min on polyimide (a) and on SiOx (b), respectively. Reproduced from [66]. Copyright (2003) American Chemical Society.

pattern that alternately attracted each component of the bcp [73, 74] were found to successfully develop perpendicular orientation in the lamella-forming bcp thin ﬁlms. Imposing an external ﬁeld that is stronger than the existing surface directed ﬁelds is another route to creating perpendicular orientations [41, 42, 62, 75–78]. In addition to the aforementioned methods utilizing elaborate manipulation of substrate surface or external ﬁelds, it should be emphasized that there is the possibility of forming perpendicular lamellae by choosing the roughness of substrate surface [79, 80]. The lamellar orientation in thin ﬁlms of a symmetric poly(styrene)-block -poly(methyl methacrylate), PS-b-PMMA, on indium tin oxide (ITO) coatings and on polyimide replicas of these was explored in detail. Although the roughness of the ITO arose from crystallization of the ITO alloy during deposition, and the relationship between the roughness and the crystallization is not well deﬁned, an ITO rougher than a certain critical roughness was found to produce perpendicular lamellae throughout the thin ﬁlm, Fig. 33(a).

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Fig. 33. Cross-sectional TEM micrographs of thin PS-b-PMMA (38k-36.8k) ﬁlm annealed at 230◦ C for 5 hours on two diﬀerent ITO substrates (ITO1 and ITO2). The roughness characteristics of the two ITO substrates are given at the bottom of the respective images. R is the root-mean-square roughness, and qs is the wave number at the maximal power spectral density obtained from the height image of the ITO. Both parameters are schematically represented in the inset on top of the ﬁgure (λs is the characteristic wavelength of the surface roughness). ITO1 is rougher than ITO2. Although parallel lamellae are formed on ITO2 (part b), perpendicular lamellae are formed on ITO1 (part a).

Substrate with Various Surface Roughness A good starting point to assess the eﬀect of the substrate roughness that dictates bcp orientation is to compare the main free energy penalties, ΔF , for each orientation, perpendicular (⊥) and parallel ( ), due to a rough substrate. The rough substrate will increase the free energy of a parallel lamellae through bending deformations of the lamellae (Fig. 34b). This cost of free energy may be circumvented by the perpendicular orientation (Fig. 34d). However, the perpendicular lamellae are accompanied by unfavorable contacts between the bcp components and the substrate. For a suﬃciently rough substrate such comparison may lead to the perpendicular orientation being more energetically favorable. We assume a simpliﬁed substrate with a single sinusoidal surface topology, having an amplitude Rs and lateral periodicity λs (≡ 2π/qs ). Other simplifying assumptions are that the mean ﬁlm thickness of both the parallel and perpendicular orientations is a multiple of lamellar periods, L, and that L is invariable, irrespective of lamellar orientation or position within the ﬁlm.

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Fig. 34. Change in the interior lamellar structure of thin ﬁlms with increasing reduced critical roughness parameter qs Rs . Parts (b) and (d) illustrate the models used to calculate the true critical roughness for parallel and perpendicular orientations, respectively. Part (a) depicts the reference state of the parallel lamellae on top of a ﬂat substrate.

The ﬁlm thickness is also considered suﬃciently large so as to neglect the surface-normal compressive forces. Referring to the theoretical treatment for the free energy (ΔF|| ) of parallel lamellae on a sinusoidal modulated substrate of roughness qs Rs by [81–83], the critical roughness, (qs Rs )c , at which the ΔF|| exceeds the free energy of perpendicular lamellae (ΔF⊥ ) is given by [80]: 3 (qs Rs )c = (γPS,PMMA /δ2 ) − 3/4 δ2 is the spreading coeﬃcient deﬁned by (γ2,PMMA − γ2,PS ), where γ2,i is the interfacial tension between the substrate and the block i(i = PS or PMMA). This expression takes into account air surface neutrality. Let us apply this criteria to the experimental results shown in Fig. 33. Taking γPS,PMMA as 1 mN/m and δ2 for ITO and polyimide substrates as roughly 0.25 mN/m, (qs Rs )c is estimated as 0.92. Although the actual (qs Rs )c (0.4) seems to be smaller than this value (parallel lamellae are observed on the substrate of smaller qs Rs ), this criterion qualitatively explains the experimental results. The critical roughness was experimentally determined by using polyimide replicas of the ITO1 surface as a substrate. The polyimide surface made from biphenyltetracarboxylic-acid and oxydianiline was not deformed during the PS-b-PMMA ﬁlm preparation process and the heat treatment around 200o C, while the roughness of this polyimide surface can be suppressed by heat treatment above the glass transition temperature of this polyimide (around 260 − 270◦ C).

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PS-b-PMMA (38k-36.8k) thin ﬁlms (thickness 250 nm) were prepared on the polyimide substrates of various roughness; the samples were then annealed at 200o C for a range of times and their structures observed iteratively with AFM between anneals. During annealing, three types of bcp surface images were observed: The ﬁrst is a perpendicular lamellae structure (see Fig. 35a) which is most frequently observed after the initial anneal on rough substrates. The second is the perpendicular lamellae with signs of surface roughening (see Fig. 35b). For a substrate with a given qs Rs , the perpendicular surface structure persisted for a time, tqs,Rs , after which the thin ﬁlm surface structure began to show macroscopic roughening. The third is a stepped surface indicative of the establishment of parallel lamellae of a diﬀerent stack number (Fig. 35c). Extended annealing of the sample then led to increased surface roughening, a disappearance of the perpendicular lamellae and the eventual development of terraced surface. It was observed that there was a longer lag time, tqs,Rs , before the thin ﬁlms on higher qs Rs polyimide substrates showed signs of surface roughening. In the case of very high qs Rs substrates this transition did not occur at all and a perpendicular surface structure was observed over the timescale covered in the experiments. Fig. 36 summarizes this behavior for the polyimide replica substrates studied and indicates, at a given time, the roughness of the substrates upon which the PS-b-PMMA (38k-37k) thin ﬁlm surface exhibits only perpendicular lamellae (ﬁlled symbols). The ﬁgure also shows the point at which each of these substrates, upon prolonged annealing, began to show signs of forming parallel lamellae (open symbols). The locus of the time at which the perpendicular lamellae start to transform to parallel lamellae (as a function of qs Rs ) is delineated by the broken line within the ﬁgure. This locus changes with time in the parameter space of qs Rs and the annealing time. It was found to become independent of time at a value of qs Rs around 0.4; this is identiﬁed as the critical roughness, (qs Rs )c , of the substrate for 38k-36.8k PS-b-PMMA on the polyimide substrate at 200o C. The experimentally determined (qs Rs )c is smaller than that expected from theoretical considerations (0.92) of this system as presented before. Here it should be noted the experimentally determined (qs Rs )c may be an underestimate of the true critical roughness of the substrate. With reference to Fig. 34c, the experimentally obtained critical roughness is redeﬁned as the criterion below which, if parallel lamellae form on the substrate, they are able to grow until they reach the air surface. Beside (or above) this AFM observed critical roughness, there is a possibility that parallel lamellae form on the substrate but do not reach the free surface, Fig 34c. It has been shown that a suﬃciently rough substrate can lead to perpendicular lamellar orientation of the block copolymer. There is a critical substrate roughness below which the parallel orientation of block copolymers is favored. Given this study, it is unsurprising that previous experiments with rough substrates did not report the means to produce perpendicular lamellae. The substrates used by [84] were not rough enough. A further experiment, which did

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Fig. 35. AFM images of PS-b-PMMA (38k-37k) bcps on a polyimide replica substrate that have been thermally smoothed to a qs Rs of 0.25. Images show the surfaces of the thin bcp ﬁlms after annealing at 200◦ C for (a) 60min, (b) 70 min, and (c) 1 day. Reproduced from [80]. Copyright (2005) American Chemical Society.

use signiﬁcantly rough substrates, also used poly(styrene)-block -poly(2-vinyl pyridine) as the model block copolymer [85]. However, this system has large diﬀerences in the attraction of its component blocks to the air and substrate surfaces. In PS-b-PMMA these diﬀerences are small and can be overcome by substrate roughness eﬀects. Formation of Patterned Surface The perpendicular lamellae system found here can load metal nano-particles preferentially into the domains constituting of certain components [54]. Palladium nano-particles were loaded into the perpendicular lamellae ﬁlm via sublimed vapor of Pd precursor (palladium acetyl-acetone complex) at elevated temperature (200◦ C). A high molecular weight PS-b-PMMA (106k-99k) was used in order to prevent disordering of the microphase-separated structure

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Fig. 36. Summary of AFM analysis of PS-b-PMMA (38k-36.8k) ﬁlms annealed on an array of polyimide substrates having a range of qs Rs values. The ﬁlled and open symbols indicate surfaces found to be in a fully perpendicular (c.f. Fig. 35a) and mixed (c.f. Fig. 35b) surface state. The dotted lines indicate the locus of (qs Rs )c from one state to the other. Reproduced from [80]. Copyright (2001) American Chemical Society.

of PS-b-PMMA during the vapor exposure process and simultaneously proceeding thermal reduction of the Pd complex. The ITO1 substrate successfully developed perpendicular lamellae of this high molecular weight copolymer. As shown in Fig. 22 (Sect. 2.2), the thin ﬁlm of PS-b-PMMA having perpendicular lamellar orientation selectively incorporates Pd nano-particles into the PS lamellae. The Pd nano-particles appear as the darkest spots and the brightest (most protruded) spots in the TEM micrograph (part a) and the AFM image (part b), respectively. The ﬁnding that a rough surface induces a perpendicular orientation in lamella-forming block copolymer ﬁlms was also demonstrated in the SVT thin ﬁlm. Fig. 37 shows an AFM height image of 200 nm thickness SVT thin ﬁlm on polyimide substrate. The thin ﬁlms was subjected to THF-vapor treatment and thereby the parallel lamellae as shown in Fig. 32a were already established in most areas of the sample. However, along a small scratch induced on the polyimide substrate before the ﬁlm preparation (indicated by the arrow), a zone in which the perpendicular lamellae form is observed. Though the scratched area may expose functional groups diﬀerent from those on the nonscratched area and thereby have altered surface energy, the orientation of the lamellae seems to be controlled by the mechanical scratch on the substrate.

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Fig. 37. The SVT triblock terpolymer ﬁlm on the polyimide substrate. There is a scratched line on the substrate surface in the area marked by an arrow. Perpendicularly orientated lamellae are observed on top of the scratched area, while the parallel lamellar orientation was observed in the surrounding area having a ﬂat substrate surface. Reproduced from [80]. Copyright (2005) American Chemical Society.

3.4 Remarks - Substrate and Thin Film Structure This section highlights the eﬀects of the substrate surface (chemistry and geometry) on the self-assembling process in lamella-forming bcp thin ﬁlms. First, it was shown that the strong short-range interactions between the substrate surface and the block segments appear as a long-range eﬀect in the non-equilibrium process through the chain connectivity of the block copolymers. The strong interaction between the bcp and the substrate leads to a special structure on the substrate surface, and this seems to signiﬁcantly inﬂuence the as-prepared (non-equilibrium) ﬁlm structure. In this study, the ﬂat substrate always led to parallel lamellae orientation in thermal equilibrium. The transformation process of the as-prepared ﬁlm structure toward

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the parallel lamellae seemed to be determined by the special structure formed on the substrate. Using the lamella-forming triblock terpolymer and the substrate preferentially attracting the middle block, these ﬁndings (the long-range substrate eﬀect) were highlighted. The discovery concerning the eﬀects of substrates on kinetic pathways and time-scale to achieve equilibrium morphology may be quite important, since in many cases systems cannot attain equilibrium. The systems rather tend to be trapped in metastable states, and hence one must ﬁnd methods to control non-equilibrium structures assisted by some basic principles of non-equilibrium phenomena to be further developed in future. Secondly, it was shown that the generally observed parallel lamellae can be switched into perpendicular lamellae by roughening the substrate surface. Here, added to the aforementioned eﬀect from the substrate, geometrical constraints induced by the substrate surface topography inﬂuence the lamellar formation process. With the substrate having a roughness above the critical one, the bcp thin ﬁlms were found to have an intriguing perpendicularly orientated lamellar structure which can be utilized in various applications. We hope the present work will contribute to some extent in nano-fabrication technology in various ﬁelds.

References 1. Fink Y., Urbas A. M., Bawendi M. G., Joannopoulos J. D., Thomas E. L. (1999) Journal of Lightwave Technology 17:1963 2. Urbas A. M., Sharp R., Fink Y., Thomas E. L., Xenidou M., Fetters, L. J. (2000) Advanced Materials (Weinheim, Germany) 12: 812 3. Urbas, A. M., Maldovan M., DeRege P., Thomas E. L. (2002) Advanced Materials (Weinheim, Germany) 14: 1850 4. Yoon J., Lee W., Thomas E. L., (2005) MRS Bulletin 30: 721 5. Leclere P. H., Parente V., Bredas J. L., Francois B., Lazzaroni R. (1998) Chem. Mater.10: 4010 6. Thurn-Albrecht T., Schotter J., Kastle G. A., Emley N., Shibauchi T., KrusinElbaum L., Guarini K., Black C. T., Tuominen M. T., Russell T. P. (2000) Science 290: 2126 7. Niu H., Zhang L., Gao M., Chen Y. (2005) Langmuir 21: 4205 8. Minelli C., Hinderling C., Heinzelmann H., Pugin R., Liley M. (2005) Langmuir 21: 7080 9. Lu J.,Q, Yi S. (2006) Lungmuir 22: 3951 10. Tjandra W., Yao J., Ravi P., Tam K. C., Alamsjah A. (2005) Chem. Mater. 17: 4865 11. Fu B. X., Lee A., Haddad T. S. (2004) Macromolecules 37:5211 12. Jain A., Hall L. M., Garcia C. B., Gruner S. M., Wiesner U. (2005) Macromolecules 38: 10095 13. Brouwer H. J., Hilberer A., Krasnikov V. V., Werts M., Wildeman J., Hadziioannou G. (1997) Synthetic Metals 84:881

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Takeji Hashimoto and Kenji Fukunaga Berkovich, Y., Garti N. (1997) Colloids and Surfaces A. 128: 91 Wagner C. (1961) Z. Electrochem 65: 581 Lifshitz I., M, Slyozov V. V. (1961) J. Phys. Chem. Sol. 19: 35 Gunton J. D., San Miguel M., Sahni P. S. (1983) In Phase Transitions and Critical Phenomena, Domb C., Lebowitz J. L., Eds. Academic Press: New York, Vol. 8, Chap. 3 Binder K., Stauﬀer D. (1974) Phys. Rev. Lett. 33:1006 Kawasaki K., Ohta T. (1978) Progr. Theor. Phys. 59:362 Oki K., Sagana H., Eguchi T. (1977) J. Phys. (Paris) C 7: 414 Horiuchi S., Fujita T., Hayakawa T., Nakao Y. Langmuir (2003) 19: 2963 Hashimoto T. (2005) Bull. Chem. Soc. Jpn, 78:1 Park M., Harrison C., Chaikin P. M., Register R. A., Adamson D. H. (1997) Science 276:1401 Hasegawa H., Hashimoto T. (1985) Macromolecules 18:589 Hasegawa H., Hashimoto T. (1992) Polymer 33:475 Russell T. P., Coulon G., Deline V. R., Miller D. C. (1989) Macromolecules 22:4600 Coulon G., Ausserre D., Russell T. P. (1990) Journal de Physique (Paris) 51:777 Fukunaga K., Ribbe A. R., Hashimoto T. (2006) Macromolecules, 39:6171 Elbs H., Fukunaga K., Stadler R., Sauer G., Magerle R., Krausch G. (1999) Macromolecules 32:1204 Fukunaga K., Elbs H., Magerle R., Krausch G. (2000) Macromolecules 33:947 Magonov S. (2001) Handbook of Surfaces and Interfaces of Materials, vol. 2, eds. Nalwa H. S., Academic Press, San Diego, CA, in pp. 393-430 Zheng W., Wang Z. G. (1995) Macromolecules 28:7215 Fukunaga K., Hashimoto T., Elbs H., Krausch G. (2002) Macromolecules 35:4406 Fukunaga K., Hashimoto T., Elbs H., Krausch G. (2003) Macromolecules 36:2852 Hashimoto T., Jinnai H., Nishikawa Y., Koga T. (2002) Polym. Symp. 190:9 Kim G. and Libera M. (1998) Macromolecules 31:2569 Kim G. and Libera M. (1998) Macromolecules 31:2670 Albalak R. J., Capel M. S., Thomas E. L. (1998) Polymer 39:1647 Russell T. P., Mayes A. M., Kunz M. S. (1994) Ordering in Macromolecular Systems; eds. Teramoto A., Kobayashi M., Norisuye T., Springer-Verlag, Berlin, 217 Huang E., Mansky P., Russell T. P., Harrison C., Chaikin P. M., Register R. A., Hawker C. J., Mays J. (2000) Macromolecules 33:80 Rockford L., Mansky P., Russell T. P., Mochrie S. G. (1999) Phys. Rev. Lett. 82:2602 Kim S. O., Solak H. H., Stoykovich M. P., Ferrier N. J., de Pablo J. J., Nealey P. F. (2003) Nature 424:411 Amundson K., Helﬂand E., Davis D. D., Quan X., Patel S. S., Smith S. D. (1991) Macromolecules 24:6546 Morkved T. L., Lu M., Urbas A. M., Ehrichs E. E., Jaeger H. M., Mansky P., Russell T. P. (1996) Science 273:931 Xu T., Hawker C. J., Russell T. P. (2003) Macromolecules 36:6178 Kim S. H., Misner M. J., Xu T., Kimura M., Russell T. P. (2004) Advanced Materials 16, 226

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Characterization of Surfactant Water Systems by X-Ray Scattering and 2 H NMR Michael C. Holmes Centre for Materials Science, Department of Physics, Astronomy and Mathematics, University of Central Lancashire, Preston, PR1 2HE, United Kingdom

1 Introduction to Surfactant Water Systems Aqueous solutions of surfactants not only have important applications as detergents, in food and cosmetic products, in oil recovery and drug delivery [1] but are now becoming important as a medium for the templating of nanostructured materials [2–7]. Surfactants are molecules which possess two diﬀerent moieties; a hydrophobic moiety and a hydrophilic moiety [8] and typically have dimensions in the range 1 to 10 nm. At very low concentrations in aqueous solution they exist as individual molecules but above the critical micellar concentration and Kraﬀt temperature they self assemble to form micelles. At such low concentrations interactions between the micelles are negligible and generally they will have a spherical shape whose radius is determined by the length of the hydrophobic tail. However as the concentration of surfactant is increased, micelles can become non-spherical (rod or disk shaped) and the interactions between them become signiﬁcant. In fact within these systems there are two important interactions determining structure; inter-molecular interactions (both head and tail groups) which to a large extent determine the aggregate size and shape and inter-aggregate interactions which can inﬂuence aggregate size and shape but more importantly can determine phase structure. These interactions can easily become strong enough to promote macroscopic order and can lead to the formation of a sequence of ordered mesophases having length scales in the range 3 to 200 nm typically. One of the key properties of these mesophases is that they divide space into two regions; hydrophobic and hydrophilic, making them ideal for templating media. The wide range of phases that can form have been reviewed elsewhere [1–9] but table 1 lists the wide range of types of phase and their space groups. Clearly there are phases with one, two and three dimensional positional order in addition to varying degrees of orientational order of the aggregates. One of the best ways of determining aggregate structure and determining the symmetry of a phase is using X-ray scattering.

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Table 1. Summary of nomenclature for lyotropic phases and for their intermediate phases. The subscripts 1 and 2 refer to normal (water continuous) and reversed (surfactant continuous) phases, modiﬁed from reference [9]. Details of the space groups can be found in reference [10] Phase

Symbols

Isotropic

Possible space groups L1 , L2 , L3 , L4 –

Micellar Cubic Nematic

I 1 , I2 Nc , N d

Hexagonal H1 , H 2 Bicontinuous cubicV1 , V2 Lamellar Lα Ribbon Rb1 , Rb2 Mesh

Mh1 , Mh2

Bicontinuous

Bc1 , Bc2

Notes L3 , L4 refer to bicontinuous isotropic and vesicle phases

Pm3n, Fd3m –

Consisting of rod or disk shaped micelles p6mm Two-dimensional space group. Ia3d, Pn3m, Im3m – pm One-dimensional space group. c2mm, p2gg Two dimensional centred rectangular space groups. 0, I4mm, R3m 0 represents the random mesh whilst three dimensional space groups I4mm, R3m represent tetragonal and rhombohedral phases mesh respectively. ? No space groups identiﬁed to date.

In addition to the structural diversity these phases contain interfacial surfaces dividing the aqueous environment from the hydrophobic alkyl chain region. The curvature of this surface is determined by the strength of the interactions between the moieties and the volume fraction of the moieties and any associated solvents. Any surface curved in three dimensions can be described by two radii of curvature at each point, Fig. 1. The curvature is de1 1 1 ﬁned in two ways: the mean curvature, H = 2 rab + rcd and the Gaussian curvature, K = r1ab r1cd . 2

H NMR is sensitive to the surface curvature in surfactant water systems and, in the case of 2 H NMR from solvent 2 H2 O it is also sensitive to the fraction of water molecules bound to the surfactant head groups. Since the surface curvature can change at a phase transition between two ordered phases, 2 H NMR can be used as a sensitive technique for establishing phase diagrams, and together with X-ray scattering can fully characterize the phases of a surfactant water system.

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Fig. 1. A region of interface whose curvature at each point can be described by two radii of curvature, rab and rcd with their respective arcs ab and cd shown.

2 Small Angle X-Ray Scattering (SAXS) There are many general works on X-ray scattering [11] and a few on scattering from speciﬁc mesophase structures, e.g. cubic phases [12, 13]. Many of the ideas given here were originally outlined in the pioneering work by Luzzati and coworkers, [13–19]. 2.1 Some Basic X-Ray Scattering Theory Scattering from a Single Electron X-rays are scattered by electrons in matter. They can be scattered both coherently and incoherently, i.e. the scattered X-ray can have the same wavelength as the incident X-ray and a ﬁxed phase relationship to it or the wavelength can be diﬀerent. The incoherent scattering arises from such eﬀects as Compton scattering, ﬂuorescence, photoelectron emission etc [11] and gives rise to background and noise. Coherent scattering arises when the electric ﬁeld vector of the incident X-ray wave causes an electron to oscillate in a direction parallel to the electric ﬁeld vector and then re-emit coherent X-rays in all directions. The intensity Ip of X-rays scattered to a point P a distance r from a single electron, mass m and charge e at point O is given by

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Ip = I0

e4 sin2 α r2 m2 c4

where I0 is the intensity of the incident X-ray and α is the angle between OP and the direction of acceleration of the electron at O. Notice that Ip ∝ 1/r2 i.e. the inverse square law is satisﬁed and Ip ∝ 1/m2 i.e. the scattering from the electrons is much stronger than that from the nucleus. This relationship, ﬁrst proposed by J. J. Thomson [11] shows that the scattered intensity depends upon angle. In general we are interested in considering the scattering from atoms or collections of atoms which contain many electrons. Two Scattering Centres We start by considering the scattering from two electrons located at O and M and separated by a distance OM, Fig. 2.

Fig. 2. The geometry of X-ray scattering from a pair of electrons.

S 0 and S are unit vectors deﬁning the directions of the incident and scattered radiation and λ the wavelength of the X-rays. The path diﬀerence between rays 1 and 2 is mMn and will determine the phase diﬀerence between rays 1 and 2. In turn this will determine whether constructive or destructive interference takes place and thereby the intensity of scattered X-rays in the direction 2θ. From Fig. 2 mM = S 0 .OM and M n = −S.OM . Thus the path diﬀerence between rays 1 and 2, is given by δ = −OM (S − S 0 ). The phase diﬀerence between rays 1 and 2 is given by φ=

2πδ (S − S 0 ) = −2π OM . λ λ

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(S−S )

where λ 0 = s is the scattering vector. s is an important parameter in scattering theory since all scattering measurements can be reduced to determining the scattered intensity I(s). The modulus of s can be easily determined from ﬁg. 2 to be 2 sin θ/λ. Scattering from a Single Atom For coherently scattered X-rays the amplitudes add so that the scattered intensity from a single atom I(s) is given by I(s) = f 2 (s)Ie (s)

(1)

Where Ie is the intensity scattered by a single electron, and f is the atomic scattering factor. The atomic scattering factor is deﬁned as f=

amplitude of the wave scattered by an atom . amplitude of the wave scattered by one electron

The scattering factor is a function of both the scattering angle θ and the X-ray wavelength λ.

Fig. 3. The atomic scattering factor for ﬂuorine, Z = 9.

When s = 0, f = Z, the atomic number of the atom and the total number of electrons in the atom. For a low density gas of well separated atoms of atomic scattering factor f the scattered intensity is given by (1) and is simply the square of the curve shown in Fig. 3. Scattering from a Group of Atoms Extending this to a group of n atoms such as a crystal, the scattering is the sum of the scattering from individual atoms modiﬁed by the interference

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between these components due to the spatial arrangement of those atoms. The scattered intensity from the group can be written

2

I(s) = F (s) where F (s) is the structure factor and is given by F (s) =

n

fj exp(−2π is.r)

(2)

j=1

fj is the atomic scattering factor of the jth atom. F (s) and I(s) not only depend upon fj but also on the spatial arrangement of atoms through the term exp(−2πs.r). This will determine the phase diﬀerence between the scattering from diﬀerent atoms and thereby the intensity of the scattering in a particular direction. Note that when s = 0, F (0) is equal to the total number of electrons in the group of atoms. F (s) will usually be a complex number containing information about the phases of X-rays scattered from diﬀerent atoms in the group. When the square of F (s) is formed to give I(s) this phase information is lost. If the number of electrons is large (2) becomes ρ(r) exp(−2π is.r)dr (3) F (s) = V

where ρ(r) is the electron density of the group of atoms and V is the volume of the atomic group. This has the form of a Fourier transform so that the electron density can be deduced from F (s) exp(2π is.r)ds ρ(r) = V

Thus in principle the intensity may be related to the electron or atomic distribution via a Fourier transform and ρ(r ) is the structure in real space whilst F (s) is the structure in reciprocal space.

3 Scattering from Surfactant Water Systems The general principles developed in the previous section can now be applied to surfactant water phases. However, caution must be exercised. An old but seldom mentioned principle in X-ray scattering is Babinet’s Principle which states that complementary structures appear identical, ie holes in a screen appear identical to opaque spots of the same size. Thus the X-ray scattering from hexagonal, H1 and reversed hexagonal, H2 phase will generate reﬂections in the same ratio to each other. Other information such as the volume fraction of surfactant must be used to distinguish between these possibilities.

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A number of assumptions also need to be made about the organization of the structure before proceeding. These are summarized in the three protocols of Luzzati [14]. 1. Aqueous and lipid components occupy distinct regions and the interface between them is ‘smooth’. Generally the aggregate is assumed to have a simple symmetry i.e. the hexagonal phase is composed of circular rods although this may not always be the case; see for example ﬁgures 7 and 9 later in the chapter. 2. Some water molecules are strongly associated with the head groups in an interfacial region. 3. The water and head groups comprise the polar region. With surfactant water systems we may distinguish three diﬀerent classes of situation. Firstly there are isotropic micellar and sponge phases, L1 , L2 , L3 , L4 which possess no long range order. Secondly there are phases that have long range positional order (i.e. more or less crystalline) in at least one direction. Examples of these phases are lamellar, Lα ; hexagonal, H1,2 : cubic, V1 and the so-called intermediate phases such as mesh phases, Mh1 (0). Finally there are nematic phases, Nc,d which consist of anisometric micelles possessing only long range orientational order. In studying surfactant water phases we are interested in structural units which consist of large aggregates of surfactant molecules for which an electron density distribution is the most appropriate form of the structure factor, (3). However these surfactant aggregates may be close enough for the scattering from individual aggregates to interfere. The observed intensity can therefore be deﬁned as I(s) = F 2 (s)J(s) (4) where J(s) is a function which describes the interference between scattering objects. 3.1 Phases with no Long Range Order Scattering from Well Separated, Non-Interacting Particles For a simple, low density monatomic gas (4) gives a very simple result J(s) = 1 and I(s) = F 2 = f 2 /v where v is the average volume available per atom (i.e. v = V /N where V is the total volume and N is the number of atoms). Thus the scattering pattern is identical to f 2 . This is the same situation that is found in dilute micellar phases where the micelles are well separated and there is no interference between particles. For spherical micelles the scattering arises simply from the electron density distribution within a single micelle (assuming monodispersity) and leads to a scattering curve with the form of a Bessel function [11]. However polydispersity in the size and shape of the micelles means that the secondary maximum

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is often very diﬃcult to observe. The equivalent functions for rod and disk shaped micelles are more complex mathematically. It is often diﬃcult experimentally to distinguish the precise micellar shape. For the central peak of the scattering there is an exponential approximation for the scattered intensity [11],

4π 2 s2 R2 I(s) ∝ exp − 3 where R is the radius of gyration of the micelle. By plotting ln I(s)as a function of the scattering angle, it is possible to obtain an estimate for R. This assumes that the particles are identical, of low density and randomly orientated. Scattering from a Liquid-Like Structure As the density of the particles (whether single atoms or spherical micelles in an isotropic solution) increases, the interference function J(s) departs from unity and starts to reﬂect the increasing local positional order measured by the radial density function n(r), i.e. the number density of particles in a thin spherical shell of radius r and thickness δr.

Fig. 4. (a) A representation of a liquid formed by spherical particles. A thin spherical shell of radius r and thickness δr is shown. (b) The radial density function, n(r) as a function of r for three diﬀerent number densities represented by the function a/v 1/3 .

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Because of the higher number density of particles, there is inter-particle interference expressed by [11] 1 J(s) = 1 + [n(r) − 1] exp(−2π i s.r)dr v V For spherical particles this becomes 2 [n(r) − 1] sin(2π s r)rdr J(s) = 1 + sv V Notice that increasing the density increases the extent of the long range order and the scattering pattern evolves to that of a liquid with a scattering maximum at about s = 1/a where a is the particle radius, Fig. 5.

Fig. 5. (a) I(s) as a function of s for increasing density of particles. (b) X-ray scattering from a micellar phase. Notice the appearance of a diﬀuse scattering ring arising from the short range positional order in the liquid.

For a dense liquid the scattering pattern shows a pronounced peak and we can write the inter-atomic interference as sin(2π sr) 1 (number of pairs at distance r) × J(s) = 1 + N 2π sr The ﬁrst maximum for

sin(2π sr) 2π sr

occurs at

1.23 2 sin θm = λ rm For a hexagonal close packed solid the ﬁrst diﬀraction peak occurs at 1.22 3 1 = sm = 2 rm rm Therefore it is tempting to associate the liquid maximum with the mean separation of particle. However, this should be treated with care particularly as the number density of the liquid decreases. sm =

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3.2 Phases with Long Range Positional Order Scattering from a Crystalline Solid In a crystalline solid the interference function is important and becomes a series of delta functions. The scattering power per unit cell for an inﬁnite crystal can now be written: I(s) = F 2 (s)J(s) = F 2 (s)δ(r∗hkl − s)

(5)

where δ is the Dirac delta function and r∗hkl is deﬁned by r∗hkl = ha∗ + kb∗ + lc∗ .

(6)

The integers h, k and l are the Millar indices of the crystal plane under consideration and a*, b* and c* are the vectors deﬁning the reciprocal lattice, i.e. a* = a −1 b* = b −1 and c* = c −1 . Here a, b, and c are the lattice vectors of the crystal. Now put ∗ Fhkl = F (rhkl ) = ρ(r) exp(−2π ir∗hkl .r)dv (7) 2 Then I(s) = Fhkl Peaks can only occur for s = r∗hkl or, according to the Bragg condition,

2dhkl sin θ = λ. Fig. 6 shows the progression of scattering patterns from a low density gas through to a crystalline solid. In ﬁg. 6(b) notice that the intensity of the scattering peaks is modulated by the single particle scattering. Lamellar Phase: 1-Dimensional Positional Order The lamellar phase, Lα is the simplest ordered structure of a surfactant water system consisting of surfactant bilayers interspersed with water layers and positional order in one direction, Fig. 7. From (5) and (6) l s0 = r∗l = lc∗ = . d0 So there are a series of lines in the scattering pattern in the ratio 1:2:3 etc. Having determined d0 other important parameters can be determined from it by using the volume fraction φhc of hydrocarbon in the sample. This can be calculated from −1 hc c φhc = 1 + ξwξ(1−c) where ξhc and ξw are the densities of hydrocarbon and water respectively and c is the concentration of hydrocarbon in the

Characterization of Surfactant Water Systems

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Fig. 6. Schematic diagrams of the scattering from (a) gas, liquid/dense gas and liquid states and (b) a crystalline solid.

Fig. 7. A schematic representation of the real space structure of the lamellar phase and the X-ray scattering pattern in reciprocal space for a powder and a single crystal sample. The graph on the right shows the scattered intensity proﬁle as a function of s.

sample. The bilayer thickness is then calculated from dhc = φhc d0 . Notice that this equation provides a scaling relationship which can conﬁrm that this phase is indeed a classical lamellar phase since dhc should remain constant as the water content changes at ﬁxed temperature and therefore d0 ∝ φ−1 hc . It will be shown below that phases with other structures and symmetries have other scaling relationships.

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The surface area per surfactant molecule, sa at the interfacial surface can also be calculated knowing the volume of the hydrocarbon, vhc from sa = 2vhc /φhc d0 Returning to (3) , we see that whilst the position of the scattering lines depends upon d0 their intensity depends upon F 2 (s) which is determined by the electron density proﬁle of the bilayer itself. Equation (3) becomes I(sz ) = F 2 (sz ) where sz is the scattering vector in the z direction parallel to the bilayer normal. Equation (7) becomes F (sz ) = [1/π sz (ρhc − ρw ) sin 2π sz tw ]+[1/π sz (ρhead − ρw ) sin 2π sz (tw + thead )] where tw is the half width of the water layer, thead is the thickness of the head group region and ρhc , ρhead and ρwater are the electron densities of the hydrocarbon, head group and water layers which typically have values of ρhc = 273 nm−3 , ρhead = 360 nm−3 and ρwater = 335 nm−3 for 2 H2 O. Fig. 8 shows a schematic representation of the electron density across the semi-bilayer section.

Fig. 8. A schematic representation of the electron density across the semi-bilayer section in the z direction. Terms are deﬁned in the text.

Using the simpliﬁed electron density proﬁle shown in ﬁgure 8, F 2 (sz ) can be calculated and compared to the intensity proﬁle of the reﬂections from the lamellae.

Characterization of Surfactant Water Systems

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Hexagonal Phase: 2-Dimensional Positional Order With the hexagonal, H1 phase we progress to a system with 2-dimensional positional order and long range orientational order. Fig. 3.2 schematically shows the structure and scattering from both powder and single crystal samples.

Fig. 9. A schematic representation of the real space structure of the hexagonal phase and the X-ray scattering pattern in reciprocal space for a powder and single crystal sample orientated with the cylinder axis n ˆ in the same plane as s and with it parallel to the incident X-ray beam. a is the lattice vector of the 2-dimensional lattice on which the cylinders are packed and dhc here is the diameter of a cylinder.

From (5) and (6) scattering maxima occur when (h2 + k 2 + hk) 2 ∗ shk = rhk = |a| 3

√ √ √ This √ generates reﬂections with s values in the ratio of 1, 3, 2, 7, 3, 12, 13 . . . . A common mistake is to assign the ﬁrst and strongest reﬂection to the lattice parameter |a|. In fact the ﬁrst reﬂection corresponds to the lowest √ |a| 3 values of Millar indices (1,0) and is d10 = 2 . The diameter of the hydrocarbon cylinders is now given by √ 2 3φhc 8φhc √ = d10 dhc = |a| π π 3 0.5 Notice that the scaling relationship between d10 and φhc is now d10 ∝ 1/φhc . 2vhc 2π√ . Once again The surface area per molecule is given by sa = | a| φ 3 hc

the intensity of the scattering peaks will be determined by the structure factor

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of the cylinders. This time a one dimensional electron density proﬁle will have to be replaced by a two dimensional map. Cubic Phases: 3-Dimensional Positional Order There are a number of phases with cubic symmetry [12, 13]. Micellar cubics comprise micelles, which may be slightly distorted into short cylinders located on lattices with cubic symmetry. In addition between lamellar and hexagonal phases there are three bicontinuous cubic phases in which both the water and hydrocarbon regions are continuous. These are illustrated in Fig. 10.

Fig. 10. The three bicontinuous cubic structures and their space groups. The distance a represents the lattice parameter. Reprinted from [12]. Copyright (1993) The Royal Society.

The structure of these phases is more complex however the same method of working applies. From (5) and (6) scattering peaks will occur when 1 2 h + k 2 + l2 . shkl = r∗hkl = |a| √ √ √ √ This √ generates reﬂections with s values in the ratio of 1, 2, 3, 2, 5, 6, 8, 3,... Not all reﬂections will necessarily be present, and will depend upon the particular symmetry and upon the structure factor from the electron density distribution. Cubic phases have been extensively reviewed by Seddon [12]. Calculating dhc and sa is possible provided the structure has been identiﬁed and a suitable structural model is used. Simple rod-box models can give reasonable estimates for these parameters (see below) however they must always

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be treated with a degree of caution. The scaling relationship for a cubic phase 0.33 although practically it is often not very useful since these will be d10 ∝ 1/φhc phases only have a narrow concentration range of only a few weight percent of surfactant. Mesh Phases: 1 or 3-Dimensional Positional Order Mesh phases are part of a group of phases known as “intermediates” since they occur between hexagonal and lamellar phases and have a number of structures and symmetries [20]. Mesh phases are essentially lamellar phases pierced by water ﬁlled pores and are analogous to hexagonally perforated phases of diblock copolymers. The random mesh phase, Mh1 (0) has the lamellae pierced by pores but these have a liquid-like arrangement and there is no correlation between the lamellae. By contrast the rhombohedral mesh, Mh1 (R3m) has pores which are arranged within the lamellae on a hexagonal lattice and are correlated from layer to layer with an ABC arrangement. Fig. 11 shows the evolution of the X-ray scattering from a lamellar phase, Lα to random mesh phase, Mh1 (0) and random mesh phase to rhombohedral mesh phase, Mh1 (R3m).

Fig. 11. (a) shows the evolution of the Mh1 (0) phase to the Lα in the C16 EO6 /2 H2 O system [21]. (b) Shows the evolution of the Mh1 (R3m) phase to Mh1 (0) in the tetramethylammonium perﬂuorodecanoate / water system as a function of temperature [22].

The diﬀerence between the scattering from Mh1 (0) and Lα phase is rather subtle. Both have 1-dimensional order and therefore produce the characteristic lamellar pattern with reﬂections in the ratio 1:2 etc. The Mh1 (0) phase diﬀers by having a broad (and consequently weaker) liquid-like reﬂection labeled

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dh in Fig. 11(a). This comes from the pores within the lamellae. It should also be noticed that d0 (Mh1 (0)) < d0 (Lα ) since as the pores open up in the lamellae they are ﬁlled by water from between the lamellae reducing d0 . In Fig. 11(b) the Mh1 (0) phase is seen at 65◦ C whilst the Mh1 (R3m) phase is seen at lower temperatures. This latter phase has long range positional order in 3-dimensions and this is reﬂected in the sharp multi-line scattering which can be indexed to the R3m space group. Calculating dhc and sa is possible provided a suitable structural model is used. Two such models are shown in Fig. 12.

Fig. 12. (a) A mean curvature model of one lamellae of the Mh1 (R3m) phase and (b) the same constructed from a series of interconnected rods (shaded) and triangular boxes.

The simple rod-box model of Fig. 12 can give good estimates of the structural parameters in these phases [21]. For example, table 2 shows rhc the hydrocarbon region semi-thickness and sa calculated for four diﬀerent phases in the C16 EO6 / water. What is noteworthy is that sa remains rather constant with phase structure but that rhc changes as the interface becomes more curved. Finally the scaling relationship for a mesh phase becomes much more like 0.5 . that of a hexagonal phase and d0 ∝ 1/φhc 3.3 Phases with Long Range Orientational Order Only Nematic Phases Lyonematics are composed of anisometric micelles, either rod shaped, Nc or disk shaped, Nd , and they usually occur between isotropic micellar phases and

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Table 2. Summary of the parameters for the 52 weight percentage C16 EO6 /water sample [23]. Model rhc (nm) Lamellar phase 0.89 Rhombohedral smoothed mesh 2.19 Cubic Ia3d 1.60 Hexagonal Phase 2.00

sa (nm2 ) 0.51 0.36 0.57 0.45

either hexagonal or lamellar phases respectively [24,25]. Typically these phases are found in short chain ionic surfactants often with an added cosurfactant or salt [24]. Both phases have a relatively low viscosity and can often be easily aligned either by an external magnetic ﬁeld or by surface forces caused by the walls of the sample container and the shear ﬂow on ﬁlling [26]. It is therefore relatively easy to obtain single crystal samples, Fig. 13.

Fig. 13. The structure of the Nd and Nc phases in real and reciprocal space shown schematically. n ˆ shows the mean orientation of the symmetry axis of the micelles (or liquid crystal director).

It is important to notice that disk shaped micelles are naturally monodispersed since the growth of the micelle is controlled by the need to increase the amount of highly curved edge, increasing its mean surface energy. In contrast rod micelles tend to be more polydispersed since after the hemispherical end caps are established the rod can continue to grow. It is for this reason that it is often impossible to observe d in the Nc phase. The scattering peaks from d in the Nd phase and d⊥ in the Nc phase are sharp relative to the scattering from an isotropic micellar solution and indicate some degree of positional

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order in those directions. In contrast d⊥ in the Nd phase is broad and comparable to that observed in the isotropic micellar phase. The value obtained for d⊥ from the maximum in the scattering peak can not be regarded as an exact value in the same way as the value obtained for d . However it is now possible to calculate the mean aggregation number for an Nd phase assuming the micelles are hexagonally arranged in layers with lattice parameter a where a = √23 d⊥ . √

The volume of a primitive cell, containing one disk, V = a2 23 d = 2 √ d2 d . 3 ⊥ The volume of one disk micelle is then = √23 d2⊥ d|| φhc and the mean aggregation number per micelle is given by 2 d|| d2⊥ φhc

n = √ 3 vhc The orientational order parameter for director, n ˆ ﬂuctuations can be obtained from the angular spread of the peak associated with d . 3.4 Summary X-ray scattering is a powerful technique for investigating the phase structure of surfactant water systems. The scattered intensity from a system is made up of two components, the scattering from the individual objects and the inter-particle interference function. One or other of these may dominate. For most concentrated surfactant water phases the degree of positional order allows us to assume that the scattering is dominated by inter-aggregate interference. Hence, they may be treated as crystals in one or more dimensions and identiﬁed. With some simplifying assumptions structural information may be easily extracted. Some phases, concentrated micellar solution, nematic and mesh show liquid-like features where it is diﬃcult to separate, the aggregate scattering and the inter aggregate interference. Scattering from dilute micellar solution where the individual particle scattering dominates, can give information about the mean radius of gyration but is rather insensitive to the shape and size distribution of the particles.

4 2 H NMR Nuclear magnetic resonance (NMR) is a very powerful technique for investigating chemical structure, phase transitions, molecular orientation and conformations, molecular dynamics and self diﬀusion in lyotropic liquid crystalline systems. Detailed reviews of these applications have been published elsewhere [27–29]. In this chapter the use of 2 H NMR will be considered simply as a technique for the identiﬁcation and phase characterization of surfactant water mixtures.

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Some Basic NMR Theory The nucleus is composed of protons and neutrons which possess ‘spin’ or intrinsic angular momentum, I. Nuclei of spin I ≥ 1 may possess non-spherically symmetrical nuclear charge distributions, resulting in their having electric quadrupole moments, which can interact with the electric ﬁeld gradient at the position of the nucleus so that a series of quantized energy levels are produced, even if there is no external magnetic ﬁeld. When a quadrupolar nucleus is placed in a magnetic ﬁeld, two energy terms need to be considered: the Zeeman and the quadrupolar contributions [30], that is 3m2I − I(I + 1) γh (3 cos2 θq − 1) (8) E = − mI B0 + e2 Qq 2π 8I(2I − 1) where γ, the magnetogyric ratio, is a constant characteristic of the particular nucleus, B0 is the externally applied magnetic ﬁeld, and mI is the nuclear spin magnetic quantum number, which has values of mI = I, I −1, I −2, ..., −I. The ﬁrst term in (8) is the Zeeman energy, and the second term is the quadrupolar contribution. eq is the electrical ﬁeld gradient, eQ is the quadrupole moment, θq is the angle between B0 and the electric ﬁeld gradient, and e2 qQ/h is the quadrupole coupling constant. For a deuterium nucleus with I = 1, (8) reduces to: EmI = −

γh 3m2I − 2 mI B0 + e2 Qq (3 cos2 θq − 1) 2π 8

(9)

If the nucleus is exposed to radio-frequency energy of the appropriate frequency, transitions will be induced between the levels given in (9) , and because the population of a lower state is slightly greater than that of an upper, there will be a net absorption of energy. A nuclear magnetic resonance spectrometer thus consists of a means of producing a strong magnetic ﬁeld, a source of radio frequency power, and a means of detecting absorption of energy by the sample. Since mI in (9) possesses three values, -1, 0, +1, there are three energy levels given by γh 1 E1 = − B0 + e2 Qq (3 cos2 θq − 1) 2π 8 2 E0 = +e2 Qq (3 cos2 θq − 1) 8 γh 1 B0 + e2 Qq (3 cos2 θq − 1) E−1 = 2π 8 Applying the selection rule ΔmI = ±1, the 2 H NMR spectrum is a doublet with a splitting given by: (E−1 − E1 )/¯h = Δν =

Δν0 (3 cos2 θq − 1) 2

(10)

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where Δν0 = 3e2¯hQq . Fig. 14(a) shows the typical doublet spectrum from a sample with a single orientation for θq whilst ﬁg. 14(b) shows the typical spectrum from a powder sample, i.e. where there is a spherical distribution of values of θq .

Fig. 14. (a) shows the typical doublet spectrum from a sample with a single orientation for θq whilst (b) shows the typical spectrum from a powder sample, i.e. where there is a spherical distribution of values of θq . This is known as a Pake powder pattern.

The two pronounced peaks in Fig. 14(b) arise from θq = 90o since in a spherical distribution this is the angle with the largest population. Measuring the peak to peak splitting from the Pake powder will give Δν =

Δν0 Δν0 (3 cos2 θq − 1) = − . 2 2

The minus sign simply refers to the fact that for θq = 90o the relative sizes of E−1 and E1 have been reversed. For practical purposes the sign can be ignored.

5 2 H NMR from Surfactant Water Systems Deuterium can be introduced into a surfactant system in two ways: either by speciﬁcally deuterating a protonated group on the surfactant molecule or by replacing the solvent water by 2 H2 O. The former type of experiment is more diﬃcult to do but can provide information about the degree of orientational and motional order of the molecule or even a group on that molecule. The

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latter type of experiment is easy to achieve and can provide information on the orientational order and surface curvature of the surfactant aggregate and also on the fraction of water bound to the interfacial region. This information can lead to the identiﬁcation and characterization of phase transitions within these systems. In the remainder of this chapter the information that can be obtained from the 2 H NMR of solvent 2 H2 O will be discussed although where appropriate results from speciﬁc deuteration will also be referred to. 5.1 Fraction of Bound Water The 2 H NMR spectra and quadrupolar splitting obtained from the 2 H of bound 2 H2 O molecules in lyotropic phases provides considerable information on their nature [31–33]. Rendall et al [31] have carried out a study of the concentration dependence of the quadrupolar splitting (Δν) recorded from 2 H2 O in non-ionic surfactants and concluded that its main component comes from the 2 H2 O bound to the ﬁrst one or two ethylene oxide groups (EO) attached to the alkyl chain. Thus it may be considered that solvent 2 H2 O can eﬀectively sit in two environments: one where they are associated with the aggregate water interface and one in free water. The bound molecules are in a dynamic exchange with the unbound molecules. If the exchange is fast, it will result in a smaller quadrupolar splitting, given by Δν = pf Δνf + pb Δνb where pi is the fraction of bound water at site i and the subscripts f and b refer to the free and bound sites for water. For free water Δνf is zero because of the rapid tumbling of the water molecule and Δν = pb Δνb . The measured splitting can be written Δν =

1 Δν0 p b (3 cos2 θq − 1) 2

(11)

5.2 Frames of Reference and Order Parameters The angle θq is not very convenient. There are three frames of reference and three associated angles which it is useful to use, Fig. 15. These are ﬁrstly the angle, θqi between the electric ﬁeld gradient, eq and the normal to the interface, ˆi, secondly the angle, θin between the normal to the interface and the symmetry axis or director, n ˆ of the phase, and ﬁnally the angle, θnB between the director of the phase and the applied magnetic ﬁeld, B 0 . In any experiment many water molecules are being considered and these will also be undergoing motional averaging so that (11) can be rewritten as 1 1 Δν = Δν0 p b Sqi (3 cos2 θin − 1) (3 cos2 θnB − 1) 2 2

(12)

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Fig. 15. Frames of reference and the associated angles used in the text.

where Sqi is the order parameter of the bound water molecules and is given by 1 Sqi = (3 cos2 θqi − 1) 2 In the case of a lamellar phase θin = 0◦ and (12) becomes 1 Δν = Δν0 p b Sqi (3 cos2 θnB − 1) 2 In contrast, for a hexagonal phase, where the director of the phase lies along the cylinder axes θin = 90◦ (12) becomes 1 Δν = −Δν0 p b Sqi (3 cos2 θnB − 1) 4 Notice therefore that there is a factor of two diﬀerence between Δν measured in the lamellar and hexagonal phases, all other parameters being equal. Where the phase has some intermediate structure, for example a mesh phase there will be diﬀusion of water molecules over the interfacial surface averaging the term 12 (3 cos2 θin − 1) and deﬁning a new order parameter Sin which reﬂects the amount of curvature in the system and the degree of departure from a lamellar phase. Equation (12) becomes 1 Δν = Δν0 p b Sqi Sin (3 cos2 θnB − 1) (13) 2 For a lamellar phase Sin = 1 whilst for a bicontinuous cubic phase Sin = 0 and no powder pattern is observed, only a single line. A mesh phase will generally have 21 ≤ Sin ≤ 1.

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Finally in nematic phases the surfactant aggregate is small and is itself ˆ and B o motionally averaged about the B o axis. The relationship between n is well deﬁned since n ˆ will align either parallel or perpendicular to B o [26]. The 2 H NMR spectrum shows well deﬁned spectra such as, in the parallel case a single doublet or, in the perpendicular case a 2-dimensional powder pattern. Δν reﬂects the orientational order parameter of the aggregate itself, SnB deﬁned by 1 SnB = (3 cos2 θnB − 1) 2 Equation (13) then becomes 1 Δν = Δν0 p b Sqi SnB (3 cos2 θni − 1) 2

(14)

In order to extract SnB assumptions must be made about the behavior of pb Sqi . The quadrupolar splitting Δν from a surfactant / 2 H2 O system reﬂects a number of factors; the degree of macroscopic order in the sample through the term (3 cos2 θnB − 1), the degree of surface curvature through Sin , and the hydration of the interface through pb . The technique is therefore useful not only to help determine phase structure but also to monitor changes of surface curvature and hydration with additives. It can also be used to investigate the macroscopic orientation of phases with the application of external ﬁelds such as electric, magnetic and shear ﬁelds. 5.3 Determination of Macroscopic Director Order The term (3 cos2 θnB −1) reﬂects the degree of director order in the sample and in all phases except nematic phases, this term is not motionally averaged. Fig. 16 shows some examples of 2 H NMR spectra taken from the C16 EO6 /2 H2 O system [34]. Fig. 16(a) shows a near perfect 3-dimensional Pake powder pattern; however notice that the θnB = 0◦ shoulders are slightly distorted indicating some departure from a true powder and probably reﬂecting partial ordering of n ˆ in this mesh phase along the ﬁeld direction. In Fig. 16(b) there is a 2-dimensional powder pattern with pronounced peaks at the θnB = 0◦ positions. Fig. 16(c) shows a near perfect pattern with a single n ˆ orientation although there are weak θnB = 0◦ shoulders and also some inﬁlling between the θnB = 90◦ peaks, denoting a second weaker external force e.g. surface orientation which is aﬀecting a small percentage of the sample. Fig. 16(d) shows a poor 3-dimensional Pake powder pattern reﬂecting an intermediate state of director order probably caused by several conﬂicting external forces acting on n ˆ . It may also reﬂect the thermal history of the sample.

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Fig. 16. 2 H NMR spectra taken from the C16 EO6 /2 H2 O system [34]. (a) a 3dimensional Pake powder pattern, (b) a 2-dimensional powder pattern, (c) a pattern with a single n ˆ orientation and (d) a poor 3-dimensional Pake powder pattern reﬂecting an intermediate state of director order.

Determination of Phase Transitions 2

H NMR provides an excellent method for determining phase transitions; both their temperatures and type. Transitions from isotropic phases to ordered mesophases are easy to see from inspection of the spectra. It is also easy to see transitions between phases that are ﬁrst order where there is a two phase region and the two phases coexist and change in proportion as the region is crossed, ﬁgures 17 and 18. In addition to the information provided by the spectra it is useful to plot the Δν taken from the inner peaks (θnB = 90◦ in (12) ) since that will reveal both ﬁrst and second order phase transitions between mesophases. Fig. 19 shows Δν plotted as a function of temperature for the C16 EO6 / 2 H2 O system [23,34]. Starting at high temperature where SAXS shows there to be a lamellar phase, there is a slow increase in Δν with decreasing temperature reﬂecting the increasing hydration of the EO head groups through the term pb in (12) (see below). At 46◦ C Δν starts to decrease and SAXS measurements

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Fig. 17. 2 H NMR spectra taken from the C16 EO6 / 2 H2 O system [34] , at the following temperatures (a) 31◦ C, (b) 30◦ C, (c) 29◦ C, and (d) 27◦ C. The sample concentration is 51% by weight of C16 EO6 and the two phases present are the V1 (Ia3d) cubic phase (single isotropic line) and hexagonal phase (Pake powder).

Fig. 18. 2 H NMR spectra taken from the C16 EO6 / 2 H2 O system [34] , at the following temperatures (a) 24◦ C, (b) 27◦ C, and (c) 28◦ C. The sample concentration is 50% by weight of C16 EO6 and the two phases present are the gel phase, Lβ (outer doublet) and hexagonal phase (2-dimensional powder).

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Fig. 19. 2 H NMR spectra taken on cooling from the C16 EO6 / 2 H2 O system [34] with a sample concentration of 50% by weight of C16 EO6 . Closed points refer to the scale on the left, whilst open points refer to the scale on the right and are a replotting of the closed squares on an expanded scale to show details of the changes in gradient. Region A is a two phase region of V1 (Ia3d) cubic phase (single isotropic line) and Mh1 (R3m) phase. Region B is a V1 (Ia3d) cubic phase.

show there to be a random mesh phase, Mh1 (0). The transition from Lα to Mh1 (0) phase is virtually continuous and is likely to be second order or only very weakly ﬁrst order. No two phase coexistence region is detected between the two phases. Similarly at 32.5◦ C where there is a transition from Mh1 (0) to Mh1 (R3m) phase, the transition is second order and not detectable by NMR. SAXS shows the transition is simply the ordering of already existing water ﬁlled pores into a positionally ordered structure. At 30◦ C the NMR clearly shows a two phase coexistence region (labeled A in Fig. 19) between Mh1 (R3m) and V1 (Ia3d) phases denoting a ﬁrst order transition associated with the signiﬁcant reorganization associated with the change in phase structure [21]. Notice that Δν for the Mh1 (R3m) phase is virtually constant in this region. In region B of Fig. 19 there is only a single isotropic line which agrees with the SAXS evidence of the V1 (Ia3d) phase. Finally below 25◦ C there is another two phase region composed of hexagonal phase, H1 and gel phase Lβ , again conﬁrmed by SAXS. 2 H NMR is the best way of delineating the phase boundaries. Although it cannot deﬁnitively identify phases as can be done with SAXS, it can precisely ﬁx the location of boundaries and two phase regions which can be diﬃcult with SAXS. The two phase regions can also be useful in delineating boundaries of phases which exist at concentrations that are too high or are beyond the scope of the study. This is achieved using the lever rule and measuring the

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relative proportions of phases and how they change with concentration in regions which are experimentally accessible. 5.4 Determination of Order Parameters It is possible to extract information from Δν plotted as a function of temperature in Fig. 19 about the order parameter, Sin . At high temperature where SAXS shows there to be a lamellar phase, (12) becomes Δν =

1 Δν0 p b Sqi 2

(15)

and the slow increase in Δν with decreasing temperature reﬂects the increasing hydration of the EO head groups through the term pb . At 46◦ C Δν starts to decrease and SAXS measurements show there to be a random mesh phase, Mh1 (0). The decrease is driven by the increase in interfacial curvature around the water ﬁlled pores. Equation (12) is now Δν =

1 Δν0 p b Sqi Sin 2

(16)

The splitting Δν can be extrapolated from the lamellar phase to lower temperature. This assumes that all the temperature change comes from the increase in hydration, pb . The extrapolated values, Δν lam are the extrapolated values of Δν in the absence of a phase change. Taking the ratio between Δν and Δν lam gives a value for Sin and a measure of curvature in the mesh phase. From (15) and (16) Δν = Δν lam

1 2 Δν0 pb Sqi Sin 1 2 Δν0 pb Sqi

= Sin

(17)

Fig. 20 shows this for the results plotted in Fig. 19. The order parameter Sin , which is 1 in the lamellar phase, falls to about 0.7 in the Mh1 (0) phase reﬂecting the increase in surface curvature. In the hexagonal phase it has dropped to 0.6 which is slightly larger than the theoretical value of 0.5 (see above). This method rests on the assumption that pb is the only factor that changes with temperature in (15) and that this is approximately linear. In a recent paper Baciu et al [35] tested this hypothesis by also examining the behavior of α-deuterated surfactants. Sin determined by 2 H NMR from water or from the surfactant was found to be identical in both C12 EO5 and C16 EO6 systems. In the C12 EO5 system the minimum value of Sin in the mesh phase was again found to be about 0.65. 5.5 Summary 2 H NMR is able to provide an accurate method for the mapping of phase diagrams in surfactant water systems. It can identify ﬁrst order phase transitions

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Fig. 20. The ﬁlled points show Δν as a function of temperature after Fig. 19. The heavy solid line is the extrapolation of Δν lam and the open points are the values of Sin calculated from (17) . Vertical lines represent the phase transitions after Fig. 19.

by the presence of two phase regions and also higher order transitions from the discontinuity in Δν when plotted as a function of temperature. Whilst the identiﬁcation of the exact nature of phases from NMR can be ambiguous, its use in conjunction with X-ray scattering or optical polarizing microscopy can provide a powerful combination. NMR not only enables the mapping of phase diagrams but also allows the state of head group hydration and the interfacial curvature to be probed. This can be of particular value when investigating the eﬀect of additives on the phase structure of these systems [36, 37].

6 Outlook Both of these techniques have been well used for many years for a wide variety of systems. However sometimes they are not used together. Recently there has been interest in using these systems as the basis for templating nanostructured materials. The addition of the metals or starting materials from which the templating takes place can disrupt the mesophase which is being used. It becomes important to be able to understand the variation of phase structure with additives which can be done easily with these techniques. A second area of interest may be the use of polar solvents and surfactants to act as templating media [38]. It will be interesting to see the development of the techniques for these novel systems.

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References 1. Fairhurst, C. E., Fuller, S., Gray, J., Holmes, M. C., Tiddy, G. J. T. In Handbook of Liquid Crystals, Demus, D., Gray, G. W., Goodby, J. W., Spiess, H. W., Vill, V., Eds., Wiley-VCH Verlagsgesellschaft mbH: Weinheim, Germany, 1998, pp 341-392. 2. Coleman, N. R. B., Attard, G. S. Microporous and Mesoporous Materials 2001, 44, 73-80. 3. Attard, G. S., Leclerc, S. A. A., Maniguet, S., Russell, A. E., Nandhakumar, I., Bartlett, P. N. Chem. Mater. 2001, 13, 1444. 4. Attard, G. S., Fuller, S., Tiddy, G. J. T. Journal of Physical Chemistry B 2000, 104, 10426-10436. 5. Attard, G. S., Edgar, M., Goltner, C. G. Acta Mater. 1998, 46, 751-758. 6. Fujii, H., Ohtaki, M., Eguchi, K. J. Am. Chem. Soc. 1998, 120, 6832-6833. 7. Yan, X. W., Chen, H. Y., Li, Q. Z. Acta Chim. Sin. 1998, 56, 1214-1217. 8. Tanford, C. The Hydrophobic Eﬀect; Wiley: New York, 1973. 9. Holmes, M. C. Curr. Opin. Colloid Interface S. 1998, 3, 485-492. 10. International Tables For Crystallography; Fourth ed., Kluwer Academic: Dordrecht, Boston & London, 1995, Vol. A. 11. Guinier, A. X-ray Diﬀraction: In Crystals, Imperfect Crystals and Amorphous Bodies; Dover Publications: 1994. 12. Seddon, J. M., Templer, R. H. Phil. Trans. R. Lond. A 1993, 344, 377-401. 13. Luzzati, V., Tardieu, A., Gulik-Krzywicki, T., Rivas, E., Reiss-Husson, F. Nature 1968, 220, 485-488. 14. Luzzati, V. In Biological Membranes, Chapman, D., Ed., Academic Press: London and New York, 1968, pp 71-123. 15. Luzzati, V., Tardieu, A., Gulik-Krzywicki, T. Nature 1968, 217, 1028-1030. 16. Reiss-Husson, F., Luzzati, V. J. Phys. Chem. 1964, 68, 3504-3511. 17. Skoulios, A., Luzzati, V. Acta Crystallogr. 1961, 14, 278-286. 18. Luzzati, V., Mustacchi, H., Skoulios, A., Husson, F. Acta Crystallogr. 1960, 13, 660-667. 19. Husson, F., Mustacchi, H., Luzzati, V. Acta Crystallogr. 1960, 13, 668. 20. Holmes, M. C., Leaver, M. S. In Bicontinuous Liquid Crystals, Lynch, M. L., Spicer, P. T., Eds., CRC Press, Taylor & Francis Group, Boca Raton, 2005, pp 15-39. 21. Leaver, M. S., Fogden, A., Holmes, M. C., Fairhurst, C. E. Langmuir 2001, 17, 35-46. 22. Puntambekar, S., Holmes, M. C., Leaver, M. S. Liq. Cryst. 2000, 27, 743-747. 23. Fairhurst, C. E., Holmes, M. C., Leaver, M. S. Langmuir 1997, 13, 4964-4975. 24. Holmes, M. C., Charvolin, J., Reynolds, D. J. Liq. Cryst. 1988, 3, 1147-1155. 25. Holmes, M. C., Reynolds, D. J., Boden, N. J. Phys. Chem. 1987, 91, 5257-5262. 26. Boden, N., Radley, K., Holmes, M. C. Mol. Phys. 1981, 42, 493-496. 27. S¨ oderman, O., Stilbs, P. Prog. Nucl. Magn. Reson. Spectros. 1994, 26, 445-482. 28. Halle, B., Quist, P.-O., Fury, I. Liq. Cryst. 1993, 14, 227-263. 29. Lindblom, G., Oradd, G. Prog. Nucl. Magn. Reson. Spectros. 1994, 26, 483-515. 30. Harris, R. K. Nuclear Magnetic Resonance Spectroscopy; Longman Scientiﬁc & Technical: 1986. 31. Rendall, K., Tiddy, G. J. T. J. Chem. Soc. ,Faraday Trans. 1 1984, 80, 33393357.

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32. Rendall, K., Tiddy, G. J. T., Trevethan, M. A. J. Colloid Interface Sci. 1984, 98, 565-571. 33. Rendall, K., Tiddy, G. J. T., Trevethan, M. A. J. Chem. Soc., Faraday Trans. 1 1983, 79, 637-649. 34. Burgoyne, J., Holmes, M. C., Tiddy, G. J. T. J. Phys. Chem. 1995, 99, 60546063. 35. Baciu, M., Olsson, U., Leaver, M. S., Holmes, M. C. J. Phys. Chem. B 2006, 110, 8184-8187. 36. Baciu, M., Holmes, M. C., Leaver, M. S. J. Phys.Chem. B 2007, 111, 909-917. 37. Wang, Y., Holmes, M. C., Leaver, M. S., Fogden, A. Langmuir, 2006, 22, 1095110957 38. Warnheim, T. Curr. Opin. Colloid Interface S. 1997, 2, 472-477.

Polyelectrolyte Diblock Copolymer Micelles Small Angle Scattering Estimates of the Charge Ordering in the Coronal Layer Johan R. C. van der Maarel National University of Singapore, Department of Physics, 2 Science Drive 3, Singapore, 117542

1 Introduction Amphiphilic diblock copolymers with a polyelectrolyte block comprise two linearly attached moieties: a charged and a hydrophobic chain part. Owing to their speciﬁc properties and the increased need of water supported polymer materials, these copolymers have found widespread applications from the stabilization of colloidal suspensions, through encapsulation and delivery of bioactive agents, to the control of gelation, lubrication, and ﬂow behavior [1,2]. Besides these technological applications, progress in this area also has implications for biophysics. Polyelectrolyte brushes are a model system for the external envelope of certain microorganisms (glycocalix) and are thought to play a role in, e.g., cell recognition and cushioning properties of synovial ﬂuid [3, 4]. The hydrophobic attachment provides a mechanism for self-assembling of the copolymers into units of mesoscopic size, which are large compared to the molecular dimensions. Major factors controlling the self-assembled structures are solvent composition, charge, ionic strength, and chemical nature and the respective sizes of the blocks. For ionic diblocks of poly(styrene-block -acrylate) (PS-b-PA) with a polyelectrolyte (PA) block length smaller than the length of the polystyrene (PS) block, a multitude of diﬀerent “crew-cut” structures has been observed by Eisenberg and coworkers [5–7]. These structures include hexagonal wormlike cylinders, lamellae, and (compound) vesicles (see Fig. 1). If the length of the core-forming block is comparable to or smaller than the length of the corona-forming block, it was observed that the copolymers associate to form spherical micelles with a hydrophobic core and a polyelectrolyte corona (Fig. 2). It should be noticed, however, that due to the high glass temperature (363 K) of the PS-block, the observed structures are in a frozen meta-stable state and they depend critically on how the samples are prepared. Once the structures are formed after cooling below the glass temperature, the functionality (i.e., aggregation number) and morphology are ﬁxed. We can use this phenomenon however to our advantage, because the frozen structures

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provide excellent model systems to investigate the properties of the interfacial polyelectrolyte brush without complications related to copolymer rearrangements and concomitant changes in morphology.

Fig. 1. Asymmetric amphiphilic block copolymers in solution. Transmission electron micrographs reprinted from [7].

Fig. 2. A schematic drawing of a polyelectrolyte copolymer micelle. The polyelectrolyte corona surrounds the hydrophobic core. The ions are trapped in the coronal layer.

Here, we review our studies of spherical micelles of PS-b-PA polyelectrolyte diblock copolymers with degree of polymerization 20 and 85 of the PS and PA blocks, respectively. At ambient temperature, the PS core is in a glassy state,

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which results in micelles with ﬁxed core size and functionality (the glass temperature of PS, Tg = 363 K). The PA corona charge is pH dependent and we will ﬁrst show with infrared (IR) spectroscopy that the corona charge can be varied between almost zero and full (100%) charge where every monomer carries an ionized group. Transmission electron microscopy (TEM) will be used to verify the micelle morphology. Then we will move on to the investigation of the core and corona size and its relation to charge, screening, and counterion distribution with small angle neutron and X-ray scattering techniques (SANS and SAXS). The main results of these scattering studies are osmotic starbranched polyelectrolyte behavior, full corona chain stretching at high charge and minimal screening conditions, and robustness of the coronal layer against the salinity generated by the addition of salt [8–10]. This behavior is strikingly diﬀerent from the situation for uncharged spherical polymer brushes and/or star-branched polymers, where the chains take a compact, coiled conformation [11–15]. At low degrees of ionization, the corona charges migrate to the outer micelle region due to the recombination/dissociation balance of the weak polyacid (charge annealing) [16]. We will show that the radial density proﬁle of the counterions is very close to the one for the corona forming copolymer segments and that most, if not all counterions are adsorbed in the coronal layer [17, 18]. We will also review SANS and SAXS experiments on our model system up to concentrations where the coronas have to shrink and/or interpenetrate in order to accommodate the micelles in the increasingly crowded volume [19, 20]. In the latter studies, it is observed that, irrespective of ionic strength, the corona shrinks with increasing packing fraction. At high charge and minimal screening conditions, the corona layers eventually interpenetrate once the volume fraction exceeds a certain critical value. Finally, we will show that this interpenetration of the arms of the micelle has a profound eﬀect on the ﬂuid rheology.

2 Small Angle Neutron and X-Ray Scattering From intensities to structure factors. The structure factors describing the density correlations of the PS and PA copolymer blocks are obtained with SANS and contrast matching in water by variation of the H2 O/D2 O solvent composition. We will also review SANS experiments in which the structural arrangement of tetramethylammonium (TMA+ ) counterions in the coronal layer of PS-b-PA micelles (PS-b-TMAPA) is investigated. The use of TMA+ counterions instead of Na+ allows contrast variation by isotope labeling of the counterion. For this purpose, a fraction of the TMA counterions is labeled by deuteration, while the contributions to the scattering related to the PA-blocks are blanked by contrast matching in water. In the case of our SAXS experiments, we have neutralized the polyelectrolyte copolymer with CsOH. Since the Cs+ ion is much heavier (atomic number Z = 55) than the organic copolymer atoms, the scattering is dominated by the counterions in

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the coronal layer. Accordingly, the SAXS intensity is directly proportional to the counterion structure factor, but there is also a small contribution from the copolymer. For a diblock PS(NP S )-b-PA(NP A ) copolymer solution, with NP S and NP A the number of monomers of the PS and PA block, respectively, it is convenient to consider the blocks as the elementary scattering units [21]. Every PS block is attached to a PA block, and, hence, the macroscopic block concentrations exactly match the copolymer concentration ρP S = ρP A = ρ. The coherent part of the solvent corrected scattering intensity is given by ¯bi¯bj Ni Nj Sij (q) I(q)/ρ = (1) i,j

where the summation runs over all structure factors Sij (q) pertaining to the density correlations among the solutes (i = j = PS, PA, and TMA; Sii is abbreviated as Si ) and solute pairs (i = j). The scattering length contrast vs , where the solute (i) and solvent (s) have scattering reads ¯bi = bi − bs v¯i /¯ lengths bi and bs and partial molar volumes v¯i and v¯s , respectively. Here, the contrast is matched either in water by variation of bs through the H2 O/D2 O solvent composition (PS and PA) or by variation of bi through deuteration of the counterion (TMA). Momentum transfer q is deﬁned by the wavelength λ and scattering angle θ between the incident and scattered beam according to q = 4π/λ sin (θ/2). The partial structure factors Sij (q) are the spatial Fourier transforms of the density correlation functions −1 ρ (q) = dr exp (−iq · r) ρi (0)ρj (r) (2) Sij V

In a selective solvent, the copolymers form spherical aggregates with a hydrophobic PS block core and a polyelectrolyte PA block corona. If the radial density of the corona is assumed to be invariant to ﬂuctuations in inter-micelle separation, the structure factor Eq. (2) takes the form −1 Sij (q) = Nag Fi (q)Fj (q) Scm (q)

(3)

with the micelle aggregation number Nag , the form factor amplitude Fi (q), and the micelle center of mass structure factor Scm (q). In the absence of interactions between the micelles and/or at suﬃciently high values of momentum transfer Scm (q) reduces to unity. The form factor amplitude Fi (q) can be expressed in terms of the radial core (i = PS) or corona (i = PA) density ρi (r) Fi (q) = dr exp (−iq · r) ρi (r) = drsin (qr)/(qr)4πr2 ρi (r) (4) Vmicelle

The scattering amplitudes are normalized to Nag at q = 0.

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Solution structure factor. For polyelectrolyte copolymer micelles, an analytic expression for the center of mass solution structure factor is not available. We have analyzed the data with a hard sphere potential and the Percus-Yevick approximation for the closure relation [22]. The solution structure factor has the form −1 (q) − 1 = 24φ [αf1 (Dhs q) + βf2 (Dhs q) + φαf3 (Dhs q)/2 ] Scm

with

2

α=

(1 + 2φ)

4

(1 − φ)

(5)

2

,

β=−

3φ (2 + φ)

4

2 (1 − φ)

(6)

and f1 (x) = (sin(x) − x cos(x)) /x3 f2 (x) = 2x sin(x) − (x2 − 2) cos(x) − 2 /x4 f3 (x) = (4x3 − 24x) sin(x) − (x4 − 12x2 + 24) cos(x) + 24 /x6

(7)

The ﬁt parameters are the hard sphere diameter Dhs and the volume fraction 3 ρmic with micelle density ρmic . The hard sphere diameter should φ = π/6Dhs be interpreted as an eﬀective diameter; its value could be smaller than the outer micelle diameter if interpenetration occurs. Furthermore, it is known that for soft objects the hard sphere potential does not correctly predict the relative amplitudes of the primary and higher order correlation peaks [23]. As will be discussed below, we have also tested a sticky hard sphere model and a repulsive screened Coulomb potential [24, 25]. However, the eﬀect of electrostatic interaction among the micelles was found to be modest, which is attributed to the fact that almost all neutralizing counterions are conﬁned in the coronal layer. Core and corona form factors. The core can be described by a homogeneous dense sphere with density ρP S and diameter Dcore . Accordingly, the radial PS block density is uniform for 0 ≤ 2r ≤ Dcore and given by 3 ρP S (r)π Dcore /6 = Nag and zero for 2r > Dcore . For such uniform proﬁle, the core scattering amplitude reads 3

FP S (q) = Nag 3 (sin (qDcore /2) − (qDcore /2) cos (qDcore /2))) /(qDcore /2) (8) Due to the relatively small core size and the mutual segment repulsion induced by the charge, the density in the coronal layer is non-uniform and varies along with the radius away from the core. To describe the corona structure we will adopt an algebraic radial PA block density proﬁle ρP A (r) = ρ0P A (2r/Dcore )−α , Dcore < 2r < Dmic

(9)

where corona chain statistics determines the value of α and ρ0P A is the density at the core - corona interface. The latter interfacial density is related to the outer micelle diameter Dmic through the normalization requirement (i.e., by integration of the radial proﬁle)

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3−α α 3 = 2 (3 − α) Nag π Dmic Dcore − Dcore

(10)

We will calculate the corona form factor amplitude with algebraic proﬁle (9) by numeric integration, although analytical expressions are available [26]. The core PS and corona PA form factors are related to the square of the scattering amplitudes and take the form PP S (q) = FP2 S (q) Nag , PP A (q) = FP2 A (q) Nag (11) The algebraic proﬁle (9) accounts for the average corona density scaling and neglects corona chain ﬂuctuations. The eﬀect of ﬂuctuations on the scattering behavior is important when the momentum transfer is on the order of the intermolecular correlation distance within the corona. Furthermore, they contribute to the corona structure factor (= SP A ) only, the cross term SP A−P S is unaﬀected due to the heterodyne interference between the amplitudes scattered by the homogeneous core and heterogeneous corona [27, 28].

3 Corona Chain Statistics The value of the density scaling exponent α is determined by the chain statistics in the coronal layer. We will gauge the corona statistics from the scaling approaches for star-branched polymers. These polymers can also serve as a model for spherical diblock copolymer micelles. The fact that the coronal region cannot extend right to the center of the micelle merely sets a certain minimum correlation length (i.e., blob size) at the core-corona interface. In the scaling approach the blob size ξ is determined by the condition that the chain remains unperturbed within the blob [29,30]. Each blob contains g monomers, each monomer with a step length l. The blob size ξ is related to the number of monomers g according to (12) ξ lg ν where the value of ν is determined by the chain statistics inside the blob, e.g. ν = 1/2 if the chain is Gaussian and ν = 3/5 with chain excluded volume interactions. For a star-branched spherical micelle, both g and ξ may vary along the radius r away from the core. The radial corona density scales as the number of monomers Nag g (r) in the shell of radius r and thickness ξ(r) ρP A (r) Nag g(r)/ r2 ξ(r) (13) and the exponent α in (9) can be derived from (12) and (13), together with a certain radial dependence of the blob size ξ(r). In the case of a neutral star-branched micelle, close packing of the blobs in the shell of radius r implies the radial dependence of the blob size −1/2 ξ(r) Nag r. In this Daoud–Cotton expanding blob model, the density (3ν−1)/(2ν) (1−3ν)/ν scaling takes hence the form ρP A (r) l−1/ν Nag r and α = 1

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or 4/3 without (ν = 1/2) or with (ν = 3/5) excluded volume interactions, respectively [11]. Borisov and Zhulina have derived scaling expressions for the size and radial density distribution of star-branched polyelectrolytes with and without screening by added salt [31, 32]. Furthermore, due to the recombination and dissociation balance of the weak polyacid block, migration of charges toward the outer coronal region (charge annealing eﬀects) might be important. Here, we will summarize the scaling results for polyelectrolyte stars, as far as they are relevant for the interpretation of the scattering data. For a derivation of the pertinent equations, the reader is referred to the papers by Borisov and Zhulina [31, 32]. Without the presence of supporting electrolyte, two diﬀerent classes of star-branched polyelectrolytes exist. When the fraction fq of ionized groups is very small, the electrostatic screening length is much larger than the micelle size and, hence, inside the corona there is no screening of Coulomb interaction. With increasing fq the majority of the counterions are trapped within the corona and, now, the concomitant osmotic pressure gives the main contribution to the corona stretching force. The transition between the unscreened and the screened, osmotic, micelle occurs at a critical charge fraction fq∗ (l/lB )

1/ν

−1/ν Nag

(14)

with lB the Bjerrum length (0.7 nm at 298 K). For micelles with a large aggregation number, the unscreened regime can only be observed if fq 1. Due to the large aggregation number (around 100), most, if not all, of our samples are in the osmotic regime, and, accordingly, we will only summarize the results for osmotic micelles. As a surprising result, the outer diameter of the osmotic micelle does not depend on the aggregation number and scales with the corona charge fraction according to Dmic N lfq1−ν

(15)

The radial scaling of the blob size ξ (r) can be derived from the balance of the elastic, conformational, stretching force and the osmotic pressure exerted by the counterions trapped inside the blob. Since the fraction of trapped counterions does not vary along the radius, the blob size in the outer region of the corona is constant and is given by ξ l fq−ν

(16)

The formation of radial strings of blobs of uniform size and, hence, uniform mass per unit length results in a density scaling ρP A (r) Nag l−1 fqν−1 r−2

(17)

and hence α = 2. Due to space restrictions, in the inner-corona region the blobs are expected to increase in size (with α on the order of unity) with increasing distance away from the core until the critical size given by (16) is

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reached. The crossover radial distance r∗ between the expanding and constant blob size regions is r∗ l fq−ν . With increasing fq , the expanding blob size region shrinks and eventually vanishes if r∗ becomes on the order of the core radius Dcore /2. Concurrently, due to the osmotic pressure of the counterions, the corona becomes uniformly extended (α = 2) and can be envisioned as strings of blobs of constant size given by (16). The presence of the core limits the range over which the blobs are allowed to expand. This is in contrast to the situation of star-branched polyelectrolytes, where the expanding blob region extends right to the center of the star. An additional screening of Coulomb interaction becomes important when the concentration ρs of added salt exceeds the concentration of counterions ρi in the coronal layer. The corona stretching force is now proportional to the diﬀerence in osmotic pressure of co- and counterions inside and outside the micelle. This diﬀerence in osmotic pressure can be obtained by employing the local electroneutrality condition and Donnan salt partitioning between the micelle and the bulk of the solution. The micelle outer radius is obtained by balancing the total osmotic stretching force with the elastic force. In the salt dominated regime (i.e., if ρs ρi ), the overall diameter of the micelle scales as 1/5 −1/5 ρs (18) Dmic Nag l2 fq2 N 3 An increase in salt concentration results hence in a gradual contraction of the micelle, because of additional screening of the Coulomb repulsion among the ionized polyelectrolyte block monomers (i.e., a decrease in electrostatic excluded volume interactions). The radial dependence of the blob size −1/3 2/3 r ξ (r) Nag l4 fq2 ρ−1 s

(19)

is obtained from the local balance of diﬀerential osmotic force and the local tension in the star arms. As each arm exhibits locally Gaussian statistics (ξ lg 1/2 ), the block density proﬁle can derived from (13) and (19) and reads 2 −2 −2 1/3 −4/3 fq ρs l r ρP A (r) Nag

(20)

The radial decay of the monomer density is described by the same exponent as in neutral star-branched polymers with excluded volume interactions in a good solvent. Accordingly, in the salt dominated regime (ρs ρi ), the corona density scaling exponent takes the Daoud-Cotton expanding blob value α = 4/3. However, in contrast to neutral stars, the elastic blobs in partially screened polyelectrolyte micelles have a blob-size scaling exponent 2/3 (19) rather than unity and hence they are not closely packed. For polyelectrolyte stars at intermediate ionic strengths, Borisov and Zhulina proposed a multiple-region-scaling model. At small distances from the core, where ρi > ρs , the corona statistics is not aﬀected by the added salt. Here, the chains are extended in the radial direction with uniform mass per unit length (α = 2, in the innermost coronal region α may be of the order of

Polyelectrolyte Diblock Copolymer Micelles

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unity due to space restrictions). With increasing distance away from the core, the local counterion concentration decreases and for r > rs the screening is governed by the salt. Hence, in the outer corona region, where ρs ρi (r), the corona density scaling exponent takes the value α = 4/3. The crossover distance 1/2 −1/2 ρs (21) rs Nag l−1 fqν is determined by the equality of the local counterion concentration ρi (r) = fq ρP A (r) in the coronal layer of the osmotic star [with radial density from (17)] and the salt concentration ρs in the bulk. With the addition of simple salt, the fully ionized micelle contracts according to (15) with a concomitant decrease in the crossover distance rs between the inner- and outer-corona scaling regimes with density scaling exponents α = 2 and 4/3, respectively. The salt penetrates the micelles and the radial decay of the monomer density scaling in the outer region is similar to the situation for neutral star-branched polymers. The blobs are not closely packed however and increase in size away from the crossover according to (19). The inner-corona region, characterized by radial strings of blobs of uniform size, remains unaﬀected until the salt concentration competes with the salinity generated by the counterions coming from the dissociation of the polyelectrolyte. PA is a weak polyacid and at low fraction of ionized monomers (fq 1) the eﬀects of charge annealing are important. In the simplest approximation, the local charge fraction fq (r) is determined by the (presumably r-independent) ionization constant K and the mass action law K = ρ(r)fq (r)2 / (1 − fq (r)) ≈ ρ(r)fq (r)2 . Because of the dissociation and recombination balance, the charge fraction is now no longer constant and increases according to fq (r) ∼ r2/(1+ν) . A remarkable result of this charge annealing eﬀect is that the local tension in the branches now increases with increasing distance away from the core. As the branches become more extended with increasing r, the blob size ξ decreases ξ (r) l r−2ν/(1+ν) and the monomer density decays faster [32] ρP A (r) l−1 r−4/(1+ν)

(22)

The density scaling exponent α takes the value 8/3 or 5/2 without (ν = 1/2) or with (ν = 3/5) volume interactions, respectively. Although the addition of salt might shift the recombination-dissociation balance and, hence, inﬂuence the ionization constant K (by replacing hydronium ions by cations from the salt), the corona scaling behavior is not aﬀected. Furthermore, in our experiments, the added salt concentration is in excess of the counterion concentration and the additional screening results in a contraction of the coronal layer according −1/5 to Dmic ∼ ρs , as in the case of highly charged micelles. The diﬀerent scaling regimes under various conditions are illustrated in Fig. 5.

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1

Absorbance

0.5 1

0

0

0.5

1

0.5 0 1 0.8 1500 0.6 1600 0.4 1700 0.2 v (cm−1) 0 1800 DN

Fig. 3. IR spectra versus degree of neutralization DN from 0.15 mole of PA/dm3 PS-b-PA solutions [16]. The inset displays the integrated peak intensities of the COO− (1570 cm−1 , open symbols) and C=O (1700 cm−1 , closed symbols) asymmetric and symmetric stretching bands, respectively, normalized to the COO-band at DN = 1.

4 Polyelectrolyte Block Ionization The ionization of the poly(acrylic acid) block can be monitored with IR spectroscopy [16]. In the range of 1500 to 1800 cm−1 two bands can be distinguished. As displayed in Fig. 3, the charge neutralization results in the appearance of the asymmetric COO− stretching band at 1570 cm−1 with a concurrent disappearance of the C=O stretching band around 1700 cm−1 . The relative peak intensities are displayed in the inset of Fig. 3. To a good approximation, the peak intensities are proportional to the molar ratio of (added) alkali and polyacid monomer (i.e., the degree of neutralization DN ). The COO− and the C=O peak intensities vanish in the limit of no (DN = 0) and full (DN = 1) charge neutralization, respectively. For fully neutralized samples, every acid group of the polyelectrolyte block is ionized. Without the addition of alkali (DN = 0), the copolymer is still weakly charged due to the auto-dissociation of the COOH group. However, as judged from potentiometry, the degree of auto-dissociation is of the order 10−3 and is beyond the accuracy of the IR experiment [33]. Accordingly, the polyelectrolyte block charge fraction fq can be tuned between almost zero and unity by adjusting the degree of neutralization DN.

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5 Association Morphology

Fig. 4. TEM replicas of PS-b-PA solutions with DN = 1 (a) and 0 (b) [16]. The DN = 1 sample was imaged directly, whereas the DN = 0 replica was obtained with freeze-fracture techniques. The bar corresponds to 100 nm. Reproduced from [16]. Copyright (2000) American Chemical Society.

The morphology of the self-assembled structures was examined with TEM [16]. Fig. 4 shows carbon replicas of solutions with full (DN = 1) and almost zero (DN = 0) polyelectrolyte block charge. The DN = 1 sample was imaged directly (after evaporation of the solvent from the grid, ﬁlm strengthening, and shadowing), whereas the DN = 0 replica was obtained with freeze-fracture techniques. For both cases, the replicas show homogeneously dispersed and well separated individual spherical micelles. With the ionization of the polyelectrolyte block, the diameter of the micelles increases due to the concomitant corona expansion from ∼30 to ∼ 50 nm. The estimated diameters are in the range of those determined by SANS (see below), and the TEM micrographs are, hence, sensitive to the physical extent of the corona. This is not a surprising result, because the carbon/platinum ﬁlm covers the whole micelle and not only the core (or the freeze-fractured surface). We have nevertheless refrained from further analysis of the TEM micrographs, because of the uncertainties

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in size estimation introduced by the fracturing and shadowing procedures. The morphology of the individual micelles is further investigated with SANS and contrast matching in the water. As will be shown below, SANS provides an in situ measurement technique without the risk of damage and/or micelle deformation caused by the TEM drying or freeze-fracturing procedures.

(a)

(b)

(c)

(d)

Fig. 5. Schematic representation of the various corona scaling regimes: (a), neutral, space ﬁlling expanding blobs with ρ (r) ∼ r−4/3 ; (b) weak charge and salt free, radial strings of blobs of decreasing size with ρ (r) ∼ r−5/2 ; (c) salt dominated, non space ﬁlling radial strings of expanding blobs with ρ (r) ∼ r−4/3 ;(d) high charge and salt free, radial strings of blobs of equal size with ρ (r) ∼ r−2 . The scaling laws pertain to locally swollen chains in good solvent.

6 Core Structure The core PS and corona PA structure factors were determined by SANS and contrast variation in the water. SANS was measured with the D22 and PAXY diﬀractometers situated on the cold nuclear reactor sources of the Institute Laue-Langevin and Laboratoire L´eon Brillouin, respectively. As an illustrative example, the PS partial structure factor pertaining to fully charged micelles without added salt is displayed in Fig. 6 (the corona structure factors will be discussed below) [20]. Here, the copolymer concentration covers the range from the diluted to the concentrated regime where the coronal layers have to shrink and/or interpenetrate in order to accommodate the micelles in the increasingly crowded volume. At the lowest micelle concentration and/or with

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excess salt, inter-micelle interference is insigniﬁcant and the core structure factor can directly be compared with the core form factor. With increasing concentration and minimal screening conditions, a primary and higher order correlation peaks emerge. As is more clearly demonstrated by the SAXS results described below, the position of the primary peak scales with the copolymer concentration ρ according to ρ1/3 . This scaling behavior is characteristic for micelles with ﬁxed aggregation number and isotropic symmetry in the local environment. There are no changes in the high q behavior of the core structure factor, irrespective charge, the presence of salt, and copolymer concentration. The lines in Fig. 6 represent the model calculations with Dcore = 9 nm (the parameters pertaining to the ﬁt of the center of mass structure factor, Dhs and ρmic , are discussed below). From the absolute normalization of the structure factors in the long wavelength (q → 0), an aggregation number Nag around 100 is derived. The aggregation number agrees with the value obtained from the core size, PS molecular weight, and PS partial molar volume. This shows that the solvent is excluded from the core and the PS is closely packed with a density close to the macroscopic density 1.05 kg/dm3 . The ﬁxed functionality is due to the fact that all experiments were done well below the glass temperature of PS (363 K).

108

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Fig. 6. Core PS structure factor versus momentum transfer for fully charged PSb-PA micelles without added salt [20]. The copolymer concentration is 44 (), 30 (♦), 17 ( ), and 4.4 (◦ ) g/L from top to bottom. The data are shifted along the y-axis with an incremental multiplication factor. The curves represent the model calculations with core diameter 9 nm.

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Fig. 7. SAXS intensity divided by copolymer concentration versus momentum transfer for fully charged PS-b-CsPA micelles without added salt [20]. The inset shows the peak position qm versus copolymer concentration ρ in the double logarithmic representation. The line represents qm ∼ ρ1/3 scaling.

7 Counterion Structure The SAXS intensities of the fully neutralized PS-b-CsPA micelles divided by copolymer concentration are displayed in Fig. 7 [20]. The synchrotron SAXS experiments were done at the BM26 “DUBBLE” beam line of the European Synchrotron Radiation Facility. Notice that the SAXS intensities are, to a good approximation, proportional to the counterion structure factor, because the scattering is dominated by the heavy Cs+ ions. As in the case of the core structure factor, the SAXS data show a primary inter-micelle correlation peak. The higher order correlation peaks are less prominent, due to the steep decrease of the counterion structure factor with increasing values of momentum transfer. As seen in the inset of Fig. 7, the position of the primary peak scales with copolymer concentration according to ρ1/3 . This result conﬁrms the isotropic local structure and ﬁxed aggregation number of the micelles, which has also been observed with SANS. At high values of momentum transfer, the SAXS intensities are seen to increase with increasing concentration. This eﬀect might be due to counterion ﬂuctuations; a ﬁt to a Gaussian background contribution yields a correlation length, which decreases from 1.2 to 0.7 nm with increasing copolymer concentration from 4.5 to 48 g/L, respectively.

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Fig. 8. Comparison of SAXS (solid curves) intensities with the reconstituted SANS intensities (symbols) [20]. The data are shifted along the y-axis with an incremental multiplication factor. The copolymer concentration is 44 (for SAXS 48, ), 30 (♦), 17 ( ), and 4.5 (◦ ) g/L from top to bottom. For the most densely concentrated set, notice the slight diﬀerence in position of the correlation peak due to a small diﬀerence in micelle concentrations. Reprinted with permission from [20]. Copyright (2005) American Institute of Physics.

It is interesting to compare the SAXS data for the PS-b-CsPA micelles with the relevant combination of the core PS and corona PA scattering contributions (there is an optimized 2% contribution from PS to the SAXS intensity and this accounts for the scattering of the core). The comparison in Fig. 8 shows a perfect agreement in both the position of the correlation peak and the variation of the structure factor with momentum transfer. We have also measured the counterion structure factor with SANS from 12 g/L samples with isotopically labeled tetramethylammonium (TMA) counterions [17]. The TMA partial structure factor is displayed and compared with the PA (corona) partial structure factor in Fig. 9. The TMA and PA structure factors show an almost perfect match in both the scaling with momentum transfer and absolute normalization. The ratio of the counterion and corona structure factors is, hence, q-independent with an average value 0.99 ± 0.10. The quantitative agreement shows that, within a 10 % error margin, all counterions are conﬁned to the coronal region. Furthermore, the distribution of the counterions along the radius is very close to that of the monomers of the corona-forming

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blocks. It is also clear that the counterions remain strongly correlated with the coronal chains with increasing packing fraction up to and including the regime where the coronal layers interpenetrate. Due to the neutralization of the coronal layer by trapping of the counterions, the micelles are almost electroneutral and the electrostatic contribution to the inter-micelle interaction potential is expected to play a minor role. The concomitant osmotic pressure exerted by the trapped counterions gives, hence, the main contribution to the corona stretching force and the micelles are in the osmotic regime. 2

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Fig. 9. Comparison of the PA corona and the TMA counterion partial structure factors. ( ): PA partial structure factor in PS-b-NaPA solutions (DN = 0.6). (•): TMA partial structure factor in PS-b-TMAPA solutions (DN = 0.5) [17]. The polyelectrolyte block concentrations are 0.1 mole of PA/dm3 . The solid curve represents the structure factor with density scaling exponent α = 2. The inset displays the radial density. The small diﬀerence in the degrees of neutralization (0.6 vs 0.5 for PS-b-NaPA and PS-b-TMAPA, respectively) has no eﬀect on the corona structure factor beyond experimental accuracy.

8 Corona Structure From neutral to fully charged micelles. We have measured the PA corona partial structure factors for diﬀerent degrees of neutralization and at a ﬁxed

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Fig. 10. PA corona partial structure factor versus momentum transfer for salt free micelles with degree of neutralisation: DN = 1 (◦ ), 0.6 ( ), 0.35 (♦), 0.1 (), 0.04 (), and 0 (∼ =) from top to bottom [16]. The data are shifted along the y-axis with an incremental multiplication factor. The copolymer concentration is 12 g/L. The solid curves represent the corona structure factor with corona density scaling exponent α = 5/2 for DN = 0, 0.04 and 0.1 and α = 2 for DN = 0.35, 0.6 and 1. The dashed curves represent the corona structure factor with α = 4/3 for DN = 0, 0.04 and 0.1 only.

micelle concentration (12 g/L) [16]. The results are displayed in Fig. 10. With decreasing degree of neutralization, the corona shrinks and, hence, the structure factor scales toward higher values of momentum transfer and the minimum at q ≈ 0.3 nm−1 becomes more pronounced (in the double logarithmic representation). Furthermore, a correlation peak develops at ﬁnite wavelengths, which is similar to the one observed in the core structure factor and is due to the ordering of the micelles. In the case of star-branched polyelectrolytes with a relatively small number of arms (∼ 12), a second correlation peak at higher values of momentum transfer has been reported [34]. The latter peak is due to ﬂuctuations and correlations among the branches within a single star. For the copolymer micelles, the number of branches (i.e., the aggregation

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number) is much higher around 100. The average segment density within the corona exceeds, say, 0.2 M. As judged from the position of the correlation peak in linear polyelectrolyte solutions at a comparable concentration, the corresponding intra-corona correlation peak is expected at ∼ 0.7 nm−1 [35]. For such high values of momentum transfer the PA structure factor is very small on the order of the error margin. Hence, the intra-corona correlation peak and the eﬀect of corona chain ﬂuctuations are beyond detection. Furthermore, the copolymer concentration is suﬃciently low in the dilute regime that the outer sections of the arms do not overlap (see below). The lines in Fig. 10 are the model structure factors calculated with an average density proﬁle Eq. (9). Rather than optimizing the density scaling exponent α, the calculation of the corona structure factor was done with α = 2 for the samples with degree of neutralization DN = 0.35, 0.6 and 1.0. For the weakly charged micelles with DN = 0, 0.04 and 0.1, we have plotted the structure factors with α = 4/3 and 5/2. The value α = 2 is relevant in the case of full stretching of the chains or the formation of radial strings of blobs of uniform size. A density scaling exponent α = 4/3 results from the Daoud–Cotton expanding blob model for uncharged star-branched polymers and α takes the value 5/2 in the presence of charge annealing eﬀects. The latter two values refer to good solvent conditions, because water is a good solvent for PA. In the calculation of the corona structure factor, the micelle diameter Dmic was optimized, whereas the other parameters were set at their nominal values obtained from the ﬁtting of the PS core structure factor. The ﬁtted diameters versus degree of neutralization to the power 3/5 are displayed in Fig. 11. For highly charged micelles with degree of neutralization 0.35 and higher, an exponent α = 2 gives good agreement between the corona structure factor and the data. Although the scaling of the structure factors toward higher q values with decreasing DN can be reproduced by a decrease in outer micelle radius, the results with α = 2 fail to describe the position of the minimum at q ≈ 0.4 nm−1 for samples with low corona charge fraction (results not shown). A Daoud–Cotton expanding blob model for neutral stars with α = 4/3 shifts the position of the minimum towards too low q values and does not give a good agreement with the data (this is most marked for our DN = 0.1 sample). We have checked that an analysis in terms of multiple scaling regimes pertaining to, e.g., expanding and constant blob size regions does not improve the description of the PA structure factor. It is necessary to increase the density scaling exponent α beyond the value two in order to predict the position of the minimum in the structure factor correctly. Indeed, with α = 5/2 nice agreement is observed, which shows the importance of the dissociation and recombination balance of the weak polyacid (which results in a migration of the charges towards the outer corona region). With increasing charge fraction, the micelle diameter ﬁrst increases, subsequently levels oﬀ above 10 % charge neutralization, and reaches the value 25 nm at full charge. The fully stretched value of the radius amounts to 25.5 nm, as estimated from the sum of the core radius 4.5 nm and the contour length

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Fig. 11. The micelle diameter versus the scaled degree of neutralization calculated with corona density scaling exponent α = 2 ( ; DN = 0.35, 0.6, and 1) and 5/2 (◦ ; DN = 0, 0.04, and 0.1) [16]. The bottom and top dashed curves represent the core and fully expanded diameters, respectively. The solid line denotes the scaling result for an osmotic salt-free polyelectrolyte star in a good solvent (ν = 3/5).

of the PA-block (85 monomers per chain with a vinylic step length 0.25 nm). As can be seen in Fig. 11, the NaPA chains in the coronas of the micelles take an almost fully stretched conﬁguration at high pH and no added simple salt. The expansion behavior can be rationalized with the scaling results for star-branched polyelectrolytes in the osmotic regime. According to (14) with Nag ∼ 100, the transition from the unscreened to the screened, osmotic, micelle occurs at a critical charge fraction fq∗ ≈ 10−4 . Even the non-neutralized (DN = 0) micelle carries a suﬃcient amount of charge to retain the major part of its auto-dissociated protonic counterions inside the corona and, hence, is in the osmotic regime. For osmotic micelles, the micelle radius scales with the charge fraction according to N l fq1−ν (15). To a good approximation, the charge fraction equals the degree of neutralization. Fig. 11 displays the micelle diameter versus the scaled degree of neutralization with the good solvent value ν = 3/5. According to (15) there should be a linear dependence, which is indeed observed within the experimental accuracy. Salt induced contraction. We will now discuss the response of the corona statistics and micelle diameter to the addition of salt [10]. As in the case of our investigation of the eﬀect of corona charge, the copolymer concentration is ﬁxed at 12 g/L. As can be seen in Fig. 12, with increasing salt concentration the corona shrinks and the structure factor scales toward higher values of momentum transfer with a concurrent sharpening of the minimum

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at q ≈ 0.3 nm−1 (in the double logarithmic representation). Again, the model structure factors are ﬁtted to the data with an average density proﬁle of the corona given by (9). Without added salt, the fully ionized PA chains are near 100 % stretched with a density scaling exponent α = 2 and outer micelle diameter Dmic = 48 nm. The core and micelle diameter as well as the aggregation number set the PA chain density at the core - corona interface ρ0P A through the normalization requirement (10) (the grafting density is 2.6−1 nm−2 ). The latter interfacial density was subsequently ﬁxed and the data pertaining to samples with added salt were ﬁtted with Borisov and Zhulina’s two-region scaling model [32]. According to this model, the corona is divided into two regions: an inner-corona region where the chain statistics is not aﬀected by the salt (α = 2) and an outer region with a density scaling governed by screened electrostatic excluded volume interactions (α = 4/3). The crossover distance rs is the only adjustable parameter and determines, together with ρ0P A , the outer micelle diameter Dmic . The ﬁtted structure factors are also displayed in Fig. 12 and the outer and cross-over diameters, Dmic and 2rs , respectively are shown in Fig. 13. In particular, the change in shape of the corona structure factor with the addition of salt is nicely reproduced. With the addition of salt, the micelle contracts with a concomitant decrease in rs . In 1 M excess salt, we almost recover the Daoud–Cotton expanding blob scaling with α = 4/3 (the optimized crossover distance between the two diﬀerent scaling regimes becomes very close to the core radius) and the outer diameter (34 nm) approaches the value pertaining to the pure acid form at minimal screening conditions 30 nm. Accordingly, despite the very high salt concentrations; the range in micelle dimension is similar to the one that can be covered by variation of the corona charge fraction (e.g., by variation of the pH of the supporting buﬀer medium). The salt-induced contraction can be rationalized with the scaling results for star-branched polyelectrolytes [31,32]. For osmotic micelles in the salt dominated regime, the crossover and the outer micelle −1/5 −1/2 and ρs [ diameters scale with the salt concentration according to ρs (18) and (21)], respectively. As shown in Fig. 13, these scaling laws are indeed observed within the experimental accuracy. The crossover occurs at an optimized distance rs , where the concentration of salt ions exceeds that of the corona segments by a factor 0.16, 0.22, and 0.52 in 0.05, 0.2, and 1 M KBr, respectively (note, however, that in 1 M salt the statistics is almost completely governed by the salt). In view of the simplicity of the model (a gradual transition between the two diﬀerent scaling regions is more likely to occur), this agreement between the counterion and salt concentrations at the crossover distance within an order of magnitude can be considered quite gratifying. Eﬀects of micelle concentration. We have also measured the corona PA partial structure factor for a range of micelle concentrations from the diluted to the concentrated regime where the coronal layers have to shrink and/or interpenetrate in order to accommodate the micelles in the increasingly crowded volume [19, 20]. The concentration dependence was investigated for samples with degrees of neutralization DN = 0.1, 0.5 and 1. Furthermore,

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Fig. 12. PA partial structure factor versus momentum transfer for fully ionized micelles (DN = 1) with KBr concentrations: ρs = 1 M (), 0.2 M (♦), 0.05 M ( ), and salt-free (◦ ) from top to bottom [10]. The data are shifted along the y-axis with an incremental multiplication factor. The curves represent the corona structure factor with an inner and outer corona density scaling exponent α = 2 and 4/3, respectively, together with the values for the crossover diameters 2rs displayed in Fig. 13.

the micelles were either salt-free or bathed in 0.04 (DN = 0.1) and 1 M (DN = 0.5 and 1) monovalent salt. As an illustrative example, the results pertaining to fully charged, simple salt free micelles are displayed in Fig. 14. There are no major changes in the high q behavior of the corona structure factor with increasing packing fraction, irrespective charge and ionic strength. This shows that the corona chain statistics is rather insensitive to inter-micelle interaction. The lines in Figs. 14 represent the model calculations. For salt-free micelles with a charge fraction exceeding 10%, the chains remain almost fully stretched and α = 2. For the 10% charged micelles, we have done the model calculations with α = 5/2 in accordance with charge annealing towards the outer coronal region due to the recombination-dissociation balance of the weak polyacid. In the presence of excess salt, the corona structure factors are compared with the

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D (nm)

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Fig. 13. The outer micelle Dmic (◦ ) and crossover 2rs ( ) diameter versus the added salt concentration for fully ionized PS-b-PA (DN = 1) micelles [10]. The top dashed curve represents the micelle diameter without added salt, whereas the bottom dashed curve denotes the core diameter. The solid lines denote the scaling results −1/5 for an osmotic polyelectrolyte star in the salt dominated regime with Dmic ∼ ρs −1/2 and 2rs ∼ ρs for the outer and cross-over diameter, respectively. Notice the robustness of the outer micelle diameter against the addition of salt.

structure factor calculated with α = 4/3 (100 and 50% charge) or 5/2 (10% charge). All ﬁtted micelle diameters are displayed in Fig. 15. With added salt and/or at low degree of ionization, the coronal layers are less extended. The ionic strength and charge dependencies of the micelle diameter agree with the results obtained for more diluted samples discussed in the previous section. With increasing packing fraction, the diameter of the micelles decreases. However, the extent to which the coronal layers shrink is modest and similar under all circumstances. The gradual decrease in size is due to the restricted free volume, increased counterion adsorption, and/or Donnan salt partitioning between the coronal layer and the supporting medium.

9 Inter-Micelle Structure Inter-micelle interference is more clearly demonstrated in Fig. 16, where the core structure factor has been divided by the core form factor (full charge and no added salt). Although the center of mass structure factor could also be derived from the corona structure factor, we have chosen to use the core structure

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Fig. 14. Corona PA structure factor versus momentum transfer for fully charged PS-b-PA micelles without added salt [20]. The copolymer concentration is 44 (), 30 (♦), 17 ( ), and 4.4 (◦ ) g/l from top to bottom. The data are shifted along the y-axis with an incremental multiplication factor. The curves represent the model calculations.

factor because the core form factor shows a smooth and moderate variation in the relevant q-range. The intensity of the correlation peaks ﬁrst increases and eventually levels oﬀ with increasing packing fraction, which shows the progressive and saturating ordering of the micelles. Notice that for the present volume fractions the position of the primary peak is mainly determined by density, whereas the respective positions of the higher order correlation peaks are most sensitive to the value of the hard sphere diameter. The lines in Fig. 16 represent the hard sphere solution structure factor convoluted with the instrument resolution and ﬁtted micelle densities ρmic and hard sphere diameters Dhs . The hard sphere model is capable of predicting the positions of the primary and higher order peaks. Furthermore, the ratio of the ﬁtted micelle densities and known copolymer concentrations provides an alternative way to obtain the aggregation number. The average value of the aggregation number Nag = 98 ± 10, as obtained from the ﬁtted densities, is in perfect agreement with the one obtained from the normalization of the structure factors. Clear

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Fig. 15. Concentration dependence of the diameter of 100 (a), 50 (b), and 10% (c) charged PS-b-PA micelles: (•), Dmic without added salt; ( ), Dmic in 1.0 M (100 and 50%) or 0.04 M (10% charge) KBr [20]. The hard-sphere diameter of salt-free micelles Dhs is denoted by ( ). The lines represent ρ−1/3 , i.e. the average intermicelle distance.

deviations between the experimental data and the hard sphere prediction are observed in the low q-range for samples with higher micelle densities. Furthermore, the model underestimates the intensity of the second order peak with respect to the primary one. We have checked that a repulsive, screened Coulomb potential does not improve the ﬁt, nor does it signiﬁcantly inﬂuence the peak positions for reasonable values of the micelle charge [24]. The minor importance of the electrostatic interaction between the micelles and the relatively small net micelle charge due to the trapping of the counterions in the coronal layer are demonstrated by similar center of mass structure factors for 10, 50 and 100% charged micelles. The failure in predicting the relative amplitude of the higher order peak is probably related to the softness of the micelles; similar behavior has been reported for interpenetrating neutral polymer stars [36, 37]. The deviations observed in the low q-range might be due to long-range inhomogeneity in density, the formation of aggregates, and/or stickiness between the micelles. We have checked that the sticky hard sphere model does not improve the ﬁt in the low momentum transfer range, but it gives a better description of the depth of the ﬁrst minimum after the primary peak (result not shown) [25]. However, the derived distance of closest

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Fig. 16. Center of mass solution structure factor for fully charged PS-b-PA micelles versus momentum transfer [20]. The copolymer concentration is 44 (), 30 (♦) and 17 ( ) g/L from top to bottom. The data are shifted along the y-axis with an increment of 1.5 units. The curves represent the hard sphere solution structure factor.

micelle approach is the same as the hard sphere diameter and we have further refrained from interpreting our data with this more elaborate model. The ﬁtted outer micelle Dmic and hard sphere Dhs diameters as obtained from the corona form and solution structure factor analysis, respectively, are displayed in Fig. 15. Dmic and Dhs are estimated within a 3 and 2% accuracy margin, respectively, which is about the size of the symbols. The hard sphere diameters were derived for salt-free micelles only, because in the presence of excess salt inter-micelle interference is eﬀectively suppressed. For the less concentrated, 15 and 17 g/l solutions, the hard sphere diameters equal the micelle diameters derived from the form factor analysis. This supports the applicability of the hard sphere interaction model in order to extract the eﬀective hard sphere diameters. At higher packing fractions and for the 50 and 100% charged micelles in particular, the eﬀective hard sphere diameters are significantly smaller than the outer micelle diameters. We take the diﬀerence as a measure of the extent to which the corona layers interpenetrate. Accordingly, the 100 and 50% charged, salt-free micelles interpenetrate around 17 g/l; for

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the smaller 10% charged micelles this happens at a higher concentration, say 25 g/l.

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Fig. 17. Viscosity versus the shear rate for 50% charged PS-b-PA micelles without added salt [20]. The copolymer concentration is 44 (), 30 (♦), 17 ( ) and 4.5 (◦ ) g/L from top to bottom. Reprinted with permission from [20]. Copyright (2005) American Institute of Physics.

Fig. 15 also displays the average distance between the micelles ρ−1/3 . Once the micelles interpenetrate, the eﬀective hard sphere diameter equals ρ−1/3 . Based on the optimized micelle densities and hard sphere diameters, eﬀective micelle volume fractions can be calculated. For interpenetrating micelles, the eﬀective volume fraction is found to be constant within experimental accuracy and takes the value 0.53±0.02. Notice that, although this volume fraction corresponds with closely packed, simple cubic order, the center of mass structure factor remains liquid-like and no long-range order in the SAXS and SANS diﬀraction patterns is observed. Interpenetration of the arms of the micelle thus occurs when the eﬀective volume fraction reaches the critical value 0.53 (and remains constant for higher micelle densities).

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Fig. 18. Frequency dependence of storage G’ (open symbols) and loss G” (ﬁlled symbols) modulus for 50% charged PS-b-PA micelles without added salt [20]. The copolymer concentration is 44 (), 30 (♦), 17 ( ), and 4.5 (◦ ) g/L from top to bottom. The lines indicate ω 1 and ω 2 scaling. Reprinted with permission from [20]. Copyright (2005) American Institute of Physics.

10 Visco-Elastic Behavior The interpenetration of the coronal layers has a profound inﬂuence on the visco-elastic properties [20]. All samples are ﬂuid and ﬂow when the test tubes are inverted. We have measured the viscosity of the solutions with 50% charged micelles. In excess salt (1 M KBr), the viscosity is in the range 1-2 mPa s, which is on the order of the viscosity of the solvent (data not shown). The viscosity versus shear rate of the samples without added salt is displayed in Fig. 17. A Newtonian plateau is observed, which increases in value by 3 orders of magnitude when the concentration is increased so that the coronal layers interpenetrate. Notice that the salt-free sample with the lowest micelle concentration has already a 10 fold higher viscosity than the ones with excess salt. For the more concentrated samples, the onset of shear thinning at high shear rates is also observed. We have also measured the dynamic moduli of the solutions with 50% charged micelles without added salt. Data are shown in Fig. 18. For the lowest concentration, the dynamic moduli show viscous liquid behavior with G (ω) ∼ ω 2 and G (ω) ∼ ω 1 . For interpenetrating micelles and in the lower

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frequency range in particular, G (ω) and G (ω) show approximately parallel scaling laws as a function of frequency with scaling exponents around unity. Such behavior has been observed for a wide variety of polymer gels and similar micellar solutions of polyelectrolytes with adhesive corona chains [2, 38]. Although we do not observe the transition to an elastic solid (G > G and almost independent on frequency), an intuitive explanation of the parallel frequency scaling behavior of G and G is the formation of an interconnected network of micelles by the interpenetration of the coronal layers as shown by the scattering experiments.

11 Conclusions and Outlook Our PS-b-PA micelles have served as an excellent model system to investigate the properties of the spherical polyelectrolyte brush under various conditions. We have found that the eﬀect of charge ordering in the coronal layer can be rationalized on the basis of scaling relations originally derived for osmotic polyelectrolyte stars. This is mainly due to the relatively large functionality of our micelles, which results in the conﬁnement of small counterions in the coronal layer. Accordingly, the main stretching force of the coronal chains is due to the osmotic pressure exerted by the trapped counterions. Due to the osmotic stretching force, the polyelectrolyte corona chains take an (almost) fully stretched conformation at high charge and minimal screening conditions. Furthermore, the coronal layer is rather robust against the salinity generated by the presence of salt and against compression forces resulting from the interaction between brushes. With increasing packing fraction, the brushes eventually interdigitate which results in the formation of a physical gel and a dramatic increase in solution viscosity. These properties are important for the design of materials for technological applications as well as for our understanding of certain biological processes involving glycoprotein brushes such as synovial lubrication. Owing to the high glass temperature of the core forming segments, our micelles have a ﬁxed functionality and the spherical morphology does not change under the various conditions. Future work can focus on polyelectrolyte block copolymers with a liquid domain of the self-assembled hydrophobic attachment under true thermodynamic equilibrium conditions. Here, as an additional level of complexity, the morphology can be controlled by the chemical potentials of the constituents, which might depend on stimuli such as temperature, pH, and ionic strength. Furthermore, so far we have studied the simplest possible morphology, i.e. spherical micelles. An obvious extension of our work is to investigate the properties of other supra-molecular assemblies of polyelectrolyte diblock copolymers such as rods and vesicles. The rod-like assemblies are particularly promising, because the asymmetric shape might result in the formation of a lyotropic liquid crystal once the packing fraction exceeds a certain critical value. The vesicular structures have already proven

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to be of interest for, e.g., the encapsulation of clone vector DNA for gene delivery [39]. Acknowledgement. I thank Wendy Groenewegen, Sasha Korobko, Wim Jesse, Stefan Egelhaaf, and Alain Lapp for their collaboration. I acknowledge the Institute LaueLangevin, Laboratoire L´eon Brillouin, and the European Synchrotron Radiation Facility in providing the neutron and X-ray research facilities, respectively. The staﬀ of these facilities is thanked for their assistance during the scattering experiments.

References 1. G. Riess, Prog. Polym. Sci. 28, 1107 (2003). 2. S. R. Bhatia, A. Mourchid, and M. Joanicot, Curr. Opin. Colloid Interface Sci. 6, 471 (2001). 3. A. Albersd¨ orfer and E. Sackmann, Eur. Phys. J. B 10, 663 (1999). 4. M. Tanaka, F. Rehfeldt, M. F Schneider, G. Mathe, A. Albersd¨ orfer, K. R Neumaier, O. Purrucker, and E. Sackmann, J. Phys.: Condens. Matter 17, S649 (2005). 5. L. Zhang and A. Eisenberg, Science 268, 1728 (1995). 6. M. Moﬃtt, K. Khougaz, and A. Eisenberg, Acc. Chem. Res. 29, 95 (1996). 7. N. S. Cameron, M. K. Corbierre, and A. Eisenberg, Can. J. Chem. 77, 1311 (1999). 8. P. Guenoun, F. Muller, M. Delsanti, L. Auvray, Y.J. Chen, J.W. Mays, and M. Tirrell, Phys. Rev. Letters 81, 3872 (1998). 9. S. F¨ orster, N. Hermsdorf, C. Bottcher, and P. Lindner, Macromolecules 35, 4096 (2002). 10. J.R.C. van der Maarel. W. Groenewegen, S.U. Egelhaaf, and A. Lapp, Langmuir 16, 7510 (2000). 11. M. Daoud, and J.-P. Cotton, J. Phys. 43, 531 (1982). 12. W. D. Dozier, J. S. Huang, and L. J. Fetters, Macromolecules 24, 2810 (1991). 13. K. A. Cogan, A. P. Gast, and M. Capel, Macromolecules 24, 6512 (1991). 14. S. F¨ orster, E. Wenz, and P. Lindner, Phys. Rev. Letters 77, 95 (1996). 15. C. M. Marques, D. Izzo, T. Charitat, and E. Mendes, Eur. Phys. J. B. 3, 353. (1998). 16. W. Groenewegen, S. U. Egelhaaf, A. Lapp, and J. R. C. van der Maarel, Macromolecules 33, 3283 (2000). 17. W. Groenewegen, A. Lapp, S. U. Egelhaaf, and J. R. C. van der Maarel, Macromolecules 33, 4080 (2000). 18. L. Belloni, M. Delsanti, P. Fontaine, F. Muller, P. Guenoun, J.W. Mays, P. Boesecke, and M. Alba, J. Chem. Phys. 119, 7560 (2003). 19. A. V. Korobko, W. Jesse, S. U. Egelhaaf, A. Lapp, and J. R. C. van der Maarel, Phys. Rev. Letters 93, 177801 (2004). 20. A. V. Korobko, W. Jesse, A. Lapp, S. U. Egelhaaf, and J. R. C. van der Maarel, J. Chem. Phys. 122, 024902 (2005). 21. Notice that this deﬁnition diﬀers from the situation for homopolymers, where the monomer is usually considered the elementary scattering unit. 22. J. K. Percus and G. J. Yevick, Phys. Rev. 110, 1 (1958).

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23. C. N. Likos, H. Lowen, M. Watzlawek, B. Abbas, O. Jucknischke, J. Allgaier, and D. Richter, Phys. Rev. Letters 80, 4450 (1998). 24. J. B. Hayter and J. Penfold, Mol. Phys. 42, 109 (1991). 25. R.J. Baxter, J. Chem. Phys. 49, 2770 (1968). 26. S. F¨ orster and C. Burger, Macromolecules 31, 879 (1998). 27. L. Auvray, C. R. Acad. Sc. Paris 302, 859 (1986). 28. L. Auvray and P. G. de Gennes, Europhys. Letters 2, 647 (1986). 29. P. G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, 1979. 30. A. Yu. Grosberg and A. R. Khohklov, Statitical Physics of Macromolecules, American Institute of Physics Press, New York, 1994. 31. O. V. Borisov, J. Phys. II 6, 1 (1996). 32. O. V. Borisov and E. B. Zhulina, Eur. Phys. J. B 4, 205 (1998). 33. This value was obtained from the experimental pK = 7.2 of simple salt-free 0.014 mole of PA/dm3 PS-b-PA solutions, unpublished results. 34. M. Heinrich, M. Rawiso, J. G. Zilliox, P. Lesieur, J. P. Simon, Eur. Phys. J. E 4, 131 (2001). 35. J. R. C. van der Maarel, L. C. A. Groot, J. G. Hollander, W. Jesse, M. E. Kuil, J. C. Leyte, L. H. Leyte-Zuiderweg, M. Mandel, J. -P. Cotton, G. Jannink, A. Lapp, and B. Farago, Macromolecules 26, 7295 (1993). 36. A. Jusuﬁ, C. N. Likos, and H. L¨ owen, Phys. Rev. Letters 88, 8301 (2002). 37. A. Jusuﬁ, C. N. Likos, and H. L¨ owen, J. Chem. Phys. 116, 11011 (2002). 38. H. H. Winter and M. Mours, Adv. Polym. Sci. 134, 165 (1997). 39. A. V. Korobko, C. Backendorf, and J. R. C. van der Maarel, J. Phys. Chem. B 110, 14550 (2006).

Structure and Shear-Induced Order in Blends of a Diblock Copolymer with the Corresponding Homopolymers I. W. Hamley1 , V. Castelletto and Z. Yang2 School of Chemistry, University of Reading, Whiteknights, Reading RG6 6AD, United Kingdom 1

Corresponding author Current address: School of Materials, The University of Manchester, Manchester M60 1QD, United Kingdom 2

1 Introduction The eﬀect of shear on soft materials is relevant to their processing, for example mixtures of polymers are subjected to shear during ﬂow and this can have a profound inﬂuence on their miscibility and rheology. Shear-induced mixing or demixing of polymer blends have both been observed. Shear-induced mixing is the typical behaviour of blends of low molar mass polymers and polymer solutions. However, shear-induced demixing can be observed for solutions of high molar mass polymers or polymer blends at high shear rates. Block copolymers are widely used to compatibilize otherwise immiscible polymers [1–4]. Approximately ten years ago, observations on polymeric microemulsions were ﬁrst reported, these being formed at low copolymer concentrations in a blend of two homopolymers with a small amount of the corresponding diblock [5–8] These systems have immense potential due to the intimate mixing of the two homopolymers that results when the interfacial tension is near zero, as it is in a microemulsion phase. The size of the phase separated domains in the microemulsion can be tuned through application of shear, and the focus of this project was to understand how shear changes the structure, in terms of enhanced mixing or demixing and also the degree of alignment. Because of the focus on microemulsions, the work is also relevant to conventional amphiphilic microemulsions, where the eﬀect of shear has been the focus of a few studies [9, 10]. Light scattering has been used by a number of groups internationally to investigate shear-induced structure formation in polymer blends and solutions [11–16]. In the case of shear-induced mixing, shear ﬂow leads to highly

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oriented stringlike domains which produce a correspondingly anisotropic light scattering pattern perpendicular to the shear direction [11–13]. For the case of shear-induced demixing in semidilute polymer solutions, so-called butterﬂy patterns have been observed; [14–16] also reported for critical polymer blends in the presence of solvent at high shear rates [12]. Light scattering is the most suitable method for investigating phase separation since this typically leads to turbidity in polymer blends, which is the result of density inhomogeneities with a size of the order of that of the wavelength of light. In addition it has been used to probe the shear-induced ordering of surfactant microemulsions (shear leads to the breakup of the microemulsion or sponge and formation of a lamellar phase [9, 10]). However, we are only aware of the work of one group on the eﬀect of shear on polymeric microemulsions, [17, 18] although it has been used to probe shear-induced order in an A/B/AB (A and B indicate distinct polymers) blends [19]. The discovery of polymer microemulsions creates opportunities to study the intricate ordering of micro- and nano-structured soft materials and is the focus of intense research activity internationally, in particular in the US and Japan. It is also of considerable commercial relevance, extending the compatibilization capabilities of present block copolymer-based systems.

2 Experimental 2.1 Materials We studied shear alignment in microemulsions comprising blends of very low molar mass polyethylene (PE), poly(ethylene oxide) (PEG, monomethoxy) and poly(ethylene)-b-poly(ethylene oxide) (PE-b-PEO). These materials were custom synthesized by Polymer Source (Canada). A poly(ethylene)-b-(d4 ethylene oxide) diblock was also purchased from the same supplier, for SANS. An additional sample of monomethoxy PEG was obtained from Fluka, and used for the SAXS experiments described herein. Table 1 lists the characteristics of the materials studied. For the polymers from Polymer Source, GPC was performed (at Manchester) to check purity. Data from the manufacturer (Fluka) were used for the monomethoxy PEG sample PEGb. Since the polymers are of low viscosity, blends were prepared by melt mixing. The densities of PE and PEO, necessary to calculate the total volume fraction of homopolymer, ΦH , were measured using a density gradient column. The density of the PE-b-PEO block copolymer was determined by adding the calculated molar volumes of the two component blocks ([molar volume of PE + molar volume of PEO]/molar mass of the block copolymer). In this way, the following values were obtained for the polymer density: ρ = 0.784, 1.034 and 0.894 g cm−3 for PE, PEO and PE-b-PEO respectively. Samples were prepared by direct mixing of weighed amounts of polymer, followed by annealing at 130◦ C (disordered phase) for 24 hs. After annealing, the sample

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Table 1. Molecular Characteristics Sample PE PEGaa) PEGbb) PE-b-PEO PE-b − d4 PEO

Mn / g mol−1 480 500 424 PE(1100)-b- PEO(1020) PE(1100)-b − d4 PEO (1120)

Mw /Mn 1.16 1.08 1.14 1.08 1.06

Estimated uncertainty: Mn to ±3%; Mw /Mn to ±0.01. a) Monomethoxy, From Polymer Source, b) Monomethoxy, from Fluka

was cooled to room temperature. All blends contain a 50:50 proportion of the homopolymers. The materials studied are similar to those previously investigated by Hillmyer et al [6] However, we have found some diﬀerences in the phase diagram (through extensive SAXS and SANS experiments, see below). 2.2 Light Scattering

Fig. 1. Light scattering instrument and control/imaging PC.

A light scattering instrument was designed and constructed in our lab. The instrument is illustrated in Fig. 1. The laser light source is at the top. It passes through a pinhole and iris onto the sample, mounted horizontally on a table. At the side of the shear cell are the control electronics blocks (platform on left-hand side). The (Rayleigh–Debye–Gans) scattering pattern is collected on a diﬀusing plate, and is focussed onto the CCD camera at the base (via a lens

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mounted in the CCD assembly). Images of scattering patterns are obtained (using the software provided with the LaVision Imager Compact CCD system purchased) on the PC shown on the right-hand side. 2.3 SAXS and SANS Small-angle x-ray scattering was performed in several separate series of experiments on station 6.2 and 16.1 at the Synchrotron Radiation Source, Daresbury Lab, UK. On both stations the wavelength was λ = 1.4 ˚ A. The wavenumber scale q = 4π sin θ/λ where 2θ is the scattering angle, was calibrated using a sample of wet collagen. The sample-detector distance was 3.3 m or 4.0 m on station 6.2 and 5.5 m or 4.5 m on 16.1. In some runs, a two-dimensional multiwire area detector was used, the data subsequently being reduced to one-dimensional form, in other runs a gas-ﬁlled multiwire quadrant detector that directly provides one-dimensional data was used. Small-angle neutron scattering was performed at the SANS-I instrument of the Swiss Spallation Neutron Source SINQ (Paul Scherrer Institute, Switzerland). The scattering data was recorded on a two-dimensional detector. A wavelength λ = 5 ˚ A and sample-detector distances 4.5 m and 18 m were used to cover an extended q range from 0.004–0.183 ˚ A−1 . The overlap between data obtained at the two sample-detector distances was reasonable although the low q data for the shorter camera length was more smeared.

3 Results As a guide, a phase diagram obtained by Hillmyer et al. for a system very similar to our own is presented in Fig. 2 [6]. They studied blends of PE (Mn = 395 g mol−1 ) and PEO (Mn = 500 g mol−1 ) with a nearly symmetric PE-bPEO diblock (Mn = 2130 g mol−1 ). 3.1 Rheology Prior to investigating shear alignment via light scattering, rheology was used to locate phase transitions in blends with compositions in the range expected to span the microemulsion region for the PE/PEO system. Representative results for a blend with ΦH = 0.65 are shown in Fig. 3. A disordering process is observed, occurring between T = 130◦ C and T = 140◦ C. There is also indication of a discontinuity in the moduli at approx T = 120◦ C. Both dynamic shear moduli have rather low values ∼ 102 Pa in the ordered melt, consistent with the observed low viscosities of the component low molar mass polymers. All of these features were also observed for samples with ΦH = 0.70 and ΦH = 0.75. The order-disorder transition in our samples is somewhat higher than that reported by Hillmyer et al. (T = 119−120◦ C for ΦH = 0 to 0.75) on

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Fig. 2. Phase diagram for a PE/PEO/symmetric PE-PEO diblock blend [6]. The total volume fraction of homopolymer is denoted ΦH . The lamellar phase region is denoted L, H denotes a hexagonal phase, and the phase separated region is labelled PS. The microemulsion region lies in between these two phases. (◦ ) Order-disorder transition determined by SANS, ( ) phase separation determined by SANS, ( ) phase separation as determined by visual cloud point measurements.

the basis of SANS data for a similar blend. The diﬀerence may reﬂect small diﬀerences in molar mass of the component polymers and/or composition of the diblock. 3.2 Light Scattering Small-angle light scattering experiments were performed using a Linkam CSS450 shear cell. This instrument has a parallel plate geometry, and steady shear was applied. Accurate temperature control is also achieved. An intensive series of experiments were performed on blends with ΦH = 0.60, ΦH = 0.80 and ΦH = 0.85. Experiments were generally conducted in the ordered melt phase at one ﬁxed temperature, although some experiments were also conducted in the disordered melt. Scattering patterns were recorded in “steady state” conditions, and also as a function of time (kinetic measurements). Fig. 4 shows typical light scattering patterns obtained shearing a ΦH = 0.80 sample, 800 μ m thick, at 5, 10, 35 and 60 s−1 . As shown in SAXS and SANS experiments (see below), this sample is a polymeric microemulsion. When a shear rate γ˙ ≤ 5 s−1 was applied to the sample, the light scattering patterns shows only a faint spinodal ring from the microemulsion structure (Fig. 4a). When a shear rate higher than γ˙ ≈ 10 s−1 was applied to the sample, the

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Fig. 3. Dynamic shear moduli (strain = 1%, frequency ω = 1 rad s−1 , heating rate 2 ◦ C min−1 ) as a function of temperature for a PE/PEO/PE-b-PEO blend with ΦH = 0.65.

scattering intensity grew in the ﬂow direction and gradually developed into a pattern with two ”wings” separated by a dark gap oriented perpendicular to the ﬂow, (Fig. 4b). The dark streak (the gap) narrowed with increasing shear rate, indicating an increase of the characteristic length in the ﬂow direction as beyond a critical shear stress, the blend underwent phase separation. As the shear rate increased, the scattering intensity increased signiﬁcantly and the dark streak disappeared gradually from lower to higher scattering vectors. Eventually, a bright-streak scattering pattern was observed (Fig. 4c). With a further increase in shear rate (γ˙ = 60 s−1 ; Fig. 4d), a dark streak emerged in the SALS pattern, dividing it into two lobes, which progressively separated and became more diﬀuse and the total scattering intensity decreased, since probably high shear rates drive the system towards remixing. Shear-induced phase separation in similar blends has previously been reported by Krishnan et al. for the poly(ethyl ethylene)/poly(dimethylsiloxane) system [17, 18]. Time-resolved measurements were also carried out. Fig. 5 shows representative data for a blend with ΦH = 0.60 during shear at γ˙ = 40s−1 at 80

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Fig. 4. Steady SALS patterns at diﬀerent shear rates for a blend with ΦH = 0.80 at 80◦ C. (a) γ˙ = 5 s−1 , (b) γ˙ = 10 s−1 , (c) γ˙ = 35 s−1 , (d) γ˙ = 60 s−1. . ◦

C. The data represented horizontal sections, illustrating the development of scattering streaks perpendicular to the shear direction. 3.3 Small-Angle Neutron and X-Ray Scattering Extensive small-angle scattering experiments were conducted, ﬁrst to determine phase behaviour (SAXS and SANS) and second to investigate chain conformation (SANS). The results for blends with ΦH = 0.65 are complex. Representative results are shown in Fig. 6a. On heating, a lamellar phase is observed, however on cooling to the same temperature (T = 80 ◦ C is chosen as a reference temperature for Fig. 6a) there is a coexistence of the lamellar phase observed on heating with an additional hexagonal phase. This observation was reproducible in three separate experiments (two of which were performed nearly one year apart). We believe the observation of a distinct structure on cooling reﬂects macrophase separation which occurs in the melt, since all samples were heated from 80 ◦ C to 160 ◦ C prior to cooling. On subsequent cooling, macrophase separated hexagonal and lamellar phases are stable. We are conﬁdent that these results are not related to crystallinity of either PE or PEO since the melting point of these polymers will be much lower than 80 ◦ C at these molar masses [6] (absence of crystallinity was conﬁrmed by WAXS). These results suggest more complex phase behaviour than suggested

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30 second

20

15 0 10 -2

-1.5

-1

-0.5

0

q / μm

0.5

1

1.5

2

−1

Fig. 5. Development of light scattering intensity proﬁle (times indicated) versus scattering vector q (calibrated using a diﬀraction grating).

by Fig. 2 in this composition range, certainly a wider composition range where the hexagonal phase is observed. Another factor to consider is the observation by Hillmyer et al. that the ordered phase structure is very sensitive to blend composition – changes of 1% leading to signiﬁcant changes in morphology near ΦH = 0.8 [6]. Since separate blends were prepared for diﬀerent experiments and compositions were only accurate to ∼ 5%, this can potentially account for some variation from run to run. The formation of a hexagonal phase in a symmetric blend containing a symmetric diblock is surprising, but has been ascribed to the asymmetry of conformation and intermolecular interactions of PE and PEO segments [6]. A recent report on microemulsions formed in the poly(ethylene-alt-propylene)/poly(butylene oxide) (PEP/PBO/PEP-b-PBO) system also mentions the observation of coexisting hexagonal and lamellar phases, and the authors speculate on the possibility of coexistence of hexagonal and other “homopolymer rich” phases in the vicinity of the microemulsion phase [8]. This speculation is consistent with our results. These authors also provide values for the conformational asymmetry parameter, which is actually rather small for the PE/PEO polymer pair, certainly much lower than for PEP/PBO. Nonetheless, our observations and those of Hillmyer et al. [6] indicate that a hexagonal phase is stable. Since the PEO block is the more ﬂexible, it is likely to be on the inside of the curved interface, i.e. the cylinders

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comprise PEO. However, this has yet to be conﬁrmed. Zhuo et al. also note that diﬀerences in solubility of the diblock in the respective homopolymers could lead to asymmetric swelling and hence the formation of non-lamellar structures, an eﬀect that can be ampliﬁed if the diblock is even slightly asymmetric [8].

Fig. 6. SAXS data for (a) ΦH = 0.65 blend at 80◦ C [•], heating; [o] cooling, (b) ΦH = 0.70 blend at 133◦ C. The arrows indicate reﬂections in the ratio 1:3 from a lamellar structure observed on heating and cooling. The bars indicate reﬂections in √ the ratio 1: 3: 2 observed for a hexagonal phase that coexists with a lamellar phase on cooling.

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A further observation for our samples is that there was some evidence from SAXS, rheology and DSC for two transitions at high temperature (see for example the rheology data in Fig. 3) – one at around T = 120 ◦ C prior to the order-disorder transition at T = 130 – 140 ◦ C. The lower transition may be a hexagonal-lamellar transition, as observed for PEP/PBO/PEP-b-PBO microemulsions, although the transition is more pronounced in the latter system [8]. A similarly complex coexistence of lamellar and hexagonal phases was observed for ΦH = 0.70 (Fig. 6b). A sample with ΦH = 0.75 had a hexagonal ordered melt as shown in Fig. 7 which shows SANS data, (SAXS data conﬁrm the structure although with much higher resolution), although the order-disorder transition was much higher than indicated in Fig. 2. It may be noted in passing that slightly diﬀerent results were obtained with the blend containing PEGa. The blends with ΦH = 0.65 and ΦH = 0.70 showed less evidence for a two phase structure and were primarily hexagonal. It is possible that the small diﬀerence in molar mass compared to PEGb could account for these diﬀerences. Blends with ΦH = 0.80 blend adopt a microemulsion

o Fig. 7. SANS √ data for a ΦH = 0.75 blend at 80 C. The bars indicate reﬂections in the ratio 1: 3.

structure. The scattering pattern comprises a single broad peak, centred at A−1 , and a broad shoulder at higher angles for temperatures in q ∗ ∼ 0.021 ˚ the range 80–110◦ C (Fig. 8). The SAXS for temperatures in the range 110A−1 (Fig.8). The 160◦ C only showed a broad peak, centred at q* ∼ 0.025 ˚ broad shoulder at higher angles had disappeared in the SAXS for 110–160◦ C probably due to a reduction in the degree of the microstructural order. For blends with compositions ΦH = 0.825 and ΦH = 0.85 the scattering data was

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Fig. 8. SAXS proﬁles for a ΦH = 0.80 blend at 80◦ C (• ) and at 120◦ C (◦ ).

less clear. There are inconsistencies between SAXS and SANS results, which may result from time-dependent phenomena due to macrophase separation, as expected for these blends based on Fig. 2. Rheology data indicated very low dynamic shear moduli, consistent with the absence of an ordered (microphase separated) structure.

4 Summary The melt morphology of blends comprised of low molar mass polyethylene, poly(ethylene oxide) and a corresponding symmetric polyethylene-bpoly(ethylene oxide) diblock copolymer has been investigated by small-angle x-ray and neutron scattering (SAXS and SANS). The order-disorder transition was identiﬁed by SAXS and shear rheometry. The focus was on the region around the polymeric microemulsion structure observed for a total homopolymer volume fraction ΦH = 0.80. Shear-induced phase separation of the microemulsion was observed by light scattering under steady shear. Blends with ΦH = 0.65–0.70 exhibited complex phase behaviour on cooling from a disordered melt, with coexisting hexagonal and lamellar structures. This is believed to be due to macrophase separation in the high temperature disordered melt. A blend with ΦH = 0.75 adopted a hexagonal microphase separated structure, below the order-disorder transition. Blends with ΦH = 0.83 and above were found to be macrophase separated at all temperatures. With the exception of the complex thermal-history dependent morphology for blends with ΦH = 0.65–0.70, our phase diagram is in good agreement with prior reports [6,8] for similar blends. Our observations of shear-induced phase sepa-

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ration of the polymeric microemulsion resemble previous reports for analogous ternary blends for the poly(ethyl ethylene)/poly(dimethylsiloxane) system. Acknowledgement. We are grateful to Mr Julio Albuerne (Universidad Sim´ on Bolivar, Caracas, Venezuela) for performing the rheology experiments whilst on a summer placement. We thank Anthony Gleeson, Chris Martin, Sasha Mykhaylyk and Petros Parras for assistance with the SAXS experiments at Daresbury, UK and Steven van Petegem for help with SANS experiments at the PSI, Switzerland. We thank Marc Hillmyer for helpful comments and for drawing [8] to our attention after completion of the original draft of this manuscript.

References 1. Polymer Blends; Paul, D. R., Newman, S., Eds., Academic: London, 1978. 2. Polymeric Blends and Alloys, Folkes, M. J., Hope, P. S., Eds., Blackie: London, 1993. 3. Datta, S., Lohse, D. J. Polymeric Compatibilizers: Uses and Beneﬁts in Polymer Blends, Hanser: Munich, 1996. 4. Utracki, L. A. Commercial Polymer Blends, Chapman and Hall: London, 1998. 5. Bates, F. S., Maurer, W. W., Lipic, P. M., Hillmyer, M. A., Almdal, K., Mortensen, K., Fredrickson, G. H., Lodge, T. P. Phys. Rev. Lett. 1997, 79, 849. 6. Hillmyer, M. A., Maurer, W. W., Lodge, T. P., Bates, F. S., Almdal, K. J. Phys. Chem. B 1999, 103, 4814. 7. Washburn, N. R., Lodge, T. P., Bates, F. S. J. Phys. Chem. B 2000, 104, 6987. 8. Zhou, N., Lodge, T. P., Bates, F. S. J. Phys. Chem. B 2006, 110, 3979. 9. Mahjoub, H. F., Bourgaux, C., Sergot, P., Kleman, M. Phys. Rev. Lett. 1998, 81, 2076. 10. L´eon, A., Bonn, D., Meunier, J. Phys. Rev. Lett. 2001, 86, 938. 11. Hong, Z., Shaw, M. T., Weiss, R. A. Macromolecules 1998, 31, 6211. 12. Kim, S., Hobbie, E. K., Wu, J.-W., Han, C. C. Macromolecules 1997, 30, 8245. 13. Kielhorn, L., Colby, R. H., Han, C. C. Macromolecules 2000, 33, 2486. 14. Wu, X.-L., Pine, D. J., Dixon, P. K. Phys. Rev. Lett. 1991, 66, 2408. 15. Milner, S. T. Phys. Rev. E 1993, 48, 3874. 16. Moses, E., Kume, T., Hashimoto, T. Phys. Rev. Lett. 1994, 72, 2037. 17. Krishnan, K., Almdal, K., Burghardt, W., Lodge, T. P., Bates, F. S. Phys. Rev. Lett. 2001, 87, 098301. 18. Krishnan, K., Burghardt, W. R., Lodge, T. P., Bates, F. S. Langmuir 2002, 18, 9676. 19. Nakatani, A. I., Sung, L., Hobbie, E. K., Han, C. C. Phys. Rev. Lett. 1997, 79, 4693.

Electric Field Alignment of Diblock Copolymer Thin Films Ting Xu1 , Jiayu Wang2 and Thomas P. Russell2 1

2

Department of Materials Science and Engineering, University of California at Berkeley, Berkeley, CA, 94720, USA Polymer Science and Engineering Department, University of Massachusetts Amherst, Amherst, MA 01003, USA

1 Introduction Block copolymers self-assemble into arrays of microdomains, e.g. lamellae, cylinders, or spheres, depending on the volume fraction of the components and χN , where χ is the Flory-Huggins segmental interaction parameter and N is the degree of polymerization [1, 2]. The sizes of the microdomains are dictated by the total molecular weight of the copolymer and, hence, are tens of nanometers in size. This size scale cannot be easily achieved using conventional photolithographic techniques and makes block copolymer thin ﬁlms ideal candidates for template and scaﬀolds for the fabrication of nanostructured materials [3–8]. For applications, controlling the microdomain orientation is essential. In a copolymer thin ﬁlm, the grains of the copolymer microdomains are found where the ordering is high locally, but globally there is no preferred orientation of the grains; on average, they are randomly arranged. Various external ﬁelds, such as shear, electric and surface ﬁelds have been used to manipulate the microdomain alignment in block copolymer thin ﬁlms. Here, we focus primarily on the electric ﬁeld alignment of diblock copolymers in thin ﬁlms a few hundred nanometers in thickness. The preferential interaction and segregation of one block to the substrate orients the microdomains parallel to the substrate surface. Electric ﬁelds normal to the surface are used to overcome these interfacial interactions and orient the microdomains in the direction of the applied ﬁeld [9–14]. We will ﬁrst discuss the inﬂuence of the interfacial interaction on the microdomain orientation in a copolymer thin ﬁlm; then the eﬀect of interfacial interaction on the electric ﬁeld alignment. We will subsequently discuss the electric ﬁeld alignment of the lamellar and cylindrical microdomains in thin ﬁlms and the electric ﬁeld induced sphere-tocylinder transition. Finally, we will discuss the eﬀect of the ionic impurities, such as lithium ions, on the electric ﬁeld alignment of copolymer thin ﬁlms. Discussion of the electric ﬁeld alignment of block copolymers in the bulk or solution can be found elsewhere [9–12, 15, 16].

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2 Interfacial Interactions: Asymmetric Diblock Copolymers Surfaces and interfaces inﬂuence the ordering properties of block copolymers [17–23]. For symmetric diblock copolymers near the microphase separation transition, the preferential interaction of blocks with the substrate produces an oscillatory concentration proﬁle normal to the substrate where the amplitude of the oscillations decay exponentially from the surface [19]. The decay length increases with increasing proximity to the microphase separation temperature, i.e. χN –(χN )s where (χN )s is the value of χN at the microphase separation transition temperature [1, 2]. The concentration of one block at the substrate interface is dictated by the diﬀerence in interfacial interactions of the blocks with the substrate and the strength of segmental interactions as shown by Menelle et al. and Mansky et al [25]. In the case of an ordered block copolymer, if the interfacial interactions between each block with the substrate are balanced, the microdomains orient normal to the surface [21, 26]. Otherwise, the preferential interactions between the surface and one of the blocks orient the microdomain parallel to the substrate [23, 27]. By anchoring a random copolymer of styrene and methyl methacrylate to the substrate, the interfacial energy, i.e. the strength of the surface ﬁeld, was precisely tuned by varying the composition of the random copolymer [28]. The inﬂuence of interfacial interactions, a surface ﬁeld, on the lamellar microdomain orientation parallel to the substrate in thin ﬁlms of symmetric diblock copolymer polystyrene-block-polymethyl methacrylate (PS-b-PMMA) was quantitatively investigated. Fig. 1a shows a cross-sectional TEM image of a ∼ 400 nm ﬁlm on a surface modiﬁed with a 58/42 (styrene/methylmethacrylate) random copolymer. In this case, the interfacial interactions between polystyrene (PS) and poly(methyl methacrylate) homopolymers with the modiﬁed surface are equal. The lamellae orient normal to the interface and this orientation propagates into the ﬁlm. At the air/polymer surface, the microdomains are parallel, due to the lower surface energy of PS. Figure 1b is a ∼ 700 nm ﬁlm on the substrate modiﬁed with a 70/30 random copolymer. The lamellae are oriented parallel to the substrate due to the preferential wetting of PS (dark) on the modiﬁed substrate. This orientation propagates through the entire ﬁlm for the thinner ﬁlms (not shown). However, for thicker ﬁlms, the parallel orientation is lost after ∼ 3 L◦ (L◦ is the equilibrium period of the diblock copolymer) from the copolymer/substrate interface, and lamellae with mixed orientations are dominant in the center of the ﬁlm. Fig. 1c shows the cross-section TEM image of a ﬁlm with a thickness similar to that of the sample in Fig. 1b prepared on a substrate modiﬁed with a 90/10 random copolymer. The parallel alignment of the lamellae propagates further into the ﬁlm (∼ 5 L◦ ). Clearly, the surface ﬁeld suppresses the ﬂuctuations at the interfaces between the microdomains in the vicinity of the interface, thereby retarding the formation of defects and promoting the persistence of the alignment of the microdomains parallel to the substrate interface.

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Fig. 1. Cross-sectional transmission electron microscopy images of annealed PSb-PMMA thin ﬁlms with diﬀerent ﬁlm thicknesses on Si substrates modiﬁed with random copolymers: (a) 58/42 (∼ 400 nm) (b) 70/30 (∼ 800 nm) (d) 90/10 (∼ 800 nm). From the TEM sample preparation, the top surface corresponds to the copolymer/substrate interface. (Used with permission, [42] Figs. 3b, 3c and 3d.).

To quantify the copolymer orientation, the linear fraction of lamellae oriented parallel to the surface of the TEM images was determined as a function of distance from the surface. Films (∼ 800 nm) on three substrates modiﬁed with 30/70 (Δγ ≈ 0.45 ergs/cm2 ), 70/30 (Δγ ≈ 0.5 ergs/cm2 ), 90/10 (Δγ ≈ 0.75 ergs/cm2 ) random copolymers were analyzed as shown in Fig. 2. The results are quite revealing. First, the shapes of the proﬁles are similar. Adjacent to the copolymer/substrate interface, the proﬁles are ﬂat, indicating a parallel alignment of the microdomains. The distance over which the orientation persists depends on the strength of the interfacial interactions. This distance is the same for the 70/30 and 30/70 cases, extending ∼3 periods from the surface. For the 90/10 case, the parallel alignment extends ∼5 periods from the interface, as would be expected from the stronger interfacial interactions. The decay of the orientation is essentially the same in all cases, i.e. the proﬁles can be superimposed. Such behavior has been predicted theoretically and indicates that the decay in orientation is a characteristic of the copolymer. Thus, there is a distance from the substrate where the parallel orientation of the microdomains persists and this distance increases with increasing strength of the interactions between the copolymer and the modiﬁed substrate surface [29].

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1.0

30/70 Dg=0.45 70/30 Dg=0.50 90/10 Dg=0.75

0.8 0.6 0.4 0.2 0.0 0

2 4 6 8 10 12 Distance from interface (period)

Fig. 2. Fraction of parallel lamellae microdomain vs. the distance from the copolymer/substrate interface. Each image was sliced into pieces of the equilibrium period thick from the copolymer/substrate interface. The fraction of the parallel lamellae was counted in each slice and plotted vs. the distance from the copolymer/substrate interface. Each point was obtained by averaging ∼ 20 μm wide cross-sectional TEM images. (Used with permission, [42] Fig. 4.)

This is also the case for the asymmetric diblock copolymer thin ﬁlms. Transmission small angle neutron scattering (SANS) was used to study the orientation of cylindrical microdomain orientation in ﬁlms of d-P-b-MMA where the PS block was perdeuterated for contrast. The incidence angle of the neutron beam was at a 45◦ angle with respect to the substrate. The schematic diagram in Fig. 3 shows a four-point scattering pattern with reﬂections at 45◦ , 135◦ , 225◦ and 315◦ indicating that the cylinders orient parallel to the surface and a two-spot pattern in the vertical direction indicates cylinders oriented normal to the surface. The rings in reciprocal space result from the random orientation of grains in the plane of the ﬁlm. Fig. 4 shows the SANS patterns of ∼ 200 nm, ∼ 300 nm, and ∼ 500 nm dPS-b-PMMA ﬁlms annealed at 170◦ C for 72 hrs. When the ﬁlm is thinner than a critical ﬁlm thickness, the interfacial interaction orients the cylindrical microdomains throughout the ﬁlm and a four-spot scattering pattern is seen (Fig. 4a and 4b). When the ﬁlm thickness is greater than this critical thickness, the cylindrical microdomains in the interior of the ﬁlm assume a random orientation, giving rise to the ring pattern in Fig. 4c. For the ∼ 500 nm annealed dPS-b-PMMA (47K) ﬁlms, the cylindrical microdomains orient parallel to the surface in the vicinity of the copolymer/substrate interface and are randomly oriented in the center of the ﬁlm.

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Fig. 3. Schematic diagram of hexagonally packed cylinders (a) parallel and (b) perpendicular to the surface in the real and reciprocal space. (c) The geometry of SANS measurement. (Used with permission, [42] Fig. 1)

Fig. 4. 2-D SANS patterns of (a) ∼ 200 nm, (b) ∼ 300 nm and (c) ∼ 500 nm dPS-b-PMMA (47K) ﬁlm annealed at 170◦ C under vacuum for 72 hrs. (Used with permission, [42] Fig. 4).

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3 Interfacial Interactions: Symmetric Diblock Copolymers Electric ﬁelds can be applied to orient the microdomain normal to the surface. A typical experimental setup is shown in Fig. 5 [30]. Copolymer ﬁlms were spin coated onto a silicon substrate. An aluminized Kapton ﬁlm was used as top electrode, where a thin layer (20-25μm) of crosslinked polydimethylsiloxane TM

(PDMS) (Sylgard ) was used as buﬀer layer between the Kapton electrode and the copolymer thin ﬁlm. The PDMS layer conforms to the electrode surface, eliminates air gaps between the top electrode and the copolymer ﬁlm, and maintains a smooth surface of the copolymer ﬁlm. The copolymer ﬁlms were heated to a temperature well above the glass transition temperature of the blocks under N2 with an applied electric ﬁeld for several hours, and then quenched to room temperature before removing the electric ﬁeld.

Fig. 5. Experimental setup of the electric ﬁeld alignment of diblock copolymer thin ﬁlms.

For block copolymer thin ﬁlms, the preferential interactions of the blocks with the interfaces represent the major impediment in achieving complete alignment of the domains. Several theoretical studies have appeared addressing the eﬀect of interfacial interactions on the electric ﬁeld alignment in diblock copolymer thin ﬁlms. A parallel orientation of the microdomains is favored when there is a diﬀerence in the interfacial energies, whereas a normal orientation of the microdomains is favored by the applied electric ﬁeld. Under certain conditions, a mixed orientation of the microdomains is predicted, where a parallel orientation of the microdomains at the surface occurs and, away from the surface, the applied electric ﬁeld dominates and the microdomains orient in the direction of the applied electric ﬁeld. With mixed orientations, however, ‘T’ junctions form when the lamellae oriented in orthogonal directions meet. These defects represent a signiﬁcant energetic penalty. A dimensionless parameter δ can be deﬁned as δ=(γAS − γBS )/γT where γAS and γBS are the interfacial energies of blocks A and B with the substrate, respectively, and γT

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is the interfacial energy between block A and block B. Theoretical calculations of Pereira, et. al. predict that, in the strong segregation limit, for δ ≤1, only a normal orientation of the microdomains is found, whereas, for δ > 1, a mixed orientation is predicted. Tsori and Andelmann, using the identical variables predicted normal orientation when δ < 2 and a mixed orientation when δ > 2 [17, 31–33]. The inﬂuence of the interfacial energies on the alignment of lamellar microdomains was also studied experimentally with PS-b-PMMA. Fig. 6a and 6b show cross-section TEM images of the thin ﬁlms after annealing under a ∼ 40 V/μm electric ﬁeld on two diﬀerent substrates onto which random copolymers with styrene fractions of 0.58 and 0.9, with corresponding δ being 0 and 0.94, respectively, were anchored. An orientation of the lamellar microdomains in the direction of the applied ﬁeld is seen in the middle of the ﬁlms. Complete alignment of the lamellae extending to the interfaces was achieved only when δ=0, i.e. when the interactions between the surface and the blocks were balanced. Mixed orientations were seen in all other cases, even when the diﬀerence in the interactions between the blocks and the substrate was much smaller than the interfacial energy between the two blocks. In comparison with the theoretical calculations, the observed range in δ to achieve complete orientation of the microdomains is much narrower [30].

Fig. 6. Transmission Electron Microscopy cross-section images of PS-b-PMMA thin ﬁlms after annealing under ∼ 40 V/μm electric ﬁeld on the substrates modiﬁed with diﬀerent random copolymers, (a) 58/42, δ ≈0 and (b) 90/10, δ ≈ 0.94. (Used with permission, [30] Figs. 2b and 2d.)

To account for the discrepancy between theory and experiment, a pathway dependence of the alignment process that is not considered theoretically

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was proposed. Fig. 7 shows the cross-sectional TEM image of the thin ﬁlm after annealing at 160 ± 5◦ C under an electric ﬁeld of ∼ 40 V/μm for 1hr. A random copolymer was anchored to the surface so that δ = 0.73. In this case, full alignment of the microdomains should occur. However, the lamellar microdomains are seen to orient parallel to the substrate initially at the substrate interface, whereas, in the center of the ﬁlm, the copolymer, while microphase separated, is not aligned. Further annealing of the sample under an electric ﬁeld produced the ﬁlm similar to that shown in Fig. 6b.

Fig. 7. Transmission Electron Microscopy cross-section images of a ∼ 800 nm PS-b-PMMA) thin ﬁlm after annealing under ∼ 40 V/μm electric ﬁeld for 1hr at 160 ± 5◦ C on the substrates modiﬁed with 80/20 (δ ≈ 0.73) random copolymer. (Used with permission, [30] Fig. 4.)

It is evident that two kinetic processes are in competition, namely the alignment of the microdomains by the electric ﬁeld and the surface induced alignment of the microdomains. While an electric ﬁeld can bias concentration ﬂuctuations of a disordered copolymer, the surface induced orientation is the stronger ﬁeld initially. The copolymers near the interface microphase separate, with an orientation that is strongly biased by the interface. The inﬂuence of the interface, of course, dissipates with increasing distance from the surface. Consequently, in the center of the ﬁlm, the copolymer microdomains are oriented parallel to the applied ﬁeld, i.e. normal to the substrate surface. This structure does not correspond to the equilibrium structure, since the formation of T-junctions is energetically costly [34, 35]. However to fully orient the microdomains, the microdomains at the interface must be rotated 90◦ . A simple rotation of the microdomains is not feasible and an undulation of the microdomains is a more plausible route. This, however, increases the interfacial area between the microdomains and also necessitates the stretching

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and compression of the chains, since the substrate is ﬂat. Thus, the energetic barrier to complete the transition to a fully aligned state is high, requiring a higher electric ﬁeld than that predicted theoretically to achieve the perfectly aligned state.

4 Interfacial Interactions: Asymmetric Diblock Copolymers For asymmetric PS-b-PMMA diblock copolymers having cylindrical microdomains, a threshold electric ﬁeld strength Et was found above which complete orientation of the cylindrical domains was achieved [13,36]. This threshold ﬁeld strength was independent of ﬁlm thickness (for ﬁlms ∼ 10–30 μm thick) and could be described by the diﬀerence in interfacial energies of the components. At ﬁeld strengths slightly below Et a coexistence of the domains parallel and perpendicular to the electrode surface was found which is consistent with the introduction of defects via undulations in the structure as one proceeds away from the surface. In the absence of an electric ﬁeld, interfacial interactions force an alignment of the cylinders parallel to the substrate. This is demonstrated by the x-ray scattering pattern obtained from an 80μm ﬁlm with α = 45◦ shown in Fig. 8a, showing that the scattering arises from grains of hexagonally packed cylinders oriented parallel to the surface.

Fig. 8. (a) SAXS pattern of a 86μm PS-b-PMMA ﬁlm annealed without an electric ﬁeld. The ﬁlm is tilted at α = 45◦ with respect to the incident beam. Scattering pattern measured at α=60˚ obtained the samples annealed under the electric ﬁeld of (b) 9.9 V/μm and (c) 14.5 V/μm. Scattering pattern in (b) shows reﬂections indicative of cylinders oriented parallel, as well as perpendicular, to the substrate. In (c) only lateral reﬂections are seen indicative of cylinders oriented normal to the surface, i.e. parallel to the electric ﬁeld lines (Used with permission, [13] Figs. 2 and 3.)

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Samples of PS-b-PMMA with thicknesses of 11, 20 and 30 μm were annealed in an electric ﬁeld having ﬁeld strengths up to 25 V/μm. Morkved et al. showed that, in the absence of competing interfacial interactions, the cylindrical microdomains aligned parallel to the electric ﬁeld lines at low ﬁeld strengths, several V/μm [5]. Here, however, an alignment of the cylinders parallel to the ﬁlm surface, i.e. normal to the ﬁeld lines, was maintained up to 11.5 V/μm. This diﬀerence is a direct consequence of the interfacial interactions of the copolymer with the electrode surfaces. Near 11.5 V/μm, as shown by the data in Fig. 8b taken at α = 60◦ for a 32 μm sample, equatorial reﬂections are evident in addition to the four oﬀ-equatorial reﬂections. Consequently, there is a range in ﬁeld strengths where both parallel and perpendicular alignment of the cylinders coexist. At higher ﬁeld strengths only two equatorial reﬂections are seen, as shown in Fig. 8c for a 32μm thick sample at 14.5 V/μm with α = 60◦ . It should be noted that experiments on PS-b-MMA with a reverse composition, i.e. PS cylinders in a PMMA matrix, showed identical alignment behavior [13]. The free energy diﬀerence in interfacial energies between the parallel and perpendicular state was calculated to be ∼ 1% of the surface energies of PS or PMMA (∼ 30 × 10−3 Jm−2 ), and is approximately equal to the diﬀerence between the surface energies of these two polymers. The observation that the surface-induced orientation of the cylinders propagates into the interior of the ﬁlm over large distances was discussed earlier. In such a system it is not possible to change the orientation across a thin ﬁlm from parallel to perpendicular to the surface without introducing defects. As long as the surface anchoring prevails, the response of the system to an electric ﬁeld is not a reorientation of the structure but rather an undulation perpendicular to the axis of the cylinders. These undulations are suppressed close to the interface but are ampliﬁed with increasing distance from the interface. A complete reorientation of the cylinders can only be realized when the applied ﬁeld overcomes the interfacial interactions which occurs at the critical electric ﬁeld strength. Close to the threshold value of the electric ﬁeld, in a situation where there are enough defects to have a ﬁnite layer of cylinders aligned by the electric ﬁeld, structural rearrangements are observed even before complete orientation.

5 Electric Field Alignment: Symmetric Diblock Copolymers The alignment of thin ﬁlms of symmetric PS-b-PMMA in an electric ﬁeld was studied in-situ as a function of ﬁlm thickness and interfacial energy using SANS and TEM. There is a competition between the applied electric ﬁeld, aligning the microdomains normal to the surface, and surface ﬁelds that tend to align the microdomains parallel to the surface. For ﬁlms with thickness t < 10 L0 , where L0 is the equilibrium period of the copolymer in the bulk, interfacial interactions are dominant and the lamellar microdomains orient

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parallel to the substrate surface regardless of the applied electric ﬁeld. If t > 10 L0 , interfacial interactions become less important, and lamellar microdomains in the center of the ﬁlms can be oriented in the direction of the applied ﬁeld, i.e. normal to the surface. Transmission electron microscopy shows that the dominant mechanism of orientation is one where the lamellae are locally disrupted and reform with an orientation in the direction of the applied ﬁeld [37]. Fig. 9 shows the evolution of the azimuthally averaged SANS intensity of a ∼ 110 nm thick ﬁlm at 170◦ C under ∼40 V/μm electric ﬁeld. The incident neutron beam is perpendicular to the substrate and the scattering intensity corresponds to the fraction of the microdomains aligned in the applied ﬁeld direction. At t = 0, the ﬁlms after spin coating show a broad, weak peak, indicating that the copolymers are poorly ordered. The evolution of the peak intensity during the in-situ SANS experiments reﬂects the competition between the applied electric ﬁeld and interfacial interactions. Initially, there is a rapid orientation of the lamellar microdomains normal to the surface, a response to the applied electric ﬁeld. However, with time, the microdomains reorient parallel to the surface, due to the preferential interactions of the PMMA with the oxide substrate. For thin ﬁlms with t ∼ 4 L0 , interfacial interactions are seen to dominate, as expected from calculations performed by Tsori and Pereira [31], [33]. As the ﬁlm thickness increases beyond the range where the interfacial interactions exert a strong inﬂuence, ﬂuctuations induced by the applied electric ﬁeld are not suppressed and the microdomains in the center of the ﬁlm orient in the direction of the applied ﬁeld. In-situ SANS experiments showed that the peak intensiﬁed rapidly (within the ﬁrst 5 minutes) and then slowly increased. While an applied electric ﬁeld is eﬀective in orienting the microdomains, the microdomain alignment is a rather slow process, taking hours to have an eﬀect. Fig. 10 shows the cross-sectional TEM image of a ﬁlm at an intermediate stage of the alignment. The overall orientation of the microdomains is along the electric ﬁeld direction, though there are still microdomains with no preferred orientation. The image is very informative regarding the alignment mechanism. In this image, there are several distinct features (labeled A, B, C) that provide insight into the alignment process. In all the A type regions, there are ellipsoidal grains with their long axes parallel to the surface. The lamellar microdomains within these grains are aligned parallel to the electric ﬁeld direction. In regions marked B, the lamellar microdomains have a random orientation. Here, grains are very small with many defects in the stacking of the lamellae. Some of the lamellar microdomains are disrupted and merge with nearby microdomains. In regions marked C, the lamellar breakup/merging is clearly seen. Fluctuations with a period similar to that of the equilibrium lamellar period are evident, in contrast to the classical Helfrich–Hurault undulations calculated for smectic and cholesteric liquid crystals. Both conformal and nonconformal ﬂuctuations in adjacent lamellae can be seen, with nonconformal ﬂuctuations prevalent in the lamellae adjacent to the substrates in

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Fig. 9. (a) The evolution of the SANS angular average intensity of the thin PSb-PMMA ﬁlm (∼ 110 nm) annealing under ∼ 40 V/μm electric ﬁeld. Solid line is the Gaussian ﬁt of the data. (b) The evolution of the peak intensity with annealing time under electric ﬁeld for the thin ﬁlms. (Used with permission, [37] Fig. 1.)

agreement with Tsori et al. and Onuki et al. [33, 38]. There are also striking similarities between the TEM images shown here and the images shown in the simulations of Zvelindovsky et al. [39–41]. The movement and coalescence of the defects, i.e. the local disruption and reformation of lamellae, appears to be the dominant pathway in achieving alignment.

6 Electric Field Alignment: Asymmetric Diblock Copolymers Controlling the cylindrical microdomain orientation in a copolymer thin ﬁlm is key to many applications. Understanding the mechanism of alignment in thin ﬁlms having diﬀerent initial states is important for process optimization and reduction in time required to achieve full alignment of the microdomains, so as to obtain ﬁlms suitable for subsequent applications. Here, we consider

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Fig. 10. Cross-sectional TEM image of PS-b-PMMA ﬁlms annealed under ∼ 40 V/μm electric ﬁeld for 6 hrs. Scale bar: 200 nm. (Used with permission, [37] Fig. 6.)

the inﬂuence of an electric ﬁeld on ﬁlms that are either poorly ordered or very highly ordered. 6.1 Initial State: Disordered/Poorly Ordered In-situ small-angle x-ray scattering studies were performed as a PS-b-PMMA asymmetric diblock copolymer was cooled from the disordered state into the ordered state under the inﬂuence of a strong electric ﬁeld. In Fig. 11a, the peak intensity, integrated over q, is given as a function of the azimuthal angle Ω. Here, q is the magnitude of the scattering vector and is given by 4π/λ sin α/2 where λ is the radiation wavelength and α is the scattering angle. Prior to the experiment, the sample was heated into the disordered state. The scattering pattern of the disordered melt without an applied ﬁeld consisted of a broad isotropic ring arising from composition ﬂuctuations. An electric ﬁeld of 30 V/μm was then applied. Fluctuations parallel to the electric ﬁeld are enhanced, while ﬂuctuations normal to the ﬁeld are suppressed. The sample was then cooled to below the disorder-to-order transition temperature. The scattering pattern became anisotropic with the application of the electric ﬁeld. The microphase-separated structure grew along the preferred direction established by anisotropic composition ﬂuctuations [36]. For disordered/poorly ordered asymmetric diblock copolymer thin ﬁlms (few hundred nanometer in thickness), the electric ﬁeld alignment process is

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similar to those of much thicker ﬁlms as shown above by Thurn–Albrecht et al. [13, 36]. The concentration ﬂuctuations in the center of the ﬁlm are biased by the applied ﬁeld, while the copolymer microdomains near the interface are strongly biased by the interface, and, therefore, oriented parallel to the plane of the interface initially. Similar to the orientation of lamellar microdomains in thin ﬁlms, the orientation of the cylindrical microdomains is a consequence of a competition between a surface ﬁeld (interfacial interactions) and the applied electric ﬁeld. Initially, the surface ﬁeld dominates [37]. Shown in Figs. 11b-d are cross-sectional TEM images of ∼ 700 nm as-cast PS-b-PMMA (120K) ﬁlms after annealing at 185 ± 5◦ C under ∼ 40 V/μm applied electric ﬁeld for 9, 12 and 16 hrs, respectively. After 9 h randomly oriented cylindrical PMMA microdomains are seen. Even with preferential interfacial interactions, the cylindrical microdomains adjacent to the copolymer/substrate interfaces were not fully parallel to the interface, and the grain size of the cylindrical microdomains was very small. After annealing under an applied electric ﬁeld for 12 hrs, ellipsoid-shaped objects are seen with their long axes oriented in the direction of the applied ﬁeld. Further annealing (16hrs) led to cylinders aligned along the ﬁeld direction [42]. 3-D TEM tomography was used to further clarify the electric ﬁeld alignment process. Fig. 11e shows a snap shot of the 3-D tomography of the ﬁlm in Fig. 10c [43, 44]. Cylindrical microdomains oriented nearly normal to the ﬁlm surface are clearly seen throughout the ﬁlm and are tilted at an angle with respect to the electric ﬁeld direction. Consequently, in this sample, the alignment is not complete and a further rotation of the microdomains in the applied ﬁeld direction will occur. Inspection of this image and those obtained at many other angles shows that the microdomains are not interconnected and that they span the entire ﬁlm. It is apparent that the electric ﬁeld has fully overcome any tendency for the microdomains to orient parallel to the planes of the interfaces. 6.2 Initial State: Ordered If the block copolymers are ordered initially with no preferred orientation of the grains of the microdomains, then a reorganization of microdomains must occur to achieve the ﬁnal aligned morphology. A simple rotation of the grains in the original structure cannot occur. The results from thicker ﬁlms (tens of micrometers) indicate that the applied ﬁeld leads to an instability that causes larger grains to be broken up into smaller sections by the ampliﬁcation of interfacial ﬂuctuations. Such a mechanism has also been postulated theoretically by Onuki and Fukuda and, more recently, by Tsori and Andelmann [33, 38]. The exact lateral length scale of this instability remains a question for further investigation. This was addressed using ﬁlms several hundred nanometers in thickness. Shown in Fig. 12a is the SANS pattern of the ∼ 300 nm pre-annealed d-PSb-PMMA (47K) ﬁlm [42]. The four-spot scattering pattern indicates that the

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Fig. 11. (a) Azimuthal angle dependence of scattering intensity, integrated over q, of a PS-b-PMMA asymmetric diblock copolymer thin ﬁlm being cooled from the disordered state into the ordered state under a 30V/μm electric ﬁeld. (b-d) show the cross-sectional TEM images of ∼ 700 nm PS-b-PMMA (120K) ﬁlms after annealing at 185 ± 5◦ C under a ∼ 40 V/μm electric ﬁeld for 9, 12 and 16 hrs respectively. (e) A snapshot of PS-b-PMMA morphology (3-D TEM tomography of c). The box size corresponds to 1029 nm × 740 nm × 236 nm. The top surface corresponds to the copolymer/substrate interface and the bottom to the copolymer/electrode interface. PMMA microdomains appear lighter color due to the lower electron density. (Used with permission [36], Fig 2 and [42], Figs. 3b, 3c, 3d and 3e).

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cylindrical microdomains are oriented parallel to the substrate throughout the ﬁlm. After annealing under an electric ﬁeld (∼ 40 V/μm) for 7 hrs, the fourspot scattering pattern is signiﬁcantly smeared out and the pattern appears elliptical in shape, arising from a deformation of the cylindrical microdomains in the direction of the applied ﬁeld. Fig. 13a shows a typical cross-sectional TEM image after annealing the ﬁlm under a ∼ 40 V/μm electric ﬁeld for 29 hrs. Within the ﬁeld of view (∼ 5.5 μm in length), only spherical objects are seen. Fig. 13b shows a series of 3-D TEM tomographic images that further support the 2-D TEM observations. Fluctuations with a characteristic period are seen along the cylindrical microdomains. With time, the ﬂuctuations grow in amplitude and, eventually, the cylindrical microdomains break up into spherical microdomains. There are striking similarities between Fig. 6a and TEM images of the order-order transition (OOT), suggesting that the electric ﬁeld induced cylinder-to-sphere transition is very similar to the thermoreversible OOT [45–48].

Fig. 12. 2-D SANS patterns of a ∼ 300 nm pre-annealed dPS-b-PMMA (47K) ﬁlm annealed under a ∼ 40 V/μm electric ﬁeld for (a) 0, (b) 7, (d) 29 hrs, respectively. (Used with permission, [42] Fig. 5.)

The late-stage of alignment was studied using a thicker ﬁlm (∼ 500 nm), where, after annealing, the cylindrical microdomains were oriented parallel to the surface at the copolymer/substrate interface and randomly oriented in the center of the ﬁlm. Fig. 14a shows a cross-sectional TEM image of the ﬁlm after annealing under a ∼ 40 V/μm electric ﬁeld for 29 hrs. In the center of the ﬁlm, where the inﬂuence of interfacial interactions eﬀects is not important, the cylinders orient along the ﬁeld direction at some tilt angle. At the copolymer/substrate interface, spherical microdomains are seen, as opposed to cylinders lying parallel to the interface. This is quite similar to the results for the thinner ﬁlms shown previously. Fig. 14b shows a series of tomographic images at diﬀerent rotation angles of the ﬁlm. The results further support the argument that the cylindrical microdomains break up into spherical microdomains under the inﬂuence of the applied ﬁeld. Subsequently, the spherical domains distort into ellipsoidal domains and then cylinders re-form, oriented at some angle with respect to the applied ﬁeld. Subsequently, the cylinders rotate into the direction of the applied ﬁeld. These observations are in full

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Fig. 13. (a) Cross-sectional TEM image of a ∼300 nm pre-annealed dPS-b-PMMA (47K) thin ﬁlm after annealing under a ∼ 40 V/μm electric ﬁeld for 29 hrs. Scale bar: 200nm. (b) 3-D TEM tomography of partial 8a. A series of snap shots of PS-bPMMA morphology (8a) viewing from diﬀerent angles. The section was tilted from -70◦ to 63◦ with 1◦ increment. The box size is 1290 nm × 940 nm × 100 nm. The top surface corresponds to the copolymer/electrode interface and the bottom to the copolymer/substrate interface. PMMA microdomains appear lighter color due to the lower electron density. (c) Zoom in image of 13(a). Scale bar: 100nm. (Used with permission, [42] Fig. 6.)

agreement with theoretical calculations and simulations by Zvelindovsky et al. [39].

7 Electric Field Induced Sphere-to-Cylinder Transition Experimental results and theoretical calculations show that the electric ﬁeld enhances ﬂuctuations along the interface between the microdomains of the two blocks due to the diﬀerence in dielectric constants of the microdomains and aligns the microdomains in the direction of the applied ﬁeld to lower the free

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Fig. 14. (a) Cross-sectional TEM images of a ∼ 500 nm pre-annealed dPS-bPMMA (47K) thin ﬁlm after annealing under a ∼ 40 V/μm electric ﬁeld for 29 hrs. Scale bar: 200nm. (b) A series of snap shots of PS-b-PMMA morphology (3-D TEM tomography of 6a) viewing from diﬀerent angles. The section was tilted from −73◦ to 57◦ with 1◦ increment. The box size is 288 nm × 288 nm × 77 nm. The top surface corresponds to the copolymer/substrate interface and the bottom to the copolymer/electrode interface. PMMA microdomains appear lighter in color due to the lower electron density. (Used with permission, [42], Fig. 8.)

energy [49]. Under suﬃciently high electric ﬁelds, spherical microdomains will be deformed into ellipsoids and, for a thin block copolymer ﬁlm with multiple layers of spheres, the ellipsoids can be suﬃciently stretched such that they interconnect to form cylinders that penetrate through the ﬁlm. Conservation of volume requires that the diameters of the cylinders be smaller than those of the spherical microdomains. Such a sphere-to-cylinder transition may oﬀer a simple route to generate cylinders with high aspect ratios from symmetric spherical microdomains. As shown by Segalman et al. [7] and Harrison et al., [50] it is possible to generate thin ﬁlms with exceptional long-range order using block copolymers with spherical microdomains. Thus, by using a sphere

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to cylinder transition, a simple route to achieve highly ordered highly aligned arrays of nanoscopic cylindrical domains having high aspect ratios is opened [51]. Fig. 15a shows the grazing incidence small angle x-ray scattering (GISAXS) pattern of a ∼ 400 nm PS-b-PMMA ﬁlm after annealing under a ∼ 40 V/μm electric ﬁeld. The peaks along qz , which reﬂect density correlations normal to the surfaces, are not observed. Along qy multiple reﬂections are evident, suggesting that microdomains are aligned normal to the substrate. A qy scan at A−1 is shown in Fig. 15b. After annealing under an electric ﬁeld, a qz ≈ 0.022 ˚ series of higher√order √ reﬂections √ √ with √ peak positions relative to the ﬁrst order are seen at 1 : 3 : 4 : 7 : 9 : 12. This indicates that the microdomains oriented normal to the surface are laterally packed into a hexagonal array. For the ﬁlm annealed without an electric ﬁeld, the GISAXS shows a ring of scattering indicating a random packing of the spherical microdomains. Only the ﬁrst order peak at q ≈ 0.0174 ˚ A with a diﬀuse shoulder is seen suggesting a morphology consistent with a disordered array of spheres or a lattice of disordered spheres (LDS). Thus, by applying an electric ﬁeld normal to the surface, the spherical microdomains were transformed into cylindrical microdomains oriented normal to the surface [51].

8 Inﬂuence of Free Ions The electric ﬁeld induced transition from spherical to cylindrical microdomains was theoretically predicted to be ∼ 70 V/μm for the PS-b-PMMA system based on the dielectric constants diﬀerence between PS and PMMA (εP S ≈2.5, εP M M A ≈6). It was argued by Tsori et al. that the presence of dissociated Li ions in the block copolymer would considerably lower the electric ﬁeld necessary to induce the sphere-to-cylinder transition. Here, the transition occurred at an electric ﬁeld strength of ∼ 40 V/μm, much lower than the calculated critical ﬁeld strength based on diﬀerences in the dielectric constants of PS and PMMA. To explain the experimental observations, they proposed an alternative mechanism where ionic impurities may be responsible for inducing the observed morphological transition. The concentration of ionic impurities required to reduce the critical ﬁeld strength suﬃciently to achieve alignment was quite low. These compelling arguments led us to undertake a systematic study of the inﬂuence of impurities on the microdomain alignment. The eﬀect of ionic impurities on the electric ﬁeld alignment of lamellar microdomains of polystyrene-block -poly(methyl methacrylate) (PS-b-PMMA) thin ﬁlms was ﬁrst studied using a copolymer containing ∼ 210 ppm lithium ions left from the ionic copolymerization [52]. After removal of the lithium impurities left from the copolymerization, only mixed orientations of the microdomains were seen after annealing under a ∼ 40 V/μm DC electric ﬁeld. This is due to the kinetic barrier in re-orienting

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Fig. 15. (a) Grazing incidence small angle x-ray scattering pattern of a ∼400 nm PS-b-PMMA ﬁlm annealed at 170◦ C under a ∼ 40 V/μm electric ﬁeld for 24 hrs. (b) The qy scan at qz ≈ 0.022˚ A of the left side of the GISAXS pattern of the ﬁlms annealed with and without electric ﬁeld. (Used with permission, [51] Fig. 2.)

the parallel lamellae at the copolymer/substrate interfaces. This is consistent with previous studies based on a dielectric mechanism [30]. However, lamellar microdomains are aligned along the applied electric ﬁeld direction throughout the entire ﬁlm if the copolymer thin ﬁlms contain 210 ppm residual lithium impurities. Fig. 16 shows a cross-sectional TEM image of a ∼ 300 nm PS-b-PMMA thin ﬁlm after annealing at 170◦ C under a ∼ 40 V/μm DC electric ﬁeld for 16 hrs. Although it has been demonstrated that lithium ion impurities remaining from the synthesis of the copolymer may amplify the inﬂuence of the applied electric ﬁeld, questions persist in terms of the direct evidence that lithium ions really improve the microdomain alignment, the nature of the interactions of the lithium ions with the diblock copolymer chains, and the origin of the enhanced alignment. Lithium chloride (LiCl) was added into a PS-b-PMMA copolymer with a number-average molecular weight (Mn) of 57 000 g/mol (57k), a polydispersity of 1.09 and a PS volume fraction of 0.60 after remov-

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Fig. 16. Cross-sectional TEM a ∼ 300 nm PS-b-PMMA thin ﬁlm after annealed at 170◦ C under a ∼ 40 V/μm DC electric ﬁeld for 16hrs. The copolymer contains 210 ppm lithium residual impurity. (Used with permission, [52] Fig. 3a.)

ing all ionic impurities. A satellite band at ∼1702 cm−1 (Fig. 17a) appeared in FT-IR absorption spectra after the addition of LiCl, due to the Li+ ←O=C coordination and the formation of lithium-polymer complexes [53]. Fig. 18 shows the corresponding cross-sectional transmission electron microscopy images of copolymer thin ﬁlms on silicon wafers with a native oxide layer after annealing at 175 ± 5o C under an applied ∼ 40 V/μm DC electric ﬁeld for 20 hrs before and after the addition of LiCl. A mixed orientation of microdomains formed in the pure diblock copolymer thin ﬁlms with the PMMA blocks preferentially segregating to the substrate surface (Fig. 18a). In thin ﬁlms with lithium-polymer complexes, the lamellar microdomains were found to orient in the direction of the applied ﬁeld, even at both interfaces of the ﬁlm, i.e. completely overcoming interfacial interactions (Fig. 18b). The microdomain alignment of diblock copolymer was enhanced by increasing the number of lithium-PMMA complexes until complete alignment of the microdomains was achieved [54]. Thus, it can be concluded that the lithium complexes in the PMMA block enhance the orientation of microdomains by the applied ﬁeld. While the ramiﬁcations of the lithium-PMMA complexes on the orientation of the microdomains are evident, the origin of the enhanced alignment is not immediately apparent. The alignment of the microdomains arises from the energetic cost associated with the misalignment of the interface between two media with the applied ﬁeld direction [9], [13], [33]. In the case of diblock copolymers, comprised of two polymer chains with diﬀerent dielectric constants, εA and εB , joined together at one end, the electrostatic contribution to the free energy is proportional to (εA − εB )2 . The dielectric constants of PMMA with diﬀerent concentrations of lithium-PMMA complexes (Fig. 19) were measured on melt-pressed disks , 300μm disks (2.0 cm in diameter) in thickness, over a frequency range from 1 MHz to 0.1 Hz at 170 and 180o C (the temperature range used in the alignment studies). As the frequency decreased from 1 MHz to 0.1 Hz at 170o C, the dielectric constant of pure PMMA increased from 4.9 to 6.3, in agreement with published values [6]. With the formation of lithium-PMMA complexes, the dielectric constant at a frequency of 0.1 Hz markedly increased from 6.3 to 11 to 29 at 170◦ C and up to 50 at 180◦ C. The dielectric constant increased even more rapidly as the frequency was decreased further. The critical electric ﬁeld strength required to orient the

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Fig. 17. IR spectra of (a) pure PS-b-PMMA ﬁlms; (b) with a molar ratio of added LiCl to the carbonyl group in PMMA equal to 1:1. (Used with permission, [54] Fig. 2.)

Fig. 18. Cross-sectional TEM images of PS-b-PMMA thin ﬁlms (a) without lithium salt and (b) with a molar ratio of added LiCl to the carbonyl group in PMMA equal to 1:1, after applying a ∼ 40 V/μm DC electric ﬁeld at 175 ± 5◦ C under N2 for 20 hrs. Scale bar: 200 nm. (Used with permission, [54] Fig. 2.)

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microdomains is related to the diﬀerence in the dielectric constants between the two blocks, the interfacial energies, and the ﬁlm thickness. The increased dielectric constant of PMMA blocks leads to an increase in the dielectric contrast between PMMA and PS microdomains, since there is only a negligible amount of lithium ions in the PS microdomains [55]. This signiﬁcantly reduces the critical electric ﬁeld strength required to overcome the interfacial interactions of the blocks so as to achieve complete microdomain alignment.

Fig. 19. Dielectric constants of PMMA with diﬀerent molar ratios of added LiCl to the carbonyl group (a) 0, (b) 1:10, and (c) 1:6 at 170◦ C. (Used with permission, [54] Fig. 3.)

9 Sequential, Orthogonal Fields The long-range orientation and ordering of microdomains in thin ﬁlms of diblock copolymers are mandatory for applications that require spatial deﬁnition of structures in three dimensions. In most cases, a unidirectional ﬁeld is used causing a high degree of orientation along the direction of the ﬁeld lines; normal to this, there is no ordering or orientation. Consequently, a second ﬁeld is required orthogonal to this ﬁeld to achieve control over 3-D order. Winey et al. investigated the inﬂuence of the thermal history on the microdomain orientation in a lamellar diblock copolymer subsequent to shearing, and showed that large grains grew from nuclei formed during the shearing process [56]. This observation points towards a route to achieve three dimensional ordering in diblock copolymer thin ﬁlms. Using a shear ﬁeld to generate oriented nuclei, a second ﬁeld applied normal to the shearing direction can

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be used to further orient the growing microphase separated morphology and, thereby, produce a near single crystal texture. A simple process is shown that combines an elongational ﬂow ﬁeld with an applied electric ﬁeld where the three-dimensional control over the orientation of lamellar microdomains in thin ﬁlms of PS-b-PMMA diblock copolymers is achieved. Roll-pressing was done below the order-disorder transition (ODT) but above the glass transition temperatures of both blocks where oriented nuclei of the microdomains form (Fig. 20). Subsequent annealing under an electric ﬁeld applied normal to the ﬁlm surface produced a lamellar microdomain morphology with a high degree of long-range order and orientation [57].

Fig. 20. (a) Schematic drawing of the roll-pressing and (b) the typical shape of copolymer thin ﬁlms after roll-pressing. (Used with permission, [57] Fig. 1.)

After roll-processing, a diﬀuse reﬂection is observed, indicating that microphase separation is not complete during the roll-pressing and a nonequilibrium morphology is frozen-in. The intensity of the SAXS along the ring of scattering, i.e. the azimuthal angular dependence of the scattering, showed weak broad equatorial peaks indicating that the oriented nuclei have formed during the roll-pressing. The SAXS pattern of the roll-pressed, thin ﬁlm af-

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ter annealing at 170◦ C under an applied electric ﬁeld (∼ 20 V/μm) shows two sharp equatorial reﬂections indicating a high degree of orientation of the lamellae normal to the surface and along the direction of ﬂow. The order parameter, f , was calculated to be 0.9 indicating that the lamellae are almost perfectly oriented in the ﬂow direction. Fig. 21 shows the cross-sectional TEM images. Fig. 21a is a typical TEM image of the copolymer ﬁlm viewed in the ﬂow direction, i.e into the XZ plane. At both surfaces, a mixed orientation can be seen. In the center of the ﬁlm, the lamellae are oriented perpendicular to the substrate (parallel to the electric ﬁeld direction). Figure 21b shows a typical TEM cross-sectional image for the rest of the ﬁlm along the same direction. Over very large areas, the orientation and order are defect-free. Only one grain was found within the ﬁeld of view. This result is quite remarkable in comparison to previous results using only an electric ﬁeld where many grains are found. Though the order and orientation extends over very large distances, defects still exist. However, from the SAXS measurements, it is apparent that the number of defects is very small.

(a) (b) Fig. 21. TEM cross-sectional images of (a) the copolymer/substrate interface and (b) the interior of the ﬁlm. (Used with permission, [57] Fig. 4b.)

Three-dimensional control over lamellar microdomain orientation can be achieved by using two orthogonal ﬁelds sequentially. The copolymer thin ﬁlm was biased ﬁrst by an elongational ﬂow ﬁeld that produced oriented nuclei during roll-pressing at a temperature below the ODT. The experiments shown here have been done with copolymer ﬁlms that were ∼ 20 − 50 μm thick. For many applications, ﬁlms with thickness ≤ 1 μm are desirable. Fundamentally, the mechanism behind the method reported here should be applicable for such thin ﬁlms also. Slight modiﬁcations in producing an elongational ﬂow ﬁeld are necessary, since such thin ﬁlms would be supported on a rigid substrate. This

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can be solved by use of novel shearing techniques, as, for example that recently developed by Chaikin and coworkers [3] to achieve long range ordering in thin block copolymer ﬁlms with spherical or cylindrical microdomains or by use of highly oriented ﬂow ﬁelds in solution, as recently discussed by Kimura et al. [48]. In either case, oriented nuclei in thin ﬁlms can be generated that can then be used to initiate the growth of microdomains under an applied electric ﬁeld.

10 Summary In this review, we have focused on the use of an electric ﬁeld to orient the microdomains of diblock copolymers in thin ﬁlms in the absence of solvent. A critical parameter that must be considered to achieve orientation is the interactions of the blocks with the conﬁning interfaces, be this the substrate or the electrode placed on the surface of the copolymer ﬁlm. To achieve uniform alignment throughout the entire ﬁlm, a suﬃcient ﬁeld strength must be used to overcome these interactions. In most cases, this is not possible if the ﬁlm is ordered and oriented prior to the application of a ﬁeld. This, however, can be overcome, to some extent, by beginning with a ﬁlm that is disordered. Enhancing the dielectric constant diﬀerence between the copolymer microdomains with the use of added salt can markedly enhance the orientation even at the interfaces. In addition, the rate of orientation and the degree of microphase separation is markedly improved. However, it is necessary to bind the salts within the microdomains by, for example, complexation of the salt with one of the components. If the ions are free, enhanced orientation by the applied ﬁeld is not observed. Finally, by coupling two ﬁelds in orthogonal directions, as, for example, a shear ﬁeld with an electric ﬁeld, exception long-range order can be achieved.

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Control of Block Copolymer Microdomain Orientation from Solution using Electric Fields Governing Parameters and Mechanisms Alexander B¨ oker Lehrstuhl f¨ ur Physikalische Chemie II, Universit¨ at Bayreuth, D-95440 Bayreuth, Germany

1 Introduction As nanotechnology increasingly gains importance in daily life, the need for novel nanoscopic structures also rises exponentially. For example, to keep up with Moore’s law, the packing density of integrated circuits has to increase on an almost daily basis. Considering the growing number of electronically stored data, it is also clear that novel data storage techniques have to be devised aiming to increase the information density on a hard disk. For such applications, the microstructures formed by block copolymers via their microphase separation present an ideal template for the fabrication of nanoscale patterns ranging from 10 − 100 nm [1]. In order to proﬁt from the self-assembly of block copolymers into various microstructures, one has to be able to control the parameters that govern this unique self-ordering process. In addition, it would be desirable to guide selfassembly via external ﬁelds to form macroscopically oriented, highly ordered structures. Block copolymer microphase separation has been studied extensively over the past two decades both experimentally (see Fig. 1) and theoretically [2–5]. In the phase separated state, these materials exhibit highly regular mesoscopic microdomain structures with characteristic length scales of the order of several tens of nanometers. Similar to polycrystalline materials, typically small grains of microdomains are formed, the size of which may be of the order of microns. As a consequence, although a single grain may have a highly anisotropic structure (e.g. in the case of cylindrical or lamellar structures), a bulk sample of a block copolymer typically exhibits isotropic materials properties. Control of the orientation of a block copolymer microstructure allows the development of polymeric materials with novel and interesting properties. Anisotropic mechanical, optical, electrical or mass transport properties can be tailored by proper orientation of the block copolymer microstructure. For example, alignment of glassy microphase-separated cylinders in a rubbery

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matrix gives a material with a glassy modulus along the cylindrical axis and a rubbery modulus along the transverse directions [6]. If the cylinders are made conductive, the material becomes a directional conductor [7]. Quantum dots or wires could be made from block copolymers with a spherical or cylindrical microstructure [8]. The birefringence inherent in lamellar or cylindrical block copolymers could be useful for optical applications. To create macroscopically anisotropic materials, various techniques aiming towards macroscopic microdomain alignment have been devised. Most prominently, shear ﬁelds [9–14], temperature gradients [15] and electrical ﬁelds [16–21] have been successfully applied to orient block copolymer microdomains from melt and solution. Flow provides a strong aligning force. ll dl

B ml

u-cic

cic cic S

sos soc

hel

M

cac

sos

ls lc

Fig. 1. Ternary phase diagram of polystyrene-b-polybutadiene-b-poly(methyl methacrylate) triblock copolymers, color according to staining with OsO4 . PS: grey, PB: black, PMMA: white [4].

However, because of boundary constraints and conditions of continuity, the potential for ﬂow-induced orientation is limited. Electric ﬁelds provide a weaker aligning force but oﬀer the advantage of local alignment control by application of spatially speciﬁc electric ﬁelds. For these reasons, electric ﬁelds may provide a unique pathway to new applications for block copolymers. In addition, electric ﬁeld alignment is scientiﬁcally interesting because the driving force for

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alignment is much simpler than that induced by ﬂow. Studies of ﬁeld alignment can be used to learn about materials properties like defect mobilities and give straightforward insight into alignment mechanisms. Recently, the high technical potential of electric ﬁelds for microdomain alignment has attracted increasing interest [18]. It has been shown that both lamellar and cylindrical microdomain structures in polystyrene-b-poly(methyl methacrylate) (PS-b-PMMA) melts could be oriented macroscopically by virtue of a DC electric ﬁeld [17, 22–26]. Due to the diﬀerences in the dielectric constants (Δε = εA − εB ) of the blocks (εPS ≈ 2.6, εPMMA ≈ 3.6) [27], the microdomains tend to orient parallel to the electric ﬁeld vector, thereby lowering the free energy of the system. The associated electric ﬁeld-induced −εB )2 2 driving force is proportional to (εA ε E [28]. Cylindrical microdomains can in principle be aligned along the ﬁeld vector resulting in a single monodomain (i.e. a block copolymer single crystal ). In a lamellar microdomain structure, on the other hand, all lamellar orientations containing the electric ﬁeld vector within the lamellar planes are energetically equivalent. Therefore, the electric ﬁeld is expected to at best favor the sub-set of lamellar orientations with the lamellar normal pointing perpendicular to the ﬁeld. So far most experiments using electric ﬁelds have been conducted in the melt. Due to the high melt viscosities, they are limited with respect to the molecular weight of the copolymers and the size of the macroscopic regions to be oriented (Mw ≈ 74 000 g/mol for thin ﬁlms of thickness 1 μm [29]; Mw ≈ 37 000 g/mol for samples of thickness 2 mm [17, 22]). In addition, temperatures close to the decomposition temperature and electric ﬁeld strengths of up to 25 kV/mm are required to achieve high degrees of orientation. These limitations render the orientation of higher molecular weight copolymers or copolymers of more complex architectures (multiblock copolymers, star copolymers, etc.) rather diﬃcult if not impossible, since their melt viscosities easily exceed the values faced in the investigations quoted above. Given the increasing interest in complex block copolymer structures and their technical potential [1], it is therefore desirable to explore alternative approaches, which circumvent the above limitations. One of these alternatives is the large scale alignment of block copolymer microdomains from concentrated polymer solutions due to the high polymer chain mobility. The mechanisms and kinetics can be investigated using small-angle X-ray scattering. This technique yields detailed insight into the microscopic structure and microdomain order with a high time resolution on the order of milliseconds. The details of this procedure as well as the parameters governing the electric ﬁeld-induced orientation will be discussed in the following.

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2 Experimental 2.1 Sample Preparation and Capacitor Setup The block copolymers were synthesized by sequential living anionic polymerization [30,31]. In the following sections, we will employ diﬀerent block copolymers such as polystyrene-b-poly(2-hydroxyethyl methacrylate)-b-poly(methyl methacrylate), polystyrene-b-poly(methyl methacrylate), and polystyrene-bpolyisoprene. We refer to the block copolymers as S47 H10 M43 82 , S49 M51100 , S50 I5080 and S50 I50100 , respectively with the subscripts denoting the weight fractions of the respective blocks and the superscript indicating the numberaveraged molecular weight in kg/mol. All polymers are rather monodisperse with polydispersities Mw /Mn ≈ 1.04. The alignment experiments were perfomed in a home-built closed capacitor with gold electrodes (electrode spacing: 0.3 − 4 mm, sample depth: 5 mm; Fig. 2). A DC voltage between 0.25 and 3 kV/mm was applied across the electrodes resulting in an electric ﬁeld perpendicular to the X-ray beam direction. Block copolymer solutions with polymer concentrations between 30 and 70 wt.% were investigated with in-situ Synchrotron-SAXS at the ID2 beamline at the European Synchrotron Radiation Facility (ESRF, Grenoble, France). 2.2 Synchrotron Small-Angle X-Ray Scattering (Synchrotron-SAXS) The diameter of the X-ray beam was 100 μm. The photon energy was set to 12.5 keV. SAXS patterns were recorded with a two-dimensional CCD camera located at a distance of 10 m from the sample within an evacuated ﬂight tube. The detector can monitor up to 120 frames (1 024 × 1 024 pixels) at a rate of 10 frames per second. Prior to data analysis, background scattering was subtracted from the data and corrections were made for spatial distortions and for the detector eﬃciency. 2.3 Calculation of Order Parameters from SAXS Data As will become clear from the experimental observations described in the following sections, domain alignment is induced by two competing external ﬁelds of diﬀerent symmetry, i.e. the interfacial ﬁeld between polymer solution and electrode surface and the electric ﬁeld, respectively. To quantify the alignment, we calculate the order parameter P2 by integrating the scattering intensity from ϕ = 0◦ to 360◦: 3 cos2 ϕ − 1 (1) P2 = 2 with

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90°

0°

E X-ray

U Fig. 2. Sketch of the experimental setup for in-situ synchrotron small angle X-ray scattering (SAXS) experiments at the ID2 beamline at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France. Reprinted with permission from Phys. Rev. Lett. [21]. Copyright (2002) The American Physical Society. 2π

2 cos ϕ =

0

dϕ Iq (ϕ) cos2 ϕ |sin ϕ| 2π 0

(2) dϕ (Iq (ϕ) |sin ϕ|)

Depending on the position of the maxima of the scattering intensity the calculation yields two diﬀerent ranges of the order parameter. For lamellar alignment parallel to the electrodes (maximum at ϕ = 0◦ ), P2 ranges from 0 to 1 with P2 = 1 corresponding to perfect lamellar alignment where all lamellar normals are oriented perpendicular to the surfaces, i.e. electrodes. For alignment of the lamellae along the electric ﬁeld direction (maximum at ϕ = 90◦ ), P2 ranges from 0 to −0.5 with P2 = −0.5 corresponding to the case where all lamellae are aligned parallel to the electric ﬁeld vector, however, with the lamellar normals being isotropically oriented in the plane of the electrodes. In order to quantify the orientation kinetics, the orientational order parameter P2 was calculated for each single scattering pattern acquired during the course of the experiment. The behavior of P2 as a function of time t has been ﬁtted by a single exponential as described by: t

P2 (t) = P2,∞ + (P2,0 − P2,∞ ) e− τ

(3)

with P2,0 and P2,∞ being the limiting values of the order parameter before application of the electric ﬁeld and at later times, respectively, and τ being the time constant. 2.4 Computer Simulation We employ the dynamic self-consistent ﬁeld theory, which describes the dynamic behavior of each molecule (modeled as Gaussian chains) in the meanﬁeld of all other molecules [32,33]. The phase separation can be monitored by

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the scalar order parameter Ψ (r, t), which is the normalized deviation of the density of a polymer component from its average value. In the case of an incompressible diblock copolymer melt the system is described by only a single order parameter. Simulating a diblock copolymer solution requires an extra order parameter for the solvent; however, we use a simpliﬁed model with only one order parameter in the present study. It was shown theoretically that a block copolymer melt can serve as a good approximation to describe general features of phase behavior of concentrated block copolymers solutions with nonselective good solvents [34]. As we have shown recently, this description is well justiﬁed and gives an excellent agreement with experiments in the case of a nonselective or almost nonselective solvent [35]. The time evolution of the order parameter in the simplest case follows a diﬀusion type equation [36] Ψ˙ = M ∇2 μ + η

(4)

with the constant mobility M , and the thermal noise η [33]. The chemical ∂ ε E2 potential in the presence of an electric ﬁeld E has the form μ = μ0 − ∂Ψ · T 8π 0 [28], where μ is the chemical potential in the absence of the electric ﬁeld, and ε is the dielectric constant of the polymeric material, which can be approximated as ε ≈ ε0 + ε1 Ψ for small Ψ . The electric ﬁeld E inside the material deviates from the applied electric ﬁeld E 0 = (0, 0, E0 ) and can be written via an auxiliary potential as E = E 0 − ∇ϕ. The potential is related to Ψ via the Maxwell equation div εE = 0. Keeping only leading terms, one can rewrite (4) in the form [37] Ψ˙ = M ∇2 μ0 + α ∇z 2 Ψ + η with α ≡ M E02

ε1 2 4π ε0

(5) (6)

The chemical potential without the electrostatic contribution μ0 is calculated using self-consistent ﬁeld theory for the ideal Gaussian chains with the mean ﬁeld interactions between copolymer blocks A and B, described by a parameter εAB [37]. The model system we study in the following is a symmetric A4 B4 - copolymer melt. The simulations have been performed in a two-dimensional box with 256 x 256 grid points and periodic boundary conditions [32]. For the simulations, the electric ﬁeld strength is parameterized by α ˜ ≡ α / (kT M v) [37,38], where v is a polymer chain volume. The samples were shear-aligned with the dimensionless shear rate γ˜˙ = 0.001, for details please see [39].

3 Role of the Dielectric Contrast As has been pointed out in the introduction to this chapter, the diﬀerence in the dielectric constants of the blocks Δε = εA − εB is a major factor

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determining the electric ﬁeld-induced microdomain orientation. Therefore, in this section, we study the orientation behavior of a S49 M51100 diblock and a S47 H10 M43 82 triblock copolymer which exhibit diﬀerent dielectric contrasts between the polystyrene and methacrylic block. 3.1 Experimental Comparison of the Diblock (S49 M51100 ) and the Triblock (S47 H10 M43 82 ) Structure Formation in Solution As an external electric ﬁeld can only align a microphase-separated structure, we studied the structure evolution of the S49 M51100 diblock and S47 H10 M43 82 triblock copolymer in tetrahydrofurane (THF) solutions with SAXS as a function of polymer concentration wp . Starting from wp = 30 wt.% and increasing wp stepwise by 2.5 wt.% up to 60 wt.% we aimed to determine the order- disorder concentration wODT of the two block copolymer systems. From these measurements we ﬁnd that for the S49 M51100 diblock copolymer the wODT is located at around 53 wt.%. For the S47 H10 M43 82 triblock copolymer, we locate wODT at around 40 wt.%. Obviously, the presence of the PHEMA middle block leads to an increased incompatibility between the blocks and thereby to a lower order-disorder concentration. As a consequence, the viscosity of solutions just above the order-disorder concentration wODT is signiﬁcantly lower for the S47 H10 M43 82 as compared to the S49 M51100 block copolymer. Reorientation Behavior After sample preparation and prior to electric ﬁeld exposure, all phaseseparated solutions exhibit a distinctly anisotropic scattering pattern with maxima located at ϕ = 0◦ and 180◦ (Fig. 3a). This pattern indicates alignment of the lamellae parallel to the electrodes. Such an alignment can be caused both by preferential interaction of polystyrene with the gold surfaces and by shear forces acting on the highly viscous solutions during the ﬁlling of the capacitor with a syringe [40]. As the lowest possible concentration (and thus viscosity) to give a phase separated polymer solution for the S49 M51100 diblock copolymer system was found at 53 wt.% in THF, the system only exhibits very slow reorientation of the microdomains at a ﬁeld strength of 2 kV/mm (see Fig. 4b). We note that the data shown here represent the fastest possible realignment kinetics achievable for the S49 M51100 system, as with increasing polymer concentration, the viscosity immediately dominates the process, rendering the reorientation impossible. Obviously, above wODT the force acting on the lamellae is not suﬃcient to lead to signiﬁcant reorientation [20], rearranging the microstructure in a highly viscous solution. In the triblock copolymer case a diﬀerent behavior is observed. As soon as an electric ﬁeld of 1 kV/mm is applied to a 40 wt.% solution, the scattering pattern changes signiﬁcantly. The peaks at ϕ = 0◦ and 180◦ decrease and new scattering maxima at ϕ = 90◦ and 270◦ grow with time (Figs. 3a–c).

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Fig. 3. (Left) Time development of the azimuthal angular dependence of the scattering intensity of a 40 wt.% solution of S47 H10 M43 82 in THF at 1 kV/mm (electrode spacing: 1 mm). The lighter regions represent the highest intensity and the darker regions indicate the lowest scattering intensity. (Right) Azimuthal scattering intensity at (a) t = 0 sec, (b) t = 100 sec, (c) t = 900 sec. Reprinted with permission from Polymer [41]. Copyright (2005) Elsevier Ltd.

Concentration Dependence The kinetics of the microdomain alignment was followed within a narrow concentration window between the order-disorder concentration wODT (≈ 40 wt.%) and wp = 50 wt.% for the S47 H10 M43 82 block copolymer. The orientational order parameter P2 was calculated for each azimuthal angular scattering intensity distribution obtained from the X-ray scattering images during the reorientation process using (1). Plots of P2 versus time yield an exponential dependence, which was ﬁtted according to (3) to give the time constants τ . As can be seen from Fig. 4 and Table 1, the time constants for the reorientation process at 1 kV/mm are in the range of some minutes (1.5 minutes for 40 wt.% and almost 6 minutes for 45 wt.%, respectively). At concentrations above 45 wt.% no electric ﬁeld induced orientation could be detected. Moreover, at 45 wt.% (Fig. 4b) the process is slowed down signiﬁcantly at an incomplete degree of alignment (P2 = 0.28), which can only be overcome by increasing the ﬁeld strength. At a polymer concentration of 50 wt.%, within the experimental time window of several minutes, no microdomain orientation could be achieved at ﬁeld strengths as high as 6 kV/mm. The results of the exponential ﬁts are summarized in Table 1. The single exponential ﬁt works quite well for all concentrations studied, as can be seen from the low χ2 values. At a suﬃciently high electric ﬁeld strength, P2 reaches about the same limiting values P2,∞ = −0.25 ± 0.02 independent of concentration. We note that the behavior displayed in Fig. 4 results from a delicate balance between an increase of both the driving force for reorientation (i.e. a larger dielectric contrast) and the viscous retardation as the polymer concentration is increased [21]. The exact behavior is diﬃcult to predict; however,

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Table 1. Time constants τ of the reorientation of S47 H10 M43 82 at diﬀerent polymer concentrations and diﬀerent electric ﬁeld strengths E obtained from least squares ﬁts using (3) (electrode spacing: 1 mm). Concentration E [wt.%] [kV/mm] 40 45 45 45

1 1 2 3

τ [sec]

P2,∞

χ2 [10−4 ]

89 348 157 151

-0.26 0.28 -0.23 -0.27

3.8 0.5 1.4 0.6

the data shown in Fig. 4 and Table 1 indicate that in the particular system studied here the increase in viscosity dominates over the increase of the driving force. Therefore, the reorientation process slows down with increasing polymer concentration. For a suﬃciently high electric ﬁeld strength the viscosity only inﬂuences the kinetics but not the ﬁnal degree of order (P2,∞ ), which is consistent with previous dielectric relaxation spectroscopy measurements on the realignment of side chain liquid crystalline polymers in their liquid-crystalline state induced by a DC electric ﬁeld [42].

-0,4

-0,2 0,0

0,0

P2

P2

-0,2

0,2

0,2 0,4

a 0

200

400 600 Time [sec]

800 1000

0,4 0

500

2kV/mm

b

1000 1500 Time [sec]

2000

Fig. 4. Evolution of orientational order parameter P2 with time (for orientation parallel to the electric ﬁeld vector). (a) 40 wt.% solution of S47 H10 M43 82 in THF at 1 kV/mm, (b) 45 wt.% solution of S47 H10 M43 82 in THF at 1 kV/mm and 2 kV/mm (). For comparison, the data for a 53 wt.% solution of S49 M51100 in THF at 2 kV/mm is added (). The solid lines represent least squares ﬁts to the data (electrode spacing: 1 mm) according to (3). Reprinted with permission from Polymer [41]. Copyright (2005) Elsevier Ltd.

If this procedure was to be used for the preparation of macroscopically aligned bulk samples via solution casting in the presence of an electric ﬁeld, the alignment process should be faster than the time needed for solvent evaporation. From the above results we conclude that concentrations between 40 wt.%

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and 45 wt.% are suitable for the preparation of dried bulk samples from solution. Figure 5 shows the feasibility of such a process for the S47 H10 M43 82 block copolymer system. Using a home-built capacitor which allows application of an electric DC ﬁeld during ﬁlm formation by solvent casting, a melt sample of 1 mm thickness (dried in the presence of an electric ﬁeld of 2 kV/mm) was prepared from a 25 wt.% solution in chloroform (comparable to 40 wt.% in THF). SAXS investigations yield an orientational order parameter of P2 = −0.4. Here we note that a similar process with the S49 M51100 diblock copolymer system did not yield anisotropic bulk samples. In the following, we shall try to give some theoretical evidence to support the experimental facts reported above.

E

Intensity [a.u.]

a

5x10

3

4x10

3

3x10

3

2x10

3

1x10

3

b

0 0

90

180

270

360

[°]

Fig. 5. (a) 2D-SAXS pattern of a ﬁlm cast in the presence of a DC electric ﬁeld (E = 2 kV/mm). The lamellae are oriented preferentially along the ﬁeld direction. (b) Azimuthal intensity distribution at the ﬁrst order reﬂection (P2 = −0.4). Reprinted with permission from Polymer [41]. Copyright (2005) Elsevier Ltd.

3.2 Theoretical Comparison of the Diblock (S49 M51100 ) and the Triblock (S47 H10 M43 82 ) Here, we will consider electrostatic arguments to discuss the diﬀerent behavior observed for the PS-b-PMMA (S49 M51100 ) and the PS-b-PHEMA-b-PMMA (S47 H10 M43 82 ) diblock and triblock copolymers with signiﬁcantly diﬀerent dielectric contrasts. We aim to estimate the electrostatic energies involved in the process of microdomain orientation. As indicated by diﬀerential scanning calorimetry (DSC), rheological and transmisssion electron microscopy (TEM) experiments, we anticipate that the PHEMA and the PMMA form a mixed phase. This leads us to treat the triblock copolymer as an AB diblock copolymer with the following composition: A: 47 wt.% PS (εA = 2.6) and B: 53 wt.% methacrylic blocks (εB = 0.81 εPMMA +0.19 εPHEMA = 4.6; with εPMMA = 3.6 and εPHEMA = 8.9 [43]). Thus the dielectric contrast for the S47 H10 M43 82 tri-

Block Copolymer Microdomain Alignment in Electric Fields

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block amounts to Δε = 2.0. For the S49 M51100 diblock copolymer the dielectric contrast only yields Δε = 1.0. As chloroform (εCHCl3 = 4.8) and THF (εTHF = 7.8) are fairly nonselective solvents for the two main components, PS and PMMA, we expect a similar swelling behavior leading merely to a dilution eﬀect with respect to the dielectric constants of each block. Therefore, with increasing solvent content in the ﬁlms, the diﬀerence of the dielectric constants is reduced and the electrostatic driving force for an alignment of the lamellae parallel to the ﬁeld is expected to decrease [26]. As has been pointed out by Amundson et al. [17] with respect to melts of PS-b-PMMA block copolymers this force is already small, so it is remarkable that its decrease still leaves a suﬃcient driving force to allow for preferential alignment of the microdomains. To estimate the driving forces for domain alignment as a function of solvent volume fraction φs , we calculate the electric energy per volume W , which is stored in a capacitor for an open setup which allows for solvent evaporation (including a layer of air in the system) and a closed system as used for the insitu SAXS studies. The model relies on two major assumptions: The dielectric constant of a mixture εmix of polymer εp and solvent εs is assumed to depend linearly on the solvent volume fraction φs : εmix = φs εs + (1 − φs ) εp

(7)

We further disregard any inﬂuence of the solvent on the partial molar volume of the polymer, i.e. the volumes of polymer Vp and solvent Vs simply add: Vmix = Vp + Vs

(8)

Film formation in the open capacitor under the inﬂuence of an external electric ﬁeld may result in signiﬁcant thickness undulations which eventually lead to the formation of column-like protrusions that connect both electrodes [20,44]. Therefore, we have identiﬁed four basic geometries to describe the open system, corresponding to a perpendicular (W⊥,col , W⊥,f lat ) and parallel (W,col , W,f lat ) alignment of the microdomains with respect to the electric ﬁeld and to a formation of columns (W⊥,col , W,col ) and a ﬂat ﬁlm (W⊥,f lat , W,f lat ), respectively (Fig. 6a). In the case of the closed capacitor, the two perpendicular (W⊥,col , W⊥,f lat = W⊥ ) and two parallel (W,col , W,f lat = W ) cases are equivalent. We calculate the energy W stored within the electric ﬁeld of the capacitor according to 1 W = ED dV (9) 2 with E being the electric ﬁeld and D the displacement ﬁeld. In contrast to the dielectric displacement D, the electric ﬁeld E along the z-direction of the capacitor is not uniform for the models, which incorporate a layered structure (W⊥,f lat and W⊥,col ). This is due to the fact that the component of the electric ﬁeld perpendicular to the interface between two materials

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is not continuous, but the one of the displacement ﬁeld is: D ⊥1 = D⊥2 but E ⊥1 = E ⊥2 . Additionally E i = D i /εi and the applied voltage U = E dz, which means that as soon as air is present as a layer in the capacitor with the applied voltage U , the electric ﬁeld in both polymer layers is reduced. This leads eﬀectively to a reduced energy stored inside the capacitor and to a reduced alignment of the block copolymer in the thinner parts of the sample. We are well aware of the fact that our calculations neglect the existence of interfacial boundary regions in concentrated polymer solutions. Therefore, the results may represent an approximation to the upper limit of the real energetic situation. In order to allow for a straightforward comparison between the open and closed system, we used chloroform for both calculations. Similar calculations using THF as solvent conﬁrmed a qualitative agreement with the results obtained for the chloroform case. a W||,col W||,flat

b

W

,flat

W||,flat - W,flat

0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

s

c

4

5 2

0.8

E

6

W||,col - W,col

1.2

0.4

,col

W [mJ/m ]

2

W [mJ/m ]

1.6

W air

2

B

W [mJ/m ]

A

4 3

WSHM - WSM

3 2 1 0

0.0 0.2 0.4 0.6 0.8 1.0

s

2 1 0

W|| - W

0.0

0.2

0.4

s

0.6

0.8

1.0

Fig. 6. (a) Four basic geometries of lamellar orientation in an open capacitor. (b/c) Free energy calculations for the lamellar orientation in an open and closed capacitor ﬁlled with polymers A and B at geometries shown in (a). (b) Energy diﬀerence between the two orientations for columns (W,col − W⊥,col ) and for a ﬂat ﬁlm (W,f lat − W⊥,f lat ) in an open capacitor which allows for solvent evaporation: S47 H10 M43 82 (—) and S49 M51100 (· · ·) block copolymers in CHCl3 . (c) Diﬀerence in calculated energy between the orientations parallel and perpendicular to the electric ﬁeld vector within a closed capacitor ﬁlled with S47 H10 M43 82 (—) and S49 M51100 (· · ·) in CHCl3 as a function of solvent volume fraction φs (geometries as depicted in (a) but with no air layer). The inset shows the overall energy diﬀerence between the two polymer systems. In all calculations the ﬁeld strength is 2 kV/mm. Reprinted with permission from Polymer [41]. Copyright (2005) Elsevier Ltd.

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Figure 6b shows the diﬀerence in energy stored per volume within an open capacitor ﬁlled with 15 vol.% polymer A (with εA = 2.6 for PS phase) and 15 vol.% polymer B (with εB = 4.6 for PHEMA/PMMA mixed phase (—) and εB = 3.6 for pure PMMA phase (· · ·)) as a function of chloroform volume fraction φs (εCHCl3 = 4.8), corresponding to the four basic geometries, at a ﬁeld strength of 2 kV/mm. The curve W,col − W⊥,col as well as the curve W,f lat − W⊥,f lat converge at a solvent volume fraction of 70 vol.% (which is the starting point of our solvent casting experiment), because at this point the capacitor is completely ﬁlled and the respective geometries are equivalent. Besides the fact that for the SHM as well as the SM system the diﬀerence in energy between the two orientations in columns (W,col − W⊥,col ) is larger than for ﬂat ﬁlms (W,f lat − W⊥,f lat ), the overall energetic diﬀerence is about three times higher for the SHM than for the SM system. In Fig. 6c, we show the results of the calculations for the diﬀerence in energy between the orientations parallel and perpendicular to the electric ﬁeld vector within a closed capacitor ﬁlled with diﬀerent solutions of our model block copolymers in chloroform with εA = 2.6 and εB = 4.6 (resembling S47 H10 M43 82 , full line) and with εA = 2.6 and εB = 3.6 (resembling S49 M51100 , dotted line) as a function of solvent volume fraction φs at a ﬁeld strength of 2 kV/mm. Compared to an open capacitor the system stores at least two times more energy at any given solvent volume fraction up to 80 vol.%. The inset depicts the overall energy diﬀerence between the above described cases, i.e. the aligned triblock terpolymer lamellae always allow the capacitor to store more energy than the diblock system. When we compare the two situations described above, we ﬁnd that: (i) the closed system stores more energy per volume, (ii) the diﬀerence in energy between the two orientations (W − W⊥ ) is always higher for the experimentally relevant concentrations in the closed capacitor, and (iii) in all cases, the SHM system is clearly electrostatically favored as it allows the capacitor to store more energy per volume. Therefore, from our experimental ﬁndings, we may conclude that, if the electric ﬁeld induced orientation of a block copolymer does not work in the closed capacitor, it will neither function in the open system, as the energetic diﬀerence is always larger for the closed capacitor setup. Furthermore, the incorporation of the high dielectric constant PHEMA middle block into the PS-b-PMMA diblock copolymer is the key to increasing the electrostatic driving force of the alignment process in order to create a well-performing methacrylate-based block copolymer system for electric ﬁeldinduced alignment from solution. Moreover, the PHEMA block enhances the microphase-separation in the block copolymer solutions compared to the PS-b-PMMA system. While the wODT for the S49 M51100 block copolymer is found at around 53 wt.%, the PHEMA containing block copolymer already microphase-separates at wp ≈ 40 wt.%. Therefore, in the latter case, the viscosity of the phaseseparated solution is considerably smaller, which promotes the kinetics of the ordering process induced by the small electric force. On the other hand,

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a lower polymer concentration also reduces the eﬀective ﬁeld strength and driving force in the capacitor as shown in Figs. 6b/c. To this point, we can not decide deﬁnitely if in this particular system enhanced phase separation or the increased dielectric contrast between the blocks is more important for the electric ﬁeld-induced ordering process to function. So far, all results point to a delicate balance between both parameters.

4 Concentration and Temperature Dependence of Orientation Mechanisms and Kinetics In this section, we describe in-situ synchrotron small-angle X-ray scattering (Synchrotron-SAXS) investigations aiming to follow the kinetics of electric ﬁeld-induced microdomain reorientation in concentrated block copolymer solutions, and thus, to elucidate the underlying microscopic mechanisms. As a model system, we employ a lamellae-forming polystyrene-b-polyisoprene diblock copolymer (S50 I5080 ) dissolved in toluene. We discuss the inﬂuence of the polymer concentration and the temperature on the reorientation behavior. In addition to the experiments, we performed computer simulations based on dynamic density functional theory (DDFT). 4.1 Structure Formation in Solution First, we investigated the ordering behavior of the block copolymer solutions in the absence of an electric ﬁeld. At room temperature, we found an orderdisorder transition (ODT) at wODT ≈ 35 wt.%, above which a well deﬁned lamellar microdomain structure is developed. A characteristic lamellar spacing of dSAXS = 39 nm is found at wp = 35 wt.%, which increases smoothly with increasing polymer concentration. After ﬁlling the capacitors with the polymer solution, the microdomain structure appears to be highly oriented parallel to the electrodes. This can be seen from the two-dimensional SAXS pattern displayed in Fig. 7a, which shows two distinct peaks around ϕ = 0◦ and ϕ = 180◦ . When an electric ﬁeld is applied across the two electrodes, the scattering pattern changes signiﬁcantly. As can be seen from the snapshots taken at diﬀerent times, the anisotropic pattern ﬁrst turns into an isotropic ring of weak intensity (Fig. 7b, t = 6 sec) before two distinct peaks are formed around ϕ = 90◦ and ϕ = 270◦ at later times (Fig. 7c, t = 34 sec). The complete set of data is displayed in Fig. 8a, where the intensity of the (1 0 0) peak is plotted as a function of ϕ for increasing time t. Clearly an almost complete destruction of the initial peaks is seen at early times followed by the buildup of new peaks around ϕ = 90◦ and ϕ = 270◦ .

Block Copolymer Microdomain Alignment in Electric Fields

a

b

0 sec

213

c

6 sec

34 sec

-0,4 d

P2

-0,2 0,0 0,2 0,4 0,6

0

15

30 45 Time [sec]

60

75

Fig. 7. Two-dimensional SAXS patterns of a 35 wt.% solution of the S50 I5080 diblock copolymer in toluene taken at 25◦ C prior to (a) and after application of an electric ﬁeld (E = 1 kV/mm) (b,c). (d) Time dependence of the orientational order parameter P2 . The solid line is a least squares ﬁt to the data according to (3) with P2,0 = 0.52, P2,∞ = −0.32, and τ = 5 sec. The arrow indicates the direction of the electric ﬁeld vector. Reprinted with permission from Macromolecules [45]. Copyright (2003) American Chemical Society.

To quantify the kinetics of the orientation process, the orientational order parameter P2 (t) was calculated from the 2D SAXS images according to (3). Using a single exponential ﬁt, we can determine the time constant of the reorientation process (Fig. 7d). 4.2 Kinetics of Microphase Orientation For the eﬀective preparation of highly anisotropic melt block copolymer samples via solvent casting in the presence of an external electric ﬁeld, it is important to ﬁnd an optimum set of parameters (i.e. degree of swelling of the block copolymer domains, electric ﬁeld strength, and temperature), which combines a maximum chain mobility (i.e. fast kinetics) with the highest possible polymer concentration. In short, the reorientation process should be faster than the rate of solvent evaporation during preparation of bulk samples from solution, i.e. it should be completed before the bulk structure “freezes”.

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Fig. 8. Time development of the scattering intensity as a function of the azimuthal angle ϕ in the presence of an electric ﬁeld of strength E = 1 kV/mm for different polymer concentrations and temperatures. (a) wp = 35 wt.%, T = 25◦ C; (b) wp = 50 wt.%, T = 25◦ C; (c) wp = 47.5 wt.%, T = 27◦ C; (d) wp = 47.5 wt.%, T = 80◦ C. Reprinted with permission from Phys. Rev. Lett. [21]. Copyright (2002) The American Physical Society.

Concentration Dependence In a ﬁrst series of experiments, we studied the reorientation kinetics as a function of polymer concentration, starting from wp = 30 wt.% and increasing wp stepwise by 1 wt.% up to 35 wt.% and then by steps of 2.5 wt.% to higher polymer concentrations. The electric ﬁeld strength E was kept constant at E = 1 kV/mm at a capacitor spacing of 2 mm. Starting from wODT , the scattering patterns changed following the behavior shown in Fig. 7 so that time constants τ (wp ) could be determined from the time evolution of P2 as shown in Fig. 9. Above wp = 50 wt.%, however, the reorientation process was so slow (τ ≈ 5 min) so that we limited our study to polymer concentrations between 34.5 and 50 wt.%. The results of the exponential ﬁts and the increase of the time constant τ with increasing polymer concentration are summarized in Table 2. The single exponential ﬁt works quite well for all concentrations studied, as can be seen from the low χ2 values. The time constants vary from τ = 0.8 sec

Block Copolymer Microdomain Alignment in Electric Fields

215

-0.4 -0.2 -0.4

P2

0.0

-0.2

0.2

0.0 0.2

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0.4 0.6

0.6 0

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0

5

10

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800

1000 1200

Time [sec] Fig. 9. Evolution of the orientational order parameter P2 with time for diﬀerent concentrations at 1 kV/mm ( = 34.5 wt.%, = 37.5 wt.%, = 42.5 wt.%, = 50 wt.% (electrode spacing: 2 mm). Reprinted with permission from Macromolecules [45]. Copyright (2003) American Chemical Society.

for the very fast processes at 34.5 wt.% to more than 3 minutes (τ = 192 sec) for the 50 wt.% solution. In addition, within some 10% scatter P2 reaches about the same limiting values P2,∞ = −0.3 ± 0.03 independent of polymer concentration. Therefore, we can conclude that the polymer concentration only inﬂuences the rate but not the ﬁnal degree of orientation. We note that the behavior displayed in Fig. 9 results from a delicate balance between an increase of both the driving force for reorientation (i.e. a larger dielectric contrast) and the viscous drag as the polymer concentration is increased [21]. The exact behavior is diﬃcult to predict; however, the data shown in Fig. 9 and Table 2 indicate that in the particular system studied here, just as found for the S47 H10 M43 82 block copolymer, the increase in viscosity dominates over the increase of the driving force. Therefore, the reorientation process slows down with increasing polymer concentration. Interestingly, along with an overall slowing down of the reorientation at higher polymer concentration, the microscopic mechanism of microdomain reorientation changes as a function of the polymer concentration. This can be seen in Figs. 8a/b, where we compare the time dependence of the scattering patterns for the limiting polymer concentrations, wp = 35 wt.% and wp = 50 wt.%, respectively. For the lower concentration (Fig. 8a) the initial peaks at ϕ = 0◦ and ϕ = 180◦ vanish almost completely (which is accompanied by a temporary drop in the normalized relative integrated intensity) as the electric ﬁeld is applied. New peaks establish at the ﬁnal microdomain orientation at ϕ = 90◦ and ϕ = 270◦ , showing a continuous growth of intensity with time. Intermediate orientations are not observed. For high polymer concentrations (Fig. 8b) a distinctly diﬀerent behavior is found. The initial peaks continuously shift from their original positions to their ﬁnal positions

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Table 2. Time constants τ of the reorientation of S50 I5080 at diﬀerent polymer concentrations obtained from least squares ﬁts using (3) (E = 1 kV/mm, electrode spacing: 2 mm). NG: nucleation and growth, R: grain rotation. Concentration τ [wt.%] [sec] 34.5 35 37.5 40 42.5 45 47.5 50

0.8 5.0 7.0 28.3 54.0 104 142 192

P2,∞

χ2 [10−4 ]

dominating mechanism

-0.26 -0.32 -0.34 -0.33 -0.33 -0.34 -0.26 -0.31

0.6 1.4 0.8 1.3 2.4 3.2 1.2 5.6

NG NG NG NG mixed R R R

at ϕ = 90◦ and ϕ = 270◦, respectively. The relative integrated intensity of the peaks increases steadily during the shift, i.e. a well-deﬁned anisotropic scattering pattern is observed throughout the entire process. At intermediate concentrations both behaviors coexist. Table 2 shows the prevailing mechanism found for the respective concentration. Temperature Dependence A qualitatively similar behavior is found when we follow the reorientation process at diﬀerent temperatures. For this purpose, a 47.5 wt.% solution was studied between 27◦ C and 80◦ C (see Figs. 8c/d). The rather high polymer concentration was chosen to access a large temperature range before reaching the order-disorder transition temperature (TODT ) of the solution. Qualitatively, we ﬁnd a “rotation” of the scattering pattern from the initial to the ﬁnal situation at low temperatures, while the scattering pattern “switches” between the two limiting situations at temperatures near TODT . The results of a quantitative data evaluation are summarized in Table 3. At the lowest temperature (27◦ C) we measure a time constant of τ = 141 sec, which gradually decreases down to 11.5 sec as the temperature is raised up to 80◦ C. The plateau values of the orientational order parameter P2,∞ seem to show a slight increase from –0.34 to –0.25 with increasing temperature, indicating a decrease of ﬁnal order. 4.3 Mechanism of Domain Alignment One of the most important aspects for the understanding of the reorientation behavior of block copolymer microdomains in solution is the knowledge of the underlying mechanisms contributing to the rearrangement of domains. In contrast to in-situ birefringence [13, 14], in-situ SANS [46], and ex-situ

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Table 3. Time constants τ of the reorientation of S50 I5080 at diﬀerent temperatures obtained from least squares ﬁts using (3) (wp = 47.5 wt.%, E = 1 kV/mm, electrode spacing: 2 mm). NG: nucleation and growth, R: grain rotation. Temperature τ [K] [sec] 300.15 308.15 316.15 324.65 333.65 343.15 353.15

141 138 106.9 86.5 52.5 40.6 11.5

P2,∞

χ2 [10−4 ]

dominating mechanism

-0.34 -0.28 -0.27 -0.28 -0.28 -0.27 -0.25

4.0 0.6 3.0 3.8 2.3 2.5 0.2

R R R mixed mixed NG NG

SAXS [10, 12, 47] measurements on block copolymer melts and solutions under shear, which lead to detailed insight into the respective mechanisms, so far only little is known about the microscopic processes during electric ﬁeld alignment. In-situ Synchrotron–SAXS combines the advantages of birefringence (high time resolution) with the detailed and straightforward information about the microscopic order characteristic of scattering methods. Indeed, the SAXS data indicate two distinctly diﬀerent mechanisms of microdomain reorientation. At low concentrations and at high temperatures, destruction of the initial peaks is followed by a build-up of scattering intensity at the ﬁnal peak positions. At high concentrations and low temperatures, on the other hand, the scattering pattern merely shifts into new peak positions accompanied by a steady increase of the integrated peak intensities. These ﬁndings point to two diﬀerent underlying mechanisms responsible for microdomain reorientation in the presence of the electric ﬁeld. Close to the order/disorder transition (ODT), i.e. at low concentrations and high temperatures, microdomains aligned parallel to the electric ﬁeld grow at the expense of those aligned parallel to the electrodes. Intermediate orientations, however, are not observed. This behavior matches the notion of the nucleation and growth (see Fig. 10a), which has previously been described for microdomain alignment under shear [47] and which was assumed to play a role in earlier electric ﬁeld experiments [17, 22]. In this case one lamella grows at the expense of another with a signiﬁcantly diﬀerent orientation by motion of a tilt boundary (wall defect) between them, leading to a direct transfer of scattering intensity between widely separated azimuthal angles ϕ. This is indeed observed in Fig. 8a. The integrated scattering intensity exhibits a temporary drop to 65 % of the initial intensity, which coincides with the decrease of the peaks of the initial orientation, indicating the formation of an intermediate structure, e.g. nucleation centers for lamellar grains oriented parallel to the electric ﬁeld. Their subsequent growth ﬁnally leads to an increase of the integrated intensity at later stages of the process.

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E

E a

b

c

E

E

Fig. 10. Schematic representation of lamellar microdomains between two electrodes. (a) Nucleation and growth and (b) grain rotation mechanism leading to microdomain orientation parallel to the electric ﬁeld vector (c).

Further away from the ODT, i.e. for high concentrations and low temperatures, the scattering pattern seems to be preserved and merely rotates into the new orientation. This observation points to grain rotation as an alternative orientation process (see Fig. 10b). In contrast to the nucleation and growth, microdomain orientations intermediate between the original and the ﬁnal orientations are observed. At the same time no increase in isotropic scattering intensity is found. We note that, in contrast to the nucleation and growth, the integrated scattering intensity in Fig. 8b is found to grow continuously. This indicates an overall growth of domains during the course of the reorientation process. In contrast to mechanical shear ﬁelds, the electric ﬁeld does not impose a preferred direction of grain rotation on the system. The fact that the ﬁnal orientation parallel to the electric ﬁeld vector is not fully reached within the experimental time frame is in agreement with the notion that the driving force for grain alignment almost vanishes as the aligned state is approached [17]. The transition from grain rotation to nucleation and growth can be explained by the fact that close to ODT we expect concentration ﬂuctuations which can be ampliﬁed using an external electric ﬁeld, i.e. the lamellar structure can easily be distorted in the direction of the electric ﬁeld vector. In addition, the mobility of defects such as wall defects at the grain boundaries is much higher. In order to further verify the above notion and to gain a more detailed understanding of the governing reorientation mechanisms, two-dimensional dynamic density functional theory simulations have been performed on lamellar block copolymer melts, which were able to reproduce the time evolution

Block Copolymer Microdomain Alignment in Electric Fields

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a

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b

5

time

2 10

5

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20

Fig. 11. Fourier transform squared as obtained by Dynamic Density Functional Theory MesoDynTM -Simulations of lamellar reorientation at dimensionless time t (in units of 1 000 timesteps). (a) System close to ODT (mean-ﬁeld interaction parameter εAB = 6 kJ/mol), (b) System further away from ODT (mean- ﬁeld interaction parameter εAB = 8 kJ/mol). Reprinted with permission from Macromolecules [45]. Copyright (2003) American Chemical Society.

of the scattering pattern observed in the experiments [37]. These simulations are based on energetic considerations namely involving electrostatic (i.e. driving force of the process) and interfacial energy (i.e. incompatibility between the block copolymer domains) arguments. The scattering functions calculated from these simulations are shown in Fig. 11 and exhibit the same characteristic features seen in the experimental scattering intensity in Fig. 8. In Fig. 12 we show a typical area of a large simulation box cropped around a newly forming grain for the system close to the ODT (mean-ﬁeld interaction parameter ˜ = 0.2. The initial structure εAB = 6 kJ/mol) at an electric ﬁeld strength α (t = 5 000 timesteps) has ﬁrst been aligned parallel to the electrodes by shear. Immediately after the application of the electric ﬁeld (at t = 5 100 timesteps), the lamellae become unstable and exhibit undulations very similar to the ones described by Onuki and Fukuda [48, 49], and, eventually, form point-like defects. These serve as nucleation centers for grains with lamellar orientation parallel to the electric ﬁeld. As the process evolves, new instabilities are formed around the new structure contributing to the growth of the grain with lamellar orientation parallel to the electric ﬁeld at the expense of all other orientations. Interestingly, the grain seems to grow faster in the direction perpendicular to the electric ﬁeld vector. On the other hand, the lower left corner of the simulation box shows a diﬀerent mechanism. Here, the reorientation, initiated by undulations, proceeds via movement of defects (disclination lines) and merely results in a rotation of the lamellar grains. This ﬁnding supports our previous notion that the two distinct mechanisms can be simultaneously found in our system. Depending on the degree of phase separation one of the two is dominant. In the presented simulation, as can be seen clearly also from larger simulation boxes, the nucleation and growth mechanism prevails [37].

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Bb

Cc

D d

Ee

Ff

Fig. 12. Real space images of a Dynamic Density Functional Theory MesoDynTM Simulation of reorientation close to ODT (εAB = 6 kJ/mol) at dimensionless time t (in units of timesteps). (a) 5 000, (b) 5 100, (c) 5 200, (d) 5 300, (e) 5 400, (f ) 5 500. The electric ﬁeld vector runs vertically. Alignment by nucleation and growth and grain rotation can be distinguished. Reprinted with permission from Macromolecules [45]. Copyright (2003) American Chemical Society.

For simulations further away from ODT, we predominantly ﬁnd that the grain rotation mechanism (induced by the movement of well deﬁned defects) governs the reorientation process. The governing mechanisms can be identiﬁed clearly from the squared Fourier transforms of the simulated lamellar rearrangement as depicted in Fig. 11 [37] which closely resemble the experimental data shown in Figs. 8a/b as well as Figs. 8c/d. Obviously, for the systems further away from ODT, the energetic cost for additional creation of defects as required for the nucleation and growth process is too high compared to the gain in energy from an aligned lamellar phase.

5 Inﬂuence of Initial Order Both lamellar and cylindrical mesophases in block copolymers preferentially align parallel to any boundary surface as such alignment typically decreases the interfacial energy of the structure with the boundary surface. In case of a plate capacitor, these eﬀects counteract the eﬀect of the electric ﬁeld which points perpendicular to the boundary surfaces. In consequence, a minimum electric ﬁeld strength (threshold electric ﬁeld strength) is required to overcome the parallel interfacial alignment [38,50–52]. Moreover, it has been shown that close to the electrodes the parallel alignment may prevail even if the bulk of the ﬁlm is oriented perpendicular to the interfaces, resulting in a mixed orientation of grains. This even holds for shear aligned samples where the eﬀective forces on the lamellae are much larger than in the electric ﬁeld case [10, 53]. While the thermodynamics of this competition is widely understood, the in-

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ﬂuence of the initial degree of order and initial orientation with respect to the boundary surfaces on the microscopic pathway leading to microdomain reorientation has barely been studied. This is even more surprising given the obvious importance of defect density and initial degree of order for the reorientation process as has been described for shear alignment experiments. Grain rotation relies on the movement of individual defects and therefore should strongly depend on the defect density. For the nucleation and growth of grains of preferred orientation on the other hand, both structural defects as well as thermal ﬂuctuations may serve as nuclei. Onuki and Fukuda have pointed out that undulation instabilities in lamellae in an oblique electric ﬁeld will only develop if the angle between the plane of the lamellae and the electric ﬁeld vector is suﬃciently large [48]. Therefore, not only the defect density but also the degree of orientation and the angle between the microdomains and the electric ﬁeld should be of importance. In this section, we investigate the inﬂuence of the degree of initial order on the microscopic route towards domain alignment. Using diﬀerent capacitor spacings (0.3 − 3.8 mm), we control the initial degree of order in the microdomain structures prior to application of the electric ﬁeld. As we have noted earlier, the lamellae are exposed both to the shear ﬁelds occurring during sample preparation (the sample solution is ﬁlled into the capacitor via a syringe inserted at one side of the cell with an intake diameter corresponding to the electrode spacing) and to the surface ﬁelds favoring parallel alignment of the lamellae [45]. In consequence, the initial microdomain orientation is not random, but preferentially aligned parallel to the capacitor plates. This pre-alignment, which improves with decreasing electrode spacing, can easily be quantiﬁed through the orientational order parameter P2 at t = 0 (P2,0 ). 5.1 Reorientation Experiments with Samples of Diﬀerent Initial Order We start our discussion with the kinetics of microdomain reorientation as followed in the center of a capacitor ﬁlled with a 35 wt.% solution of S50 I50100 in toluene (we note that due to the higher Mw , this sample is further away from ODT than S50 I5080 and is thus expected to exhibit the grain rotation mechanism) with the lamellae oriented parallel to the electrodes exhibiting an initial degree of orientational order of P2,0 = 0.50. Figure 13a shows the time dependence of the azimuthal scattering intensity at 1 kV/mm. Clearly, at short times an intermediate orientation is observed pointing to grain rotation as the dominant reorientation process. A major part of the scattering intensity is rotated continuously from the initial orientation at ϕ = 0◦ and ϕ = 180◦ , respectively, to the ﬁnal orientation at ϕ = 90◦ and ϕ = 270◦ . This situation changes signiﬁcantly if we turn to samples with higher initial alignment. In Fig. 13b we show the reorientation behavior of a polymer solution with P2,0 = 0.83. In contrast to the data shown in Fig. 13a, only

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Fig. 13. 3D representation of the azimuthal angular dependence of the scattering intensity for the reorientation of a 35 wt.% solution of S50 I50100 in toluene under an electric ﬁeld of 1 kV/mm. (a) P2,0 = 0.50; (b) P2,0 = 0.83. Reprinted with permission from Langmuir [54]. Copyright (2005) American Chemical Society.

two distinct domain orientations are observed resulting in scattering peaks at ϕ = 0◦ and ϕ = 180◦ (initial) and at ϕ = 90◦ and ϕ = 270◦ (ﬁnal).

2,0

S50 I50100

Fig. 14. (a) Kinetics of a 35 wt.% solution of in toluene at 1 kV/mm for diﬀerent initial degrees of order. Evolution of the orientational order parameter: : P2,0 = 0.04 (R), : P2,0 = 0.50 (R), : P2,0 = 0.66 (R), •: P2,0 = 0.83 (NG). (b) Experimental time constants τ as a function of the initial order parameter for solutions of S50 I50100 in toluene at 1 kV/mm: = 55 wt.%, = 50 wt.%, = 35 wt.% from (a). Open symbols relate to grain rotation as the dominant mechanism, while full symbols refer to nucleation and growth. Reprinted with permission from Langmuir [54]. Copyright (2005) American Chemical Society.

At intermediate times both orientations coexist, while almost no intermediate orientations are observed and only a negligible portion of the sample rotates. The time dependent plot of the azimuthal scattering intensity clearly shows the switching between the initial and the ﬁnal orientation. This scattering behavior is indicative of nucleation and growth of grains of the ﬁnal

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orientation. Grain rotation seems to be almost completely suppressed in this situation. It is interesting to quantify the kinetics of reorientation as a function of P2,0 . In Fig. 14a we show the time dependence of the P2 -values calculated from the raw data according to (1). Fitting a single exponential to the data using (3) yields time constants τ , which increase from τ = 7.8 sec at P2,0 = 0.04 to as much as τ = 81 sec for P2,0 = 0.83 (see Table 4). In Fig. 14b, we plot the Table 4. Time constants τ of the reorientation of S50 I50100 at diﬀerent values of P2,0 obtained from least squares ﬁts using (3) (wp = 35 wt.%, E = 1 kV/mm). NG: nucleation and growth, R: grain rotation. P2,0

τ [sec]

dominating mechanism

0.83 0.66 0.55 0.50 0.04

81.0 29.6 15.0 9.0 7.8

NG R R R R

kinetic data (τ -values) as a function of the initial degree of order P2,0 of the microdomains for three diﬀerent block copolymer solutions. Open symbols relate to grain rotation as the dominant mechanism, while full symbols refer to nucleation and growth. Only at suﬃciently low degrees of order grain rotation is observed. We identify a minimum degree of initial order characterized by a P2,0 -value between 0.6 and 0.7 (grey area in Fig. 14b), at which we observe a switch from rotation of grains to nucleation and growth. It therefore seems reasonable to assume that the degree of alignment (and in turn: the defect density) has signiﬁcant inﬂuence on the microscopic pathway for microdomain reorientation in the presence of the electric ﬁeld. 5.2 Simulating Systems with Diﬀerent Degrees of Initial Order The above interpretation is strongly corroborated by analysis of the real space data provided by computer simulations. We have performed two-dimensional dynamic self-consistent ﬁeld simulations on lamellar diblock copolymer melts starting from two diﬀerent initial conditions. In all cases the microdomain structure was ﬁrst exposed to a shear ﬁeld resulting in alignment of the lamellae in a well-deﬁned direction to the electric ﬁeld vector. Diﬀerent shearing times and directions (Figs. 15a/b: perpendicular to the electric ﬁeld vector, Fig. 15c: parallel to the electric ﬁeld vector) were chosen in order to produce initial states of diﬀerent degrees of alignment. The copolymer system studied here has mean ﬁeld interactions εAB = 6 kJ/mol. It was found earlier to exhibit nucleation and growth as the dominant reorientation mechanism [37,45].

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a

c

b

A

360

B

360

d

270

180

180 90

90

0

Time

5000

0

f

270 [°]

[°]

[°]

270

0

360

e

180 90

0

Time

5000

0

0

Time

5000

Fig. 15. (a–c) Self-consistent ﬁeld theory simulation of initial microdomain structure, prealigned using shear: (a) highly aligned sample (7 500 timesteps of shear perpendicular to electric ﬁeld), (b) less aligned sample (2 500 timesteps of shear perpendicular to electric ﬁeld), (c) least aligned sample (2 500 timesteps of shear parallel to electric ﬁeld). (d–f ) Fourier transform squared of 2D simulated structures at dimensionless time t: (d) Nucleation and growth mechanism for highly aligned sample, (e) Combination of nucleation and growth mechanism and some grain rotation for the less aligned sample. (f ) Solely grain rotation mechanism for the least aligned sample. The electric ﬁeld strength is α ˜ = 0.2. The arrow indicates the direction of the electric ﬁeld vector. Reprinted with permission from Langmuir [54]. Copyright (2005) American Chemical Society.

As the electric ﬁeld is applied to the microdomain structures described above, a distinctly diﬀerent behavior is observed depending on the initial degree of alignment parallel to the electrodes. In the highly aligned sample with a largely dominant microdomain orientation perpendicular to the electric ﬁeld vector (Fig. 15a), the reorientation process is rather slow and proceeds exclusively via nucleation and growth. The scattering functions calculated from these simulations are shown in Fig. 15d and exhibit the same characteristic features seen in the experimental scattering intensity in Fig. 13b. In the less aligned sample (Fig. 15b) reorientation is found to proceed faster and grain rotation increasingly contributes to the reorientation. This is seen in the scattering functions shown in Fig. 15e, which resemble the experimental data found at intermediate P2,0 -values (Fig. 13a). In the case of a structure even less aligned parallel to the electrodes (Fig. 15c), the same copolymer system exhibits only grain rotation via movement of individual defects perpendicular to the lamellae. The scattering function in Fig. 15f exclusively shows a shift of the peak, with no signs of the nucleation and growth mechanism present. This is very similar to what was observed for the lowest initial degree of order

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in this study, P2,0 = 0.04, as depicted in Fig. 16, showing that the lamellae are less ordered and tilted towards the electric ﬁeld lines. Obviously, consistent with the theory of Onuki et al. [48], the initial angle between the lamella plane and the electric ﬁeld vector is not suﬃcient for the instabilities in the structure to grow. Such instabilities are needed to nucleate grains of an orientation parallel to the external ﬁeld. In this case, the only possible route to follow for the system is the rotation of grains, as it proceeds via individual defect movement. Moreover, at these intermediate orientations, the electric ﬁeld induced torque acting on the lamellae increases with increasing lamellar misalignment with respect to the electrodes and reaches its maximum at a tilt angle of 45◦ [28].

Fig. 16. 3D representation of the azimuthal angular dependence of the scattering intensity for the reorientation of a 35 wt.% solution of S50 I50100 in toluene under an electric ﬁeld of 1 kV/mm and P2,0 = 0.04. Reprinted with permission from Langmuir [54]. Copyright (2005) American Chemical Society.

A detailed analysis of the real space view of the above described processes reveals that even in the absence of defects in the initial structure in Fig. 15a, grains of the new phase nucleate due to the growth of instabilities. The number of nuclei depends on the initial defect density. The speed of the reorientation processes can be monitored by P2 plots which qualitatively follow similar trends as the experimental curves as a function of initial alignment. In addition, we have also performed simulations for the copolymer system with mean ﬁeld interactions εAB = 8 kJ/mol, which was found earlier to reorient via the grain rotation mechanism only [37]. Here, we ﬁnd a total suppression of the rotation in the better aligned samples. The system is trapped kinetically and only a few defects typical for nucleation and growth are slowly generated. This ﬁnding is consistent with the interpretation of the experiments mentioned earlier, that the rotation will be increasingly suppressed when the lamellae are initially oriented more parallel to the electrodes, i.e. with increasing P2,0 .

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Judging from the simulation results, we may indeed assume that the different degrees of initial order are responsible for the observed diﬀerences both in the kinetics and in the mechanism of reorientation. As can be seen from Fig. 14b, the process of reorientation becomes slower with increasing alignment and decreasing defect density. This is accompanied by a switch in mechanism from rotation of grains to nucleation and growth. Once the degree of initial order reaches a value of P2,0 > 0.6, grain rotation seems to be largely suppressed. For P2,0 > 0.7 grain rotation is no longer observed. Obviously, the number of defects in the microstructure becomes insuﬃcient to support grain rotation by defect movement. Here, structural defects exclusively serve as nucleation centers for new domains. Comparing the rather diﬀerent time constants for the nucleation and growth process at the highest degrees of initial order, one is led to the assumption that in these cases the number of structural defects is too small to initiate suﬃcient nucleation centers. Therefore, thermal ﬂuctuations of the lamellae ampliﬁed by the external electric ﬁeld are needed for nuclei to be formed. In summary, it has been shown in this section that both the reorientation pathway and the reorientation kinetics for lamellar microdomains in an external electric ﬁeld strongly depend on the degree of order present prior to the application of the ﬁeld. Samples of the same concentration but diﬀerent initial order not only exhibit diﬀerent mechanisms of orientation but also proceed at diﬀerent rates. We observe consistently that for all systems rotation of lamellae by defect movement is faster than reorientation by nucleation and growth of new domains. Based on our results, we may conclude that above a certain initial orientation parallel to the electrodes the defect density and electric ﬁeld-induced torque are too low to allow for rotation of grains. In addition, the pressure on the well-aligned lamellae (as already pointed out by Onuki and Fukuda) [48] is larger than for less aligned samples and therefore leads to undulation instabilities which ﬁnally serve as nucleation centers for the growth of grains oriented parallel to the external electric ﬁeld. This leads to a switch in orientation mechanism with increasing initial microdomain orientation from rotation to nucleation and growth.

6 Outlook In this chapter, we have shown that polymer concentration, dielectric contrast of the blocks and temperature as well as the initial degree of order in the system are some of the key parameters governing the electric ﬁeld-induced alignment of block copolymer microdomains from solution. The interplay of these parameters determines the kinetics as well as the mechanisms of the overall orientation process. Gaining control over these factors turns electric ﬁelds into a powerful tool to direct block copolymer self-assembly. While the microscopic mechanisms and their determinig factors seem clear, the underlying mechanism at the level of individual macromolecules or even

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monomer units remains elusive. Therefore, one future area of research in this ﬁeld could be concerned with the investigation of conformational changes of the block copolymer chains induced by the external electric ﬁeld, e.g. a signiﬁcant stretching or compression of the Gaussian chains. In addition, it has been anticipated theoretically, that a strong electric ﬁeld may induce a shift in the order-disorder transition temperature (TODT ) at ﬁeld strengths close to the dielectric breakdown of polymeric materials (E > 30 kV/mm) [55, 56]. Such eﬀects have already been found to be induced by shear ﬁelds in lamellar block copolymer solutions [46]. For electric ﬁelds, Wirtz and Fuller reported on ﬁeld-induced structures in homopolymer solutions near the critical point [57, 58]. Tsori et al. recently found that electric ﬁeld gradients of moderate ﬁeld strength (4 kV/mm at 1 kHz) lead to demixing of blends consisting of low molecular weight PDMS and squalane [59]. So far, the above described phenomena and their impact on the morphology formation have not been studied for block copolymers. All these investigations will contibute to a deeper understanding of the interactions of polymer microstructures with external electric ﬁelds thus enhancing the high potential of the described directed self-assembly processes for future technological applications of block copolymer templates such as data storage, nanoelectronics or lab-on-a-chip devices [1]. Acknowledgement. I would like to thank all coworkers from the University of Bayreuth and the European Synchrotron Radiation Facility (ESRF) who have contributed to the work presented in this chapter, in particular: Volker Abetz, Hubert Elbs, Franz Fischer, Helmut H¨ ansel, Armin Knoll, Georg Krausch, Axel H.E. M¨ uller, Kristin Schmidt, Frank Schubert, Volker Urban, Thomas M. Weiss, and Heiko Zettl. In addition, I would like to acknowledge Agur Sevink and Andrei Zvelindovsky for performing the MesoDynTM simulations. This work is ﬁnancially supported by the ESRF and the Deutsche Forschungsgemeinschaft (DFG) in the framework of the Sonderforschungsbereich SFB 481 (Teilprojekt A2). Finally, I am grateful to Wolfgang H¨ afner for patiently introducing me to LATEX and much helpful technical advice. Moreover, I thank him, Kristin Schmidt and Heiko Schoberth for proof-reading of the manuscript.

References 1. 2. 3. 4. 5.

C. Park, J. Yoon, and E. L. Thomas. Polymer, 44:6725–6760, 2003. F. S. Bates and G. H. Fredrickson. Annu. Rev. Phys. Chem., 41:525–557, 1990. F. S. Bates and G. H. Fredrickson. Physics Today, 52:32–38, 1999. V. Abetz and R. Stadler. Macromol. Symp., 113:19–26, 1997. S. Ludwigs, A. B¨ oker, V. Abetz, A. H. E. M¨ uller, and G. Krausch. Polymer, 44:6815–6823, 2003. 6. R. J. Albalak, E. L. Thomas, and M. S. Capel. Polymer, 38:3819–3825, 1997. 7. T. L. Morkved, P. Wiltzius, H. M. Jeager, D. G. Grier, and T. A. Witten. Appl. Phys. Lett., 64:422–4, 1994.

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Structure and Dynamics of Cylinder Forming Block Copolymers in Thin Films Larisa A. Tsarkova Physikalische Chemie II, Universit¨ at Bayreuth, D-95440 Bayreuth, Germany

1 Introduction Over the last decades thin block copolymer ﬁlms have been the subject of intensive research, both experimental and theoretical. Nanoscale block copolymer structures have been used for nanolithography [1,2], fabrication of hybrid metallic [3–5] and inorganic [6] nanocomposites, semiconducting nanomaterials [7–10], and high-density information storage media [11, 12]. Comprehensive reviews on the physics of the phase behavior [13–16], characterization techniques [17], control over long-range order and orientation of block copolymer patterns [2, 18] and on their nanotechnological applications [19–21] are available. While compositionally asymmetric block copolymers [22,23] are the most promising materials for the fabrication of functional surface structures, most of the reported research concerns lamella forming systems. However, these ﬁndings have only been partially conﬁrmed for cylinder-forming block copolymers. An alignment of the bulk structures parallel or perpendicular to conﬁning surfaces as well as macroscopic structuring of the free surface was observed ﬁrst in lamellar systems [24–26] and later in cylinder- [27–32] and sphereforming [33, 34] block copolymer systems. The formation of novel structures which are not stable in the bulk of the respective materials is speciﬁc to thin ﬁlms of compositionally asymmetric block copolymers [16, 35]. Various theoretical models have been applied to describe this complex morphological behavior [36, 37]. However, the reported experiments and theoretical studies agree qualitatively only in part, and the underlying mechanisms determining the structure formation under given conditions are still poorly understood. In this chapter, we present fundamental aspects of the phase behavior and ordering dynamics in thin ﬁlms of compositionally asymmetric block copolymers. Cylinder-forming AB diblock and ABA triblock copolymers are used as model systems. In the introduction we brieﬂy describe the physics which commonly governs block copolymer phase behavior both in bulk and in thin

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ﬁlms; then we deﬁne characteristic features of thin ﬁlm behavior with special emphasis on the parameters that are readily controlled in experiments. Along with the discussion of surface ﬁelds and conﬁnement eﬀects in Sect. 2, we focus on the issues that are seldom discussed in the literature such as molecular weight and molecular architecture eﬀects on the phase behavior and on microdomain dynamics (Sect. 2), and the analysis of the characteristic microdomain dimensions (Sect. 3). We present measurements of the characteristic spacings and ﬁlm thickness with sub-micron resolution and demonstrate conﬁnement induced distortions of microdomain dimensions in thin ﬁlms relative to the respective parameters in bulk. In Sect. 4 we describe time-resolved details of structural ordering of cylinder microdomains which indicate that the dynamics of defect annihilation in block copolymer thin ﬁlms is considerably more complex than anticipated so far. 1.1 Block Copolymers Block copolymers consist of two or more covalently bonded immiscible components (blocks) and belong to the class of ordered ﬂuids, which display crystallike order on a mesoscopic length scale and ﬂuid-like order at a microscopic scale [16,40–44]. The mesoscopic structure formation in such systems is driven by competing interactions. The incompatibility of the monomers of diﬀerent blocks provides the short range repulsion which drives the phase segregation of the blocks into microdomains with mesoscopic length scales of 10 – 100 nm (microphase separation). A macroscopic phase separation is prohibited due to the covalent bond between the blocks. In mean ﬁeld theory, two parameters control the phase behavior of diblock copolymers: the volume fraction of the A block fA , and the combined interaction parameter χAB N , where χAB is the Flory–Huggins parameter that quantiﬁes the interaction between the A and B monomers and N is the polymerization index [40]. The block copolymer composition determines the microphase morphology to a great extent. For example, comparable volume fractions of block copolymer components result in a lamellar structure. Increasing the degree of compositional asymmetry leads to the gyroid, cylindrical, and ﬁnally, spherical phases (Fig. 1). If there are no strong speciﬁc interactions between A and B monomers like hydrogen bonding or charges, the interaction parameter χAB is usually a positive value and is small compared to unity. Positive values of χAB indicate a net enthalpic repulsion of the monomers. If χAB N is large enough the system minimizes the nonfavorable contacts between A and B monomers by microphase separation (stronger segregation). The induced order incorporates some loss of translational and conﬁgurational entropy. χAB is inversely proportional to the temperature of the system. Therefore mixing of the blocks is typically enhanced at elevated temperatures. When the temperature of the system increases (χAB decreases), the entropic factors eventually dominate

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and the system becomes disordered. This process is called order-disorder transition (ODT) and the respective temperature is the order-disorder transition temperature (TODT). Since the enthalpic and entropic contributions scale as χAB and N −1 , respectively, the product χAB N controls the phase state of the polymer (Fig. 1).

Fig. 1. Schematic representation of a phase diagram of self-assembled structures in an AB diblock copolymer melt, predicted by self-consistent mean ﬁeld theory [45] and conﬁrmed experimentally [46]. The MesoDyn simulations demonstrate morphologies that are predicted theoretically and observed experimentally in thin ﬁlms of cylinder forming block copolymers under surface ﬁelds or thickness constraints. Dots with related labels within the area of the cylinder phase indicate the bulk parameters of the model AB and ABA block copolymers discussed in this work (Table 1).

1.2 Physics of Block Copolymers in Thin Films A block copolymer in a conﬁned environment exhibits certain properties which can be characterized as thin ﬁlm behavior. This behavior is primarily dictated by the enhanced role of surface/interfacial energetics, as well as the interplay between the characteristic block copolymer spacings and the ﬁlm thickness. Surface ﬁelds are deﬁned as diﬀerences in the surface/interfacial energies between A and B blocks of the copolymer at the ﬁlm surfaces. They determine to a great extent the ﬁlm composition near the surfaces, i.e. the wetting conditions. The interfacial energetics and related wetting conditions for lamella forming systems are explicitly delivered in the recent review by Fasolka at al [47]. Symmetric wetting occurs when the same block is located at both interfaces. Experimentally this case can be realized in free standing ﬁlms [28,29,48]

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b

Free surface

A

B B

A

A

B

Film thickness ho

Symmetric wetting ho ~ Lo

Asymmetric wetting ho ~ 1½Lo

A

AS BS

Substrate

Substrate

Substrate

Fig. 2. (a): Schematics of parameters aﬀecting the phase behavior in block copolymer ﬁlms. The sketches of ABA triblock (left) and AB diblock (right) copolymers indicate possible molecular organizations in microdomains such as looping of the majority block in ABA block copolymer. (b) Schematic representation of a symmetric and an asymmetric lamella-forming ﬁlm with A (light) and B (dark) segments. The width of the A-B interface depends on the segregation power [39, 40].

and in the ﬁlms conﬁned between two identical surfaces [49,50]. Alternatively, block copolymer ﬁlms with diﬀerent blocks at each surface, termed antisymmetric ﬁlms in the case of lamellar systems, exhibit asymmetric wetting conditions. Typically in substrate supported ﬁlms, the block with the lowest surface energy will preferentially accumulate at the free surface, while the component with the lowest interfacial energy will be enriched at the supporting substrate. Preferential attraction of one of the blocks to the surface breaks the symmetry of the structure and results in layering of microdomains parallel to the surface plane through the entire ﬁlm thickness. The energetically favored ﬁlm thicknesses are then quantized with the characteristic structure period in the bulk (denoted here as Lo for lamella-forming systems and ao for compositionally asymmetric block copolymers). In the case of symmetric ﬁlms the thickness was shown to scale as n Lo , where n is an integer. If diﬀerent blocks are preferentially located at the interfaces, the thickness of stable the ﬁlms scales as (1/2 + n)Lo . As a result, the surface topography depends on the initial ﬁlm thickness h0 . When the thickness h0 of an as-cast substrate-supported, for example, symmetric lamella-forming ﬁlm deviates from nLo , topographical features such as islands, holes or bicontinuous surface patterns with two distinct thicknesses nL0 and (n + 1)L0 are formed to satisfy this constraint (Fig. 3). This phenomenon is most clearly manifested in ﬁlms with a prepared thickness gradient [51] or at the edges of spin-coated samples where ﬁlms are slightly thicker than in the center (inset to Fig. 5b). Nucleation and subsequent growth of surface relief structures have been investigated most for lamellar systems as a function of surface ﬁelds [52–54], molecular architecture [27], ﬁlm thickness [55, 56], and annealing conditions [27, 56, 57]. Evolving macro and microstructures have been probed with real and reciprocal space analysis techniques [17] such as optical microscopy, phase measuring interference microscopy, scanning force microscopy (SFM), transmission electron microscopy (TEM), neutron or x-ray reﬂectivity and

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grazing incidence small-angle x-ray scattering (GISAXS). For example, the area and height of islands and holes depend on the interfacial energy between the ﬁlm and the surface. At low surface ﬁelds, islands or holes have a smaller average area and smaller step height than islands or holes on strongly interacting substrates [52]. Also the dynamics of surface patterns formation and ordering was found to proceed faster in the case of strong surface ﬁelds. This result was attributed to the large-scale cooperation between adjacent regions of the polymer ﬁlm [52]. The step proﬁle at equilibrium depends on the absolute terrace heights. Both for lamella [33, 55] and cylinder-forming block copolymers [30, 31] as the ﬁlm becomes thick, the shape of the edge of the island or hole structure changes from a step function to a tanh function; relatively thick ﬁlms remain smooth even upon long-term annealing [33]. A more detailed summary of these studies can be found in recent reviews [15, 18, 58]. Dynamic studies of surface structure formation have shown that in addition to the thermodynamic parameters, kinetic factors often determine the experimentally observed morphologies or topography of the surface layer [34, 60, 61]. In particular it was shown that the layering of spherical domains parallel to the substrate or free surface does not imply a long-range lateral ordering of microdomains within each layer [33]. The latter process appears much more kinetically limited. Also a strongly interacting substrate was shown to retard the development of parallel lamellae compared to that at the air interface [62]. Formation of abnormal surface structures such as hole-in-hole and island-on-island [63] or interconnected islands and fractal holes [64] has been attributed to the stress induced by residual solvent. In addition, diﬀering kinetic mechanisms of lamella ordering have been related to certain ﬁlm thicknesses (spinodal decomposition for thick ﬁlms versus nucleation of the domains in thinner ﬁlms) [60]. Substrate supported thin ﬁlms with the same block at both ﬁlm surfaces can not be strictly considered as symmetric due to the diﬀerences in the absolute interfacial energies at the ﬁlm’s surfaces. Recent investigation of thin homopolymer ﬁlms demonstrated that a noninteracting interface such as air, inert gas, or vacuum introduced undulational instabilities [65] and accelerates segmental motion [66]. Furthermore, conﬁnement and strong surface ﬁelds are known to aﬀect both thermodynamic properties (glass transition temperature Tg , enthalpic interactions between the monomers) and dynamic properties (viscosity, diﬀusivity) of a polymer [67]. For example, depending on the substrate the Tg of ultrathin PMMA ﬁlms was found either to increase or, on the contrary, to decrease with decreasing ﬁlm thickness [68, 69]. Although these phenomena are not completely understood, the prevailing interpretation is based on the concept that a polymer ﬁlm on a solid substrate consists of three layers [68–71]. The polymer layer near the free surface is characterized by enhanced mobility and reduced packing density as compared to the bulk polymer. The behavior of the middle layer is similar to that of a bulk. The properties of the layer next to the substrate depend on the strength of the polymer-substrate interaction. As the ﬁlm thickness decreases, the contribu-

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tion of the top and bottom layers becomes increasingly signiﬁcant and determines the thickness dependence of the apparent Tg . Thus, strong polymersubstrate interactions restrict the chain mobility and lead to an increase of Tg of the polymer with decreasing ﬁlm thickness. On weakly interacting substrates, the free surface eﬀect dominates, and Tg decreases with decreasing thickness. The above eﬀects are likely to be general to polymeric materials. However, the substrate and conﬁnement modiﬁed physical parameters of block copolymers and their role in the morphological and dynamic behavior in thin ﬁlms are still open research areas. The development of experimental tools that directly access dynamics in thin ﬁlms will lead to new insights.

Islands

Bicontinuous

Holes

50 nm

2 μm ho~(n+1/3)ao

4 μm ho~(n+1/2)ao

4 μm ho~(n+2/3)ao

Film thickness h Fig. 3. SFM height images representing examples of surface macrostructures in thermally annealed SB1 ﬁlms with increasing ﬁlm thickness. The z-scale (50 nm) is exaggerated.

The intrinsic 3D interfacial curvature in the compositionally asymmetric block copolymers provides extra degrees of freedom for the phase behavior in hexagonally structured microdomains. It is now well established that conﬁnement of a cylinder-forming block copolymer to a thickness other than nao together with surface ﬁelds eﬀects can cause the microstructures to deviate from that of the corresponding bulk material. Surface structures in Fig. 1 are examples of simulated [72–74] and experimentally observed morphologies [28, 30–32, 75, 76] that are formed in thin ﬁlms of bulk cylinder-forming block copolymers. Additional frustration to the cylinder structure arises when the minority component is driven to the free surface or to the substrate due to interfacial interactions. At the same time, entropically driven surface segregation eﬀects have been observed in block copolymer ﬁlms. For example, neutron scattering measurements in lamella forming ﬁlms demonstrated that the conformationally smaller block preferentially segregated to both the solid and air interfaces [77]. In accordance with experiments, Monte Carlo simulations suggest that an energetically neutral surface exhibits a slight entropic preference to the shorter block due to the enrichment of chain ends near the hard wall [78].

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The published theoretical and experimental work on thickness and surface ﬁelds dependent structures in thin ﬁlms of asymmetric block copolymers indicates a much richer phase behavior than that produced by bulk melts. Despite the extensive research on the phase behavior of block copolymers in thin ﬁlms, there are only a few conclusive observations which can be applied to block copolymer ﬁlms in general. Most of the reported observations are restricted to the particular systems under investigation. It is especially diﬃcult to make general conclusions about the eﬀect of the surface ﬁelds as there is no well determined quantitative measure of this parameter in experiments. Often it is the molecular weight, composition, molecular architecture and annealing/quenching conditions which determine the unique thin ﬁlm behavior of a given block copolymer.

2 Surface Structures in Thin Films of Cylinder Forming Block Copolymers Here we summarize the results of a complex study of the phase behavior and pattern formation in thin ﬁlms of cylinder forming diblock and triblock copolymers which have compositionally/structurally analogous architecture. We focus on substrate supported ﬁlms, which are easily prepared and are mostly demanded in nanotechnology. The polystyrene-block -polybutadiene copolymers reported here form in bulk hexagonally ordered glassy polystyrene (PS) cylinders embedded in a soft polybutadiene (PB) matrix and are therefore ideally suited for thin ﬁlm characterization by scanning force microscopy (SFM). The diﬀerences in the molecular architecture and physical parameters are indicated in Table 1 and Fig. 1. With Tapping Mode SFM, the glassy PS cylinders that are buried underneath a rubbery PB surface layer can be imaged. The amount of the tip indentation and height artifacts have been quantiﬁed by Knoll et al. [79]; this allowed reliable identiﬁcation of the recorded microstructures. Equilibration of the phase-separated microdomains was performed by thermal annealing [32] or controlled swelling in the vapor of a non-selective solvent [31]. The solvent acts as a plasticizer and provides mobility to the ﬁlm without a signiﬁcant increase in temperature. Additionally, the solvent aﬀects polymer-polymer and polymer-surface interactions, thereby allowing both the ﬁlm thickness and the strength of the molecular interactions to be varied in a controlled way. 2.1 Development of Surface Relief Structures The aim of processing of block copolymer ﬁlms is to produce long-range ordered nanoscale features. Commonly, polymer ﬁlms are prepared by spincoating. Since the solvent usually ﬂashes oﬀ quickly, the resulting microphase separated structures are highly disordered, while the ﬁlm surface is smooth. Equilibration of microdomains is frequently done by annealing at elevated

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Larisa A. Tsarkova Table 1. Copolymers of polystyrene(S) and polybutadiene(B) blocks Label Structure Mw , kg/mol fP S ,wt% a0 ,nm SB1 S13 B34 47.3 SBS S14 B73 S15 102.0 SB2 S26 B70 96

26.1 26.0 24.5

30 40 70

temperatures or by exposing the ﬁlms to the solvent vapor which is common to block copolymer components. Quite often, during annealing undesired dewetting leads to the rupture of the polymer ﬁlm (Fig. 5a). Dewetting of polymer ﬁlms is extensively discussed in the literature (see recent review [80]), and we will not cover this issue here. The kinetic behavior of the surface relief structures in cylinder-forming block copolymers is likely to be similar to that of lamellar systems. The coarsening of the surface macrodomains is typically followed with time-resolved optical micrographs or SFM topographs which are used to determine the averaged macrodomain radius as a function of annealing conditions. The development of the surface roughness on an initially smooth ﬁlm starts during the early stages of annealing. The relative and absolute terrace heights increase with time until the characteristic interlayer distance is achieved. On a longer time scale, the pattern of terraces is still coarsening. This process is driven by the eﬀective line tension of the two-dimensional islands (holes) which tends to minimize with time the total length of the terrace edges. However, on a smaller length scale (over a few microdomain spacings), the ﬁlm thickness and the step proﬁle can be considered constant. According to our observations of three block copolymer systems (Table 1), the terrace formation was typically faster at higher vapor pressure/annealing temperature and for thinner ﬁlms. The dynamics of macrodomains depends more strongly on the block copolymer molecular weight and molecular architecture. Equilibration of SB1 and SBS ﬁlms by solvent annealing was achievable on reasonable experimental time scales (from tens of minutes to several hours depending on the ﬁlm thickness and on the solvent vapor pressure). The SB1 diblock copolymer, which is half of the SBS triblock copolymer, always moves faster due to its higher intrinsic diﬀusivity. The long SB2 diblock copolymer showed very limited movement during even a day-long annealing at the highest solvent content. Comparison of the microdomain dynamics in SB1 and SB2 ﬁlms suggests that the reduction of the chain mobility in SB2 ﬁlms is stronger than would be expected from the increased viscosity due to higher molecular weight. Although no quantitative analysis on the coarsening kinetics has been performed for SB and SBS ﬁlms, the macrodomains that formed at low polymer volume fraction in a ﬁlm were typically several times larger than in concentrated polymer ﬁlms. This observation emphasized the important contribution of diﬀusive transport during the grain growth. Conversely, at low vapor pres-

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sure and at low annealing temperatures, the step region in SB1 ﬁlms on silicon oxide substrates was typically characterized by a sequence of morphologies as shown in Fig. 4 d, which introduced additional interfacial energy at the phase boundary [32]. Surprisingly, in these cases the macrodomain average size was quite small (about 1 μm) corresponding to an extremely high total area of the terraces edges. Signiﬁcantly for the following discussion, the stage of terrace development when the step proﬁle does not change with time was chosen as a measure of the thermal equilibrium, as was suggested by Knoll at al [31]. The terrace heights and microdomain morphologies in such ﬁlms have been measured and analyzed to construct the phase diagrams of surface structures (Sects. 2.3 and 2.4). 2.2 Surface Fields and Wetting Conditions Variation of the preferential interaction of the block copolymer components with the substrate provides important insight into the microdomain ordering in thin ﬁlms. Several routes have been developed to modify the interfacial interactions involving the utilization of chemically diﬀerent substrates [32, 81], the adsorption of random block copolymer layers composed of identical monomers to the diblock copolymers [82], or the grafting of endfunctionalized random copolymers (brushes) [83–85]. These can be applied to both interfaces of the ﬁlm [49, 86] and also with diﬀerent surface coverage [84, 87] which allows the variation of the interactions between the copolymer and the substrate in a systematic manner. Growing polymer chains from a surface using anchored initiators [88] and chemical modiﬁcation of selfassembled monolayers (SAM) [52,89–92] have been used to control the aﬃnity of polymer segments to a surface and thus to manipulate the orientation of morphologies parallel or perpendicular to the surface. Another approach to diminish the attractive potential for one component of the block copolymer to a substrate is to increase the roughness of the substrate. This was shown to inhibit the growth of substrate- directed parallel lamellae formation and resulted (in conjunction with a neutral air/polymer interface) in the formation of perpendicularly oriented lamellae [93, 94]. Recently we demonstrated that the phase behavior and interlayer period of cylindrical domains depends on the surface ﬁelds and related wetting conditions [32]. Thin ﬁlms of SB1 (Table 1) have been equilibrated by thermal annealing on two chemically diﬀerent substrates and used as models of symmetric and asymmetric wetting. On both substrates the free surface of the ﬁlm is covered by a 10 nm PB layer as a lower surface energy component [79]. Evaporated carbon coating was used as a low energetic surface and nearly neutral substrate for the block copolymer components. In the other system, the majority polybutadiene phase segregated both to the free surface and to the silicon oxide substrate due to the strong speciﬁc interaction. Strong pinning of the polybutadiene block to the silicon oxide substrate was also noticed

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SiOx/Si

Carbon/SiOx/Si

a

h = 14 nm ~ 1/2 ao

h = 17 nm ~ 1/2 ao

b

10

o

0o

100 μm

T1/2

c

T0 T21/2

d T2

SiOx

1 μm

T11/2

T1

Fig. 4. (a,b): SFM phase images of surface structures in ultrathin SB1 ﬁlms formed upon annealing in vacuum (at 120o C for 20 hours) on indicated substrates. White corresponds to PS phase.(c,d): 3D pictures representing coexisting terraces and surface structures formed upon thermal equilibration. White stripes are PS cylinders in a PB matrix; white featureless area in image d is the PS lamella sheet; hexagonally perforated lamellae (PL) are visible in the step between terraces T11/2 and T21/2 as ordered dark dots (c); randomly perforated lamellae are formed in terrace T1 (d).

by Harrison at al [85], who observed retarded ordering dynamics of a single cylinder layer on the silicon oxide. Fig. 4a,b shows microdomain patterns in ultrathin (∼ 1/2a0 ) ﬁlms on each of the studied substrates. The shape of the microdomains on the carbon coated substrate (Fig. 4a) can be described as half-cylinders, while the disordered pattern on the silicon oxide can not be attributed to any distinct morphology (Fig. 4b). Also in thicker ﬁlms, the phase behavior depends on the substrate (Fig. 4c,d). On silicon oxide substrates the lamella-like non-bulk morphologies appear in thick ﬁlms of up to four layers (Fig. 4d). Comparison of the surface structures in Fig. 4 suggests diﬀerent segregation regimes on each of the substrates. For lamellar systems it has been shown earlier that the degree of interfacial segregation of the block copolymer is proportional to the surface potential [87]. The above results suggest that strongly interacting surfaces aﬀect the enthalpic interactions of the adsorbed monomers and thus enlarge the incompatibility of the block copolymer components, which in turn initiates phase transitions to non-bulk morphologies.

Structure and Dynamics in Thin Block Copolymer Films

Carbon/SiOx/Si

SiOx/Si

ho = 30 nm ~ ao

ho = 30 nm ~ ao

a

b

40 mm

c Film thickness h [nm]

241

40 mm

d

150

100

50

h ~ (1/2+n )aca

h ~ n asi

}wo 0

1

2

3

4

5 0

1

2

3

4

5

Number of layers [n]

Fig. 5. (a,b): Optical micrographs of the surface topography of SB1 ﬁlms that were equilibrated thermally in vacuum for 20 hours on the indicated substrates. The inset in (b) demonstrates terrace formation at the edge of the sample. (c,d): Plots of the absolute ﬁlm thickness (measured with Metrology SFM) versus the number of layers on carbon coated (c) and silicon oxide (d) surfaces. The sketches illustrate the presumed inner structure of the ﬁlm.

Fig. 5 summarizes the observations of the substrate dependent terrace formation for the wetting conditions described above. Upon thermal annealing, the structures align parallel to the ﬁlm plane forming up to ﬁve distinct layers [32]. Optical micrographs in Fig. 5a,b demonstrate substrate determined diﬀerences in the surface topography of SB1 ﬁlms with the same initial thickness. In Fig. 5c-d, absolute terrace heights are plotted versus the number of layers. The equilibrium terrace heights hi are hi = n csi for symmetric and hi = (n + 1/2) cca for asymmetric wetting, where csi = 31.1 ± 0.2 nm and cca = 27.4 ± 0.2 nm. The results have been attributed to the substrate induced half-period shift in the layering of cylindrical domains. Thus, when the wetting conditions are dictated by the major component of the cylinder phase, the equilibrium terrace thickness obeys laws established for lamellar systems.

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When a minor component is driven to the substrate/free surface of a ﬁlm, a brush-like layer is formed via anchoring of the short block to the substrate [85]. In this case the response of the surface topography and the phase behavior to the thickness constraint is anticipated to be strongly dependent on the strength of the surface ﬁelds. A few indications to the distinct dependence of the surface topography on the block copolymer composition were reported in [61]. In this study, diblock copolymers containing a majority of the wetting component formed islands and holes more readily, while those having a minor fraction remained ﬂat even for long annealing times. 2.3 Phase Behavior in Block Copolymer Melts For cylinder forming block copolymers, surface ﬁelds and conﬁnement eﬀects have primarily been considered theoretically [72, 74, 95–98]. Simulations using dynamic density-functional theory (DDFT) suggest that the surfaces may initiate transitions from a hexagonal to a lamellar phase in the case of suﬃciently strong surface interactions [72, 74]. For weak surface interactions, the parallel or perpendicular alignment of the cylinders is predicted to be dependent on the commensurability of the ﬁlm thickness to the bulk spacing of the cylindrical domains [98]. The parallel oriented cylinders either terminate with half cylinders or with a wetting layer of the block which is preferentially attracted to the surface [74, 99]. Recent calculations indicate that a signiﬁcant change in the ﬁlm morphology occurs even for a small change in external parameters if the block copolymer composition is close to the borderline between two diﬀerent bulk phases [96]. In the self-consistent mean ﬁeld approach, the surfaces are allowed to modify the Flory-Huggins interaction parameter and the chemical potential in the adjacent copolymer layer [100]. Most of the above predictions on the surface ﬁeld-regulated phase behavior in cylinder forming block copolymers have not been supported by experiments due to the limited number of studies utilizing chemically diﬀerent substrates. Harrison at al [85] have used a layer of a random block copolymer and a polymer brush to tune the wetting conditions of a single layer of cylindrical microdomains. The variation of the interfacial interactions at the polymer/substrate interface in solvent-swollen ﬁlms of a triblock copolymer [31] and a terpolymer [101] seems to be insuﬃcient, as the solvent signiﬁcantly screens the presence of the substrate. Mapping of the microdomain morphologies to the local ﬁlm thickness was used to construct the phase diagram of the surface structures in SB1 melts as a function of the ﬁlm thickness and the strength of the surface ﬁeld at the substrate (Fig. 6). Gray areas mark the averaged borders of the equilibrium terrace heights, where the favorable morphologies developed. The white regions indicate the transition thickness (steps between the terraces). The black curves are drawn as a guide to the eye in line with the phase boundaries predicted by DDFT simulations [72–74, 97]. Phase behavior in thin SB1 ﬁlms shows clear dependence on the type of the substrate. On the neutral

Structure and Dynamics in Thin Block Copolymer Films

Carbon/SiOx/Si

243

SiOx/Silicon surface field at the substrate

T1½

120 T4½

PL 4

CII

CII/PL Film thickness (nm)

T2½

90

T3½

CII

T3

L

PL

3

CII T2½

CII

PL

2 PL T1½

CII

CII

30

T1½

L 1 CII

T½ 0

T2

L

60

CII½

Carbon/SiOx/Si

T0

dis

T1

SiOx/Si

10o

T½

0o

500 nm

T1

Fig. 6. The phase diagram of the surface structures on a weakly interacting surface (carbon coating) and under a strong surface ﬁeld with a preference for the major component (silicon oxide). Gray areas mark the borders of the energetically favored ﬁlm thickness. The SFM phase images (1x1μm) present examples of the surface structures on carbon coated (left column) and silicon oxide (right column) substrates in the indicated terraces. Black curves schematically contour the phase boundaries and are drawn as a guide to the eye in line with the DDFT simulations. Reproduced with permission from [32] (with adaptions). Copyright (2006) American Chemical Society.

substrate (carbon coated silicon) under asymmetric wetting conditions, cylinders are aligned parallel to the surface both at the energetically favored and the transition thicknesses. The interlayer distance is consistent with the bulk spacing of the SB cylindrical microdomains. Half-cylinders are stabilized at the bottom of the ﬁlms reﬂecting no preference of the SB components to the substrate. Under a strong surface ﬁeld and symmetric wetting conditions, the perforated lamellar (PL) phase is typically developed in up to 4 layers. The cylinder phase was found in only the ﬁrst layer of structures at the favored ﬁlm thickness and at all transition regions. Thickness induced phase transition from the cylinder to the lamella (L) structures is attributed to the increased

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incompatibility of the SB components when the PB block is strongly attracted to the substrate. In agreement with the simulated phase diagrams, the border lines of the PL and L phases shift into the higher surface ﬁelds with increasing ﬁlm thickness. The stabilization of the non-bulk structures in the uppermost layer on the silicon oxide, as well as a discrepancy of 10% in the terraces’ period on the two substrates (Fig. 5), suggest that the strong surface ﬁeld induces the phase transition throughout the SB ﬁlms. Earlier Matsen predicted [102] that the cylinder to PL phase transition is accompanied by an increase of 10% in the microdomain spacing. Recently, the non-bulk PL phase has been observed in up to six layers of structures in bulk-cylinder forming polystyrene-block poly(methyl methacrylate) copolymer [103]. The authors postulated that the substrate-induced reorganization may be responsible for the formation of the PL structure which templates from the ﬂat surface. Radzilowski at al [28] have studied cross sections of free standing ﬁlms of a cylinder forming polystyreneblock-polybutadiene diblock copolymer (with ao ∼ 22 nm). These ﬁlms exhibited symmetric wetting by the majority polybutadiene component, however with both ﬁlm sides facing air. The authors established a PL phase at the minimum ﬁlm thickness, which was 20% higher than the interlayer thickness in a cylinder phase. The higher characteristic domain spacing of the PL structure is consistent with our observations. In thicker free standing ﬁlms, the cylinders were aligned parallel to the surface plane. We emphasize that in symmetric substrate supported thin ﬁlms the non-bulk PL phase propagates through several layers of structures, emphasizing the strong surface ﬁeld eﬀect at the substrate. The depth-proﬁling of multilayered block copolymer ﬁlms is indispensable for the control of nanostructured materials and their applications. Over the past few decades experimental methods for in-depth proﬁling of polymer ﬁlms have utilized labeling of components with deuterium. The major disadvantage of these techniques [17] which include forward recoil spectrometry [104], nuclear reaction analysis, secondary ion mass spectrometry (SIMS) [33, 105], neutron reﬂectivity [17, 106, 107] and small-angle neutron scattering (SANS) [108, 109] is that the composition-depth proﬁle is averaged over large lateral distances and does not address lateral patterning on the nanoscale. Simulations show a cylinder phase in the bulk of the ﬁlm below the PL at the surface [72–74, 97]. Also experimentally observed non-bulk phases in the topmost ﬁlm layer have been identiﬁed as surface reconstructions that are modulated by conﬁnement eﬀects [76, 97, 110]. Using transmission electron microscopy (TEM) to image cross sections of thin ﬁlms, although very useful in identiﬁcation of the orientation of lamella or cylinder domains [12, 111], does not unambiguously distinguish between in-plain cylinders and the PL morphology [28, 103]. Nanotomography [112, 113] and depth proﬁling with nonselective etching [114,115] or UV radiation [116] have proved to be power-

Structure and Dynamics in Thin Block Copolymer Films

245

ful tools for the study of structural details in the interior of block copolymer ﬁlms, although experiments of this kind are still rare. Recently, buried structures in block copolymer ﬁlms have been probed with GISAXS [117,118]. The authors were able to identify hexagonal packed cylinder structures, hexagonally PL, and the gyroid phase in block copolymer ﬁlms and to establish preferential orientations of microdomains. With GISAXS the limitations of conventional SAXS with respect to extremely small sample volumes in the thin-ﬁlm geometry have been overcome. As with SFM, the length scales between molecular and mesoscopic distances are detectable with this surface-sensitive scattering method. 2.4 Phase Diagram of Surface Structures in Swollen Films

a

b

C||,2 C^

PL||,1 C||,1 C^ 500 nm

Fig. 7. (a) Phase diagram of the surface structures in SBS ﬁlms (Table 1) versus the partial chloroform vapor pressure. Horizontal dashed lines represent the energetically favored terrace thickness. Data are given for equilibrium ﬁlm thickness of lying cylinders (circles), disordered structures (stars) and for upper and lower bounds (open and closed symbols, respectively) of up-standing cylinders (squares) and the perforated lamellar phase (triangles). Reprinted with permission from [76]. Copyright (2002) the American Physical Society. (b) Phase SFM images representing the indicated surface structures along the vertical dashed line in (a).

The equilibration of block copolymer ﬁlms by exposure to solvent vapor is an alternative to the thermal annealing procedure. It has been shown that the resulting morphologies are strongly sensitive to the solvent evaporation rate [119–121], the selectivity of the solvent [121–127] and to the volume fraction of the polymer in a swollen ﬁlm [31]. Block copolymer ﬁlms in the presence of selective solvents exhibit structural polymorphism which is very promising in terms of fundamental understanding and practical applications [128].

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As established by Knoll at al [31,76], exposing thin ﬁlms to well-controlled vapor pressure with a subsequent fast quench provides reproducible data on the phase behavior in swollen thin ﬁlms. This experimental approach was used recently for SBS triblock copolymer [31]. The degree of one-dimensional swelling was calibrated by in-situ spectroscopic ellipsometry. The phase behavior, the absolute terrace heights and the border lines between the phases were determined with Metrology SFM. The phase diagram of the surface structures was constructed by plotting the observed phases as a function of the ﬁlm thickness and the solvent partial vapor pressure (Fig. 7). At favored ﬁlm thicknesses, cylinders orient parallel to the ﬁlm plane, while a perpendicular orientation is formed at intermediate ﬁlm thicknesses. In ﬁlms thinner than 1.5 domain spacings and at high polymer concentration, the cylindrical microdomains reconstruct to the PL structure. These experimental results are in excellent agreement with computer simulations based on dynamic density functional theory (DDFT) [73, 74]. We have used the same experimental ap-

a

b

C||,3 C^

140

100 80 60 40 20

PL

C||,5 lateral order-disorder transition

h [nm]

120

0 0.8

C||,4

PL||,2 C||,2

PL

C||,3

L

PL C||,2

PL||,1

L PL

C||,1

C||,1 C^

C||,½ dis 0.6

p/ps

0.4

200 nm

Fig. 8. (a) Phase diagram of the surface structures in SB1 ﬁlms (Table 1) versus the partial chloroform vapor pressure. Horizontal dashed lines represent the energetically favoured thickness of the terraces. Data are given for equilibrium ﬁlm thickness of lying cylinders (circles), disordered structures (stars), up-standing cylinders (squares), the perforated lamella (triangles) and lamella (diamonds) phases. (b) Phase SFM images representing the indicated surface structures along the vertical dashed line in (a).

proach to study thin ﬁlms of concentrated solutions of SB1 diblock copolymer. The comparison of the phase diagrams of surface structures in Figs. 7 and 8 suggests that the general phase behavior of the SBS triblock copolymer is very

Structure and Dynamics in Thin Block Copolymer Films

247

similar to that of the SB diblock copolymer. In particular, the same sequence of structures with increasing ﬁlm thickness is observed (Figs. 7b and 8b), including deviations from the bulk structure, such as the PL and a wetting layer in ultrathin ﬁlms. This conﬁrms the universality of the reported phenomena. However, the details in the phase diagram of SB1, such as the position of the phase boundaries, the structure of steps between terraces, and the stabilization of the lamellar phase diﬀer from those of the triblock copolymer. The observed diﬀerences in the phase behavior are presumably the consequence of the molecular architecture eﬀects, which originate from the larger molecular weight of the middle majority block and from the speciﬁc chain arrangement in triblock microdomains (Fig. 2). Eﬀect of the Molecular Architecture In the bulk of block copolymer melts, diﬀerences in molecular architecture at constant chain length and composition have been shown to aﬀect thermodynamic properties, such as microphase separation, ODT regime, chain conformation and ﬂuctuation eﬀects [42], as well as viscoelastic behavior [129]. The need to locate the two block-junctions of ABA triblock copolymer at a domain boundary signiﬁcantly reduces the conformational entropy of its ordered state and hence increases the stability of the disordered phase relative to that of an AB diblock copolymer of the same composition and chain length. If one considers a melt of symmetric ABA triblock copolymer and that of a hypothetical melt formed by snipping the triblock copolymer in half, then pulling the middle B block of the ABA copolymer out of its domain costs approximately twice as much Gibbs energy as pulling the B block of the AB copolymer from its domain. Consequently, in this case the melt of the triblock copolymer will remain ordered up to a higher temperature [129]. The asymmetry in the microphase-separation boundary has been conﬁrmed experimentally [130]. Fluctuation eﬀects are predicted to be greater in the disordered triblock melt compared to a diblock copolymer of the same chain length, while the interaction parameter χ and its temperature dependencies are found to be the same for all molecular architectures. Furthermore, theory predicts that near the ODT the additional segregation will amplify chain stretching and produce a larger domain spacing in the ordered triblock melt. Experimentally, it was shown that lamella-forming triblock copolymers are stretched 10% more than their corresponding diblock copolymers [130]. There are only a few reported studies concerning structurally analogous tri- and diblock copolymers [27, 33, 98, 131] and molecular weight effects [33, 132] in thin ﬁlms. Lamellar structures in thin ﬁlms can be ordered faster than the cylinder phase formed by the block copolymer with the same molecular weight [27, 131] which indicates diﬀerent transport mechanisms in these phases. On the other hand, the layering of spherical domains propagates a surprisingly long distance (nearly 100 layers of spheres) perpendicular to the surface [33], while not more than 8 ordered layers of cylinders have

248

Larisa A. Tsarkova T2

T1

C||,1

C||,2

T2

T3

C^

C||,3

a 1 μm

b

6 mkm

400 nm

Fig. 9. SFM phase images of the surface structures in SB2 (a) and SB1 (b) ﬁlms with increasing thickness, annealed at a chloroform vapor pressure of 100% and 80%, correspondingly.

been reported in thin supported ﬁlms [103]. Yokoyama at al [33] have convincingly demonstrated that in the low molecular weight regime, the layering is controlled by thermodynamics. Decreasing the temperature of annealing and increasing values of χ increase the number of layers of spherical domains in substrate supported ﬁlms. However, at large χN (further from the ODT) the kinetics becomes very slow due to slow diﬀusion. In this regime the number of layers of spheres increases with increasing annealing time, and block copolymers with large χN show a smaller number of ordered layers at a given annealing time [33]. Busch at al [132] studied the inﬂuence of molar mass on the lamellar orientation in thin ﬁlms and concluded that for weak surface interactions, molar mass is the key factor which determines lamellar orientation in spincoated ﬁlms. In the low molecular weight regime, lamella ordered parallel with respect to the free ﬁlm surface, whereas for high molar mass (above 100 kg/mol), a perpendicular orientation of the lamella was observed. The latter eﬀect was explained by entropic contributions, such as the nematic eﬀect and the chain end eﬀect. The above considerations and our observations suggest that it is not feasible to maintain similar preparation conditions and thermodynamic parameters when comparing molecular architecture eﬀects in thin ﬁlms. As mentioned above, AB diblock copolymers with a molecular weight around 100 kg/mol show very limited movement both under elevated temperatures and in swollen ﬁlms, even at the lowest achievable polymer volume fraction φ of about 0.3. Under these conditions, a comparison with diblock copolymers of half the length is not possible due to the diﬀerences in χN , which strongly aﬀects microphase segregation and phase behavior. Fig. 9 depicts surface structures with increasing thickness of SB2 (a) and SB1 (b) ﬁlms which have been annealed under controlled chloroform vapor. Films of SB2 showed topographical features and long-range lateral microdomain order, such as lying cylinders in terraces and perpendicular oriented cylinders at the transition thickness.

Structure and Dynamics in Thin Block Copolymer Films

249

The degree of one-dimensional swelling of SB1 and SB2 ﬁlms was calibrated by in-situ spectroscopic ellipsometry, and was found to be similar to that of the SBS triblock copolymer reported earlier by Knoll et al. [31]. Nevertheless, SB2 ﬁlms already show a disordered pattern at a vapor pressure of about 80% (φ ≈ 0.65), which indicates the vicinity of the ODT under these conditions. To conclude, the reported studies and our observations strongly suggest that the layering of microdomains perpendicular to the ﬁlm plane and formation of long-range lateral order in block copolymer ﬁlms is possible only in a narrow regime of χN . Improvement of structural order can be achieved by ﬁne tuning the molecular architecture, adding solvents or modifying the surface interactions.

3 Characteristic Dimensions of the Microdomain Structures For an in-depth understanding of the self-assembly process in block copolymer ﬁlms a quantitative measure of the characteristic distances and their dependence on relevant physical parameters (temperature, chain length, etc.) is indispensable. In bulk such measurements have typically been done by small angle X-ray scattering (SAXS) and the results are widely in line with the theoretical predictions [39]. In thin ﬁlms, however, the situation is considerably more diﬃcult as conventional scattering experiments are barely possible due to the insuﬃcient amount of material. In lamellar forming block copolymers, neutron reﬂection experiments have successfully been used to determine the characteristic lamellar spacing in thin ﬁlms [50,133]. Up to now no such quantitative experiments on characteristic lengths have been reported for compositionally asymmetrical block copolymers. This is most likely due to the experimental diﬃculty of accurately measuring lateral distances in the presence of a large number of structural defects [108]. The microdomain structure of cylinder forming or sphere forming block copolymer thin ﬁlms is typically imaged using SFM. However, the experimental error involved in a conventional SFM length measurement is of the order 10-15% due to non-linearities of the piezoelectric elements and thermal drifts. Recently, Knoll [134] has presented an algorithm to quantify the characteristic lateral spacings in thin ﬁlms of cylinder block copolymers. In order to reliably determine the lateral spacings, the imaging has to be performed with a SFM optimized for metrological measurements (Dimension 3100M SFM; Veeco Instruments). This instrument is based on piezo elements operated in active feed-back mode in all three dimensions; in consequence, length measurements are possible with an experimental error smaller than one nanometer. Figure 10a presents a typical phase image of cylinder structures. The white contour lines are taken from the height images (not shown) and are superimposed onto the phase images. They mark the borders of the areas with the favored ﬁlm thickness (terraces). Between two neighboring terraces the

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Larisa A. Tsarkova

a

1 2 34

b

3

34

4

Spacings [nm]

33

500 nm

32 31

12

30 29 -10

-5

0

5

10

Height [nm]

Fig. 10. (a) SFM phase image of the surface structures in an SB1 ﬁlm equilibrated under 70% of the partial chloroform vapor pressure. The white contour lines are taken from the height image (not shown) and mark the borders between the ﬂat regions, as well as the transition region between the terraces. (b) Spacings of cylindrical microdomains as a function of the local ﬁlm thickness.

ﬁlm thickness increases gradually. The result of the analytical procedure is shown in Fig. 10b, where the characteristic distances between the cylinders are plotted as a function of the local thickness of the ﬁlm. Analysis of this kind of data as a function of the polymer concentration and of the local and overall thickness of SBS and SB swollen ﬁlms has been recently reported [134, 135]. Quite strikingly we observed a systematic variation of the lateral spacing between the cylinders as the ﬁlm thickness increases from n to n+1 layers of cylinders. On increasing local ﬁlm thickness, the lateral distances ﬁrst decrease, then increase rather abruptly, and eventually decrease again (Fig. 10b). Note that the overall variation of the lateral distance amounts to only some 10% of the characteristic spacing, which would be hidden by the experimental error if a conventional SFM had been used for the imaging. The abrupt change of the microdomain spacing on inclusion of an additional layer of cylinders qualitatively resembles the ﬁndings of Lambooy at al [133] for conﬁned lamellar systems and indeed can be explained along similar lines. We start from a single layer of cylinders and steadily increase the ﬁlm thickness. If we assume an aﬃne and volume conserving deformation of the microdomain unit cell, the increasing ﬁlm thickness will ﬁrst lead to a stretching of the unit cell, thereby resulting in a decrease of the lateral distance between neighboring cylinders. Eventually, the excess energy needed for this deformation will exceed the energy needed to accommodate an additional layer of cylinders at the given ﬁlm thickness, which will be accompanied by an overall compression of the unit cell and consequently an increase of the lateral spacing. Despite the similarities, however, we note that in the system described here the ﬁlm thickness is not imposed on the system by external constraints but spontaneously adjusts itself in the region between adjacent terraces.

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Below we give further examples of spacings analysis in thin ﬁlms and demonstrate that this approach can be widely used to understand and to control nanopatterned surfaces. In particular, microdomain dimensions can be changed in a systematic manner by varying ﬁlm preparation. Temperature Dependent Microdomain Spacings Using the same analytical approach, we studied thermally equilibrated ﬁlms of SB1 melts. Fig. 11a shows the behavior of both the bulk spacing and the spacing within a single layer of SB1 cylinders as a function of the annealing temperature. The bulk data were obtained by small angle synchrotron radiation at the European synchrotron radiation facility (ESRF) at Grenoble, France [32, 136]. With increasing temperature the bulk characteristic spacing decreases. We observe a similar and even more pronounced eﬀect in a single layer of cylinders. Again, the size of the eﬀect is rather small and potentially hidden within the experimental error in conventional SFM experiments [136]. The eﬀect of temperature on the characteristic spacing in a phase separated

Spacings [nm]

34

a

34

33

33

32

32

350 375 400 425 0 Temperature of annealing [K]

b

1

2 3 4 Number of layers

5

Fig. 11. (a) Characteristic microdomain spacing versus the temperature of annealing. Open symbols indicate SAXS measurements, ﬁlled symbols represent SFM measurements. The solid lines are linear ﬁts. (b) Characteristic microdomain spacing evaluated using home-built software versus the number of cylinder layers (ﬁlm thickness) at 105o C (squares) and 120o C (circles).

block copolymer system can be easily rationalized within the framework of the strong segregation theory. Here, the characteristic spacing is determined by the competition of enthalpic (interfacial energy) and entropic (chain stretching) contributions to the free energy of the system. The entropic term favors a random coil structure of the chains while the enthalpic term favors rather stretched chains and thereby an overall reduced interfacial area between the diﬀerent blocks. As the entropic term is proportional to temperature, a temperature increase will lead to less chain stretching thereby reducing the interdomain spacing. Importantly, the diﬀerences in the slope of the temperature

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Larisa A. Tsarkova

dependence of spacings in the bulk and in thin ﬁlms (Fig. 11a) indicates a stronger segregation conditions in thin ﬁlms as compared to the bulk of the respective material [136]. Film Thickness Dependent Microdomain Spacings Additionally to the spacing variation with the local ﬁlm thickness at the step between the terraces (Fig. 10), we also found that the spacing between the cylinders measured within the individual terraces increases with increasing terrace number. Fig. 11b summarizes this eﬀect for thin ﬁlms of SB1 diblock copolymer equilibrated by thermal annealing at two diﬀerent temperatures. The data indicate that the average spacing between neighboring cylinders is smallest in the thinnest ﬁlms and eventually approaches the bulk value at ﬁlms consisting of three or more layers of cylinders. The large error bars in Fig. 11 arise from the averaging over a large number of diﬀerent measurements performed on diﬀerent samples, while the error of an individual measurement is as small as in Fig. 10. The overall scatter between diﬀerent samples results in somewhat larger uncertainty. However, the eﬀect itself is clearly measurable and points to one-dimensional stretching of a unit cell in thin block copolymer ﬁlms [135]. Shear Induced Deformation of Microdomains Processing of polymer ﬁlms often results in the formation of dewetting patterns which are undesirable for thin ﬁlm applications where generally smooth and defect-free polymer coatings are required. As was mentioned in Sect. 2.1, the physical aspects of wetting and dewetting of a solid substrate by liquid polymeric ﬁlms are quite well understood [80]. Typically, the dynamics of dewetting and the shape of the dewetting rims in simple polymer liquids are characterized. Systematic studies of dewetting of block copolymer ﬁlms are much rarer. Recently, alignment and orientation of cylindrical domains in the artiﬁcially nucleated annuli of a diblock copolymer ﬁlm was demonstrated by Hahm at al [137]. However, the question of shear-induced deformation of microdomain during the development and propagation of the dewetting front has not been addressed so far. Figure 12 presents diﬀerent characteristics of a snapshot of a dewetting rim in an SB2 ﬁlm. The series of SFM phase images in Fig. 12a captures surface structures starting from the front of the rim (from the bare silicon surface) up to a distance of about 20μm apart. The zoomed-in areas in the bottom phase images clearly demonstrate a high degree of the orientational order next to the front (left image), which gradually transforms into a poly-grain structure typical for these annealing conditions of lying cylinders with vertically oriented cylinders at grain boundaries (right image). According to the results of the digital spacing analysis in plot (c), the spacing between the nearest cylinders at the front of the rim is about 30% smaller than the typical spacing in the absence of shear deformations.

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253

Height [nm]

a

150 100 50 0

2

4

6

8 Distance [μm]

10

12

14

16

Spacing [nm]

b 75 70 65 60

400 nm

55 0

5 10 15 Distance [μm]

Fig. 12. SFM phase images in SB2 ﬁlms presenting ordering of cylinder domains in a snapshot of a dewetting rim. The ﬁlm has been equilibrated under a chloroform vapor pressure of 100%. Plot (b) is a cross section of the height image of the area displayed in (a). The areas highlighted by the white squares in (a) are shown in the respective phase images to the left and to the right of the plot (c). Plot (c) presents cylinder spacings versus the distance from the dewetting front.

Accumulation of this kind of data at diﬀerent stages of the rim development can be used for quantitative analysis of stress distribution within the moving front and for obtaining new insights into rheological properties of thin polymer ﬁlms under conditions that prevent the use of other techniques. To summarize, the analytical approach described in this section and the examples of its application present a new type of quantitative characterization of thin structured ﬁlms. These results suggest novel routes to tune and control the microdomain structures without changing the composition of the ﬁlm.

4 Dynamics The growing number of applications of nanopatterns in technology is a strong incentive to develop an improved understanding of ordering dynamics with the aim of controlling the resulting nanopatterned surfaces. The high degree of order relative to the ﬁlm surfaces, as well as long-range in-plane order of the ﬁlm are imperative to applications of nanopatterned surfaces [18].

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Dynamic behavior in bulk of block copolymer melts has been extensively studied with SAXS, SANS, forced Rayleigh scattering (FRS) and other techniques [39,138]. However the experimental data on associated dynamics in the bulk is limited to the ODT regime, below which local relaxation and overall chain diﬀusion are signiﬁcantly suppressed due to the presence of the interface. This limits the application of scattering techniques to probe the dynamics of block copolymers below the ODT. Moreover, with scattering and other spatially averaging techniques only the average dynamics can be accessed. Very little is known about the dynamics of individual defects. The key role of the generation and annihilation of topological defects in block copolymer systems has been emphasized in studies of structural phase transitions [139–143], transport mechanisms [144], long range alignment [145–153] and reorientation of microdomains under shear [154–156] and in electric ﬁelds [157, 158]. During the last decade, studies on dynamics in thin block copolymer ﬁlms focused on well deﬁned defects in highly ordered layers of cylindrical [146, 147, 149, 159] or spherical microdomains [148, 150, 160, 161]. These studies demonstrated strong similarities between the ordering of block copolymer microdomains and the ordering of two-dimensional smectic systems [146, 149, 150, 152, 160] or even solid crystalline materials [110]. However, cyclic annealing and snap-shot imaging of the same spot was utilized in this kind of experiment and limited the timescales of dynamic observation to tens of minutes or even hours [146, 159]. While topological defects in polymer thin ﬁlms indeed resemble those commonly observed in other forms of ordered matter, block copolymers exhibit morphological and dynamic properties that are speciﬁc to their polymeric nature [104, 144, 151, 162–164]. As described in Sect. 2, cylinder forming block copolymers in conﬁned geometries frequently exhibit non-bulk structures or hybrid morphologies in response to thickness constraints and surface ﬁelds. The dynamics of the phase transition from the cylinder to the PL phases in thin ﬁlms of concentrated SBS solutions has been imaged with in-situ SFM [165]. In this work, quantitative analysis of defect motion led to an estimate of the interfacial energy between the cylinder and the PL phases. Recently we have reported on the fast defect dynamics in thin ﬁlms of SB1 diblock copolymer that was imaged directly in a ﬂuid state at elevated temperatures [166]. The SFM movie was compiled from continuously captured phase images. The high temporal resolution of scanning uncovered elementary dynamic processes of structural rearrangements on time scales not accessible so far. We observed short term interfacial undulations, fast repetitive transitions between distinct defect conﬁgurations and unexpected defect annihilation pathways via formation of temporal excited states [167]. According to the phase diagram of surface structures (Fig. 6), a two-layer thick SB1 ﬁlm reliably exhibits a cylindrical morphology. In-situ dynamic measurements in similar SB1 melts demonstrated that annihilation of topological defects in many instances proceeds via local patches of non-bulk PL and lamellar phases, with lifetimes ranging from one minute to hours.

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Figure 13 presents snap-shots of the four-hour long evolution of a cluster of the PL rings at the boundary between the cylinder grains. Initially, an array of PL rings was aligned along the grain boundary and grouped around a “horse-shoe” defect (Frame 48). The elementary processes of the PL phase evolution, such as additional connections between cylinders (Frame 137), the movement of kinks parallel to the cylinder/PL border (Frame 233) are very similar to those observed earlier during the cylinder to PL phase transition in a swollen ﬁlm of a triblock copolymer [165]. The ﬁnal annihilation into the +1/2 disclination (Frame 249) proceeded in less then 5 minutes, much faster than the life-time of this temporal phase. The low interfacial tension of about 2.5 mNm−1 between the cylinder and the PL phase [165] is likely to account for the energetically favorable pathway of structural rearrangements via temporal phase transitions and explains the stabilization of PL patches under long term annealing in highly defective areas of the cylinder phase. We remark that the phase transitions from metastable to equilibrium morphologies as well as thermally reversible orderorder transitions have been extensively studied before [140,142,143,168], while we described local exited states in the equilibrium phase that are induced by energetically unfavorable defect conﬁgurations.

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Fig. 13. Phase SFM images showing the transient PL phase at the boundary between the cylinder grains. In Frame 48, an array of PL rings is aligned along the grain boundary and grouped around a “horse-shoe” defect. In Frame 249 the transient phase is annihilated into the +1/2 disclination. The total evolution of the PL phase is about four hours. Reproduced with permission from [167]. Copyright (2006) American Chemical Society.

The simulations based on the DDFT conceptually match the experimentally observed ordering and growth of grains of cylinders via temporal phase transitions. Fig. 14 captures the reorientation of a cylindrical grain via the formation and annihilation of the PL phase. The initial ﬁlm thickness was chosen to be 1.5 microdomain spacings in order to accelerate terrace forma-

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tion in a natural way similar to the experiment. First, the undulations of interfacial walls and developing dislocation are visible (Fig. 14a). The newly forming connections between neighboring cylinders serve as nucleation centers for the PL domains (Fig. 14b). The PL patch grows via the undulations in cylindrical domains which eventually connect to form the PL lattice sites (Fig. 14c). Presumably, the domains of the PL phase support redistribution of local densities. Finally, the microdomains reorganize into the new cylinder orientation (Fig. 14d). a

b

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Fig. 14. Snapshots of the MesoDyn simulations, which model a thin supported ﬁlm of a A3B12A3 cylinder forming block copolymer in a 128 x 32 x 26 simulation box. Crops of the middle layer, visualizing the reorientation of cylinders via the transient PL phase are shown after (a) 56 000, (b) 57 200, (c) 58 400 and (d) 59 600 time steps. The thin ﬁlm morphology is shown by the isodensity surface of component A for a threshold value of A =0.33.

High temporal resolution of in-situ imaging allowed us to capture the elementary steps of defect motion, such as the rapid opening and closing of a connection between cylindrical domains (Fig. 15). The structure marked with ﬂuctuates mainly between the conﬁguration with three“open ends” (Frame 166) and the conﬁguration with two “open ends” (Frame 167). The neighboring defect (at a distance of about three microdomains) marked by in Fig. 15 switches between several conﬁgurations with one “open end” (A and F) and several with two “open ends” (B, C, and D), however each conﬁguration diﬀers in the position of the junctions between the cylinders. To analyze the temporal evolution of the microdomain oscillations we associate the conﬁgurational energy with the number of open ends in the structures (1-2) and (A-F) (upper and lower plot in Fig. 16, correspondingly). The thick gray curves highlight the periods of prevailing appearance of conﬁgurations with one open end and with two open ends. Within each of these periods there are short-time transitions into other conﬁgurations. The similarity of the averaging curves suggests that the events at the two neighboring sites are correlated on the scale of at least several domain spacings and on a time scale of seconds. The velocity of the observed fast transitions was compared with the selfdiﬀusion coeﬃcients in an SB1 melt. The two time scales diﬀer by several orders of magnitude. The discrepancy between the observed and expected

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diﬀusivity in SB1 ﬁlms could be an indication of an enhanced diﬀusion in the regions with defects in the microdomain structure. Another possibility is that the transport during the closing/opening of a connection between the cylinders is not diﬀusive. It might be either hydrodynamic ﬂow or a cooperative motion of clusters of polymer chains enhanced by defect strain ﬁelds. Similar enhanced defect dynamics were observed in ﬁlms of sphere-forming block copolymers on patterned substrates [34]. The transport of spherical domains from islands was much faster than would be expected if it were controlled by the diﬀusion of chains. By measuring depth proﬁles with SIMS in thin ﬁlms of asymmetric diblock copolymer, the hopping frequency of a chain between the layers was determined. The diﬀusion coeﬃcient estimated from this hopping frequency was in agreement with that determined from measurements of diﬀusion over much longer length scales [105]. 1

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Fig. 15. Crops (250 x 250 nm) from selected frames of the SFM movie showing the oscillations between distinct defect conﬁgurations. and mark the open ends of the cylinders. The structure marked with ﬂuctuates mainly between the conﬁguration with three ”open ends” (1) and the conﬁguration with two ”open ends” (2). The structure marked with ﬂuctuates between conﬁgurations with one (A,F) or two ”open ends” (B, C, D). Reproduced with permission from [166]. Copyright (2006) American Chemical Society.

The reported experimental and theoretical work on block copolymer dynamics indicates that dynamic aspects in thin ﬁlms are more problematic than equilibrium ones, in part because of the experimental and theoretical challenges faced when probing time-dependent changes in systems with many controlled parameters, high-energy barriers between local free-energy minima and a wide spectrum of time- and length scales. As mentioned above, the

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Fig. 16. Temporal evolution of the defect conﬁgurations displayed in Fig. 15. The conﬁgurations are sorted and grouped along the conﬁguration coordinate according to the number of “open ends”. The gray curves suggest correlations of defect dynamics. Reproduced with permission from [166] (with adaptions). Copyright (2006) American Chemical Society.

microdomain dynamics are further inﬂuenced by interfacial interactions and by the topographical and chemical heterogenity of the substrate. Most of the theoretical models of block copolymer dynamics consider mainly diﬀusion driven processes [39]. For example, simulations based on dynamic density functional theory (DDFT) in detail match experimentally observed phase transitions and demonstrate that on large time scales the microdomain dynamics can be described with a mean- ﬁeld approach [165]. Theoretical approaches based on dissipative particle dynamics (DPD) correctly simulate the (compressible) Navier Stokes behavior and thus account for hydrodynamic interactions in a block copolymer melt [169]. Comparison with Brownian dynamics (BD) simulations demonstrated that the hydrodynamic forces play a critical role in the kinetics of microphase separation into the hexagonal phase. These simulations also showed that hydrodynamics appears not to be crucial for the evolution of symmetric block copolymers. Consequently, the lamella phase formed an order of magnitude faster than the hexagonal phase, which was attributed to a higher viscosity for the hexagonal phase than for the lamellar phase [169]. The inconsistency in the experimental observations of the transport velocities in structured block copolymer ﬁlms strongly suggests that along with diﬀusion, other transport mechanisms are involved in microdomain dynamics and may inspire future theoretical research.

5 Control of Nanostructure in Block Copolymer Thin Films: Long-Term Prospects With advanced physical characterisation techniques and theoretical analysis, the scientiﬁc understanding of the structure, dynamics and interfacial behavior of compositionally simple, model block copolymers is rapidly expanding.

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The interest is supported, above all, by their growing importance in nanotechnology. The development is possible due to conjugation of new synthetic capabilities, processing methods and self-assembly paradigms. Currently, the attention is shifted to new structural motifs for designing systems capable of hierarchical self-assembly into complex, well ordered, functional mesostructures. Recent remarkable progress in polymerisation techniques allows the preparation of well deﬁned tailor-made macromolecules with precise control over the molecular weight, structure, architecture and placement of functional groups [170]. A comprehensive review by Granick at al [67] describes recommended synthetic directions and approaches including the creation of organopolymeric composites (light emitting devices) and bio-related hybrid materials. Increasing the structural complexity and functionality of new block copolymer materials while introducing additional hierarchy into self- assembling inevitably implies more complex dynamic routes to achieve the desired order on mesoscopic and macrososcopic scales. A detailed comparison between morphological studies and theoretical predictions will develop better understanding and control over the structure of functionalized block copolymers and will allow the tailoring of optical, mechanical, conducting and other functional properties. However, with the expanding spectrum of complexity and functionality of new polymer-based hybrid materials, no general routes and recipes to process polymer-assisted materials should be expected. The eﬀorts to establish new approaches to guide self-assembly of complex functional materials is the key to further technological application. Acknowledgement. This work was supported by the Deutsche Forschungsgemeinschaft (SFB 481) and the State of Bavaria (HWP-Program). Helpful discussions with R. Magerle, G. Krausch and A. Knoll are acknowledged. The MesoDyn simulations images in Figures 1 and 14 are the courtesy of R. Magerle and A. Horvat. The phase images in Fig. 7b are courtesy of A. Knoll, who is also thanked for providing the software for performing digital spacing analysis.

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Mathematical Description of Nanostructures with Minkowski Functionals G.J.A. Sevink Leiden Institute of Chemistry, Leiden University, PO Box 9502, 2300 RA Leiden, The Netherlands

1 Introduction Modern three-dimensional (3D) simulation methods have become valuable tools for the understanding of (collective) phenomena originating from processes at length and/or time scales beyond experimental resolution, i.e. in (soft) nanotechnology. An example of such a process is the spontaneous formation of vesicles from biological or synthetic molecules. The advances in simulation capacity have confronted simulators with several new and interesting problems, for instance in data-handling and analysis. How does one retrieve the important (higher-level) information from the enormous amount of detailed data that one generates? Here, we consider a speciﬁc simulation method developed on parallel supercomputers in the late nineties. It generates a few GB of data for each run. The community using this type of software on standard desktop PCs is growing, and subsequently so is the amount of generated data. Eﬃcient methods for the analysis of this data will therefore only become a more pressing requirement in the future. The purpose of this chapter is to discuss the basic concepts of this method (DSCF), and to formulate a solution to the ‘data problem’ in terms of integralgeometry morphological image analysis (MIA). However, even for those not interested in DSCF, this contribution can be valuable. The particular MIA Minkowski functionals - is very general and can be used for the analysis of any complex 3D pattern or image, at any length scale. In the remainder, both the MIA and our simulation tool are introduced. The potential of both methods is demonstrated in a single example. We have chosen to concentrate on ‘nanostructures’ and ‘Minkowski functionals’ and adapt a light narrative tone in part of the introduction, in particular for the properties of human eye-sight. Of course, we risk the scientiﬁc curse of being too sketchy or even incomplete. The technical details are indeed a lot more elaborate and complex, but would divert the story too much. We salvage our choice by referring to some useful books here. The reader searching for a profound understanding of human image processing and recognition

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is referred to a number of well written books: ‘The Language of Shape’ by S. Hyde and ‘Solid Shape’ by J.J. Koenderink. 1.1 Why is Pattern Analysis Important? Patterns and/or shapes (we will not make a distinction further on, and use the shorthand patterns) play an important role in everyday life. In particular, the recognition of and quick response to certain patterns is of huge inﬂuence on our daily functioning. For instance, after getting up, eating breakfast and drinking a cup of dark liquid, the majority of the individuals making up earth’s population have to manoeuvre through cities that look like gigantic ant-colonies from the viewpoint of a bird’s eye. The ﬂux in this world consists of networks of pedestrians, bicycles, busses, cars, trains and subways, all conﬁned to very limited spaces and adapting (to) diﬀerent and sometimes interacting patterns. What happens when a collision occurs in one of these networks, or between diﬀerent networks? Close to the site of the collision one or more networks almost instantaneously come to a stand still, further away they become chaotic. The eﬀect on the individual depends on the momentum gathered in each speciﬁc pattern, and may eventually be lethal. Avoiding collisions with certain networks is therefore of great importance to us. This importance has several eﬀects on the modern human being, one of them being the ability to see and process patterns on various levels. In an unknown city we are able to participate in a network and read and analyze signs in connection to ‘mental’ maps at the same time. Moreover, we can do this at more or less the same pace as people who are completely familiar with the place. This ability is not restricted to certain special persons: we all become experienced (except for a happy few). At some point we want to do all of these simultaneous processes without spending too much of our brainactivity (or capacity), since there are always many more important things to process in parallel (money problems, fantasies etc). All this boils down to the observation that the brain is capable of observing objects and situations in the blink of an eye, and responding to it (semi)automatically. How do we do this? Or alternatively: What are the limitations? Let’s suppose that evolution has equipped the brain with all kinds of functions (ﬁlters) to do all kinds of complex observation (ﬁltering), for instance to (roughly) determine the speed of moving objects. From all possible patterns, what are the patterns that are important for our survival? And what is the relation between the observation of single patterns and similarities between diﬀerent patterns (which are equivalent to diﬀerences)? What is the inﬂuence of scale? Birds and ﬁsh, for instance, create rather regular patterns and are able to react to the motion of the whole pattern without any knowledge of the pattern itself, only by very local interaction. Common sense tells us that the human eye is more capable of observing regular patterns than irregular ones. One could think of ‘observing’ single objects as comparing (ﬁltering) the observable to certain templates stored in our brains. These templates depend

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on the person, but the most common ones share regularity, and as a result the eye may experience diﬃculties in determining speciﬁc properties of unfamiliar irregular objects. For instance, we immediately distinguish a car from a tree, but it is much less straightforward to distinguish the diﬀerences in a small collection of either cars or trees. While most people have this ability to distinguish between like objects, this property is in most cases limited. Many people can rather easily distinguish familiar faces, a process which also focusses on tiny diﬀerences, but experience diﬃculties when these faces hide in a crowd. In order to compare diﬀerent objects that are either complex or alike we have to focus on details. We somehow use measures equipped to deﬁne an object, or similarities between objects, in just a single or few fast and easily computable numbers. Apparently our brains have evolved such that they have not developed these ﬁner measures for general purposes. The MIA does just that: it provides a small amount of numbers that uniquely identify any object. The rise of computers and recent advances in computational power have had several eﬀects on the way we experience the world surrounding us. Most of all, it has had an dramatic eﬀect on our understanding of complex physical phenomena that were too diﬃcult or hazardous to tackle either by experimental approaches or by theoretical simpliﬁcations. In the early days of science, most theoretical problems were considered by deriving analytical solutions to (partial) diﬀerential equations that were either linearized or could be mapped to known (and solved) equations. The rise of computers and the development of numerical analysis allowed for the solution of a much wider range of problems, including nonlinear equations and problems with multiple time and length scales. In the early digital age, most of these solutions still lived in 1D due to computational limitations. Advances in theory and parallel computing, allowing for new algorithms to do distributed computations on a large number of processing units, brought 2D and 3D solutions to complex (partial) diﬀerential problems within reach. At ﬁrst, solutions to this new type of problem were restricted to relatively small subsets of 3D (in combination with boundary conditions). These subsets were not necessarily small in absolute size; they were small in terms of the number of free variables considered, such as number of grid points (in ﬁnite diﬀerence methods), or maximum polynomial degree (in ﬁnite element methods). For these relatively small scale problems, a large number of speciﬁc numerical methods were developed. Only very recently it was found that an increase in size may introduce extra diﬃculties. First of all, physical phenomena acting on a larger scale in multi-scale problems cannot be simulated on smaller scales. If they are present, they only appear in large scale simulations. Secondly, the presence of boundary conditions often induces a certain regularity to the simulated patterns. For larger simulation volumes, the inﬂuence of boundary conditions is reduced and structures will in general be much less regular. This blurring of the simulated patterns counteracts the understanding of the simulated patterns in easy rules (suggesting regularity). Distance measures of the simulated patterns from regular patterns become a prerequisite. Finally, when one simulates a pattern that evolves in time, it is

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important to have an overall measure that signals when a certain marked event takes place (for instance, the El Nino current). Only then can one analyze the data (i.e. ﬁnd the cause of the event) eﬃciently. 1.2 Polymer Systems Complex spatio-temporal patterns are produced by many phenomena in nature. Although the interactions due to which these patterns are formed can be simple, the dynamics of patterns and the patterns themselves can be quite nontrivial. Examples of such systems are cellular automata, superconductors of 1st and 2nd type, Rayleigh–Bernard cells in liquids, Belousov–Zhabotinski chemical reactions, and block copolymers. Block copolymers are long, often ﬂexible, molecules consisting of two or more chemically diﬀerent blocks. In a melt or solution block copolymers tend to micro-phase separate (bringing alike blocks together) under certain conditions. The scale at which the patterns appear is determined by the blocks’ characteristic length, and is often referred to as the mesoscopic regime [1]. Micro-phase separation leads to the formation of periodic structures similar to crystals. The symmetry groups of these ’soft’ crystals depend critically on the requirement of the pattern to ﬁll the available space (entropy), the energetic interaction of the blocks with their surrounding (enthalpy) and the conditions under which the ‘soft’ crystals are formed. For the simplest case of unconﬁned diblock copolymers, which are linear polymers composed of two diﬀerent subchains (A and B blocks), a variety of ordered bulk phases, including lamellae, hexagonally packed cylinders, bodycentered-cubic spheres, and a bicontinuous network structure called a gyroid, is observed [2]. In a physically conﬁned environment, structural frustration, conﬁnement-induced entropy loss, and surface interactions can strongly inﬂuence molecular organization. In particular, it is possible that conﬁnement can lead to unusual morphologies which are not accessible in the bulk, thus providing opportunities to engineer novel structures [3, 4]. In systems with linear multi-block (for instance, ABC) or star chain topologies, even the unconﬁned bulk phases can become very complex [5, 6]. In reality, perfect ‘soft’ crystals are mostly found in systems that are either constrained by the presence of solid surfaces (thin ﬁlms) or inﬂuenced by external forces (shear or electric ﬁelds). These phases are so-called equilibrium phases that are stable in time against ﬂuctuations. They can be predicted theoretically by mimimization of free energy, containing both entropic and energetic contributions. However, upon formation from an initially unstructured situation, the evolving structures/patterns are often highly defected and go through many stages that are far from perfect ‘soft’ crystals. Since most polymer systems are ﬂuids with high overall viscosity, the resulting characteristic times for defect movement and annihilation can be very long. Processing conditions such as shear [7, 8] or applied electric ﬁelds can inﬂuence this eﬀect seriously and determine to a large extent how the phase separation can proceed. Electric ﬁelds were shown

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to even give rise to phase transitions (for instance, from spherical to cylindrical phases in AB-systems) for ﬁeld strengths that are high enough [9]. The growing potential of perfect ‘soft’ crystals in nanotechnology is a strong incentive to develop an improved understanding of the ordering process, aiming at controlling the resulting pattern in particular in thin ﬁlms (so-called nanopatterned surfaces) [10–12]. The key role of the generation and annihilation of topological defects in block copolymer systems has been emphasized in studies of structural phase transitions [13–17], transport mechanisms [18], long range alignment [19–26], and reorientation of microdomains under shear [27–29] and in electric ﬁelds [30, 31]. Some studies demonstrate strong similarities between the ordering of block copolymer microdomains and the ordering of two-dimensional smectic systems [20, 23–25, 32] or even solid crystalline materials [33]. However, the resolution in these observations (timescales of dynamic observations with a lower bound of tens of minutes or even hours) were limited by the imaging procedure consisting of cyclic annealing and snap-shot imaging of the same spot [20, 34]. While topological defects in polymer thin ﬁlms indeed resemble those commonly observed in other forms of ordered matter, block copolymers exhibit morphological and dynamic properties that are speciﬁc to their polymeric nature. Determining the features of evolving block copolymer systems (that is: how certain structural elements evolve in time, and what their nature is) is often a diﬃcult task both experimentally and theoretically. In computer simulations one obtains information on the three dimensional microstructure; simple visual inspection is not suﬃcient, especially in larger systems. Fourieranalysis is a common procedure in this case, and provides useful information for almost periodic structures but is less useful for aperiodic patterns that do not resemble ‘soft’ crystals. In the initial stages of micro-phase separation the structure is often topologically reminiscent of the periodic one, but highly deformed - stretched, squeezed, etc. This is in particular the case if the system is subjected to external ﬁelds. In this situation the analysis of topological and geometrical quantities, followed by an expression for the similarity measure with respect to perfect structures, would be of great help. 1.3 Minkowski Functionals The huge scales in space and time covered by our (parallel implemented) simulation technique, hamper us from grasping the important features from imaging the 4-D data alone. To give an idea: for each of the simulations considered here, the amount of data is as large as 64 × 64 × 64 = 262144 (the number of positions on the grid) double-precision (8 bytes) spatial data times 600 (every 50 time steps for a total of 30000 time steps), resulting in a total amount of data for each simulation of almost 1.3 Gbyte. An eﬃcient tool for the analysis of 3D patterns is an image functional. Image functionals [35] are numerical functions deﬁned on a two valued (black-and- white) representation of the patterns or images [36–39]. They perform a measurement of certain

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properties or features in the image, such as brightness, object locations, surface, perimeter, size distribution, etc. The digitization process required prior to the application of an image functional should include an accurate (with a minimum information-loss) mapping of the spatial and functional (color, gray-level, intensity) information contained in the pattern/image of the form: i) division of the pattern/image into subsets or pixels, each with a single value, ii) some algorithm applied to the set of all pixels that determines if a pixel is black (object) or white (background). Additive image functionals φ possesses the property that for any two subsets P1 and P2 in a digitized image φ(P1 ∪P2 ) = φ(P1 )+φ(P2 )−φ(P1 ∩P2 ). This property can be seen as avoiding double counting in determining the properties of a black-and-white image; it is certainly satisﬁed for a large number of image functionals (volume etc). However, a collection of non-additive image functionals exist [36]. We will discard these non-additive functionals here, as we are primarily interested in MIA and their completeness in terms of morphological information that can be retrieved from 3D images. Integral-geometry morphological image analysis (MIA) employs additive image functionals to assign numbers to the shape and connectivity of (digitized) patterns. A fundamental theorem of Hadwiger [40] states that under certain conditions each image functional can be written as a linear combination of 3 (2D) and 4 (3D) quermassintegrals or Minkowski functionals, meaning that all morphological information present in an image is essentially contained in these 3 or 4 Minkowski functionals. An appealing feature of them is that they are easily and eﬃciently computable, and correspond to intuitively simple geometrical and topological measures: area, boundary length and Euler (connectivity) number (2D), and volume, surface area, integral mean curvature and Euler (connectivity) number (3D). Minkowski functionals were shown to be very valuable for the description of complex morphologies in many areas of science, ranging from phase separating (block co)polymer systems [41–48] like the one considered here, complex ﬂuids [49–52], composite materials [51], reaction-diﬀusion systems [52, 53], to large-scale structures in the universe [52, 54, 56]. An extensive review, including many examples of application, was recently published [57]. This ﬁeld is still growing; a new and very promising vectorial Minkowski functional was recently developed by Klaus Mecke. In our calculations the patterns are concentration ﬁelds on an equidistant grid. The pixels naturally coincide with the grid cells or discretization subsets (where the grid nodes are in the centre of each pixel). The remaining question is how to choose the algorithm that determines whether a pixel is inside or outside a pattern in the digitization procedure prior to the calculation of the Minkowski functionals. For ﬁelds, a standard choice is thresholding (choose object values above a certain number, the threshold) which has physical relevance, as it is the equivalent of choosing the position of interfaces in a microstructure. Usually one calculates Minkowski functionals for a set of threshold values. In the remainder we demonstrate that a limited amount of

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threshold values are suﬃcient for our purpose, based upon physical considerations. We show that this choice is crucial for the correct interpretation of the dynamical pathway of a pattern.

2 Methods 2.1 General Issues We start by reviewing a number of rather basic mathematical ingredients that are important for the understanding of this chapter. These include the notion of sets, measures, topology, geometry and the distinction between continuous and discrete. In the review of sets, we will restrict ourselves to naive set theory, were sets are introduced and understood via a rather self-evident concept. In each paragraph we start with a simple deﬁnition. For interested readers, we expand our deﬁnitions mathematically more rigorously, and will not always be able to circumvent some mathematical notation. What is a Set? Deﬁnition A set is a collection of objects considered as a whole. The objects themselves are called elements or members, and can be anything. Sets are conventionally labeled by capital letters. The standard notation for a set is A={elements}; each element a of A is denoted as a ∈ A. Two sets A and B are equal, A=B, if they have the same elements. If every member of the set A is also a member of the set B, then A is a subset of B, A ⊆ B; if A = B, then A is a proper subset and A ⊂ B. If a set has no elements, the set is called the empty (or null) set and denoted by ∅.A set A is called a closed set if the limit a of any converging sequence ai ∈ A is an element of A itself. If this property does not hold, A is called an open set. Simulation Intervals In the simulations, we consider intervals or subsets V syst of the Euclidian space R3 : V syst ={r = (x, y, z) ∈ R3 |x ∈ [0, a], y ∈ [0, b], z ∈ [0, c]} (a, b, c ∈ R). Moreover, we discretize this interval using an equidistant spacing Δx (in m3 ), such that the simulation volume is Vd ={˜ r= (x, y, z) ∈ R3 |x ∈ {0, .., Lx −1} ⊂ N, y ∈ {0, .., Ly −1} ⊂ N, z ∈ {0, .., Lz −1} ⊂ N}. A function f deﬁned in this simulation volume has a ﬁnite carrier, at the boundaries we therefore use periodic boundary conditions: f (r + nL) = f (r) with n ∈ Z, r ∈ Vd and LT = (Lx , Ly , Lz ). The dimensionless coordinate vectors ˜ r are related to spatial positions r by r = Δx˜ r. Convex Bodies Convexity is a useful notion in integral geometry. A set K of points in the Euclidean d-dimensional space Rd is called convex if for every pair of points in K the entire (straight) line segment connecting the two points ⊂ K. A convex set K = ∅ is called a convex body. A single point x ∈ Rd is also a convex body. We will only consider compact sets (meaning that they are bounded and closed). The class of all compact convex sets is denoted by

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K. The parallel set Kr of a compact convex set K∈ K at a distance r is the union of all closed balls, deﬁned by the points b such that b − a ≤ r, a ∈ K. As a simple example, the parallel set of a 2D circle with radius a is given by a circle (with the same center point) of radius a + r. One can show that Kr ∈ K [58]. For an arbitrary parallel set Kr of a convex body K, the Steiner formula gives that the volume v (d) (for arbitrary dimension d) of the new set Kr is related to the geometrical and topological properties of K. In 3D (d=3), this relation is given by 3 d (3) (1) Wν(3) (K)rν v (Kr ) = v ν=0

(d=3)

(3)

(3)

with Wν the 3D Minkowski functionals, W0 = V (K) (volume V), W1 = (3) (3) S(K)/3 (surface area S), W2 = 2πB(K)/3 (mean breadth B) and W3 = 4πχ/3 (Euler number χ). [40] Convex rings The convex ring R is the class of all subsets A of Rd which I can be expressed as ﬁnite unions of compact convex sets A = Ki ; Ki ∈ K. i=1

If A1 and A2 both belong to R then so do A1 ∪ A2 and A1 ∩ A2 . What are Measures? General The shorthand deﬁnition of a measure is a function that assigns a number to subsets of a given set A. Examples of such numbers are size, volume or probability. These measures are only valuable when one is able to assign such a measure (or size) to every subset of A. We will need the axiom of choice, that states that if A is a set of non-empty sets, there exists a function f deﬁned in A, such that for each set a in A, f(a) ∈ a. This axiom of choice implies that when the size under consideration is the standard length for subsets of the real line, there exist sets known as Vitali sets for which no size exists. In general, we will not experience this problem, but formally instead of all subsets one considers a smaller collection of privileged subsets of A whose measure is deﬁned; these sets constitute the σ-algebra. We will not discuss these here, and refer the reader to the general mathematical literature. Euclidian space A speciﬁc σ-algebra of importance on the Euclidean space Rn is that of all Lebesgue measurable sets. The Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. It is used throughout real analysis, in particular to deﬁne Lebesgue integration. Sets which can be assigned a volume are called Lebesgue measurable; the volume or measure of the Lebesgue measurable set A is denoted by λ(A). What is Topology? General The motivating insight behind topology is that some geometric problems do not depend on the exact shape of the objects involved, but rather on

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the way they are connected. One of the ﬁrst papers in topology was the demonstration, by Leonhard Euler, that it was impossible to ﬁnd a route through the town of K¨ onigsberg (now Kaliningrad) that would cross each of its seven bridges exactly once. This result did not depend on the lengths of the bridges, nor on their distance from one another, but only on connectivity properties: which bridges are connected to which islands or riverbanks. This problem, the Seven Bridges of K¨ onigsberg, is now a famous problem in introductory mathematics, and led to the branch of mathematics known as graph theory. Topological Equivalence In order to deal with those problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on. From this need arises the notion of topological equivalence. The impossibility of crossing each bridge just once applies to any arrangement of bridges topologically equivalent to those in K¨ onigsberg. Formally, two spaces are topologically equivalent if there is a homeomorphism between them. For a formal deﬁnition we refer to the literature. The informal idea of a homeomorphism is that one space can be deformed into the other without cutting it apart or gluing pieces of it together. An interesting example of this equivalence are a coﬀee cup and a doughnut, which both contain one hole. In that case the spaces are said to be homeomorphic, and they are considered to be essentially the same for the purposes of topology. Topological Invariants A topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. In particular: a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. A common problem in topology is to decide whether two topological spaces are homeomorphic or not. To prove that two spaces are not homeomorphic, it is suﬃcient to ﬁnd a topological property which is not shared by them. Euler Number One of the topological properties or invariants is the Euler number, a number that describes one aspect of a topological space’s shape or structure. It is commonly denoted by χ. The Euler characteristic χ was classically deﬁned for polyhedra, according to the formula χ = V - E + F, where V,E and F are respectively the numbers of vertices (corners), edges and faces in the given polyhedron. The most basic deﬁnition is for convex sets: for convex K ∈ K the Euler number is deﬁned as χ = χ(K) = 1 for K = ∅ and χ = χ(K) = 0 for K = ∅. Moreover, due to the fact that the Euler number is a topological invariant we also have that if M and N are any two topological spaces that are a subspace of a larger space X, then so are their union and intersection, and χ(M ∪ N ) = χ(M ) + χ(N ) − χ(M ∩ N ). What is Geometry? General Geometry is a branch of mathematics that is concerned with the properties of conﬁgurations of geometric objects - points, (straight) lines and circles are the most basic geometrical objects. Geometrical quantities (or mea-

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sures) such as the ones considered in this chapter, ie volume, surface area and integrated mean curvature, allow us to distinguish between objects that are topologically (from a viewpoint of connectivity) equal. For instance, a sphere and a cube cannot be distinguished from a topological point of view (they have the same Euler number, are topologically equivalent), but are in no way equal from a geometrical point of view. Geometrical quantities are invariant under simple transformations, such as rotation and translation. Continuum Versus Discrete Polymer Physics Focussing on modelling in polymer physics, the distinction between discrete and continuum is sometimes rather subtle. For instance, a number of methods (like, for instance, Lattice-Boltzmann [59]) solve continuous equations by ﬁctitious discrete events on a grid. Theories such as the molecular dynamics or mesoscopic particle methods deal with discrete objects (particles or chains) moving and/or colliding with time according to fundamental laws. Continuum theories can contain an explicit representation of a polymer chain (ie a microscopic Hamiltonian) like the self-consistent ﬁeld theory used here, but this information may also be inserted via coeﬃcients in a free energy expression [60]. The important properties are given in terms of functional dependencies (for instance, a free energy as an expression containing concentration ﬁelds). In the best case, these functionals result from coarse-graining (smearing) some of the lower level (discrete) properties of a system, although most methods rely on the determination of some eﬀective parameters from experiments. The advantage of all mentioned methods is that the fastest degrees of freedom in a system are averaged out, allowing for longer time and length scales to be modelled (depending on the degree of averaging). A disadvantage is that one has to sacriﬁce some of the detail at a lower level (interaction, space and time resolution). Eventually, the nature of the event itself and the space and time scales at which the event takes place determine which model is best used [61]. SCF For the sluggish block copolymer systems the length (10 nm up to μm) and time (seconds up to hours) scales are well modelled by mesoscopic SCF methods [62]. The interactions are often not very speciﬁc, and can therefore be well described by some mean-ﬁeld (Flory–Huggins) interactions [63]. But then, if we choose the self-consistent method and some kind of expression for the free energy of the system in terms of (concentration) ﬁelds, how do we solve these equations in a computer (assuming that they are too complex to be solved analytically, as they often are)? We cannot solve continuous equations in a computer, and have to map these equations in discrete time and space along the way. We introduce a discretization in time and space (see Sect. 2.1), and rely on ﬁnite diﬀerence schemes to do this in the most eﬃcient and accurate way possible. But then again, there are choices. One can think of the underlying chain as being discrete (beads linked to each other with a ﬁxed size) or continuous [64] (worm-like, allowing for a discretization at a

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later stage). Moreover, the chain location can be restricted to the grid nodes (so-called lattice methods), allowing for a speed-up of the calculations but restricting the directions in which a chain can move [65]. Restrictions In our DSCF method we use chain of beads connected by springs. After making them dimensionless, we numerically solve the diﬀerential equations on the simplest equidistant grid. The grid spacing Δx (sometimes also referred to a mesh size) limits the value of the ﬁeld gradients: steeper gradients cannot be represented on the grid. This limits the degree of microphase separation that the model is able to describe. Moreover, the bead spacing is related to the grid spacing for reasons of numerical accuracy [66]. The discretization in time is a matter of choice, but limited by the length that a bead can diﬀuse in between discrete time intervals (the dimensionless time interval DΔx−2 Δt ≤ 0.5, with D the diﬀusion coeﬃcient). The dynamics itself is however not restricted to the grid. The values of the concentration ﬁeld are only calculated on the grid nodes. Away from the calculation grid nodes these ﬁelds are continuous and (twice) diﬀerentiable, but undeﬁned in the strict sense. Using a constant ﬁeld value in the calculation of the Minkowski functionals for the grid cells can therefore be considered as an approximation of the Lebesque integral, and not necessarily a much worse choice than any type of interpolation. 2.2 Modelling of Pattern Formation in Polymer Systems We give a short outline of the theory used in the simulations; for more details see [67,68] and references therein. We model the pattern formation that occurs when a block copolymer melt or solution is brought into a state where the chemically diﬀerent blocks phase separate on a mesoscopic level (1-1000 nm). In our model, a block copolymer molecule is represented by a Gaussian chain, consisting of N beads. Each bead typically represents a number of chemical monomers. Diﬀerences in monomers gives rise to diﬀerent bead species (for example, ANA BNB , for a diblock copolymer, ANA /2 BNB ANA /2 , for a symmetric ABA-triblock copolymer; N = NA + NB ). The three-dimensional volume of the simulated system is denoted by V syst , and contains n Gaussian chains. Solvents are incorporated as single beads [69]. The inter-chain interactions are incorporated via a mean ﬁeld with interaction strength controlled by the Flory–Huggins parameters χIJ . The microstructure patterns are described by the coarse grained variables, which are the bead concentration (in m−3 ) or density ﬁelds ρI (r) of the diﬀerent species I; in the calculations we introduce dimensionless ﬁelds θI (r) = νρI (r), with ν the constant volume of a single bead. Given these density ﬁelds a free energy functional F [ρ] can be deﬁned as follows [67–69] Ψn F [ρ] = −kT ln UI (r)ρI (r)dr + F nid [ρ] . (2) − n! I V syst

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Here Ψ is the partition function for the ideal Gaussian chain in the external ﬁelds UI , and F nid is the contribution due to the non-ideal mean-ﬁeld interactions. The external potentials UI and the density ﬁelds ρI are bijectively related in a self-consistent way via a density functional for Gaussian chains. Several methods can be employed to ﬁnd the minimum of free energy (2). They can roughly be divided into static and dynamic methods, although a number of hybrids exist which are generally referred to as quasi-dynamic methods (for instance [70]). In this article, we use a dynamic scheme that has the advantage that it intrinsically considers dynamic pathways towards a free energy minimum, including visits to long-living metastable states. In this sense, the model can be seen to mimic the experimental reality when compared to static schemes, which are optimizations, based upon mathematical arguments. The thermodynamic forces driving the pattern formation in time are the gradients of the chemical potential μI (r) = δF/δρI [67–69] giving rise to ∂ρI (3) = MI ∇ · ρI ∇μI + ηI ∂t where MI is a constant mobility for bead I and ηI (r) is a noise ﬁeld, distributed according to the ﬂuctuation-dissipation theorem. In the presence of a steady shear ﬂow, with velocity vx = γy, ˙ vy = vz = 0, an extra convection term is added to the right hand side of the diﬀusion equation (3) equal to −γy∇ ˙ ˙ is the shear rate (the time derivative of the shear strain γ) x ρI . Here γ and sheared boundary conditions apply [67, 71–74]. 2.3 Minkowski Functionals We have discussed the importance of Minkowski functionals in the introduction and brieﬂy mentioned the important theorem of Hadwiger. Here we consider the explicit procedure to calculate these functionals for any (irregular) pattern resulting from our numerical simulation method. Minkowski functionals are a special case of image functionals that are: i) Motion invariant, ii) C-additive, and iii) Continuous. The theorem of Hadwiger [40] now formally states that any functional φ over K satisfying properties i)-iii) can be written as d φ(K) = aν Wν(d) (K) (4) ν=0 (d)

where d ∈ {1, 2, 3} is the dimension, Wν are the Minkowski functionals for this dimension and aν ∈ R suitable coeﬃcients. Motion invariance (the image functional is invariant under rotations and/or translations) and continuity (for any sequence of compact sets {KI } with limI→∞ KI = K in the Hausdorﬀ sense, limI→∞ φ(KI ) = φ(K)) is satisﬁed for most image functionals of interest. C-additivity is a special case of additivity (see introduction) limited to the set K of convex bodies, which has an extra requirement, as the union of two convex sets (for instance, two touching spheres) is not necessarily a convex set

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itself. The restriction to convexity is the important issue here, as black-andwhite images are by no means automatically convex sets (see Sect. 2.1 for the deﬁnition of convexity). We would like to satisfy Hadwiger’s theorem for all our images/structures, as it basically states that the Minkowski functionals represent all information about an image for a large class of image functionals. One can circumvent this problem for arbitrary (non-convex) images by going to unions of convex sets or convex rings R and extending the Hadwiger theorem to sets in R instead of K. In R the Euler number satisﬁes all requirements in a natural way. The additivity for the remaining Minkowski functionals in R is satisﬁed by deﬁning (d) Wν (A) = χ(A ∩ Eν )dEν , ν = 0, .., d − 1 (5) for A ∈ R, where Eν is a ν-dimensional plane (manifold) in Rd . This deﬁnition reproduces the functionals obtained from Steiner’s formula in Sect. 2.1. In this way, the not necessarily convex image can be decomposed into a union of convex black pixels. The Minkowski functionals for the total black-and-white image can then be calculated from the values for the convex pixels and the (multiple) intersections of diﬀerent convex pixels [52] Wν(d) (A)

=

I (d) W ν ( Ki ) i=1

=

I

Wν(d) (Ki )

i=1

+(−1)I+1 Wν(d) (K1 ∩ .. ∩ KI )

−

I

Wν(d) (Ki ∩ Kj ) (6)

i,j=1;i<j

(7)

We can even avoid the exhausting calculation of the parts containing the intersections in this equation by decomposing each convex pixel K further into a union of disjunct n-dimensional convex sets that have no elements in ˜ (n = 2), open edges common: the interior body C˜ (n = 3), interior faces Q ˜ ˜ L (n = 1) and vertices P (n = 0). The interior of a set A is deﬁned by A˜ = A/∂A, where ∂A is a mathematical notation for the boundary of A. The values of the Minkowski functionals of the open interior A˜ of an equalor lower-dimensional (n) body A ∈ R embedded in Rd (n ≤ d) are related by [55] ˜ = (−1)d+n+ν W (d) (A) , ν = 0, ..., d Wν(d) (A) (8) ν which follows from the addivity of the Minkowski functionals, equation (5) ˜ = (−1)n , with n the dimension of A˜ (remember that and the deﬁniton χ(A) for convex A ∈ K ⊂ R χ(A) = 1 for A = ∅ and χ(A) = 0 for A = ∅). For the cubic grid (pixels) with discretization distance Δx used in the simulations, one can easily calculate the values of the Minkowski functionals for ˜ Q, ˜ L ˜ and P˜ from (8) and the Steiner formula in Sect. 2.1 as: W (3) = C, 0 (3) (3) Δx3 (1, 0, 0, 0), W1 = Δx2 /3(−6, 2, 0, 0), W2 = πΔx(1, −2/3, 1/3, 0) and (3) ˜ Q, ˜ W3 = 4π/3(−1, 1, −1, 1), where we employ the shorthand notation (C, ˜ P˜ ). Since the Minkowski functionals (7) are additive and the subsets C, ˜ L,

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˜ L ˜ and P˜ of each pixel do not overlap, we can calculate the Minkowski Q, functions for the black-and-white picture P(r) as ˜ ˜ + Wν(3) (Q)n ˜ ˜ + Wν(3) (L)n ˜ ˜ + Wν(3) (P˜ )n ˜ Wν(3) (P) = Wν(3) (C)n C Q L P

(9)

where nI˜ (I ∈ {C, Q, L, P }) is the number of open bodies of the particular kind in the picture. The actual nI˜ are calculated by considering the change ΔnI˜ obtained after sequentially adding one object pixel to an initially white background, starting from nI˜=0, and ΔnI˜ > 0 for the intersections of the open bodies of type I˜ of the newly added pixel with the pattern at that moment = ∅. This may sound like magic, but it can be derived from the additivity of the Minkowski functionals. If P is the set of active pixels at some instance of the addition process described above, and p is the added pixel with ˜i ∪ L ˜ j ∪ P˜k p = C˜ ∪ Q

(10)

i = 1, .., 6, j = 1, .., 12 and k = 1, .., 8, the change in the Minkowski functionals ΔW = W (P ∪ p) − W (P ) (dropping sub and superscripts for a moment) is according to the additivity given by W (p) − W (P ∩ p), where W (p) can be calculated for each subset in (10) separately since the intersection of these subsets is empty. If P and p overlap (let us say for just one open body I˜ in (10); the case of more bodies is similar), the intersection P ∩ p = I˜ and ˜ = 0. Finally, we end up with very simple formulas for the geometrical ΔW (I) and topological quantities: V = nc , S = −6nc + 2nq , 2B = 3nc − 2nq + nl and χ = −nc + nq − nl + np , with ni the number of open elements of kind i in the picture P. Details of the actual numerical calculation procedure can be found in the work of Michielsen et al [57, 75]. The Calculation Procedure Here, the starting point is the output of our simulations at timestep TMS: the 3-D dimensionless concentration ﬁelds θI (r, t = T M S) or shorthand θI (r). Due to the compressibility and/or the strong solvent preference for one specie, it is enough to consider the evolution of the concentration ﬁeld for only one bead species (A or B). We call this ﬁeld θ(r) in the remainder; all pattern dynamics information is contained in this ﬁeld. The black-and-white picture P is build up from the reference ﬁeld as P(r) = Θ(θ(r) − h). Here, Θ(x) is the Heaviside step function, giving 1 (x ≥ 0) or 0 (x < 0). In other words, the picture P is build up from the concentration ﬁeld by thresholding, and setting the values of the thresholded ﬁeld to binary valued pixels. One should keep two things in mind: i) the Minkowski functionals that are related to geomet(3) rical measures (Wν , ν = 0, .., 2) are the result of a digitization procedure. The volume, surface area and mean breadth are calculated for a digitized picture, which is build up from cubic black-and-white pixels. Due to their ﬁnite resolution, the smoothness of the boundary of any object may depend on

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its orientation with respect to the computational grid, thereby introducing a small violation of the motion invariance. For the box size under consideration here, we found that this digitization eﬀect is negligible. ii) the resulting picture P and its corresponding Minkowski functional values automatically depends also on the choice of the threshold value h: P(r) is in fact P(r, h). However, the picture P(r, h) can be completely described in terms of Minkowski functionals. The implementation explained in the preceding subsection is very straightforward, and should only be optimised with respect to double counting. As our simulations are carried out with periodic boundary conditions, we update boundaries prior to the Minkowski functional calculation by a common procedure: we add an extra layer on all sides of the original grid with the correct (periodic) boundary values. Structure Information and Minkowski Functionals (3)

The Minkowski functional W3 ∼ χ is the same as the Euler characteristic deﬁned in algebraic topology. Using this equality, the Minkowski functional χ can be understood as the number of connected components minus the number of tunnels (holes) plus the number of cavities. For instance, χ = 1 for a solid sphere, χ = 2 for a hollow sphere, χ = 0 for a torus, and χ = −1 for a ∞-shape which has 2 holes. Due to this additivity, we can use this knowledge for the determination of the topology of the major part of the local structures from the Euler characteristic. For AB and ABA block copolymers, the amount of amenable mesostructures is limited to micellar, cylindrical, bicontinuous or lamellar morphologies. We calculated the Minkowski functionals for standard ’perfect’ morphologies in a 323 (L = 32) simulation volume V syst (the total number of disjunct elementary structures is speciﬁed): spheres (9), cylinders (5), gyroid (8 unit cells) and lamellae (2). The gyroid phase (generated by the formula f (r) = sin x cos y + sin y cos z + sin z cos x) is a special structure with a so-called minimal surface f (r) = 0, meaning that the principle curvatures are equal but opposite in sign at every point. We performed thresholding with h = 0.5 (see elsewhere in this section for details). Here, and in the remainder, ˜ integrated mean curvature (B) ˜ the considered volume (V˜ ), surface area (S), and Euler number (χ) are related to the actual Minkowski functionals by (3) (3) ˜ = B/L (B=3/2πW (3) ) and χ V˜ = V /L3 (V=W0 ), S˜ = S/L2 (S=3W1 ), B 2 (3) (=3/4πW3 ). These values are found as V˜

S˜

˜ B

χ

functional form

spheres 0.026 1.32 13.25 9 θ(r) = 1 − |r − ri |2 /r2 for |r − ri | ≤ r cylinders 0.12 3.13 15.71 0 θ(r) = 1 − |r − ri |22 /r2 for |r − ri | ≤ r gyroid 0.50 9.94 -9.42 -32 θ(r) = 1/3(f (r · π/8) + 3/2) lamellae 0.56 4.0 0.0 0 θ(r) = 1/2(sin(x · π/8) + 1) with r = 4 the radius, midpoints ri and | · | and | · |2 the length of the vector in three and two dimensions, respectively. For inﬁnite cylinders and lamellae at

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least one of the principle curvatures is zero, hence the zero value of the Euler number. For the gyroid system, the Euler number is negative (=-4 for one unit ˜ should be equal cell) showing the eﬀect of connectivity in this structure; B zero, but deviates as a consequence of the digitization process. With increased ˜ (= −0.59) is much closer to the resolution (single unit cell, twice as large) B theoretical value. We compared our calculation results to results obtained by a method that interpolates the grid values by a marching cube algorithm. [76] We found almost perfect correspondence with only one h-independent scaling factor. We end up with a few very simple rules-of-thumb for the interpretation of structures: very positive χ can be interpreted as a majority of micellar (spherical or cylindrical) structures, very negative χ as highly connected structures with many tunnels. From the Euler characteristic it is impossible to distinguish between spherical and cylindrical micelles; we therefore will refer to these structures as micelles. For cylindrical systems, a Euler characteristic χ = 0 can be interpreted as highly-oriented cylindrical domains, which are, due to the periodic boundary conditions, topologically equivalent to tori. Diﬀerence Measures As mentioned before, we would like to have a diﬀerence measure to recognise the evolving ’soft’ crystal symmetries present in the defected structures. The diﬀerence D between any perfect structure Q and the picture P could be deﬁned as (d = 3) D(Q, P) = |Wν (Q) − Wν (P)|. However, the Minkowski ν

functionals are not scale invariant. In the case of diblock AB copolymer systems, the interesting features of a pattern are indiﬀerent of individual χAB , the Flory–Huggins interaction parameter, and N , the total length of the chain, as long as χAB N is constant [2]. The pattern spacing, however, does depend on the length N (and topology) √ of the underlying polymer chain (the radius of gyration of a chain Rg ∼ N ). Consequently, the number of structures (for instance, spheres) in a compact simulation volume V d depends on N . Moreover, the width of the A − B interfaces in the continuous θ ﬁeld depends on the value of χAB . This has an eﬀect on the digitization procedure (the interface location, discussed above). As a result, we cannot take just a representative perfect structure Q; we have to match both properties somehow, by scaling Q to the speciﬁc polymer system at hand. In the remainder we will not use the diﬀerence measure explicitly, but concentrate on the time evolution of the Minkowski functionals. We will show that most interesting information is contained in the Euler number. Considerations About the Threshold All calculations (we numerically solve (3)) start from uniform density ﬁelds θI (r) = θI0 , with

Mathematical Description of Nanostructures with Minkowski Functionals

θI0 =

1 θI (r) Lx Ly Lz d

285

(11)

r∈V

the average density or average concentration of block I. This reﬂects the case where all components are completely mixed. During the simulation, the total concentration of all species I remains constant. Let us consider a system over time. The ﬁrst step of simulation corresponds to a quench of the system into an ordered phase. Locally, deviations of the average value θI0 start to develop, in time leading to a ﬁnal fully phase-separated melt or solution with values θI (r) between the natural extremes 0 and 1. The starting and ﬁnal states of the system both have distinct features. A schematic illustration can be found in Fig. 1. Phase separation consists of two simultaneous processes: the amplitude of the deviation of the concentration ﬁelds from their average value grows in time, and domains of density inhomogeneity change their shape and size. Let us consider the ﬁrst process. In ﬁgure 1 we sketch the growth of density inhomogeneity for a 1-D system. If we choose the threshhold value equal to the average density (solid straight line), the picture P will have the same features (connectivity, domain size) for the upper and lower sketches. For an arbitrary threshold value (dashed line) the features will be very diﬀerent: the lines cross the graphs in diﬀerent positions, therefore both the connectivity and domain shape (and even the number of domains) will be diﬀerent. In this case, the picture P is a view of “the tip of the iceberg”. For the second process, where the domains change as well, “the tip of the iceberg” view is very sensitive to small changes in inhomogeneity. Changes in domain shape and connectivity will be in particular seen under the inﬂuence of externally applied shear ﬂow. Therefore by combining two threshold choices, from which one is equal to the average density value (h = θ0 ), one can separate information originating from the two processes that contribute to phase separation. In the applications we will therefore consider two choices for the systems under consideration: h = 0.5 and h = θ0 .

3 Application We analyze the dynamics of pattern formation of two diﬀerent block copolymer systems under an applied shear ﬂow. One system is a diblock copolymer melt, the other is a solution of triblock copolymer. The particular aim of this exercise is to study the inﬂuence of the chain architecture and presence of the solvent on the pattern kinetics. Shear is often used in processing experimental and/or industrial polymer systems. Getting an understanding of the fundamental properties and underlying mechanisms is of some importance. Although the systems diﬀer in several respects, the end structure is visually the same. Both systems form a cylindrical microstructure, which in equilibrium would be a perfect array of hexagonally packed cylinders. In absence of the applied shear, the cylinders would be hexagonally packed on a local scale,

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x

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Fig. 1. Schematic view of the inﬂuence of the threshold choice in the 1-D case. From top to bottom: development of density inhomogeneity ρ(x) (nonmonotonic line) as a function of spatial coordinate x over time due to the progress of phase separation. Straight solid line: level of average density; dashed line: an arbitrary level. Reprinted from [78]. Copyright (2004) by the American Institute of Physics.

but the orientation on a larger scale would be isotropic, and the structure would have many defects of relatively low energy [69]. 3.1 Shear Induced Structure Reformation We study two shear scenarios, which diﬀer in the moment that shear was applied to the systems. This allows us to clarify the inﬂuence of shear on both processes occurring during phase separation: the growth of concentration inhomogeneity and change of domains. The evolution of both structures in the ﬁrst shear scenario is shown in Fig. 2. Shearing of the second system in the second shear scenario was previously published in [77]. Visual inspection of images conﬁrms the development of hexagonally arranged cylinders from an initially poor structure. Initial stages (the ﬁrst two images in each row) do not exhibit easily spotted diﬀerences, while the more developed structure is clearly more defected in the case of a melt. Two mechanisms play a role in the formation of a structure: microphase separation is dominant in the initial stages, (re)orientation of domains is predominant at later stages. To deduce the details of the processes and their interplay, one needs to examine a tremendous number of images in three dimensions. Some guiding is obviously very desirable. As mentioned in the introduction, Fourier transformation (often the only experimental information) gives some grip in later stages of the alignment process, but is not conclusive in the initial stages [77]. Although visual inspection suggests that there is a diﬀerence in the development of well aligned cylinders between the melt and the solution, no decisive conclusion is possible. A standard way of characterizing pattern formation in a polymer system is to consider the evolution of the degree of microphase separation for the cylinder forming component [69] 1 P = θ(r)2 − (θ0 )2 ∈ [0, P max ]. (12) Lx Ly Lz d r∈V

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Fig. 2. Top: snapshots of the A3 B7 diblock copolymer melt. Bottom: snapshots of the 55% solution of A3 B9 A3 triblock copolymer in a one-bead solvent. The block interaction parameters for the solution were previously published in van Vlimmeren et al [69]; for the melt AB = 7.5 kJ/mol. The shear parameter was chosen γ˜˙ = 0.001 (see [67] for details). The snapshots are taken at dimensionless time steps (from left to right): 200, 2000, and 25000 TMS. For both systems, shear was applied from TMS=0. The snapshots show isosurfaces of the cylinder forming component at θI = 0.33. Reprinted from [78]. Copyright (2004) by the American Institute of Physics.

In a homogeneous system P = 0, while in totally segregated systems P max = θ0 − (θ0 )2 (provided that the sum concentration of all components is chosen to be 1). For the melt system under consideration θ0 = NA /(NA + NB ) = 0.3, while for the solution θ0 = 0.55 ∗ NB /(NA + NB ) = 0.33 (polymer concentration 55%). There is some uncertainty, however, because the eﬀective average concentration in the smaller polymer-rich regions (since we may have solvent/polymer separation on a larger scale) may be higher, since the cylinder forming component is hydrophobic. However, from the average values we have roughly the same P max in both cases. Fig. 3 shows the time evolution of the segregation parameter for the two systems. The segregation parameter for the triblock copolymer solution is an order of magnitude lower than for the diblock copolymer melt. The reason for this diﬀerence is that for the considered Flory–Huggins interaction parameters the degree of segregation is higher for the melt than for the solution. This clariﬁes our choice of systems: we can study the kinetics of phase separation in systems having very diﬀerent degrees of phase separation but the same equilibrium microstructure (cylin-

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ders), and, moreover, roughly the same average concentration of the cylinder forming block. If we were to choose one polymeric composition, say a diblock copolymer melt, and vary the degree of phase separation by varying the Flory– Huggins parameters to have the same large diﬀerence in separation, we would necessarily shift into the phase space where the system experiences another symmetry, diﬀerent from cylinders. As one would expect from a global parameter such as P , it monitors the degree of phase separation rather well, but does not give any information on local rearrangements in the structure. To this aim, we consider the Minkowski Functionals as a function of time. We consider two diﬀerent choices of the threshold value (see the discussion in the previous sections): h = θ0 and h = 0.5. 0.15

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Fig. 3. The segregation parameter P as a function of time for the cylinder forming component. Shear is applied from TMS=0. The noisiness of the lines is a reﬂection of the noise in the dynamic equations. Reprinted from [78]. Copyright (2004) by the American Institute of Physics.

From all Minkowski functionals, the Euler characteristic is the most illustrative. For the solution (Fig. 4, left) we observe the large inﬂuence of the choice of the threshold: choice h = θ0 shows a very positive Euler characteristic, while for h = 0.5 this number is very negative. For both choices, the limiting behaviour of the Euler characteristic with increasing time is zero, which is also reached at the same instant in time. This value can be associated with the state of well aligned cylinders (see subSect. 2.3), as we can also see from Fig. 2. The fact that the Euler characteristics at later stages coincide is therefore expected, as the equilibrium morphology of aligned cylinders is reached at an early stage (around TMS=10000) and the degree of phase separation and the position of the interfaces in space no longer signiﬁcantly change. The Euler characteristic for the melt (Fig. 4, right) is distinctively less sensitive to the diﬀerent choices of the threshold. For both choices, the Euler characteristic is negative at the initial stages, although the Euler characteristic is signiﬁcantly lower for h = 0.5. At later stages, the two curves approach

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Fig. 4. The Euler characteristic as a function of time for the cylinder forming component in the solution (left) and in the melt (right). The shear was applied starting from TMS=0. The Euler characteristics were calculated for two choices of the threshold parameter: h = θ0 (◦), and an arbitrary one, h = 0.5 (). Reprinted from [78]. Copyright (2004) by the American Institute of Physics.

and coincide at the end. Based on the Euler charateristic, the two polymer systems would have completely diﬀerent kinetic pathways of phase separation, depending on the choice of the threshold. However, the diﬀerence is not so surprising as it might look at the ﬁrst glance. If the threshold is chosen at an average concentration value ( in the graphs) both systems develop themselves starting from the initially highly interconnected network (very negative Euler number) towards inﬁnite cylinders (tori), slowly reducing the number of connections and therefore holes. This threshold value “sees” all concentration deviations around θ0 , even very small ones. As is clear from the sketch in the Fig. 1 the topological picture of higher concentration modes will be diﬀerent. If the arbitrary value h is higher then θ0 (as in our case), less interconnections will be seen, as they have lower ﬁeld values then “tops of the iceberg”. Due to this reason the Euler number for both systems is higher in the case of h = 0.5 (◦ in the graphs). Moreover, if the system has a lower degree of phase separation (as the solution in our case, Fig. 3) the number of “seen” interconnections is even less. In this case, the concentration deviations overshooting the h = 0.5 value will be mostly seen as topological micelles, and the Euler number will be positive (Fig. 4, left, ◦). As the micelles grow and merge into the cylinders (tori), the Euler number levels out. As a result, by combining information from the evolution of the Euler characteristic for two choices, we conclude that there are two simultaneous processes in the kinetic pathway of the structure rearrangement in a ﬂow. One is the removalt of interconnections (defects) between cylinders, and another is the merging of micelles into cylinders. The relative contribution of these processes to the pathway depends on the degree of phase separation. Fig. 5 shows the volume, surface area and mean curvature for one of the polymer systems (the solution) for the two diﬀerent choices of threshold. The volume and surface area for h = 0.5 are lower than for h = θ0

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Fig. 5. The volume, surface area and mean curvature as a function of time of the cylinder forming component for the solution. The shear was applied starting from TMS=0. The Minkowski functionals were calculated for two choices of the threshold parameter: h = θ0 (◦), and an arbitrary one, h = 0.5 (). Reprinted from [78]. Copyright (2004) by the American Institute of Physics.

simply due to the fact that there are always less regions with high densities than with average ones. The fact that the volume and surface area are noisy for h = 0.5 shows that the high concentration ﬁeld values are much more sensitive to the breakage and reformation of local structures. Partial melting of the already phase separated structures gives rise to a reduction of the local concentration to values below the threshold. As a consequence the structure itself drops out of the image P. Then they emerge again, ﬁrst as micelles. As the number of structures with high concentration values is lower, the noise level in the graphs is higher. The volume for h = θ0 decreases quickly at the very beginning of the phase separation and then stays constant. The most drastic drop in volume corresponds to the time when phase separation shoots up (see Fig. 3). At that stage the system microphase separates from the initially homogeneous state, decreasing the contact between diﬀerent blocks and therefore lowering the enthalpic contribution to the free energy. The slight increase of the volume for h = 0.5, however, is much slower. It corresponds to the fact that regions with high concentration are still growing while phase separation continues, as is seen on slight increase of the segregation parameter P on the same time scale (Fig. 3). As the volume value in this case is smaller than for h = θ0 this increase does not contradict the decrease of the total free energy. The surface area for h = θ0 is decreasing, which suggests that the surface tension of such an interface is positive. As the surface area levels out at the same time as the Euler characteristic, this suggests that the main mechanism of reducing surface area is the removal of interconnections in the structure. The surface area of high concentration domains (h = 0.5) is roughly constant (after averaging over the noise). The volume in this case is slightly increasing, suggesting that the domains adopt a rounder shape in the cross-section; a mechanism that indeed occurs with cylinders in a ﬂow, see Fig. 2. The mean curvature for both threshold choices decreases (apart from the very ﬁrst stages of phase separation for the choice h = θ0 ). The monotonic decrease after the initial stages suggests positive bending constants of

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the interfaces. The two graphs of the mean curvature are qualitatively very diﬀerent in the very ﬁrst stages of phase separation (see Fig. 3 as well). At that stage the interfaces are only developing. The mean curvature for the average concentration choice h = θ0 grows rapidly at the very beginning. In this case the system starts to develop from the homogeneous concentration θ0 , and initially consists of a network of interconnections with very diﬀuse interfaces, induced by the noise. This network of interconnections is rich in saddle points which have low mean curvature. While phase segregation progresses and the network of interconnections coarsens, the interconnections become longer and develop substantial cylindrical parts between them. As a result, the mean curvature increases. As the interface develops, the process continues mostly by breaking interconnections (therefore reducing the number of saddle regions), and the mean curvature drops. For the high concentration domains (h = 0.5) the decrease is persistent during evolution and is much more drastic due to the fact that the system for this threshold choice consists initially of spherical micelles with higher curvature than that of the ﬁnal cylindrical micelles. In this case the interface is only “seen” starting from h = 0.5 > θ0 and therefore will be simply absent during ﬁrst few time steps. This is not observed in the graph, because the system is ﬁrst stored after 50 timesteps. 0

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Fig. 6. The Euler characteristic as a function of time of the cylinder forming component for two systems. The melt is denoted by , the solution by ♦. Shear was applied at TMS=0. Reprinted from [78]. Copyright (2004) by the American Institute of Physics.

In the remainder, we will concentrate on the Euler characteristic as a function of time. We have seen already that the Euler characteristic is a valuable means to distinguish the dominant mechanisms in kinetic pathways. If we compare the melt and the solution (Fig. 6) for the threshold h = θ0 , we see that the topological pathways are distinctly diﬀerent. In the melt, initially there are less connections than in the solution, and most of the connections are easily removed. The remaining connections are very long living. The growth

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of the Euler number for the solution is initially slower, and has a small characteristic plateau around the ﬁrst thousands time steps. After this temporary stagnation, the Euler number continues to grow, overshoots the values for the melt, and reaches the state of perfect cylinders, much more perfect than the melt system (compare also ﬁnal images in Fig. 2). This diﬀerence can be explained bearing in mind the results for the second choice of the threshold value, Fig. 4. The solution is a much less segregated system than the melt (see Fig. 3). High concentration regions appear as micelles in the ﬁrst stages of phase separation (Fig. 4, left, ◦). The micelles will be seen also at lower threshold values, in reduced quantity, among newly emerging structures. In the very beginning the number of micelles grows (increase in Fig. 4, left: ◦). The same process may be expected at other threshold values. This, together with breakage of interconnections, contributes to the initial fast growth of Euler number at h = θ0 . Consequently, the number of micelles decreases, as they merge into the cylinders. For h = θ0 the two processes (a decrease of micelles and breakage of interconnections) therefore balance each other, resulting in a short plateau in the Euler number graph. Finally, when most of the micelles have disappeared, the second process takes over and the system proceeds towards a perfect cylindrical phase. One should bear in mind that as the solution is much less segregated than the melt, new micelles will appear and coalesce all the time, which makes the initial slope of the curve smaller than the one for the melt system. As we have two processes involved in the 0.007

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Fig. 7. The segregation parameter P as a function of time of the cylinder forming component for the solution (left) and the melt (right) at diﬀerent shear scenarios. The label 1 (◦) refers to the situation where the shear is applied from the beginning; label 2 (•) refers to the case where the shear is applied at TMS=5000 (melt) and TMS=10000 (solution). The inset in the left ﬁgure focusses on the enhancement of phase separation by shear at the very early stages in the melt system. The noisiness of the lines is a reﬂection of the noise in the dynamic equations. Reprinted from [78]. Copyright (2004) by the American Institute of Physics.

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phase separation, namely development of interfaces and domain rearrangement, one should study to what extent shear aﬀects either of them. In the preceding paragraphs we discussed the case of shear applied from the start. Here, we proceed with a discussion of the case where the shear was applied well after the interfaces were formed, so that we can separate the two processes. The signiﬁcance of the time at which the shear is applied can already be seen in the segregation parameter, as shown in Fig. 7. For both systems we have studied two cases: case 1 where the shear is applied from the beginning, and case 2 where the shear is applied to an already phase separated structure at a later instance. The inﬂuence of shear is stronger for a weakly separated system (solution, ﬁg 7, left). In both melt (see inset in Fig. 7, right) and solution we see enhancement of the phase separation by shear at the very ﬁrst stage, when the interfaces are formed. After the ﬁrst thousand time steps the shear starts to suppress the phase separation in both systems. That could be due to the fact that, at this stage, the domain rearrangement starts to play a major role. The shear breaks some domains such that they can reconnect in the ﬂow direction [73, 77]. This phenomena is equivalent to partial melting of the microstructure, and the segregation parameter is therefore lower. This region is however relatively short for the solution when compared to the melt (for the melt this region extends until the instance where shear is applied in case 2, TMS=5000). This could be explained by the fact that, as the solution is a much weaker segregated system than the melt, the domain breakage by shear occurs easier in the solution. By the time most of the interconnections are removed, the system consists of cylinders in the direction of ﬂow. In general, the system without interconnections is in true equilibrium (without shear), and has a lower free energy than the system with interconnections. Therefore, if the system reached that state of perfect cylinders in the ﬂow direction, it continues to enhance the interfaces, and has a higher segregation parameter than the system without shear, full of structural defects like interconnections. The much stronger segregated melt system did not reach the perfect cylinder state even after longer shear, so it is simply not yet in the state just discussed for the solution. The kinetics of defect removal in the stronger segregated system is simply slower. When shear is applied at a later instance, to an already well separated system, partial melting occurs (drop in P in Fig. 7). The weaker the phase separation in the system the more the structure melts. This melting consists of two contributions, one of which is due to overall partial melting of the interfaces, and the second and most profound is due to the breakage of domains like interconnections and cylinders. Both systems recover and reach the same segregation parameter value as in the scenario where the shear was applied from the beginning. Therefore, neither system has a long memory of the shear history. The Euler number gives more information of the kinetic pathways for the above mentioned shear scenarios. We further elaborate on the eﬀect of diﬀerent shear instances and threshold choices in Fig. 8. In this ﬁgure, the left column shows the Euler characteristics for choice h = θ0 of the threshold and diﬀerent instances of applied shear; the right column shows

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Fig. 8. Euler characteristics as a function of time of the cylinder forming component for the solution (top) and melt (bottom). Left: h = θ0 ; the shear initiated from the beginning () and at later times (), which is TMS=10000 for the solution and TMS=5000 for the melt. The inset in the bottom ﬁgure focusses on the very early stages of the Euler characteristic for the melt system. Right: h = θ0 (◦), and h = 0.5 (); the shear initiated at TMS=10000 for the solution and TMS=5000 for the melt. Reprinted from [78]. Copyright (2004) by the American Institute of Physics.

the eﬀect of diﬀerence choices of the threshold for the second shear scenario (where the shear was applied at a later time). The Euler graphs in the left column of Fig. 8 are remarkably similar to the graphs of the segregation parameter P for the same systems (shown in Fig. 7). All the conclusions which have been just drawn on the basis of the segregation parameter P in Fig. 7 and previous knowledge derived from the visual inspection of many 3-D images [77] can be also made solely on the basis of Euler number graphs in Fig. 8 (left). Moreover, the information contained by the graphs of the Euler characteristic is much richer. The enhancement of the phase separation by the shear in the very initial stages after TMS=0, as well as partial melting and breakage mechanism after the application of shear at TMS=10000 (solution) and TMS=5000 (melt) (see discussion of Fig. 7), are strongly correlated with the removal and creation of interconnections that can be deduced from the

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Euler characteristic for the average concentration threshold choice, h = θ0 (Fig. 8, left). In particular, shear from the beginning leads to an enhanced removal of connections (see Fig. 8, top left, and inset in bottom left). We see that our interpretation, that partial melting prior to reformation of structures proceeds via ﬁrst breakage of domains and then recombination of them in the ﬂow direction, is not complete. A drop in the Euler number at the instance where shear was applied ( in Fig. 8, left) manifests that the sequence can be reversed. First new interconnections are formed (in the direction of ﬂow presumably), and then only unfavorable interconnections (in the way of the ﬂow) break. We conclude, without looking into 3-D images, that the ﬁnal structure consists of perfect cylinders. We also see that the instance at which the shear is applied to the stronger separated system (melt) has no dramatic eﬀect on the topological dynamics of the structure, which does not conﬂict, however, with the interpretations based on Fig. 7 (right). The Euler characteristics for higher concentration values (h = 0.5) give us addition information, Fig. 8 (right column). The behaviour is very diﬀerent for the solution and the melt. At the very beginning in a weakly separated system (solution) the high concentration modes (◦) form an interconnected network without shear. Later on this network breaks into micelles. On the contrary, in the presence of shear (Fig. 4, left, ◦) micelles are formed at the very beginning. Therefore the shear suppresses interconnections in the initial stages of phase separation in solution for both choices of threshold value (notice that the initial Euler numbers for h = θ0 (squares) are much lower without shear as well). When shear is applied at TMS=10000, the high concentration values, h = 0.5, show breakage of cylinders into spherical micelles, while in the case of the average concentration threshold interconnections are formed (opposite bumps in graphs in Fig. 8, right top). Both structural changes lead to aligned cylinders at the end. Remarkably, breakage into micelles is not seen for the melt, when shear is applied at TMS=5000 (Fig. 8, right bottom, ◦). This suggests why the less segregated solution system has less defects at the end than the stronger segregated melt (see Fig. 2). The solution system has a rather ﬂexible structure, on which shear, applied at a later instance, has a generic eﬀect: it recombines the high-concentration micelles and breaks up the connections at the average concentration level that are not in the shear direction. The absence of the intermediate micellar phase (at least in noticeable quantity) for high densities in the melt makes it much more diﬃcult to reorient in shear ﬂow. The suppression of high-concentration micelles by shear in the melt is also seen in another striking diﬀerence in Euler number graphs for the two systems. The high-concentration modes of the melt system at the very ﬁrst stages of phase separation in the absence of shear are spherical micelles (Fig. 8, right bottom, ◦), contrary to interconnections in the solution. These micelles are absent if shear is applied from the very beginning (Fig. 4, right, ◦), although it is possible that the structure is a collection of interconnections and some spheres, as the total Euler number is not very low. In the absence of shear, the micelles promptly form an interconnected network and the evolution follows

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the average concentration modes (Fig. 8, right bottom). This diﬀerence could be due to the compositional diﬀerence between diblock copolymer melt and triblock copolymer solution and is beyond the scope of this chapter.

4 Concluding Remarks We have shown how one can use modern techniques to structure and interpret large amounts of simulation data. The Minkowski functionals are a powerful tool, as all motion invariant, additive and continuous image functionals can be written as linear combinations: an important ’class’ of information about the patterns is contained in these Minkowski functionals. A very important step is the thresholding procedure that is applied to the simulation data prior to the Minkowski functional calculation. The important question is: what threshold value or values contain redundant information? Using a priori knowledge of our system, we make a physically motivated choice for the two threshold values that we need for our analysis. We ﬁnd that a minimal set of two threshold values is suﬃcient to unraﬄe the phase separation kinetics from a few graphs of time dependent numbers alone, without even considering visual information. This approach enhances the eﬃciency of the morphological analysis and minimizes both the computational eﬀort and amount of data enormously. Acknowledgement. The supercomputer resources were provided by a grant of NCF at the High–Performance Computing Facility (SARA) in Amsterdam. The author thanks Dr. Andrei Zvelindovsky (University of Central Lancashire) for fruitful discussions and collaborations.

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Scaling Theory of Polyelectrolyte and Polyampholyte Micelles Nadezhda P. Shusharina and Michael Rubinstein Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599-3290, USA

1 Introduction Polymer solutions have been extensively studied for the past three decades [1–3]. Owing to the successful application of scaling theory [1] the solution properties of uncharged polymers are now reasonably well understood. However, many practically important polymers, both synthetic and natural, are charged in polar solvents, most commonly in water. The added complexity of charged systems stems from their long-range electrostatic interactions. The additional emerging length scales make the scaling approach to charged systems much more challenging than for neutral ones. At the same time, research into the functional materials, drug delivery formulations and stabilization of colloidal systems has led to the development of new types of polyelectrolytes, including charged polymeric surfactants. Study of these new polymers is of great industrial importance and provides an excellent opportunity for the introduction and validation of theoretical approaches. Therefore the theory of solutions of charged polymers remains a quickly developing area of polymer physics and material science [4–6]. In contrast to low-molecular weight compounds, polymers have a very important structural degree of freedom called molecular architecture. Speciﬁcally, linear polymer chains can be linked together in diﬀerent fashions forming a single macromolecule. The control of molecular architecture is a widely used approach in the development of polymers with desired properties [7]. The simplest examples of the chain arrangement are block copolymers, where chemically diﬀerent chains are linked together end to end, and polymer stars, where several chains are linked at one point. The conformations of polymer subchains in a branched molecule depend on its architecture. For example, the interaction of monomers in a star is stronger than in a linear chain because of the additional interactions between the monomers belonging to diﬀerent chains (arms). If, for example, the monomers along the chain repel each other, the arms in a star will be more extended than the equivalent linear chains [8].

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The structure of polyelectrolyte stars is even more complex [9]. An accumulation of a large charge in a small volume of the star leads to a non-uniform distribution of counterions in solution. The non-uniformity is due to an interplay of the electrostatic energy of the star and the entropy of the counterions. To lower the electrostatic energy, counterions tend to be conﬁned within the volume of the star. However, complete conﬁnement (condensation) of counterions would lead to a signiﬁcant loss of their entropy, so the minimum of the free energy is achieved when a fraction of counterions resides within the star, while the remaining counterions are spread throughout the surrounding solution [10,11]. The uncompensated charge of the star leads to a larger extension of the star arms as compared to neutral stars. Moreover, this extension is not uniform because of the existence of two characteristic regions in the star. In the center the concentration of monomers is so high that the short-range monomer-monomer interactions are stronger than the electrostatic long-range interactions. Hence, in the center the extension is the same as in a neutral star. In the outer region the electrostatics dominate, making the extension much larger than that in the corresponding part of a neutral star. The general models developed for star-branched macromolecules can be successfully applied to macromolecular aggregates of similar size and geometry. For example, star-like structures may be formed not only by grafting of polymer chains to a small colloidal particle, but as a result of the self-assembly of linear chains. The most familiar case of self-assembly is the formation of micelles, driven by the surface tension at the boundary between solvophobic and solvophilic parts of amphiphilic molecules [12, 13]. Block copolymers [14,15] form micelles when monomers of diﬀerent blocks have a diﬀerent aﬃnity for the solvent. In this case, blocks tend to segregate, forming ﬁnite-size micelles in solution. A micelle usually has a spherical dense core made of insoluble blocks and a corona consisting of soluble blocks. Many types of water-soluble polymers are charged, so polyelectrolyte blocks are a convenient choice for stabilizing aqueous solutions of polymeric micelles. As part of a block copolymer chain, each charged block is grafted onto the surface of the micellar core, so the structure of the corona resembles that of a polyelectrolyte star. However, the number of arms in the corona is not ﬁxed as in a star, but is equal to the aggregation number of the micelle. It is determined by the balance between the attraction of the uncharged hydrophobic blocks and the repulsion of the charged ones. The ﬁrst systematic study of polyelectrolyte micelles was published more than twenty years ago [16]. Since then, micellar solutions of various block copolymers have been investigated. For example, the hydrophobic block could be polystyrene (PS) or poly-tert-butylstyrene (PtBS), and the polyelectrolyte block could be poly(acrylic acid) (PAA) or poly(sodium styrenesulfonate) (NaPSS) [17–20]. Theoretical predictions of the equilibrium structure of the micelles have been described in several works [21–24]. While in polyelectrolytes all dissociated groups are like-charged, polyampholytes bear groups of opposite charge on the same chain. There has been

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considerable interest recently in polyampholytes both in solution and adsorbed on surfaces [25]. The major feature of polyampholytes is the controllable magnitude and sign of the charge of these molecules. The charge of polyampholytes is determined by the relative fraction of positively and negatively charged groups and can be adjusted during synthesis and, in the case of weak polyelectrolytes, by the variation of solution pH [26, 27]. The net charge governs the conformation of a polyampholyte: highly charged chains are in their extended conformations, whereas weakly charged chains form compact colloidlike globules. Increasing polymer concentration in solutions of such globules leads to aggregation of the chains and precipitation. Most of the studies concerning polyampholytes deal with randomly distributed oppositely charged groups within a chain. Much less attention has been paid to the case of block polyampholytes where the opposing charges are grouped in long sequences. Several recent experimental works report micelle formation in solutions of diblock polyampholytes (diblock copolymers with a positively and a negatively charged block) [28, 29]. In such micelles, oppositely charged blocks collapse in the micellar core. The blocks carrying the uncompensated charge are in the corona, stabilizing the micelle in solution. Micellization in diblock polyampholyte solutions has been considered theoretically [30, 31]. -

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Fig. 1. Sketch of (a) a diblock copolymer with one uncharged and one polyelectrolyte block; (b) a diblock polyampholyte; (c) a spherical star-like micelle or a star of radius R and its counterions.

Below we review a scaling description of polyelectrolyte micelles in solutions of (i) diblock copolymers with an uncharged and a polyelectrolyte block, and (ii) diblock polyampholytes, see Fig. 1. In both cases the solvent is a polar medium (e.g., water) with dielectric permittivity ε at temperature T . The shell (corona) of these micelles is composed of polyelectrolyte chains and therefore the energy of the shell is the same as the energy of a polyelectrolyte star (as long as the corona is much larger than the micellar core). The equilibrium micellar aggregation number is deﬁned from the free energy balance between the core and the corona. We will start our review with the theory of polyelectrolyte stars and then continue with the speciﬁc features resulting from diﬀerent interactions of the core.

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The chapter is organized as follows. In the next section the scaling description of a single polyelectrolyte chain will be brieﬂy outlined [1, 32]. The main results for polyelectrolyte stars will be presented in Sect. 3. In Sect. 4 the theory for polyelectrolyte micelles will be reviewed according to the papers [21, 22] and Sect. 5 will be dedicated to the theory of polyampholyte micelles [30, 31].

2 Scaling Description of Polyelectrolyte Chains Let us brieﬂy recall the scaling description of a single polyelectrolyte chain in a dilute salt-free solution. These chains are stretched due to the intra-chain electrostatic repulsion. The conformation of the chain can be described as a linear array of electrostatic blobs [1, 32]. The size of the electrostatic blob ξel is determined by the condition that the energy of electrostatic interaction between two neighboring blobs along the chain is of the order of the thermal energy: 2 e2 f 2 gel ≈ kB T. (1) εξel Here e is the elementary charge, gel is the number of monomers in a blob, kB is the Boltzmann constant, and T is the absolute temperature (we keep our discussion at the level of scaling approximation and omit numerical coeﬃcients). The statistics of the chain is unperturbed by the electrostatic interactions on a length scale smaller than the electrostatic blob size. It is determined by the short-range interactions between monomers in solution. We will consider the case of a θ-solvent [1] where the monomers interact via three-body repulsion. The statistics of a chain in a θ-solvent is almost Gaussian, hence the size of the electrostatic blob ξel is related to the number of monomers gel in it as 1/2 ξel ≈ bgel and substituting this relation into (1) one concludes that ξel ≈ b(uf 2 )−1/3 ,

(2)

where u = lB /b is the ratio of the Bjerrum length lB =

e2 εkB T

(3)

to the Kuhn length b [2, 3]. The Bjerrum length is a measure of the strength of the electrostatic interactions. It is equal to the distance at which the interaction between two elementary charges in the medium with the dielectric constant ε is of the order of kB T . For ﬂexible polymers in an aqueous solution, b ≈ 1nm, T ≈ 300K, ε ≈ 80, and the parameter u is approximately equal to unity. The energy of a polyelectrolyte in the elongated conformation is of the order of the thermal energy kB T per electrostatic blob

Scaling Theory of Polyelectrolyte and Polyampholyte Micelles

N Fpolyel ≈ ≈ N (uf 2 )2/3 . kB T gel

305

(4)

The length of the polyelectrolyte chain is estimated as Rpolyel ≈ ξel N/gel and, using (2) we get (5) Rpolyel ≈ bN (uf 2 )1/3 . In the next section we will consider the conformations of a star polymer consisting of a ﬁxed number of polyelectrolyte arms, see Fig. 1c.

3 Polyelectrolyte Stars 3.1 Free Energy of the Star Let us consider a dilute solution of polyelectrolyte stars composed of p identical arms. We will assume that the arms are ﬂexible chains of N Kuhn segments of length b, and the non-coulombic interactions between the segments correspond to a θ-condition. The net charge of an arm (counted in the elementary charge units e), or the valence is Znet = N f,

(6)

where f is the fraction of charged monomers in an arm. Correspondingly, there are pN f monovalent counterions per star in the solution. The counterions are distributed between the volume of the star and the volume of solution not occupied by the stars, i.e. fraction α of them is freely ﬂoating in the solution and fraction 1 − α is conﬁned within the volume of the star, reducing its charge. This conﬁnement is one of two types of counterion condensation called spherical condensation [33]. The counterions within a sphere surrounding a star remain osmotically active. If the linear charge density along the arms of the star is higher than approximately one elementary charge per Bjerrum length (3), the counterions condense into the close vicinity of each arm and practically lose their translational entropy [34, 35]. This case, called cylindrical (Manning) condensation, can be achieved for polymers with a higher fraction of charged monomers f in solvents with lower dielectric constant, or for multivalent counterions. In the following description we will restrict our consideration to the case with only spherical condensation and assume no cylindrical condensation. The radius of the star R depends on the uncompensated (eﬀective) charge of the star eZef f = epZnet α. Both the equilibrium radius and the eﬀective charge of the star can be found by minimizing the free energy of the star with respect to R and α. The electrostatic self-energy per arm of the star with eﬀective charge eZef f and radius R is Fel pZ 2 α2 ≈ lB net . (7) kB T R

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The contribution to the free energy per arm due to the elasticity of an arm is related to conformational entropy losses in a stretched chain [3, 11] Fstr R2 ≈ . kB T N b2 The entropic part of the free energy per arm due to counterions is pZnet (1 − α) Fcount ≈ Znet (1 − α) ln + Znet α ln(f αcb3 ), kB T R3

(8)

(9)

where c is the average monomer concentration in the solution. The two terms in (9) are the entropic part of the free energy per arm due to counterions conﬁned into the volume of the star and to free counterions, respectively. 3.2 Polyelectrolyte and Osmotic Regimes The limit α ≈ 1 corresponds to the polyelectrolyte regime where most of the counterions are in the surrounding solution and the eﬀective charge of the star is very close to the net charge. Then, the electrostatic energy per arm of the star is determined by the eﬀective charge of the star, which is approximately equal to its total net charge epZnet : pZ 2 FelP E ≈ lB net . kB T R

(10)

The equilibrium size of the star is determined by minimizing the sum of the energy of electrostatic repulsion between arms (10) and the elastic energy (8) with respect to the star size R: RP E ≈ bp1/3 N (uf 2 )1/3 .

(11)

The extension of the star arms is larger than the extension of a free polyelectrolyte chain Rpolyel , see (5), because of the additional stretching of the arms in the star caused by the inter-chain electrostatic repulsion. Each arm of the star can be represented as a sequence of “tension” blobs [3] of the size: ξt ≈

ξel b2 N ≈ bp−1/3 (uf 2 )−1/3 = 1/3 . RP E p

(12)

The fraction of free counterions α in the polyelectrolyte regime depends on the star concentration. Indeed, the volume accessible to free counterions is not inﬁnite but of the order of the cube of the distance between the stars in solution. In the polyelectrolyte regime the following expressions for α can be written [21]: α = 1 − φstar eU if 1 − α 1, (13) where φstar ≈ R3 c/pN is the volume fraction occupied by stars in the solution and the dimensionless potential of the star U is equal to

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307

pZnet . (14) R The physical interpretation of the parameter U is the energy of electrostatic interaction between the charged star and a counterion located at the periphery of the star (speciﬁcally at the distance R from the center of the star) divided by the thermal energy kB T . The counterions can be considered as “free” if U 1, i.e. if the entropy loss due to conﬁnement of a counterion inside the star is larger than the energy gain due to the reduction of the electrostatic repulsion. The scaling estimate of the size of the charged star in the polyelectrolyte regime (11) assumed uniform stretching of the star arms. The conformation of an arm is thought of as a string of tension blobs of equal size given by (12). To calculate the radial distribution of monomers in the star more precisely, one has to consider the balance of forces acting on the section of a star arm. Such calculations have been done for an individual polyelectrolyte chain [32,36,37]. It was shown that in a chain of ﬁnite length the monomers are distributed non-uniformly and the chain has a trumpet-like conformation with blob size increasing from the middle point to the ends [36, 37]. The inhomogeneity occurs because the electrostatic potential is smaller near the ends where the charges have fewer neighbors. The increase of the blob size towards the ends is however slow and the correction to the size of the chain obtained by simple scaling estimations is logarithmic. These predictions of the scaling model have been conﬁrmed by Monte Carlo simulations [37]. The technique developed in [32, 36] can be applied to model non-uniform stretching of arms in an individual polyelectrolyte star [33]. Here, the arms have a conformation of a half trumpet with the tension blob size logarithmically increasing from the center to the periphery of the star. At the crossover between the regime of mostly free counterions (α ≈ 1) to the regime of mostly conﬁned counterions (α 1) the electrostatic potential of the star is of the order of the thermal energy kB T , i.e. the parameter U ≈ 1. For this value of the electrostatic potential the tension blob size ξt (12) is equal to the distance between charges along the chain. Thus, the free energy loss due to chain stretching is of the order of the thermal energy kB T per charge, i.e., the same as the entropy gain per counterion released from the star. Above this crossover the excess counterions start condensing on the star and the solution of stars is said to be in the osmotic regime [11]. The parameter α in the osmotic regime can be approximated as [21] U ≡ lB

α=

1 1 ln U φstar

if α 1

(15)

The stretching force in the osmotic regime is dominated by the entropy of the conﬁned counterions and is usually referred to as the osmotic contribution to the free energy of the star OS pZnet b3 Fcount ≈ Znet ln . (16) kB T R3

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Balancing this conﬁnement entropy of counterions with the elastic energy of the arm, see (8), we obtain the equilibrium size of the star in the osmotic regime: (17) ROS ≈ bN f 1/2 , which is independent of the number of arms in the star. The size of the tension blob in the osmotic regime corresponds to one charge per blob: ξt ≈ bf −1/2 .

α

α

(18)

Fig. 2. (a) The fraction of counterions released from the star α, and (b) the radius of the star R as functions of the number of arms in the star p; (c) the fraction of counterions released from the star α, and (d) the radius of the star R as functions of the fraction f of charged monomers in an arm. The ﬁxed parameters are N = 250, u = 1.5, f = 0.5 in (a) and (b), p = 30 in (c) and (d).

Let us comment on the monomer distribution in the osmotic regime. An exact solution of the Poisson–Boltzmann equation coupled to the chain elasticity is available for the planar osmotic brush [11]. It was demonstrated that at equilibrium the counterions are conﬁned within the brush in the osmotic regime preserving the local electroneutrality. The loss of the counterion entropy is compensated by the gain of the electrostatic energy. The local electroneutrality is achieved by the local concentration of counterions proportional to the concentration of polymer segments. In the spherical geometry, similar eﬀects have been observed by solving the Poisson–Boltzmann equation numerically [38] and by Monte Carlo simulations [39]. These theoretical approaches predict that the density proﬁles of the monomers and counterions closely follow each other inside the star. This ﬁnding is conﬁrmed by experiments using small angle neutron scattering [40].

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The dependencies of α and R on the number of arms p and the fraction of charged monomers f are shown in Fig. 2. These dependencies are obtained by minimizing the free energy of the star with the contributions given by (7), (8), and (9). The fraction of free counterions decreases with increasing bare charge of the star at the expense of increased fraction of conﬁned counterions. At the same time the slope of the dependence of the star radius changes indicating the crossover from the polyelectrolyte to the osmotic regime. The crossover values p∗ and f ∗ can be obtained from the condition U = 1 in (14) p∗ ≈ u−1 f −1/2

(19)

f ∗ ≈ u−2 p−2 .

(20)

The polyelectrolyte regime is realized at p < p∗ (Fig. 2a,b) or at f < f ∗ (Fig. 2c,d). The osmotic regime appears at higher values of p and f .

4 Polyelectrolyte Micelles The results obtained in the previous section can be used to calculate the equilibrium aggregation number of spherical star-like micelles made of diblock copolymers with one uncharged and one polyelectrolyte block. Let us consider a diblock copolymer with N0 monomers in the uncharged block and N monomers in the polyelectrolyte block. Both blocks are ﬂexible polymers with the same Kuhn segment length b. To ensure micellization, monomers of the neutral block are solvophobic (hydrophobic) and their interaction is described by the temperature-dependent excluded volume parameter [3] v≈

T −θ 3 b , T

(21)

where the deviation from the θ-temperature is negative in a poor solvent indicating eﬀective attraction between monomers. The non-electrostatic interactions of monomers of the polyelectrolyte block are θ-like and the fraction of charged monomers in it is f , so the chain carries the net charge eZnet = eN f . 4.1 Unimers The uncharged block in a selective solvent collapses into a globule leading to a tadpole conformation of the whole chain with a neutral globular head and a polyelectrolyte tail. The conformation of the collapsed block is described by a dense packing of the thermal blobs of size [3] ξT ≈

b4 |v|

(22)

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The density of the globule is proportional to the excluded volume parameter v (21) and its size is 1/3 N Rgl ≈ b 01/3 . (23) |v| Each thermal blob inside the globule is attracted to a neighboring blob with the energy of the order of the thermal energy kB T . Thus, in the framework of the scaling approach the bulk contribution to the free energy of a globule Fgl is proportional to the number of thermal blobs in the neutral block N0 /gT ≈ N0 |v|2 /b6 . In addition to the bulk free energy, the globule has a surface free energy contribution due to the polymer-solvent interface. The origin of this term is the diﬀerence between the number of nearest neighbors of a blob inside the globule and on its surface. The surface energy can be estimated as the thermal energy kB T per blob at the surface of the globule [3]: uni 2 Rgl Fsurf 2/3 ≈ 2 ≈ N0 |v|4/3 . kB T ξT

(24)

The electrostatic self-energy of the tadpole tail and the length of the tail are given by (4) and (5), respectively. 4.2 Micelles Aggregation in solutions of neutral-polyelectrolyte diblock copolymers is driven by the attraction between the heads of the tadpoles. By bringing the heads of two tadpoles together, one reduces the energy of the polymer-solvent interface. The stabilizing factor which prevents the formation of an inﬁnite aggregate with increasing polymer concentration is the electrostatic repulsion between tails. The possibility of micellization is determined by the entropy of the tadpoles. The micelles are formed at solution concentrations higher than the critical micelle concentration (cmc) [12]. The aggregation number of micelles is controlled by the balance between the surface energy of the uncharged micellar core and the electrostatic repulsion between the charged blocks in the corona. The radius of the core is (cf. (23)) Rcore ≈ bp1/3 N0 |v|−1/3 , 1/3

(25)

and the surface energy per chain of a micelle formed by p diblock copolymers is R2 Fsurf 2/3 ≈ core ≈ p−1/3 N0 |v|4/3 . (26) kB T pξT2 As in dilute solutions of polyelectrolyte stars, both polyelectrolyte and osmotic regimes can be observed in the micellar solution depending on the net charge of chains forming the micelle.

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Polyelectrolyte Regime (Free Counterions, α ≈ 1) Substituting (11) into the expression for the electrostatic energy of the corona (10) we obtain the electrostatic contribution to the total energy of the micelle per chain: FelP E ≈ p2/3 N (uf 2 )2/3 . (27) kB T The aggregation number in the polyelectrolyte regime is determined by the balance of this electrostatic free energy of the corona and the surface free energy of the core (26): 2/3

pP E ≈

N0 |v|4/3 f . Znet (uf 2 )2/3

(28)

Note that all contributions to the total free energy of the micelle, namely Fsurf (26), Fel (27) and Fstr (8) are balanced at equilibrium (Fsurf ≈ Fel ≈ Fstr ). Osmotic Regime (Condensed Counterions, α 1) The aggregation number in the osmotic regime is estimated by balancing the surface energy of the core (26) and the osmotic term of the free energy, see (16), which leads to N 2 |v|4 (29) pOS ≈ 0 3 . Znet At the crossover between the polyelectrolyte and osmotic regimes the two free energy contributions FP E (27) and FOS (16) are of the same order of magnitude and the micellar aggregation numbers given by (28) and (29) are ∗ per one chain of the micelle at this crossover is equal. The net valence Znet ∗ ≈ Znet

4/5

N0 u2/5 |v|8/5 . N 1/5

(30)

In Fig. 3 we have plotted the aggregation number p as a function of the net valence of the diblock copolymer chain. Diblock copolymers with a low valence ∗ (Znet < Znet ) form polyelectrolyte micelles (regime P E), whereas strongly ∗ ) attract more counterions into the corona and charged chains (Znet > Znet the equilibrium micelles formed by these highly charged copolymers are in the osmotic regime (OS). As the charge of polyelectrolyte block is increased, aggregation of unimers into micelles becomes unfavorable. For the micelle to be formed, the surface energy of the unimer head has to be larger than the entropy of the counterions per chain being conﬁned inside the osmotic micelle. Micelles would not form if bringing block copolymers together led to the loss of counterion entropy larger than the decrease in the surface energy of the aggregate. The maximum net charge at which the aggregation is still possible is determined by the condition

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312

Fig. 3. The aggregation number of micelles formed by neutral-polyelectrolyte diblock copolymers as a function of the net valence per chain Znet = N f . The parameters are: N0 = 1000, f = 0.01, u = 1. 2/3

uni Znet ≈ N0 |v|4/3 .

(31)

The aggregation number in the polyelectrolyte regime increases with decreasing polyelectrolyte block charge eZnet leading to the increase of the QN , see Fig. 3, nonmonomer concentration in the corona. At Znet = Znet electrostatic short-range interactions between monomers become more important than the long-range electrostatic interactions. The aggregation number of the micelles at small Znet is determined by the minimization of the surface free energy and the energy of the arm stretching due to the short-range interarm repulsion [8]. However, such quaisi-neutral micelles (regime QN in Fig. 3) may become unstable. Since the neutral blocks in the core are “grafted” onto the surface of the core they become stretched. If the stretching free energy of a block in the core is of the order of the diﬀerence between the free energy of the corona and the surface free energy per chain, the spherical geometry of the core is unfavorable leading to shape transitions into cylindrical and planar aggregates [24, 41]. 4.3 Critical Micelle Concentration Experimental data suggest that micelle formation in solutions of block copolymers is a reversible process with a rather sharp transition between the two states: the solution of unimers and the solution of micelles [42–45]. There is a phase equilibrium (coexistence) of the two states at the critical micelle concentration (cmc). Therefore, the chemical potentials of a free chain (unimer) and

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of a chain in the micelle of equilibrium size are equal. Below we consider the equilibrium between unimers and the micelles of the two types, polyelectrolyte and osmotic. In the polyelectrolyte regime, aggregation is governed by competition between the loss of entropy of chains aggregated into the micelle and the energetic gain due to reduction of the surface energy of the micelle (the entropy of the counterions is the same in the solution of unimers and micelles because there is no condensation in the polyelectrolyte regime). The diﬀerence in the surface energies of the unimer, see (24), and of the equilibrium micelle, see (26), can be approximated by the unimer contribution, since −1/3 uni and pP E 1. The cmc in the polyelectrolyte regime Fsurf ≈ pP E Fsurf can be estimated as: uni F surf 2/3 (32) ≈ exp −N0 |v|4/3 ccmc b3 ≈ exp − kB T In the case of the osmotic micelles, the main contribution of the free energy change is related to the entropy loss of counterions, as they are localized within the micellar volume. In this case the counterion entropy loss is compensated by the surface energy gain due to the aggregation of chains into micelles. This leads to the following expression for the critical micelle concentration 2/3 uni pN 1 Fsurf N0 |v|4/3 3 ccmc b ≈ cmic exp − ≈ 3 exp − , (33) Znet kB T R Znet where cmic = pN/R3 is the monomer concentration inside the micelle. The fact that the counterions become conﬁned when unimers aggregate to form the osmotic micelle greatly increases the cmc (note the factor 1/Znet in the exponential). In a more precise consideration one should take into account a small but ﬁnite fraction of counterions (1- α) 1 localized in a star in the polyelectrolyte regime. If the number of localized counterions per chain in a micelle Nf(1-α) is larger than unity then the critical micelle concentration is controled by the entropy of counterions, since the entropy loss due to conﬁnement of the counterions exceeds the entropy loss due to conﬁnement of free chains into a micelle. 4.4 Eﬀect of Added Salt Salt ions screen the electrostatic interactions and thus an addition of lowmolecular-weight salt essentially inﬂuences the equilibrium properties of solutions of polyelectrolyte micelles, see [46] and references therein. The eﬀect of salt on solutions of aggregated polyelectrolytes is more complex than that on solutions of free polyelectrolytes due to strong inter-chain interactions and the phenomenon of counterion condensation in micelles.

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Addition of salt to grafted polyelectrolyte layers leads to a decrease in the layer thickness because of weakening of the chain stretching [11, 23]. The stretching reduces upon screening of the inter-chain electrostatic repulsion between densely grafted chains. The screening eﬀects in micellar systems result in increasing aggregation number [23]. Since the repulsion between the corona chains weakens, the free energy per chain in a micelle decreases and therefore larger micelles can be formed. The size of the micellar corona decreases slower than the thickness of the layer with a ﬁxed number density of grafted chains due to the increase of the aggregation number with salt concentration. The increasing aggregation number leads to decreasing inter-chain separation and thus to stronger chain stretching. At the same time, however, the stretching weakens due to an enhanced screening. These predictions, based on the scaling arguments, [23] are in good agreement with the experimental data [46].

5 Polyampholyte Micelles Star-like micelles can be formed in solution of diblock copolymers with oppositely charged blocks. Both aggregation and stabilization of these micelles are governed by the electrostatic interactions. Consider a diblock polyampholyte chain consisting of N segments, N+ in the positively charged block and N− in the negatively charged block, N = N+ + N− . The short-range non-electrostatic interactions between segments are assumed to be θ-like. The fractional charge of the two blocks is assumed to be the same, f+ = f− = f . Therefore, the net charge of the chain eZnet is related to the block length asymmetry ΔN = N+ − N− according to eZnet = eΔN f

(34)

For deﬁniteness, we assume that N+ > N− , so that ΔN > 0. 5.1 Unimers The conformation of a diblock polyampholyte chain depends on the block charge asymmetry, or the net charge Znet . Chains with high charge asymmetry (ΔN ≈ N ) are elongated similar to polyelectrolyte chains, see Sect. 2. On the contrary, if the diblock polyampholyte chain is nearly charge-symmetric (ΔN N ), it collapses into a globule, see Fig. 4a. The collapse is driven by the charge density ﬂuctuation-induced attraction [47, 48] between oppositely charged blocks. The equilibrium density of the globule is determined by the balance between ﬂuctuation-induced attraction and three-body repulsion (in the case of a θ-solvent). There is an important length scale in the globule called the correlation length ξ. Polymer statistics at the length scale smaller than the correlation length is unperturbed by the ﬂuctuation-induced attractive interactions. This leads to the usual θ-solvent scaling relation between the

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315

correlation length ξ and the number of monomers g in the correlation blob, ξ ≈ bg 1/2 . At length scales larger than the correlation length, attractive interactions result in dense packing of the correlation blobs. The local structure of the “melt” of blobs resembles that of a concentrated solution of positively and negatively charged polyelectrolytes with each blob being predominantly surrounded by oppositely charged blobs, see Fig. 4. The electrostatic interactions between any two neighboring blobs are of the order of the thermal energy kB T : lB f 2 g 2 ≈ kB T. (35) kB T ξ

ξ ξ

ξ

Fig. 4. Blob picture of (a) a charge-symmetric polyampholyte globule. A correlation volume (blob) containing a section of positively charged chain is predominantly surrounded by blobs containing sections of negatively charged chains. (b) A tadpole conformation of a charge-asymmetric unimer with head size Rgl and tail length Rtail . ξ ≈ ξel is the correlation blob size. Reproduced from [31]. Copyright (2005) American Chemical Society.

The blob size ξ is, therefore, determined by a condition similar to that controlling the size of an electrostatic blob, cf. (1). However, in the case of charge-symmetric polyampholytes, the neighboring blobs are predominantly oppositely charged, reﬂecting the attractive nature of intra-chain interactions. Qualitatively, the typical length scale of the repulsive electrostatic interactions in strongly charge-asymmetric block polyampholyte chains – electrostatic blob size ξel (2) is the same as the length scale of the attractive interactions in charge-symmetric polyampholytes ξ ≈ ξel .

(36)

The correlation blobs inside the globule are space ﬁlling, therefore the local monomer concentration is cgl b3 ≈

b b3 g ≈ ≈ (uf 2 )1/3 , ξ3 ξ

(37)

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which determines the size of the globule Rgl ≈

N cgl

1/3

≈ bN 1/3 (uf 2 )−1/9 .

(38)

The volume contribution to the free energy of the globule is proportional to the number of its constituent blobs: Fgl N N b2 = − ≈ − 2 ≈ −N (uf 2 )2/3 . kB T g ξ

(39)

This free energy is of the same order of magnitude as the electrostatic energy of a polyelectrolyte chain (4) and the minus sign reﬂects the attractive nature of the intra-chain interactions. The surface energy can be estimated as the thermal energy kB T per correlation blob on the surface of the globule [3]: uni 2 Fsurf Rgl ≈ 2 ≈ N 2/3 (uf 2 )4/9 . kB T ξ

(40)

At the intermediate charge asymmetries, block polyampholyte chains adopt a tadpole conformation, see Fig. 4b. The head is formed by all of the negatively charged monomers and a compensating amount of the positively charged monomers. The remaining ΔN monomers form the tail. The charge of the head is nearly zero, so the total net charge eZnet of the chain is carried by ΔN f charged monomers in the tail. For a high enough valence of the diblock Znet (see the discussion in [31]) the radius of the head, given by (38) is much smaller than the length of the tail (cf. (5)) Rtail ≈ bΔN (uf 2 )1/3 .

(41)

Next, we will consider micellization in dilute solutions of the tadpoles and discuss how the net charge per chain Znet inﬂuences micellar structure. 5.2 Star-Like Micelles The aggregation in solutions of block polyampholytes is driven by the charge density ﬂuctuation-induced attractive interactions between oppositely charged blocks. The unimers start to aggregate above the cmc to reduce the surface tension at the boundary between the collapsed blocks and the solvent. A novel feature of the polyampholyte micelles is the possibility of partitioning of the chains between the core and the corona. Indeed, in the case of micelles formed by diblock copolymers with one hydrophobic and one polyelectrolyte block, the core is formed exclusively by the hydrophobic blocks. In the micelles of diblock polyampholytes, both positively and negatively charged blocks form the core. The corona is formed by the excess of blocks with higher charge (in our model, positively charged), but this excess can be partitioned

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317

in diﬀerent ways. One possible structure is realized when the total net charge of the micelle epZnet is carried by p equivalent tails in the corona, see Fig. 5a. All chains in such micelle are in a similar conformation contributing the same portion of the longer block to the core of the micelle [30]. Such arrangement of polyampholyte chains in the micelle is called the restricted model. The structure of such micelles is similar to the structure of the micelles made of diblock copolymers with a hydrophobic and a polyelectrolyte block, described in the previous sections. The main diﬀerence is that in the polyampholyte micelles both blocks participate in the formation of the core, and the corona consists not of the entire positively charged block but only of an outer part consisting of ΔN monomers. In the unrestricted model [31] one releases the equipartition constraint allowing some chains to be completely conﬁned to the core of the micelle while the others have the entire stronger charged block in the corona, see Fig. 5b. + +

+

+

+

+

+

-

+

+

- -+ +- -++ + + - + +- + - -+ - + + + -+ + + +

+

+

-

+

+

+

+

+

+

-

+ +

+

-+ -

+

- - -+ +

+

+

+

+ +

- + -+ +- + - -+ - + - + + - -+ - +

+ + +

+

+

+

+

+

+

+

+

+

+

a)

b)

+ + +

Fig. 5. Sketch of star-like micelles of diblock polyampholytes: (a) a micelle according to the restricted model; (b) a micelle with disproportionated chains according to the unrestricted model. Reproduced from [31]. Copyright (2005) American Chemical Society.

The cores of the micelles with aggregation number p have the same size in the two models (42) Rcore ≈ bp1/3 N 1/3 (uf 2 )−1/9 , and correspondingly the same surface free energy per chain R2 Fsurf ≈ core ≈ p−1/3 N 2/3 (uf 2 )4/9 . kB T pξ 2

(43)

In order to compare the two models one needs to calculate the electrostatic energy associated with the total charge epZnet of the aggregate [31]. Each disproportionated micelle in the unrestricted model is composed of p chains with p positively charged blocks in the corona and p − p chains

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completely conﬁned to the core, see Fig. 5b. The charge balance in the core: pN− ≈ (p − p )N+

(44)

gives an estimate for the number p of positively charged blocks in the corona of the micelle Znet ΔN ≈p . (45) p ≈ 2p N + ΔN Nf Note that if Znet N f , the number of blocks in the corona for the unrestricted model p is much smaller than in the restricted model (p p). Thus in the restricted model the net charge epZnet is on short sections of all stronger charged blocks. In the unrestricted model the same net charge epZnet is placed on few entire blocks in the corona which makes the electrostatic energy of the disproportionated micelle lower than the electrostatic energy in the restricted model. In the osmotic regime the diﬀerence between the two models is also determined by the size of the corona. In the unrestricted model the size is larger than that in the restricted model. This creates a larger volume for the counterions conﬁned in the disproportionated micelle and therefore the entropy loss is smaller compared to the restricted model. Below we evaluate the energy of the micellar corona according to the unrestricted model in the two regimes of the counterion distribution and calculate the equilibrium aggregation number. Polyelectrolyte Regime (α ≈ 1) The interactions between corona chains in the polyelectrolyte regime are purely electrostatic and the equilibrium size of the corona is determined by the balance of the energy of electrostatic repulsion between chains in the corona, see (10) and their elastic energy due to stretching of these chains to the corona size R. The number of monomers in each of p corona blocks is N+ = N (1 + ΔN/N )/2 ≈ N . The elastic energy of a star-like micelle per chain (cf. (8)): Fstr p R 2 ≈ (46) kB T p N b2 is equilibrated by the electrostatic energy (10) leading to the equilibrium size of the corona 2/3 N RP E ≈ bp1/3 ΔN (uf 2 )1/3 . (47) ΔN Substituting (47) into the expression for the electrostatic energy of the corona (10) we obtain the electrostatic contribution to the total free energy of the micelle per chain in the unrestricted model: FelP E ≈ p2/3 ΔN kB T

ΔN N

2/3 (uf 2 )2/3 .

(48)

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319

The aggregation number in the polyelectrolyte regime is determined by the balance of the electrostatic free energy of the corona (48) and the surface free energy of the core (43): pP E ≈

N 4/3 f 5/3 5/3 Znet (uf 2 )2/9

.

(49)

Osmotic Regime (α 1) The aggregation number in this regime is calculated by minimizing the sum of the surface energy of the core (43) and the osmotic term of the free energy (16) which leads to (uf 2 )4/3 N 2 pOS ≈ . (50) 3 Znet At the crossover boundary between the polyelectrolyte and osmotic regimes the two free energy contributions FP E and FOS are of the same order of magnitude and the micelle aggregation numbers, given by (49) and (50) are equal to each other resulting in the expression for the crossover between the polyelectrolyte and osmotic regimes ∗ ≈ Znet

N 1/2 (uf 2 )7/6 . f 5/4

(51)

The polyelectrolyte regime is realized if the net valence per chain is larger ∗ than this crossover value, i.e. at Znet > Znet , and correspondingly the osmotic regime takes place at smaller Znet . This situation is opposite to the one occurring in solutions of micelles with a hydrophobic core, see Fig. 3. The opposite order of appearance of the regimes is explained by the behavior of the electrostatic potential of the micelle ≈ lB pZnet /R as a function of Znet . The major factor determining the potential is the total charge of the micelle epZnet since the dependence of the micellar radius R on Znet is weak. In both types of micelles the decrease of the valence per chain Znet leads to larger micellar aggregation number p. In polyelectrolyte micelles the aggregation number p grows slower than 1/Znet resulting in decrease of the potential. In the case of polyampholyte micelles the aggregation number grows faster than 1/Znet , and the micellar charge increases with decreasing net charge per chain. Therefore the larger values of the potential and the osmotic regime correspond to smaller net charge per chain. 5.3 Other Types of Micelles Star-like micelles in the polyelectrolyte and osmotic regimes are stable over a wide range of the parameters, such as the fraction of charged monomers on the blocks and the net charge per chain. In addition to these two types of micelles, other micellar structures can be stable in solution. The unrestricted model predicts the existence of unusual micelles. Due to the disproportionation of chains

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in the polyampholyte micelles, small aggregates need not be star-like [31]. If the net charge of the micelle epZnet is smaller than the charge of the longer block eN+ f , the core of the micelles is stabilized by only two blocks partially extended into the corona. However, the regime of such “double-tailed” micelles is very narrow and could be diﬃcult to observe experimentally. The aggregation number of double-tailed micelles increases with decreasing net charge of a chain eZnet and eventually the net charge of the micelle epZnet becomes equal to the charge of the two longer blocks. The corona of the micelles with higher net charge consists of more than two chains corresponding to the star-like polyelectrolyte regime, see Sect. 5.2. If the polyampholyte diblocks are weakly charged (f is small), the decrease of the net charge per chain results in an increase of the core size rather than in the increase of the electrostatic potential because of the larger number of chains that are completely contained within the core. In this case the micellar core is of the order of the micellar size and the micelles are called “crew-cut”. The electrostatic potential of these micelles is smaller than kB T , therefore their counterions remain free in solution and the crew-cut micelles are in the polyelectrolyte regime. Micelles made of diblock polyampholytes with strongly charged blocks are larger than those with weakly charged blocks due to the stronger attraction between oppositely charged blocks. The decrease of Znet leads to an increase of the electrostatic potential and therefore to the counterion conﬁnement within ∗ given by (51) star-like micelles. At a net charge per chain smaller than Znet osmotic micelles are present in solution. In the framework of the unrestricted model, the micellar cores are spherical. For the morphological transitions to other geometries (cylindrical micelles or bilayers) to occur, the core blocks have to be strongly stretched [24, 41]. Since the chains in the micelle are disproportionated between the core and corona, the core blocks are not stretched because most of the chains are completely conﬁned to the core and are therefore not deformed. This implies that the shape transitions do not take place for disproportionated block polyampholyte micelles. This feature places the polyampholyte micelles into a unique class of aggregates. 5.4 Stability of the Micelles Solutions of charge-symmetric diblock polyampholytes are unstable and undergo phase separation [47]. The polyampholyte micelles are made of chargeasymmetric diblocks, so the uncompensated charge ensures their stability. Nevertheless, the micelles can precipitate at non-zero net charge if the chemical potential of a chain in the micelle is higher than the chemical potential of a chain in the sediment. A chain in the sediment interacts with other chains via ﬂuctuation-induced attraction. The bulk contribution to the chemical potential, see (39) is the same for all aggregated structures, so it does not aﬀect the stability boundary.

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The other two contributions are the entropy of a chain and the entropy of ΔN f counterions per chain conﬁned to the sediment. The former term is much smaller than the latter one and can be neglected. Therefore, the stability of the solution of polyampholyte micelles is mainly determined by the entropy of counterions. Polyampholyte micelles (both star-like and crew-cut) in the polyelectrolyte regime are always stable. This stability is due to the entropy of the counterions which is much larger in the dilute supernatant than in the dense sediment. Star-like osmotic micelles are also stable because the counterions conﬁned to a star-like micelle still have a larger volume to explore than the ones in the sediment. The star-like osmotic micelles in the unrestricted model remain stable even if the net charge of a block polyampholyte is as low as one elementary charge. This remarkable stability is due to the partitioning of the chains and can be explained by the fact that even in this limiting case the corona of the equilibrium micelle is much larger than the core [31]. The unrestricted model predicts that charge-asymmetric block polyampholytes will not precipitate, but rather form micelles with disproportionated chains. This prediction is, however, applicable to the case of monodisperse charge distribution, i.e., if all chains carry the same net charge. Charge polydispersity will shift the region of stability of a homogeneous micellar solution. For example, a situation with an average of one elementary charge per chain can be realized by mixing positively charged, negatively charged and uncharged chains. In this case neutral and mutually neutralized fraction of oppositely charged chains would precipitate and only the fraction carrying the total net polymer charge will remain in soluble micelles.

6 Summary and Potential Applications The increasing number of theoretical and experimental studies concerning micellization of charged block copolymers reﬂects growing interest in these systems both academically and in view of industrial applications. In the present chapter we have reviewed the scaling models of micelle formation in dilute solutions of diblock copolymers consisting either of hydrophobic and polyelectrolyte or of oppositely charged blocks. Most commonly the micelles are spherical star-like aggregates with a dense core and a solvated corona. A certain number of diblock copolymers are held together in the micelle by the attractive interactions in the micellar core. The origin of this attraction is diﬀerent in the two systems. In the case of uncharged-polyelectrolyte diblocks the eﬀective attraction arises from the selectively poor solvent quality for the uncharged blocks. Diblock polyampholytes interact via the charge density ﬂuctuation-induced electrostatic attraction between oppositely charged blocks. The driving force for micellization is a reduction of the surface tension between the collapsed blocks in the core and the solvent. Stabilization

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of the micelles which prevents them from unrestricted growth is assured by the electrostatic repulsion between like-charged blocks in the micellar corona. The interactions in the corona are similar in the two systems and resemble the interactions in a polyelectrolyte star. In our review we have demonstrated that diﬀerent types of interactions in the micellar core result in a number of properties diﬀerent in the two types of micellar solutions. The main structural feature of polyampholyte micelles is a disproportionation of chain conformations. The chains in a micelle divide into two populations: some of them are completely conﬁned to the core and others having entire strongly charged blocks in the corona. Such arrangement of the chains corresponds to the largest distance between like-charged monomers and therefore to the lowest electrostatic contribution to the total free energy of the micelle. As a consequence of this disproportionation, the dependence of the micellar aggregation number on the net charge per chain eZnet is stronger in polyampholyte micelles. As Znet decreases the electrostatic repulsion between chains in the corona weakens, leading to an increase of the aggregation number for the two types of micelles. In polyampholyte micelles, however, there is an additional factor increasing the aggregation number with decreasing Znet , namely an increase of the number of chains completely conﬁned to the micellar core. As a result of faster growth of polyampholyte micelles their electrostatic potential increases with decreasing Znet in contrast to a decrease of the potential of the polyelectrolyte micelles. Such behavior of the potential results in the diﬀerent order of the polyelectrolyte and osmotic regimes in the diagram of states: in a solution of polyelectrolyte micelles an increase of net charge per chain leads to the crossover from the polyelectrolyte to the osmotic regime whereas in a solution of polyampholyte micelles the sequence of regimes is reversed. The disproportionation also inﬂuences the stability of spherical micelles. The large number of blocks in the core of the polyampholyte micelle are not grafted to the surface of its core and therefore they are not stretched even in micelles with a very large aggregation number. In contrast to polyampholyte micelles, core blocks in polyelectrolyte micelles become stretched causing shape transitions to cylindrical and planar aggregates. Solutions of micelles made of uncharged-polyelectrolyte block copolymers have been intensively investigated theoretically and experimentally during the past decade. Most of the studies have focused on the micellar structure as a function of molecular characteristics such as the fraction of dissociated groups in the polyelectrolyte block, the degree of polymerization, and the relative lengths of the blocks. Modern synthesis allows us to obtain block copolymers with well deﬁned characteristics, hence the relationships between the chemical composition and the micellar properties can be well established. The theory operating with the molecular characteristics as parameters helps to steer the experimental studies which would lead to important technological applications. These applications will be based on the fact that various materials can

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be placed inside the micellar core. For example, micelles can be utilized for a delivery of a drug to regions disrupted by a tumor [49]. Micelles can also be used as “nanoreactors” to produce metal or semiconductor particles [50]. While polyelectrolyte micelles have been widely studied and a good agreement between theory and experiment has been demonstrated, experimental data on block polyampholyte micelles are still limited to a few studies. Lack of data is explained by the diﬃculties in the synthesis of block polyampholytes with a controlled charge on each block, and by the large number of parameters governing the system behavior. Nevertheless, this ﬁeld is rapidly developing, driven by the growing scientiﬁc and industrial interest in these systems. The interest is based on the unique properties of block polyampholytes which lead to a variety of applications, most importantly biomedical [51,52]. The fact that both blocks of the polyampholyte are water soluble and the core of the micelles contains electrical charges makes polyampholyte micelles a perfect vehicle for drug delivery, because various charged substances such as ions, proteins, and nucleic acids could be selectively concentrated in the core held there by electrostatic interactions. Enzymes inside the aggregates are segregated from the outer phase and therefore remain active even under unfavorable conditions outside. The other principal advantage of polyampholyte block copolymers is that it is possible to regulate the charge and, therefore, the association and stabilization behavior of the micelles. The enhanced stability of the micelles against precipitation in solutions away from the isoelectric point may be advantageously used in industry.

7 Perspectives In the present review we have described the structural properties of spherical micelles made of linear block copolymers. Nevertheless, the possibilities for the design of novel structures are endless. Developments in synthetic chemistry and the increasing number of characterization techniques open perspectives for studying micellization in solutions of comb copolymers, H-shape, and starshape copolymers as well as combinations of these architectures. Block copolymer micelles are very promising systems because of the variety of possible ways to control their structure [25]. The practical way to regulate the magnitude and sign of charge in the polyampholyte solutions is to change the solution pH. Polyampholyte diblock copolymers with blocks that are hydrophobic in the absence of charge form “reversible” micelles [53, 54]. Depending on whether pH range is acidic or basic, one block is uncharged while the other is either positively or negatively charged. Therefore, the charge of the corona can be reversed by changing pH. Even richer solution behavior is observed in triblock polyampholytes with a neutral hydrophilic block [55–58], or with an oppositely charged block between two like-charged blocks [59]. The new molecular structures lead to micelles with a core, a shell and a corona, and to networks with reversible cross-links.

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The notion of disproportionation, introduced for micelles formed by diblock polyampholytes can be applied to other polyampholyte systems. A simpler, yet conceptually similar, object is a polyampholyte star made of a ﬁxed number of diblock polyampholytes [60]. Such stars may be obtained by micellization in aqueous solution of triblock copolymers with two oppositely charged blocks and one neutral hydrophobic block attached to one of the charged blocks. At a temperature lower than the glass transition temperature for the neutral block the core of the micelle is frozen and the aggregation number remains unchanged. In the polyampholyte star, the core-shell structure can be formed within it. Depending on the block charge asymmetry, solutions of polyampholyte stars may exhibit the properties of either stable colloidal solutions, or solutions with a precipitation threshold. The existing model for micellization of diblock polyampholytes [31] explains many of their properties and describes the micellar structure in great detail. However, the simplifying assumption of equal charging of the two blocks does not allow the direct comparison with experiment. Many polyelectrolytes are weak acids or bases, so their degree of dissociation is a function of the solution pH. To improve the model one needs to consider the aggregation behaviour in the solution of block polyampholytes with the blocks having different degree of dissociation. In fact, the problem is more general and can be traced back to the solutions of oppositely charged polyelectrolytes. Despite a number of experimental studies, see [61] and references therein, the stability of soluble complexes of polyanions and polycations solutions at ﬁnite concentrations still remains an open question. To summarize, extensive theoretical eﬀort is desired to elucidate the interactions resulting in stable multimolecular structures in the solutions of polyelectrolytes and polyampholytes. Extensive computer simulations would test the analytical models and help to analyze the properties of real systems. Acknowledgement. N. S. would like to thank Prof. P. Linse for a critical reading of the manuscript. M.R. acknowledges ﬁnancial support of the NASA University Research, Engineering, and Technology Institute on Biologically Inspired Materials award NCC-1-02037, National Science Foundation awards CHE-0616925 and NIRT CBET-060987, as well as National Institutes of Health award 1R01HL077546-01A2.

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The Latest Development of the Weak Segregation Theory of Microphase Separation in Block Copolymers I. Ya. Erukhimovich Chair of Physics of Polymers and Crystals, Department of Physics, M. V. Lomonosov Moscow State University, Russia

1 Introduction One of the most interesting phenomena occurring in copolymer systems is the so-called order-disorder transition (ODT) or microphase separation, i.e. formation of ordered morphologies possessing the symmetry of a crystal lattice [1–5] which occurs with changing (typically decreasing) temperature T . With further decrease of T the ODT is often followed by various order-order transitions between the diﬀerent ordered morphologies1 . Obviously, the physical reason for this ordering is competition between the short-range segregation and long-range stabilization tendencies. More precisely, with decreasing T the energy gain upon local segregation grows as compared to the loss of the translational entropy accompanying such segregation whereas the immiscible blocks can not separate fully because of their covalent bonding. As a result, an ordered pattern of alternating domains, which are ﬁlled preferably by monomers of the same sort, arises. Block copolymer melts with diﬀerent structural and interaction parameters are known to form diﬀerent morphologies at the order-disorder and order-order transitions so that the ultimate goal of the theory is to determine the symmetry and geometry of the most stable ordered phases for a copolymer melt or blend given its composition, architecture and temperature as well as predict the thermodynamic, scattering and mechanical properties of the phases. The milestones in achieving this goal would be i) a general understanding of the relationships between the block copolymer architecture and the symmetry of the occurring ordered morphologies as well as scenarios of the order-disorder and order-order transitions; ii ) a quantitative description and calculation of the stable ordered phases structure; and iii ) calculation of various parameters of the phases. Therewith, having in mind the huge variety 1

In what follows we use the terms “morphology”, “ordered phase” and “symmetry class” as synonyms.

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of block copolymer architectures chemists could synthesize, it is very important that the theory is capable of providing express information (even though semi-quantitative) about new copolymers rather than only explain observed systems. As stated by Reimund Stadler, “the large number of a priori imaginable combinations in ABC triblock copolymers makes it necessary that theory and experiment have to be closely linked. By such a close feedback it may be avoided that the development of new materials is merely based on accident.” The ﬁrst ever attempt to theoretically describe the microphase separation transition was done by Meier [6] within the so-called strong segregation approximation (SSA). According to the SSA, the alternating domains forming the ordered morphologies are just microphases, which consist of the corresponding chemically homogeneous blocks, the width d of the inter-domain transient layer (domain interface) being rather narrow as compared to the domain size L. Therefore, the SSA is based on the idea that the ODT is controlled by a balance between an entropic loss due to conﬁnement of the polymer blocks within (or outside of) the domains (micelles) and energetic gain (as compared to the uniform state) under formation of these micelles. Since within the SSA the unlike monomers contact and interact within the domain interfaces only, the energetic gain is basically determined by the values of the total domain interface and interface (surface) free energy. The SSA, which was further elaborated by Helfand et al. [7–10] and, ﬁnally, by Semenov et al. [11–14] results naturally (in the limit of zero interface width) in the idea that the morphology of the ordered phases is fully determined by the requirement that the total inter-domain surface is minimal. Very transparent and appealing from the geometrical point of view, the minimal surface approach is rather popular among mathematicians and experimentalists [15–18]. However, the condition of the narrow interface d L is practically never fulﬁlled in the real block copolymer melts above the glass transition temperature Tg . E.g., in the binary AB diblock copolymers the condition d L is fulﬁlled for χ ˜ ≥ 100 [19] where χ ˜ = χN is the reduced Flory interaction parameter [20] and N is the total degree of polymerization of the diblock copolymer chain. In the case the Flory parameter χ is related to the temperature T via the simple relationship 2χ = θ/T , θ being the Flory temperature, the condition holds for rather low temperatures T ≤ θN /100 ∼ TODT /10, where TODT is the ODT temperature calculated by Leibler [21] in the opposite so-called weak segregation approximation (WSA). Besides, the minimal surface approach disregards the long-range contribution [11] due to micelles’ ordering. The WSA, on which this review is focused, is related to the situation at the very onset of ordering. The physical idea the WSA is based on was ﬁrst stated by Landau [22] as early as 1937 as a toy model of the 2nd order phase transition accompanied by an explicit symmetry change. In the contemporary designations the original Landau Hamiltonian reads:

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ΔF (r, {Φ (r)}) d r. T (1) Here the Fourier transform c (q) = d r c (r) exp (iqr) has a minimum at a ﬁnite q = q∗ , the order parameter Φ is assumed to be scalar for a while, the speciﬁc excess free energy of the system at the point r F ({Φ (r)}) 1 = T 2

Φ (r1 ) c (r 1 − r 2 ) Φ (r2 ) d r 1 d r 2 +

ΔF (r, {Φ (r)}) α β = Φ3 (r) + Φ4 (r) T 3! 4!

(2)

is determined by the value Φ(r) at the same point only and T is the temperature measured in energy units (the Boltzmann constant kB = 1). Consistent with the general principles of statistical physics, [23] the system described by the Hamiltonian (1) is in a state described by a proﬁle {Φ (r)} with a probability w ({Φ (r)}) ∼ Z −1 exp (−F ({Φ (r)})/T ) ,

(3)

the normalization constant Z being the partition function, which can be written as the corresponding functional integral over all proﬁles {Φ (r)} : Z = exp (−F ({Φ (r)})/T ) δΦ (r) (4) Thus, the total free energy of the system and any observable quantity a, which, evidently, is just a thermodynamic average, i.e. the average taken with the probabilistic measure (3), read F = −T ln Z = −T ln exp (−F ({Φ (r)})/T ) δΦ (r) (5) a ({Φ (r)}) exp (−F ({Φ (r)})/T ) δΦ (r) (6) a = a ({Φ (r)}) = exp (−F ({Φ (r)})/T ) δΦ (r) The most important observables are, of course, the average order parameter ¯ (r) = Φ (r) Φ

(7)

and the correlation function ¯ (r1 ) Φ (r2 ) − Φ ¯ (r2 ) S (r1 − r2 ) = Φ (r1 ) − Φ

(8)

as well as the scattering factor G (q) = S (r 1 − r 2 ) exp (i q (r 1 − r 2 )) dr 1 dr 2 V,

(9)

where V is the total volume of the system. For τ > 0, a minimum (at least, a metastable one) of the virtual free energy (1) is provided by the order parameter proﬁle:

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Fig. 1. Typical behavior of the plane perturbations in weakly ordering systems. The curves correspond to diﬀerent values of the reduced (positive) temperature τ , the values of τ are decreasing and the system is approaching ODT in the downward direction.

Φ (r) ≡ 0.

(10)

¯ (r) = 0, whereas to calculate the correlation function and scatterThen Φ ing factor one can keep only the ﬁrst non-vanishing (quadratic) terms in the expansion of the Hamiltonian (1) appearing in the integrals (5)-(8). In this approximation, which is referred to as the random phase approximation (RPA), the desired expressions read: 2 −1 , G (q) = 1/c (q) ≈ τ + C q 2 − q∗2

(11a)

where the second approximate equality holds near the minimum of the function c(q). If the value of the minimum τ = c (q∗ ), which plays the role of a reduced temperature, measuring how close the spatially uniform (disordered) system is to the loss of its stability, is small enough, then the correlation function reads (in an approximation valid for τ q∗4 ) ⎧ # # √ ⎨ sin (q∗ r) exp −r τ /C (2q∗ ) r τC # # √ S (r) ∼ ⎩ cos (q∗ r) exp −r τ /C (2q∗ ) q∗ τ C

d=3 (11b) d=1

The behavior of the weakly ordering systems described by the correlation function (11b), which is shown in Fig. 1, is rather diﬀerent from that of simple

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liquids2 . Since the correlation function is proportional to the response function, [23, 24] the equation (11b) implies that any perturbation arising in the weakly ordering systems propagates as a harmonic wave (spherical, cylindrical or plane depending on the perturbation’s dimensionality), which oscillates with the period L = 2π/q∗ and decays with distance rc ∼ q∗ C/τ . The waves’ amplitude is inﬁnitely increasing (within the RPA) as τ → 0. At τ ≤ 0 the perturbations become inﬁnite. Were there no other restrictions, it would mean that the spatially homogeneous state becomes absolutely unstable with respect to the growth of the order parameter waves with the wavelengths L (henceforth we refer to such waves as critical waves) and the ¯ (r) = 0 corresponds to a maximum RPA is not valid anymore. In this case Φ of the virtual free energy, whereas the minimum of the latter is provided by ¯ (r) = 0. a ﬁnite thermodynamic equilibrium order parameter proﬁle Φ In fact, however, the local order parameter cannot exceed some ﬁnite value (e. g., for diblock copolymer melts Φ is a linear function of the volume fraction of the monomers A), which implies that with increase of ﬂuctuations they become strongly correlated even in the disordered state, the correlation resulting in a decrease of the correlation function S(r) and, thus, a stabilization of the disordered (spatially homogeneous) state as compared to the RPA results. Quantitatively this eﬀect is addressed and explained by so-called Brazovskii (Hartree) ﬂuctuation corrections [25] to the mean ﬁeld approximation, which are shown to transform the order-disorder transition into the 1st order one even at the critical point. In the mean ﬁeld approximation, i.e. under assumption that the ﬂuctua¯ (r) = 0 can be tions around the new equilibrium order parameter proﬁle Φ neglected, which implies calculation of the integral (4) via the deepest descent method within the so-called pre-exponential accuracy, we arrive at the well-known expression for the free energy $ % ¯ (r) F = −T ln Z = F Φ (12) It follows from the presented discussion that at the onset of ordering the ¯ (r) is expected to be built basically by all of the critical equilibrium proﬁle Φ waves. However, since the critical waves are coupled via the non-linear excess free energy (2), the critical waves both interfere and generate some new waves (so-called higher harmonics). As a result, only some discrete sets of these waves survive, each stable set of the standing order parameter waves corresponding to a crystal lattice. ¯ (r), If the symmetry of a morphology is that of a spatial lattice , Φ generally, is an inﬁnite series in Fourier harmonics corresponding to the set of the points of the lattice −1 reciprocal to : [23] 2

Remember [23] that the correlation function of simple liquids is described by the Ornstein–Zernike correlation function S (r) ∼ exp (−r/rc )/r, where rc is the correlation radius.

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¯ (r) = Φ

A (qi ) exp i (qi r + ϕi )

(13)

qi ∈ −1

But close to the critical point τ = 0, α = 0, where α is the coeﬃcient appearing in the cubic term of the Landau expansion (2), the coupling generating the higher harmonics is small, so that one can keep in the expansion (13) the main (or primary) harmonics only, i.e. those whose wave numbers are equal to the critical value q∗ : ¯ (r) = AΨ (r) , Ψ (r) = Φ exp i (qi r + ϕ (qi )) (14) |qi |=q∗

Then the spatially periodic order parameter can arise (in the mean ﬁeld approximation) as the 2nd order phase transition or the 1st order phase transition close to the 2nd order one, which is referred to as weak crystallization. [22,26,27] It was shown [21,28,29] based on the universal (Gaussian) conformational behavior of the long polymer blocks that the Landau instability does occur in block copolymer systems. E.g., for the binary incompressible AB copolymers the function c(q), which appears in (1) and according to (11a) is just the inverse scattering intensity, was shown [21,28-30] to read c (q) =

gAA (q) + gBB (q) + 2gAB (q) 2

gAA (q) gBB (q) − (gAB (q))

− 2χ,

(15)

where g (q) = gij (q) is the so-called structure matrix (see Sect. 3), which is determined by the macromolecules’ architecture only. The plots c(q) for molten diblock copolymers which were ﬁrst calculated by Leibler [21] are presented in Fig. 2 for various χ (temperatures). It is easy to see that the main prediction of the theory in the disordered state is that the only change of the shape of the curve c (q) ∼ I −1 (q) is its overall downward shift. As is seen from Fig. 3, where the experimental data [31] for I −1 (q) are presented, this prediction holds very well for T > 180◦ C, i.e. whilst systems are in the disordered state. It is worth noticing that the way the experimental data are presented in Fig. 3 is the most natural from a theoretical point of view (it provides a direct opportunity to identify the region of the disordered state as the whole) but extremely rare in the experimental literature. Moreover, it was shown by Leibler [21] via a strict microscopic consideration that the Landau toy model perfectly describes the phase behavior of slightly asymmetric molten diblock copolymers An Bm . His seminal theory of microphase separation in diblock copolymers [21] became a real paradigm for both building the phase diagrams of the ordered phases’ stability within the WSA, given the phenomenological coeﬃcients in the Landau expansion of the free energy of the weak segregating systems in powers of an order parameter, and microscopic calculation of the coeﬃcients for block copolymers

Theory of Microphase Separation in Block Copolymers

Fig. 2. The curves c(q) labeled by numbers n correspond to the values of χ ˜ = 2 (n − 1). Based on [21].

333

Fig. 3. Inverse intensity for diﬀerent temperatures. The curves correspond to T of 240,220,200,180,160 and 100◦ C (from top to bottom). (Fig. 5a of [31]).

Fig. 4. The conventional phase diagrams of the molten diblock copolymers An Bm (left) [21] and star block copolymers (An )4 (Bm )4 (right) [35, 36]. The phase transition lines DIS-BCC, BCC-HEX and HEX-L are the lower, middle and upper curves, respectively.

with a given architecture. In particular, Leibler [21] found that the thermodynamically stable ordered morphologies for diblock copolymers are the bodycentered cubic lattice (BCC) and the structures possessing hexagonal (HEX) and lamellar (L) symmetries, the sequence of the transitions being the disordered phase (DIS)–BCC–HEX–L (see Fig. 4). Further we refer to the phases BCC, HEX and L as the conventional or classic ones. Phase diagrams with

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the same topology were found for a variety of molten AB block copolymers with diﬀerent architectures [32–36] (see Fig. 4). For some special sets of parameters, all the phase transitions lines (DISBCC, BCC-HEX and HEX-L) merge at the critical point where the 2nd order phase transition from the disordered to lamellar phase occurs. For An Bm diblock copolymers the critical point corresponds to the symmetric diblock copolymer in case the repeated units of both blocks have the same excluded volumes v and Kuhn lengths a. Later Fredrickson and Helfand [37] incorporated the Brazovskii ﬂuctuation corrections into the Leibler theory and showed that they considerably shift the ODT towards higher temperatures as compared to the mean ﬁeld WST. It is worth noticing that according to the WSA the ODT in block copolymers shares a common physical background (the Landau weak crystallization) with various physical phenomena like the blue phase appearance in liquid crystals [38, 39] charge-density waves generation upon addition of an ionic solute to a solvent in its critical region [40, 41] and microphase separation in weakly charged polyelectrolyte solutions, [42, 43] the polymer speciﬁc features of this approach appearing at the stage of microscopic calculation of the Landau expansion coeﬃcients only. The WSA also provides description of a rather special type of ordering predicted within the WSA also for random copolymers [44–51]. Thus, the weak segregation theory of microphase separation in block copolymer systems provides a unique opportunity to test the general phenomenological concepts of the statistical theory of solid-liquid transitions via a rigorous microscopic consideration. Unfortunately, the region of the WSA applicability corresponds to a rather narrow vicinity about the critical point. Besides, the WST employs the socalled ﬁrst harmonics approximation which we discuss in more detail below and it is often believed [52] that within this approximation “the predictions about ordered structures are limited to classical phases of lamellar, hexagonal, and body-centered cubic structures, and consequently the possibility of other structures such as bicontinuous structures, e.g., double gyroid, is excluded.” The double gyroid (G) phase mentioned here is an important phase characterized by Ia¯ 3d space group symmetry, which was ﬁrst discovered in lipid-water and surfactant systems [53, 54] and has attracted much interest during the last decade due to the bi-continuous morphology characteristic of this phase. So, during the last decade the so-called self consistent ﬁeld theory (SCFT) by Matsen, [55–57] which is free of these shortcomings, became dominant in understanding the behavior of the ordering block copolymer systems. The SCFT, which is considerably more polymer-speciﬁc than the WST, is based on the Edwards [58]–Helfand [7–10] idea that inhomogeneities in the density proﬁles of chemically diﬀerent polymer repeating units are caused by some eﬀective (self-consistent) ﬁelds, which, in turn, are themselves determined by the arising density proﬁles. The new powerful trick elaborated by Matsen and Schick [55, 56] was to seek for the desired density proﬁles and the self-consistent ﬁelds some series in the eigen functions of the correspond-

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ing diﬀusion equation with due regard for their space symmetry. In contrast to the WST, which involves only a few (1 to 12, depending on the lattice symmetry) primary harmonics, the SCFT series involves many hundreds of eigen functions, and, thus, provide a much broader region of the SCFT applicability. It enabled Matsen and Schick [55, 56] to succeed in building the SCFT phase diagrams of molten AB diblock and star copolymers revealing the stable G phase in reasonable agreement with experiment. [59, 60] A speciﬁc feature of these phase diagrams in the plane (f, χ) ˜ is existence of two triple points fA = f1t < fcrit , T = T1t and fA = f2t > fcrit , T = T2t , where three phases HEX, G and L coexist. (As usual, fA = n/N is the composition of the A monomers, n, m and N = n + m are the total numbers of the A and B units per block copolymer chain and the total degree of polymerization of the chain.) Therewith, the conventional sequence DIS-BCC-HEX-L and non-conventional one DIS-BCC-HEX-G-L hold for compositions within and outside of the interval(f1t , f2t ), respectively. The SCFT has been successfully applied to describe the stable ordered phases in various block copolymer systems (see, e.g., [61–63] and the references in other chapters of the book) and it became dominant in understanding the behavior of the ordering block copolymer systems. Accordingly, the WST has been considered for some time as a sort of old-fashioned and outdated technique. In fact, however, the areas of expertise of the SCFT and WST are rather complementary rather than overlapping. In particular, it is worth noticing that the experimental [60] and SCFT [55] phase diagrams are in qualitative agreement only and there is a notable upward shift of the experimental phase transition lines in the plane (f, χ) ˜ as compared to the SCFT ones. The shift is due to the Brazovskii–Fredrickson–Helfand (BFH) ﬂuctuation corrections [25–27, 37] neglected within the SCFT, which are far from being minor. However, these corrections are easily incorporated into the WST [37, 64–71] and it is within the WST that this upward shift of the phase transition lines was explained and quantitatively described3 . The application of the WST requires the calculation of cumbersome expressions for the so-called higher structural correlators; still, the SCFT is no less technically involved, whereas the corresponding numerical calculations are much more time consuming than those needed for the WST (the same is valid for other numerical methods discussed in this book). Besides, unlike the SCFT, a considerable part of the calculations necessary to build the phase diagrams within the WST can be done analytically. All these advantages enabled the WST analysis [73–76] of the ternary ABC block copolymers, which resulted in understanding of many peculiar properties of these systems. In particular, it is easy to incorporate into the framework of the WST the eﬀects 3

Besides, the WST with due regard for the so-called ﬂuctuation caused q∗ renormalization [67] provided an explanation of the shift of the scattering peak location towards lower values of q with temperature decrease, which is noticeable in Fig. 3 and was ﬁrst reported in [72].

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of non-Flory interactions, which resulted in development of WST theories of the ODT in reversibly associating [52, 77–80] and compressible [81–83] block copolymers as well as the WST analysis of the ODT phase diagrams with the lower critical ordering temperature [84, 85].

Fig. 5. The phase diagram of the molten diblock copolymers [92] including the Fddd phase. Full lines are the phase boundaries calculated from the WST. Points denoted by + are the lamellar-Fddd boundary points and ♦ are the Fddd -hexagonal boundary points calculated using the numerical SCFT method. Reprinted with permission from [92]. Copyright (2006) by the American Physical Society.

Finally, it was demonstrated via a general WS analysis [26,76,86] that under certain conditions the most stable phases around the critical point are not necessarily the classic ones. Instead, some other cubic phases such as gyroid G, simple cubic (SC), face-centered cubic (FCC), the so-called BCC2 , also called single or alternated gyroid (we refer to all the phases but BCC, HEX, and L as the nonconventional ones) are stable, whereas the classic phases are metastable only. Moreover, the phase diagrams of the ternary ABC block copolymers (both linear and miktoarm) built via a generalization [76] of the Leibler WST were found to reveal all the aforementioned non-conventional stable phases. The list of the non-conventional phases shown to be stable within the WST (we describe the phases in more detail in the Appendix) is to be appended by a new phase belonging to the symmetry class Fddd (O70 ) and called orthorhombic. The phase, which was found by Bates et al. [87–90] in the ternary linear ABC block copolymers, is strongly degenerate to allow weak segregation as discussed by Morse et al. [91, 92] Remarkably, the phase was shown to be stable within the WST even in the vicinity of the critical point [92] (see Fig. 5). One more recent achievement of the WST is the discovery of the socalled “structure-in-structure” morphologies or two-scale microphase sepa-

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Fig. 6. Schematic illustration of the self-organized structures of PS-block P4VP(MSA)1.0 (PDP)1.0 . The local structures are indicated; macroscopically, the samples are isotropic. (A) Alternating PS layers and layers consisting of alternating one-dimensional slabs of P4VP(MSA)1.0 and PDP for T < TODT ( [14]). (B) Alternating two-dimensional PS and disordered P4VP(MSA)1.0 (PDP)1.0 lamellae for TODT < T < TOOT . (C) One-dimensional disordered P4VP(MSA)1.0 (PDP)x (with x 1) cylinders within the three-dimensional PS-PDP medium for T > TOOT . Reprinted with permission from [93]. Copyright (1998) by the AAAS.

ration. It was found experimentally by ten Brinke, Ikkala et al. [93] who studied self-assembling supramolecular structures in poly(4-vinylpyridine)-block polystyrene (P4VPb-PS) diblock copolymer with side chains (e.g., pentadecylphenol, PDP) attached by hydrogen bonds to the P4VP block. The idea of the resulting morphologies is given by the schematic and TEM micrograph presented in Fig. 6 and Fig. 7. Another (and quite diﬀerent) example of 2-scale morphologies was found by Goldacker, [94] Abetz and Stadler in blends of polystyrene-b–polybutadiene-b–poly(tert-butyl methacrylate S33 B34 T33 (total

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M = 1.6×105 ) and polystyrene-b –poly(tert-butyl methacrylate S47 T53 (total M = 1.03 × 105 ), which contains 10 wt% triblock copolymer (see Fig. 8).

Fig. 7. Transmission electron micrograph of PSblock -P4VP(NPD)1.0 , where nominally one nonadecylphenol (NPD) has been hydrogen bonded with each pyridine group. The long period of the alternating PS (light grey) and P4VP(NPD)1.0 (dark grey) lamellae equals LD ∼ 550 ˚ A. The number-averaged molecular masses of the P4VP and PS blocks were 49500 and 238,000 daltons, respectively. The P4VP-(NPD)1.0 lamellae are further ordered into alternating lamellae of nonpolar nonadecyl tails of NPD molecules and polar P4VP backbones. The long period of this structure is LC ∼ 40 ˚ A. The two sets of lamellar structures are, as expected, mutually perpendicular. Reprinted with permission from [93]. Copyright (1998) by the AAAS.

Fig. 8. Transmission electron micrograph of the mixture of copolymers S33 B34 T33 and S47 T53 ; dark-grey and light-grey bands correspond to ST lamellas, and black bands refer to B layers. Reprinted from [97] with permission.

Despite an apparent visual diﬀerence of the structures shown in Fig. 6 and Fig. 7, it was shown [95–97] that within the WST both systems possess a common feature. Namely, for these systems the function c(q) appearing in (1), (1.11a) and (15) is rather sensitive to details of the block copolymer ∗ ∗ and q = qmax , the architecture and could have two minima4 at q = qmin 4

Note that Landau himself has strongly emphasized [22] that “it is absolutely improbable” that the function c(q) vanishes simultaneously at more than one minimum and, thus, the disordered state becomes unstable simultaneously for order parameter waves with diﬀerent wave lengths. More precisely, the two-scale instability corresponds to a strongly degenerate situation. But the wealth of the

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∗ ∗ lengths Lmax = 2π/qmin and Lmin = 2π/qmax corresponding to the larger and smaller characteristic scales of the morphologies (see Fig. 9).

Fig. 9. The q -dependence of the inverse scattering intensity for the poly (A)m block-poly(A-graft-B )n polymer shown in Figs. 6,7 and analyzed within the RPA in [95]. Here the curves labeled I, II, III, IV and V correspond to (n, m) = (21, 3), (20, 5.5), (20, 5.69), (20, 6) and (1, 44), respectively (n is the number of the h h side chains per the comb-like block, m = NA /d, d and NA being the degrees of polymerization of the side chains and homopolymer A block, respectively. Reprinted from [95] with permission.

Similar behavior was also found for some block copolymer solutions [98,99] which stimulates us to seek some properly designed block copolymer solutions capable of serving as photonic crystals. The WST phase diagrams of the two-length-scale morphologies were built [100–103] and supported by the SCFT calculations [104–107] of the phase diagrams in a broader temperature interval. Such a two-scale behavior is closely related to formation of nonconventional morphologies [76, 100–103] and is expected to be, along with using multi-component block copolymers, a new rather eﬃcient route towards tuning the phase behavior of self-assembling block copolymer nanostructures. The purpose of the rest of this chapter is to provide a better understanding of the latest advancements in the WST. Since the basic features of the WST in binary AB block copolymers are well described in the original papers as well as reviews, [1–3] in what follows we skip the derivations as well as discussion of the BFH ﬂuctuation corrections and focus only on the latest results, which are obtained in the mean ﬁeld approximation. The subsequent parameters controlling the block copolymer ﬂuctuation behavior makes it possible to realize such a degenerate and “improbable” situation in some properly designed systems.

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presentation is organized as follows. In Sect. 2 we consider a simple weak segregation model enabling us to understand the physical and mathematical bases, which cause the non-conventional phases’ stability. Here we introduce also the 2nd shell harmonics approximation, which gives an example of important distinction between the strongly and weakly ﬂuctuating ﬁelds. In Sect. 3 we outline a generalization of the Leibler WST to the multi-component block copolymer systems, which emerges due to the broader application of the distinction between the strongly and weakly ﬂuctuating ﬁelds. Some applications of the generalized WST to the ternary ABC block copolymers are described in Sect. 4. In the Conclusion we summarize the current state and most urgent problems of the WST. Finally, in the Appendix the most typical conventional and non-conventional weakly segregated ordered morphologies are described in detail.

2 The WST and the Non-Conventional Phases’ Stability Non-Locality (Fourth Vertex Angle Dependence) Eﬀects. The stability of the non-conventional morphologies in weakly segregated systems is determined by the degree and character of non-locality of their free energy as a function of a speciﬁed proﬁle of the corresponding order parameter (we refer to such function as the virtual free energy). Indeed, let us start with a generalization of the original Landau Hamiltonian (1), which is to take into account that for polymer systems the speciﬁc excess free energy (2) is not local. Namely, it takes the form n 1 & ΔF (r, {Φ (r)}) = Φ (ri ) d ri (16) Γn (r1 − r, ..rn − r) n! n=3,4 i=1 where some continuous functions Γ3 (R1 , R2 , R3 ) , Γ4 (R1 , R2 , R3 , R4 ) describe how much the speciﬁc excess free energy at the point r depends on the whole proﬁle of {Φ (r)} in the vicinity of the point r rather than on the local value of Φ (r) at the point r only. Obviously, the expressions (2) and (16) coincide in the limit Γ3 (R1 , R2 , R3 ) = α δ (R1 ) δ (R2 ) δ (R3 ) , Γ4 (R1 , R2 , R3 , R4 ) = β δ (R1 ) δ (R2 ) δ (R3 ) δ (R4 ) . The meaning of the non-locality becomes clearer when rewriting the free energy (1), (16) in the Fourier-representation:

2

c (q) |Φq | dq + ΔF3 + ΔF4 , 3 2 (2π) n n & Φ (qi ) dqi 1 qi Γn (q1 , . . . , qn ) δ ΔFn = . 3 n! (2π) i=1 i=1 F =

(17) (18)

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Hereafter, we refer to functions and their Fourier transforms as the same functions in r- and q-representations, respectively, and distinguish them only by the choice of the letters used to denote their arguments; this convention is not expected to cause any misunderstandings due to the context. It is worth noticing that the functions Γn appearing in the cubic and quadric free energy contributions (16) depend on the structure of the system. Substituting the characteristic of the WST expression (14) for the equilibrium order parameter into the mean ﬁeld free energy (12) with due regard for expressions (1), (16)-(18) and minimizing the ﬁnal expression with respect to the amplitude A we get [21] f (τ ) =

ΔF = VT

3 |α | +

2 − 32τ β 9α

3

|α | −

2 − 32τ β 9α

3 212 β

(19)

Here we introduce the cubic and quadric vertices 3 k −3/2 Γ3 (q1 , q2 , q3 ) exp i ϕi = γ (1) C , α = i=1 3! q1 + q2 + q3 = 0, |q1 | = |q2 | = |q3 | = q∗ C =

3 2

k β = 4!

(20) cos

(3) Ωj k 3/2

4 Γ4 (q1 , q2 , q3 , q4 ) exp i

ϕi

i=1

q1 + q2 + q3 + q4 = 0, |q1 | = |q2 | = |q3 | = |q4 | = q∗ ' ' (4) λ0 (0) k λ0 (hi ) + 4 λ (h1 , h2 ) cos Ωj = + . 4k k2 (3)

(21)

(4)

In (20), (21) the phases Ωj , Ωj are the algebraic sums of the phase shifts ϕ appearing in (14) for triplets and non-coplanar quartets of the vectors {qi } involved in the deﬁnitions of α and β, respectively. In addition, we use the designations and parameters of Leibler [21]: 2

γ(h) = Γ3 (q1 , q2 , q3 ), q12 = q22 = q∗2 , q32 = (q1 + q2 ) = hq∗2 , (22) λ(h1 , h2 ) = Γ4 (q1 , q2 , q3 , q4 ), 2

h1 = (q1 + q2 )

/q∗2 ,

|qi | = q∗ , i = 1, ..4 2

h2 = (q1 + q3 )

(23)

/q∗2 ,

2

h3 = 4 − h1 − h2 = (q1 + q4 ) /q∗2 . λ0 (h) = λ (0, h) = Γ4 (q, −q, p, −p), '

(24) 2

h = (q + p)

/q∗2 .

(25)

The symbol n appearing in (20), (21) implies summation over all sets of n vectors for given morphology . The ﬁrst summation in (21) is over all

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pairs of non-collinear vectors qi and qj , 2k is the number of the vectors in the reciprocal space belonging to the coordination sphere with the radius q∗ . The phase transition lines τ 1 / 2 (γ (1)) between the morphologies 1 and 2 are determined by the equation f 1 (τ, (γ (1))) = f 2 (τ, (γ (1))) .

(26)

In particular, if the cubic vertex vanishes due to the symmetry for both morphologies 1 and 2 the phase transition line between the morphologies is determined by the equation β 1 = β 2

(26a)

The topology of the obtained phase diagrams is inﬂuenced by the explicit form of the function λ (h1 , h2 ), which appears in the expression (21) for the fourth vertex β. . Following Leibler [21] we refer to the h -dependence of the vertex λ as the angle dependence since the values of the parameters hi depend on the angles between the vectors qi . For diblock copolymers the angle dependence is rather weak [21], which enabled Fredrickson and Helfand [37] to propose the following commonly accepted approximation: λ (h1 , h2 ) ≈ λ (0, 0) = λ0 (0) .

(27)

However, the approximation (27) was shown [76] not to stay true for any polymer systems. To estimate the eﬀects of this angle dependence on the phase diagram of the weakly crystallizing systems we approximate the function λ (h1 , h2 ) as follows [76, 86]: 3 3 3δ 2 2 4 − f (hi ) = λ0 1 − hi λ (h1 , h2 , h3 ) = . (28) 32 i=1 i=1 The approximation (28) keeps only the ﬁrst non-constant term in the expansion of λ (h1 , h2 ) in powers of hi , the positive (negative) sign of δ corresponds to a disadvantage (advantage) of the lamellar structure as compared to all other ones. The resulting phase diagrams in the plane (τ, γ (1)) are sets of parabolas (29) τ 1 / 2 = 9 τ˜ 1 / 2 γ 2 (1) (32λ0 ), where the reduced temperatures τ˜ 1 / 2 depend onlyon the angle dependence strength δ. The reduced phase diagram in the plane τ˜ 1 / 2 , δ demonstrated [76,86] that increase of the strength of the model angle dependence (28) results in increase of stability of the G phase (as compared to HEX and L) and that of the various non-conventional cubic phases as compared to L. Therewith, all competing phases are to be taken into account. As shown in Fig. 10, the

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a)

343

b)

Fig. 10. The reduced phase diagram for the model angle dependence (28) in the plane (the reduced temperature τeﬀ - the angle dependence strength δ). a) the phase diagram [86] calculated without taking into account that the orthorhombic phase Fddd could exist; b) the phase diagram calculated with due regard for the phase Fddd.

stability of the newly discovered orthorhombic phase Fddd (see [87–92] and Appendix) reveals a non-monotonous behavior with increase of δ, the region of the Fddd phase stability turns out to be as big as that of the G phase. It was claimed, [86] based on the reduced phase diagram shown in Fig. 10a, that the conventional phase transition sequence DIS-BCC-HEX-L occurs only for δ < δ0 = 0.362. Now, after the Fddd phase was discovered, one can assert that this sequence never holds. Instead, the following non-conventional sequences occur: i) the sequence5 DIS-BCC-HEX-Fddd -L for δ < δ12 , δ12 = 94 ; ii) the sequence DIS-BCC-HEX-Fddd -SG for δ1 > δ > δ12 , δ1 = 0.61005; iii) the sequence DIS-BCC-HEX-G-Fddd -SG for δ2 > δ > δ1 , δ2 = 0.61684; ii) the sequence DIS-BCC-HEX-G-SG for δ23 > δ > δ2 , δ23 = 23 ; iii) the sequence DIS-BCC-HEX-G-FCC for δ3 > δ > δ23 , δ3 = 0.822 ; iv) the sequence DIS-BCC-G-FCC for δ34 > δ > δ3 , δ34 = 56 ; v) the sequence DIS-BCC-G-SC for δ45 > δ > δ34 , δ45 = 0.891 ; vi) the sequence DIS-BCC-SC for δ2 > δ > δ45 , δ2 = 43 ; vii) the sequence DIS-BCC for δlim > δ > δ2 , δlim = 1.538 ; Finally, βBCC vanishes at δ = δlim , which is characteristic of the socalled tricritical point where the 2nd order phase transition line terminates and the 1st order phase transition line starts. For δ > δlim the original weak segregation approximation does not hold anymore and the next (5th, 6th etc.) terms of the Landau expansion have to be taken into account. The approximation (28) could seem to be too academic but it provides a good idea of the phase behavior of some real systems. In particular, the reduced phase diagram of the molten ternary linear ABC block copolymers 5

For δ =0 and for the angle dependence characterizing the Leibler molten diblock copolymers this sequence was found ﬁrst by Morse et al. [91, 92].

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Fig. 11. The reduced phase diagrams [76] along the critical line fA = fC = fB /2 calculated with due regard for the actual angle dependence of the ternary ABC block copolymers in the plane (the non-selective block composition fB , the reduced temperature τeﬀ ) for linear (a) and miktoarm (b) ABC block copolymers. The regions of stability of the disordered state and body-centered cubic, hexagonal, lamellar, double gyroid, single gyroid, face-centered cubic, simple cubic and BCC3 lattices are labeled by the numbers 0, 1, 2, 3, 4, 5, 6, 7 and 8, respectively. Reprinted from [76] with permission.

with the non-selective middle block (see Fig. 11a) calculated by the author [76] before the Fddd phase was discovered is rather similar to that presented in Fig. 9a, the composition of the middle block playing the role of the angle dependence strength parameter δ. Thus, the region of the Fddd phase stability for the ternary linear ABC block copolymers with the non-selective middle block is expected to neighbor with that for the G and SG phases in qualitative agreement with experiment [84, 85]. Remarkably, the phase behavior of the molten ternary star (miktoarm) ABC block copolymers with one of the arms non-selective with respect to two others diﬀers strongly both from that of molten diblock copolymers and the linear ABC block copolymers middle block (see Fig. 11b). The physical origin of the non-conventional cubic phases’ stability for ternary linear ABC block copolymers with a long non-selective middle block is obvious: short strongly incompatible side blocks prefer to aggregate into small micelles rather than into thin layers. This tendency, which is “coded” into the angle dependence of the eﬀective non-local vertices Γ4 (q1 , q2 , q3 , q4 ) becomes apparent already at the very onset of ordering when segregation is still weak. The 2nd Shell Harmonics Approximation. Thus, the ﬁrst harmonics approximation (14) along with due regard for the angle dependence of the 4th vertex provides a reasonable explanation of the stability of the non-conventional phases. However, during the last decade some authors [71, 108–110] queried the reliability of the approximation (14). In particular, Hamley and Podnek [71] suggested that the existence of gyroid morphology is

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due to the anomalously large (by modulus) but negative contribution of the second harmonics with h = q2 q∗2 = 4/3 characteristic for the G morphology to the total free energy. Of course, this suggestion by itself is not suﬃcient to explain the G phase stability in diblock copolymers since the second harmonics with h = 4/3 are also characteristic of the BCC3 and G2 morphologies, which all belong to the G family. Moreover, it has no relation to the problem of the G stability at the very critical point, which depends only on the strength of the angle dependence of the fourth vertex as shown above. However, if the angle dependence is not strong enough to provide the G stability at the critical point, we are to deal with two closely related problems: i) which phases are stable at the triple points (if any) existing near the critical point; and ii ) which factors determine location of the triple points. In general, the contribution of many harmonics (rather than that of the only second ones) determines the location of the triple point in question, which is shown via direct calculation by Matsen and Schick [55, 56]. But it is natural to expect that only a certain ﬁnite number of the higher harmonics are really relevant if the triple points are close enough to the critical one. To single out such selected higher harmonics, instead of the ﬁrst harmonics approximation (14) we chose the trial function as follows [47, 76]: ¯ (r) = Ψ (r) + Φ ψh (r) , (30) h =1

where

Ψ (r) = A

exp i (qi r + ϕi )

(31)

qi ∈ −1 , q2i =q∗2

and

ψh (r) =

aqi exp i ((qi r) + ϕ(qi ))

(32)

qi ∈ −1 , q2i =hq∗2

are the sums of the main and higher harmonics, respectively. After substituting the trial function (30)-(32) into (17), (18) the virtual free energy takes the form F = Fmain + Fhigh + Fcoupling , (33) where Fmain = F ({Ψ (r)}) is the contributions of the dominant harmonics (31), ΔFcoupling is that due to coupling between the higher dominant and harmonics, generated by the cubic term of the original Hamiltonian: (3)

ΔFcoupling VT

=

A20 2

qi

∈ −1 ,

γ (q1 , q2 , −q1 − q2 ) aq1 +q2 exp (i (φ1 + φ2 ))

q2i =q∗2

(34) and the contribution of the higher harmonics is determined as follows:

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Fhigh =

VT 2

qi

∈ −1 ,

2

q∗4 (h − 1) + τ

2

|aqi | .

(35)

q2i =hq∗2

All other terms we skipped in (33) are irrelevant. Indeed, minimization of the free energy (33) with respect to the complex amplitudes aq , q2 q∗2 = h = 1, gives ' A20 γ (h) q exp (i (φi + φj )) aq = − (36) 2 2 q∗4 (h − 1) + τ where Σq means summation over all pairs of main harmonics given the condition (37) q i + qj + q = 0 According to (36), (37) the second harmonics induced by coupling (34) belong to all the coordination spheres of the corresponding reciprocal lattice, the radius of which does not exceed double the radius of the dominant coordination sphere, their amplitudes being of the 2nd order of magnitude with respect to the main harmonics amplitudes A0 . The number of coordination spheres of diﬀerent radii satisfying (37) depends on the lattice symmetry and varies from 1 (for the LAM) to 8 (for the G) and 10 (for the BCC3 ). We refer to all of the corresponding harmonics as the 2nd shell harmonics. For lattices ' with non-zero phase shifts like SG, G and G2 , the factor q exp (i (φi + φj )) appearing in (36) and, thus, the corresponding higher harmonics vanish identically, which gives a natural derivation of the extinction rules [111] within the WS theory. Substituting (36) into (33)-(35) results in the ﬁnal expression for the free energy including the higher harmonics contribution up to the order of O A40 : (38) ΔF = V T τ A20 + α A30 + β¯ A40 , where β¯ is related via eqs (21), (23)-(25) to the fourth vertex Γ¯4 renormalized with due regard for the 2nd shell higher harmonics: Γ¯4 (q1 , . . . , q4 ) = Γ4 (q1 , . . . , q4 ) − (B (q1 ,.q2 ; q3 , q4 ) +B (q1 ,.q3 ; q2 , q4 ) + B (q1 ,.q4 ; q2 , q3 )) γ 2 (h) p2 B (q1 ,q2 ; q3 , q4 ) = , p = −q1 − q2 = q3 + q4 , h = 2 (39) 2 q∗ q∗4 (h − 1) τ is omitted in the deﬁnition (39) since |τ | q∗4 is a condition of the WSA validity (see Appendix 2). It is worth remembering now that the reduced phase diagrams built in Fig. 10 and Fig. 11 describe the stability of the weakly segregated phases at the critical point only, i.e. in the limit γ → 0. It follows from (39), (39) that the phase transition lines are aﬀected by the actual dependence γ(h) within a ﬁnite vicinity of the critical point where the cubic vertex is small but ﬁnite.

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Fig. 12. The phase diagram of the diblock copolymer in the 2nd shell harmonics approximation (without taking into account the Fddd phase existence). The designations of the phases are the same as in Fig. 11. a) comparison with the Leibler [21] phase diagram (shown by the dashed lines); b) extrapolation to the region of comparatively high values of χ, ˜ which demonstrates possibility of the G2 lattice (labeled by the number 9) stability. Reprinted from [76] with permission.

The Advantages and Limits of the 2nd Shell Harmonics Approximation. The phase diagram of the molten diblock copolymer calculated in [71] with taking into account the 2nd shell harmonics contribution to the 4th vertex of the virtual free energy as described above is presented in Fig. 12. One can estimate both the advantages and deﬁciencies of our approximation comparing it with those of Leibler [21] and Matsen and Schick [55]. As is seen in Fig. 12a, as far as the conventional phases are concerned our phase diagram almost coincides with that of Leibler [21] precisely approaching the latter in the vicinity of the critical point. The only diﬀerence would be some broadening of the BCC phase stability region (basically at the cost of the HEX phase) with increase of the diblock copolymer asymmetry. The situation changes drastically as soon as we include in the list of competing phases those of the G family which were not taken into account in the original paper [21]. All three phases of the family described in Sect. 2 become stable when the asymmetry |f − fc | increases and the 2nd shell harmonics effect is taken into account. It is worth noticing that when we calculate the free energy of the ordered phases within the conventional 1st harmonics approximation the phases of the G family only turn out to be metastable. Therewith, in our approximation the triple point LAM-HEX-G is located at f = 0.462, χ ˜ = 10.88, which is rather close to the result f = 0.452, χ ˜ = 11.14 obtained by Matsen and Schick [55] within the SCFT using many more harmonics. Comparing the presented numerical results for the triple point we conclude that our 2nd shell harmonics approximation somewhat overestimates the eﬀect of

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the higher harmonics but, nevertheless, is in a reasonable agreement with the SCFT results obtained using the whole series of the higher harmonics [55]. Other clear evidence of such anoverestimation is that the phase transition ˜DIS-BCC3 (f ) → −∞ when f → fdi = line DIS-BCC3 drops oﬀ sharply χ 0.4183. At the ﬁrst sight this result seems meaningless, but, in fact, its physical meaning is rather clear. It can be understood by analogy with that of the spinodal of block copolymers with respect to microphase separation. The latter was deﬁned [21, 28, 29] as the line (surface) where the inverse scattering factor appearing in the quadratic term of the free energy (2.2) vanishes. Accordingly, the correlation function (1.11b) calculated within the RPA diverges here and, thus, the uniform state of the systems becomes absolutely unstable. However, taking into account the ﬂuctuation corrections [25–27,37,64–71] shows that the function S (r) stays ﬁnite and the uniform state stays stable (at least metastable) even beyond the RPA spinodal. Thus, the latter should be now understood as a crossover line between the regions with diﬀerent temperature scaling of the correlation radius and the exact border of the region where the RPA does not hold even qualitatively. Similarly, the sharp dropping oﬀ of the phase transition line DIS-BCC3 when f → fdi is shown [76] to be determined by the fact that the minimal quadric vertex β ¯ = min β changes sign at the point f = fdi due to the 2nd ¯ = BCC3 for molten shell harmonics renormalization of the vertex, therewith diblock copolymer. So, βBCC3 (f ) < 0 for f < fdi and, therefore, the expansion of the Landau Hamiltonian in powers of the order parameter Φ up to the 4th term only becomes inapplicable. As in the spinodal case, the unphysical divergence of the leading term is removed by including in the expansion the terms of higher order than that causing the divergence. In our case this means taking into account at least the terms of the 5th and 6th powers in Φ as well as the 3rd shell harmonics contributions. The corresponding generalization of the WST is expected to smooth (not eliminate!) the sharp phase transition ¯ shown in Fig. 12. It is natural to refer to line DIS-BCC3 (in general, DIS-) the line β ¯ = 0 as the WS border line since beyond it the higher harmonics eﬀect becomes so important that the system can not be described properly even within the 2nd shell harmonics approximation of the WS theory. Two more interesting features of the modiﬁed WS phase diagram shown in Fig. 12 are the phase transition lines G–BCC3 at f = 0.4343 and G–G2 (see Fig. 12b) situated at relatively high values of χ. ˜ It is important to stress that the WS theory can not claim responsibility for prediction of the precise location of both these phase transition lines. Indeed, they lie so far from the critical point that the stability of the phases BCC3 and G2 could be only an artifact of the WST extrapolation beyond its validity region. Nevertheless, these phase transition lines are interesting as indications of the fact that the stability of the double gyroid phase G is only caused by a moderate development of the 2nd shell harmonics, whereas a further increase of the

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degree of segregation results in increase of stability of other cubic phases (in our case, BCC3 and G2 ) at the cost of the G phase.

Fig. 13. The phase diagrams of the molten symmetric triblock (a) and trigraft (b) Am Bn Am copolymers in the 2nd shell harmonics approximation (without taking into account the Fddd phase existence). The designations of the phases are the same as in Fig. 11. Reprinted from [75]. Copyright (2003) American Chemical Society.

The described features of the phase diagrams of molten diblock copolymers are characteristic of binary AB block copolymers with various architectures as is exempliﬁed by the phase diagrams of molten symmetric triblock and trigraft Am Bn Am copolymers we calculated within the 2nd shell harmonics approximation (see Fig. 13). Summarizing, the WST in the 2nd shell harmonics approximation provides rather reasonable accuracy in locating the triple point HEX-G-LAM and interesting (much less reliable, though) hints as to stability of some other non-conventional cubic phases.

3 WST Applications to Multi-Component Block Copolymer Systems The order parameter for these systems is the n-component vector of the local deviations Φi (r) = φi (r) − φ¯i of the partial local volume fractions φi (r) = vρi (r) of the repeated units (monomers) of the i-th sort (here v is the excluded volume, supposed to be the same for all sorts of monomers) from their values φ¯i averaged over the whole volume of the system. As common for all polymer systems consisting of ﬂexible macromolecules, their virtual free energy takes the form [112, 113] F = Fstr ({φi (r)}) + F ∗ ({φi (r)}) .

(40)

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where the ﬁrst term is the so-called structural free energy Fstr corresponding to the entropy of the inhomogeneous ideal system of the copolymer macromolecules under consideration, with certain speciﬁed spatial proﬁles of the volume fractions of the repeated units of the i -th sort, and the second is the interaction contribution we discuss in more detail later. Within the WS theory the free energy (40) is evaluated by its expansion in powers of Φi (r) $ % F ({φi (r)}) = F φ¯i + ΔF2 + ΔF3 + ΔF4 + . . . (41) with the contributions ΔFn deﬁned as follows: ΔF2 1 Γij (q) Φi (q) Φj (−q) dq = , 3 T 2 (2π) n n & 1 Φαi (qi ) dqi ΔFn = δ qi Γα(n) (q ,.., q ) , 1 n 3 1 ,..,αn T n! (2π) i=1 i=1 n = 3, 4

(42)

(43)

In the disordered state the thermodynamically average values of the ﬂuctuations Φi (r) are zero. Thus the only observable quantities are the correlation functions (n Φi (0) Φj (r) exp (−ΔF ({Φl (r)})/T ) l=1 δΦl (r) (n Sij (r) = Φi (0) Φj (r) = exp (−ΔF ({Φl (r)})/T ) l=1 δΦl (r) (44) and their Fourier transforms (scattering factors) Gij (q) = Sij (r) exp (iqr) dr. Random Phase Approximation. Let us assume that the ﬂuctuations are small and keep in the free energy expansion (41) the quadratic contribution (42) only. Then, as ﬁrst shown by the author [114] (also [115–117], the matrix G = Gij (q) can be expressed in terms of two independent matrices characterizing the connectivity and interaction eﬀects: G−1 = Γ = g−1 − c,

(45)

where the matrices g−1 and -c are contributions to the matrix Γ appearing in (42) from the structural and energetic addendums, respectively (see (40)), matrices G−1 and g−1 are inverse to the matrices G and g, respectively, and the so-called structural matrix g is deﬁned [114–116] as follows: (S) nS γij (q) (46) g = gij (q) , gij (q) = S

In the second of the deﬁnitions (46) nS is the number density of the (macro)molecules with structure S, summing up is carried out over all the structures S present in the system including the monomers, and the molecu(S) lar form-factors γij (q) read

Theory of Microphase Separation in Block Copolymers (S)

γij (q) =

exp iq (r (li ) − r (nj ))S ,

351

(47)

where r(li ) is the vector-radius of the l-th repeated unit of type j and the symbol < . . . > s implies averaging over all Gaussian conformations of the (1) macromolecule S. For monomers, obviously, γij (q) = δij . The matrix g is just the matrix of the correlation functions for the ideal polymer system i.e. a system with the same structure as that under study but with no interactions between its repeated units, whereas (48) cij (r) = −δ 2 F ∗ ({φl (x)}) (δρi (0) δρj (r)) is the matrix of the direct correlation functions, which is well known in the theory of simple liquids [118]. The uniform (disordered) phase stays thermodynamically stable (at least, metastable) with respect to micro- or macrophase separation when the quadratic term (42) is positive deﬁnite [29]: min Λ (q) = Λ (q∗ ) > 0

(49)

where Λ(q) is the minimal of the eigenvalues λi (q) , i = 1, .., n of the matrix Γ of rank n and wave number q∗ of the critical order parameter waves is the location of the absolute minimum of the function Λ(q). Therewith q∗2 > 0

(50)

is the condition that it is micro- rather than macrophase separation which occurs after the uniform state becomes unstable. Accordingly, the spinodal line (surface), which delineates the region in the space of the structural and interaction parameters of the system under study where the spatially uniform (disordered) state is absolutely unstable within the RPA, reads [29] min Λ (q) = Λ (q∗ ) = 0,

(51)

The interaction term F ∗ is naturally determined assuming that it does not depend on the polymer structure of the system. (This natural requirement is, in fact, rather subtle. Strictly speaking, it is correct only when the Lifshitz number Li = v/a3 , which plays the role of the Ginzburg parameter, is small (here a is the Kuhn length). If Li is not small then the eﬀective monomer-monomer interaction is strongly inﬂuenced by the correlation between the neighboring (along the chain) monomers and renormalized accordingly. It is this phenomenon for which Khokhlov [119,120] coined the term “quasimonomers” and which is quantitatively addressed by the PRISM theory [121,122]. Henceforth we assume that the interaction term F ∗ describes the properly renormalized interaction.) In the simplest case of the compressible Flory–Huggins lattice model (when some cells of the lattice are not occupied by any monomer repeating units) the interaction term F ∗ reads

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F ∗ ({φi (r)}) = T

⎛ ⎞ dr ⎝ 1 (1 − φ (r)) ln (1 − φ (r)) + χij φi (r) φj (r)⎠, v 2 i =j

(52) 'l where φ (r) = i=1 φi (r) is the total volume fraction of all sorts of monomers. Then the addendums in (52) describe the compressibility and van der Waals interaction eﬀects, respectively. In the incompressibility limit 1 − φ (r) → 0 the ﬁrst addendum in (52) vanishes and we get the well known quadratic expression [62, 117] χij φi (r) φj (r)dr/v. (53) F ∗ ({φi (r)}) = (T /2) i =j

Thus, in this limit the higher vertices in the virtual free energy (40) expansion in powers of Φi (r) are determined by the structural entropic term only. But in general (beyond the incompressible Flory–Huggins model) the enthalpic (interaction) contribution to the free energy can aﬀect either the cubic and quadric terms as found for molten diblock copolymers taking into account the “quasimonomer” renormalization [123, 124] and using the equation of state model. [81–83]. It is worth noticing that the characteristic scales of the structural and interaction contributions into the higher vertices are rather diﬀerent (of the order of the macromolecule and monomer size, respectively). Thus, the main source of the strong angle dependence (if any) of the 4th vertex, which may lead to stability of the non-conventional phases as discussed in Sect. 2, is expected to be the structural contribution determined by the so-called higher structural correlators as shown in detail in [21, 47, 76, 117]. Critical Points in the Multi-Component Block Copolymer Systems and the Strongly and Weakly Fluctuating Fields. Violation of the (meta)stability condition (49) is suﬃcient but not necessary to guarantee crystal ordering. Typically, a ﬁnite order parameter proﬁle (13) arises via a discrete 1st order phase transition when the condition (49) still holds and the disordered phase is at least metastable. However, the span of this proﬁle decreases when the relative magnitude of the cubic terms decreases and, ﬁnally, the ordering transforms into a continuous 2nd order phase transition (within the mean ﬁeld approximation only!) at the critical point where the cubic terms vanish. Thus, the WST certainly holds (at least in the same sense as the SCFT does) in the vicinity of the critical point(s). It is easy to locate the critical point(s) (if any) for the scalar WST describing the ODT in the incompressible binary block copolymers, where only one cubic term exists (see eq (2)). But the virtual free energy of the n-component block copolymer systems contains ∼ n3 cubic terms (that of incompressible polymers contains (n − 1)3 terms), so that it could appear [62] that it is hardly possible at all to apply the WST to many-component systems. Nevertheless, [47, 76] it is possible to reduce consideration of the n-component systems to that presented in Sect. 2 for those with a scalar order parameter

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via distinguishing the minimal Λ(q) and all other (positive) eigenvalues λi (q) (i=2,. . . n) of the matrix Γ appearing in (42) as well as the corresponding (s) eigenvectors Ei (q) and ei (q). The projections of the vector order parameter Ψ (q) = Ei (q) Φi (q) ,

(s)

ϕs (q) = ei (q) Φi (q) ,

s = 2, .., n.

(54)

into the corresponding eigenvectors play the role of the strongly and weakly ﬂuctuating ﬁelds, respectively, the account of the latter being carried out similarly to the treatment of the 2nd shell harmonics in Sect. 2. As a result, one gets the eﬀective scalar free energy of the weakly segregated multi-component block copolymer systems in the form (38), where the eﬀective cubic vertex α is deﬁned by (20) with (q1 ,.., q3 )Eα1 (q1 ) Eα2 (q2 ) Eα3 (q3 ) . Γ3 (q1 ,.., q3 ) = Γα(3) 1 ,α2 ,α3

(55)

All other cubic terms either renormalize the eﬀective quadric vertex β described by a cumbersome expression [76] we skip here for brevity, or contribute to the terms of the order of magnitude O (Ψ 5 ) and thus exceed the accuracy of the WST.

4 The WST Predicted Peculiarities in the Multi-Component Block Copolymer Systems The RPA Structure Factors in the Disordered State. To calculate the structure factor matrix G for the Flory–Huggins model we ﬁnd the matrix c for ﬁnite compressibility from (48), (52): −1 (56) cij (q) = v (1 − φ) + χij /2 , substitute (56) into the r.h.s. of the general RPA equation (45), invert the resulting matrix and, ﬁnally, take the incompressibility limit 1 − φ (r) → 0. In this (or an equivalent) way the structure factor matrices G(q) were found and the spinodal conditions analyzed [73–76, 97–99] for some ternary ABC block copolymer systems. Werner and Fredrickson [74] studied the spinodal conditions (49), (50) for molten linear and comb-like ABC (monodisperse and statistical) block copolymers and found the spinodal lines as well as the q∗ dependence on the values of three independent Flory parametersχAB , χAC , χBC . They found that increase of one of these χ -parameters (given that the two others are ﬁxed) could result in a non-monotonous ordering tendency called the reentrant ODT. A similar result was found by the author et al. [73] Namely, the spinodal ODT temperature Ts in the linear ABC block copolymers, in which one of blocks is much shorter and more incompatible than the two others, changes non-monotonously with the increase of the short block incompatibility, the minimal value of Ts and the period of the arising ordered

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structure being less than those for the corresponding diblock copolymer without any third strongly incompatible block. Thus, the peculiar situation that “more incompatibility results in less segregation” is explained by an additional entropic loss related to conﬁnement of the ABC macromolecules in a lattice with smaller periodicity. Two-Length-Scale Behavior in Molten ABC. Cochran, Morse and Bates [75] considered in more detail the scattering behavior of the linear ABC triblock copolymer melts and found that by tuning the values of the architecture, scattering contrast and interaction parameters one achieves a reasonable agreement between the two-peak proﬁle of the SAXS indicatrix observed in the disordered poly(isoprene-b-styrene-b-dimethylsiloxane) (ISD) [125] and the RPA structure factor, the height of one of the peaks increasing when approaching the ODT (see Fig. 14). The authors concluded that “the RPA structure factor is representative of the true structure in disordered ABCs” and attributed such a two-peak proﬁle “to the natural existence of multiple length scales in ABCs”. An additional insight into the nature of such an unusual behavior is provided by the RPA analysis [76] for symmetric An Bm Cn block copolymers with non-selective middle block (χAB = χBC = χ). For such a symmetric ABC the RPA free energy (42) reads 1 d3 q 2 2 ΔF2 = (Φ (r)) + λ (Φ (r)) λ , + + − − 3 4N (2π) ˜AC − 4χ, ˜ Φ+ (r) = ΦA (r) + ΦC (r) , (57) λ+ = b (q) + χ λ− = a (q) − χ ˜AC , Φ− (r) = ΦA (r) − ΦC (r) where χ ˜ = χN , χ ˜AC = χAC N , N = m + 2n and the functions a(q) and b(q) deﬁned in [76] both for linear and miktoarm ABC depend on the reduced squared wave number Q = q 2 a2 N/6 only. It turns out that both the character of the weakly segregated morphology occurring in such a symmetric system and the very possibility to describe it within the WS theory depends crucially on the values of the interaction pa˜ is divided by the lines λ− (χ ˜AC , χ) ˜ =0 rameters. Namely, the plane (χ ˜AC , χ) ˜AC , χ) ˜ = 0 into i) the stability region (λ− > 0, λ+ > 0), where the and λ+ (χ ﬂuctuations of both order parameters Φ+ (r), Φ− (r) are ﬁnite and the uniform state is stable (or at least metastable) with respect to these ﬂuctuations; ii ) the AC-modulation region (λ− < 0, λ+ > 0), where the uniform (disordered) state is unstable with respect to formation of a certain proﬁle Φ− (r) = 0, the order parameter Φ+ (r) being weakly ﬂuctuating; iii ) the B-modulation region (λ+ < 0, λ− > 0), where the uniform state is unstable with respect to formation of a certain proﬁle Φ+ (r) = −ΦB (r) = 0, the order parameter Φ− (r) being weakly ﬂuctuating; and iv ) the region (λ− < 0, λ+ < 0), where the uniform state is unstable with respect to ﬂuctuations of both order parameters Φ+ (r) and Φ− (r).

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Fig. 14. Computed RPA scattering of a compositionally symmetric model, N = 100 ISD block copolymer, at six temperatures approaching the spinodal limit for three diﬀerent choices of the scattering contrast vector c. The temperatures shown in each instance are 1.0324Ts , 1.0171Ts , 1.0121Ts , 1.0072Ts , 1.0020Ts , and 1.0002Ts , where Ts = 302.4 K. Temperature decreases with increasing peak intensity. Reprinted from [75]. Copyright (2003) American Chemical Society.

The lines λ+ = 0 and λ− = 0 in the plane (χ ˜AC , χ) ˜ are the straight lines χAC − 4χ = − min b (q) = −b (q+ ) , −1 −1 b (q) = min g ˜ (q) + g ˜ (q) , 11 12

(58a)

χAC = min a (q) = a (q− ) , −1 −1 a (q) = g ˜ (q) − g ˜ (q) , 11 12

(58b)

the critical wave numbers q+ and q− characterizing the periods of the proﬁles Φ+ (r) and Φ− (r), respectively, being the locations of the absolute minima of the function a(q) and b(q), sought within the semiaxis 0 < q 2 < ∞. The lines

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Fig. 15. The spinodal behavior of the symmetric ternary ABC copolymers: a) the fA -dependences of the reduced squared critical wave numbers q− (curves 1) and q+ (curves 2) for both the linear (solid) and miktoarm (dashed) symmetric ABC copolymers; b) the classiﬁcation of the spinodal instability regions in the (χ, χAC )plane. The solid lines satisfy (58a), (58b) for fA = 0.245, the numbers 0, 1, 2 and 3 label the stability, AC-, B-modulation and two-length-scale regions, respectively; the dashed lines describe the temperature evolution of the systems with χAC /χ > k0 (I) (I) (a), χAC /χ = k0 (b) and χAC /χ < k0 (c), where k0 = χAC /χ(I) and χ ˜AC , χ ˜(I) are the coordinates of the point of interSect. of the solid lines; c) the fA -dependences (I) of the coordinates χAC (curves 1) and χ(I) (curves 2) for both the linear (solid) and miktoarm (dashed) symmetric ABC copolymers; d) the fA -dependences of the ratio (I) k = χAC /χ(I) for both the linear (solid) and miktoarm (dashed) symmetric ABC copolymers. Reprinted from [76] with permission.

(58a) and (58b) intersect at the point with the co-ordinates (I) χAC = χAC = a (q− ) ,

χ = χ(I) = (a (q− ) + b (q+ ))/4.

(59)

As is seen from Fig. 15a, the values of the reduced squared critical wave numbers q+ and q− for both the linear and miktoarm ABC copolymers are rather diﬀerent. ˜ into regions with diﬀerent types A typical separation of the plane (χ ˜AC , χ) of spinodal instability is shown in Fig. 15b. The dependences of the coordinates (I) (I) χ , χ(I) and the ratio k = χ χ(I) of f are plotted in Figs. 15c,d, AC

AC

respectively. As seen from Fig. 15c, the solid curves 1 and 2 do intersect and for the compositions corresponding to the intersection point the ABC should reveal the 2-scale behavior we discussed in the introduction, which is in good agreement with the results of [75]. Two-Length-Scale Behavior in Blends ABC and AC. Another way to realize 2-scale behavior is to blend the triblock ABC and diblock AC copolymers. The author et al. [97] analyzed the spinodal stability of such blends within the RPA, the famous Hildebrand approximation for the χ-parameters being used for simplicity: χij = v(δi − δj )2 /(2T ),

(60)

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357

where δi is the solubility parameter of the i-th component supposed to be temperature-independent. It is convenient [73] to take as the two independent interaction parameters characterizing the ternary systems in the approximation (60) the following: 2

χ = χAC = v (δA − δC ) /(2T ),

x = (2δB − δA − δC ) / (δA − δC ) . (61)

χAC characterizes incompatibility of the side blocks in the ABC triblock copolymer whereas the selectivity parameter x describes how much is the middle block B is selective with respect to the side blocks. Remarkably, in this approximation the spinodal condition (49) takes the simple form [73] −1 (62) τ˜ = (2χAC N ) max W (Q, x) where N is the total degree of polymerization of the triblock copolymer and the function W depends on the reduced squared wave number Q = q 2 a2 N /6, selectivity x and structural parameters involved in the deﬁnition of the structural matrix g(Q) as described in [73, 97]. For the blends of ABC and AC the function W is shown [97] to have one or two maxima at Q = Qmin and Q = Qmax depending on the blend structural parameters and selectivity and, thus, to be capable of revealing two-length-scale behavior. A new interesting feature of these blends is that Qmin can reduce to zero6 . Information concerning the localization of various modes of instability in the space of parameters of the ABC/AC blends is provided by the so-called phase portrait in the (φABC , x) plane as shown in Fig. 16. For blends where the middle block of ABCs is not long enough the function W (Q) only has the maximum Q = Qmax , which can be located either on the boundary of the interval 0 ≤ Q < ∞ of the permissible values of Q, i.e. at Qmax = 0 (in this case the blend would undergo macrophase separation at low temperatures) ) or within the interval, i.e. at Qmax > 0 (in this case the blend would undergo microphase separation at low temperatures). The line separating these two regions is referred to as the Lifshitz line (see Fig. 16a). The Lifshitz curve has two vertical asymptotes at φABC = 0 and φABC = φL (φL = 0.2 in Fig. 16a). Thus, the macrophase separation of the ABC/AC blend may occur only in the interval 0 < φABC < φL . For mixtures with φABC > φL only the ODT (microphase separation) is possible. For the longer middle blocks B the function W (Q) can have two maxima and, accordingly, the Lifshitz line splits into two lines (see the bold dashed and solid lines in Fig. 16b), which display the set of parameters providing reducing to zero the 6

The situation is reminiscent of that found by Holyst and Schick [126] who carried out RPA analysis of a symmetric ternary mixture of A and B homopolymers and AB diblock copolymers. They discovered that in some situations one of the components of the matrix of the structure factors Gij (q) for this mixture could exhibit two equal maxima. However, this situation could be observed in the disordered phase only and both maxima never diverge simultaneously in the A/B/AB mixture as occurs in the ABC/AC blend.

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Fig. 16. The phase portraits of the AN CN /AhN BmN ChN for h = 2; a) m = 4 and b) m = 10. The region corresponding to the macro- and microphase separation instabilities are labeled by the numbers 1 and 2, respectively. a) the only Lifshitz line is plotted; b) the bold solid and dashed lines are stable and metastable Lifshitz lines, respectively, within the region conﬁned by the dotted thin lines the function W (Q) has two maxima; on the thin short dashed line both maxima are equal and the system reveals two-length-scale behavior.

locations of the stable or metastable maxima, respectively. A typical phase diagram for this situation is presented in Fig. 16b. An important prediction made in [97] is that it should be possible to produce regular superstructures reminiscent of the pattern shown in Fig. 8 via a subtle tuning of the selectivity, composition φABC and structural parameters of the ABC/AC blends. A similar RPA analysis was carried out for solutions of the di- tri- and regular polyblock copolymers in non-selective solvents, [98] where the conditions for two-length-scale behavior were also found. The Critical Lines for ABCs. For simple temperature dependence χAC (T ) = ΘAC / (2T ), χ (T ) = Θ/(2T ) the states of a ternary ABC sys˜ tem with diﬀerent temperatures are located on the straight line χ ˜AC = k χ, ˜AC , χ). ˜ As shown in Fig. 15b, the system leaves the k = ΘAC /Θ in the plane (χ stability region crossing either the line (58a) or (58b) depending on the value of k. In the ﬁrst case (B-modulation), which occurs, e.g., for ABA copolymer (χAC = 0), the eﬀective cubic vertex reads (3) (3) (3) (3) γ(1) = 2−3/2 Γ111 + Γ222 + 3Γ112 + 3Γ221 , (3)

(3)

Γijk (1) = Γijk (q1 , q2 , q3 ) ,

|qi | = q∗ ,

(63)

q1 + q2 + q3 = 0.

The straightforward calculation as consistent with [32, 33, 36, 73] shows that there is a unique critical point where the cubic vertex (63) vanishes for the symmetric triblock (miktoarm) copolymer An Bm An . The point is located at fB = 0.49 (fB = 0.557). On the contrary, in the second case (AC-modulation) the cubic vertex vanishes identically for symmetric copolymer with any composition of the non-selective block since it reads

Theory of Microphase Separation in Block Copolymers

γ (1) = 2−3/2 Γ111 − Γ222 + 3 Γ221 − Γ112 (3)

(3)

(3)

(3)

359

.

(64)

So, the ternary ABCs belonging to the AC-modulation class are expected to undergo much smoother ODT than those belonging to the B-modulation class. In the Hildebrand approximation the symmetry assumption χAB = χBC = χ holds if the middle block is non-selective with respect to both side blocks (x = 0), which occurs, e.g., for poly(isopren-b–styrene-b-2-vinylpyridine) triblock copolymers [127]. But the continuous ODT transition in the ternary block copolymers occurs not only for fA = fC and x = 0. The critical lines were built by the author [76] via numerical solution of the equation γ (1, fA , fC ) = 0 for diﬀerent values of 0 < x < 1 (see Fig. 17).

Fig. 17. The critical lines for the linear ABCs in the Hildebrand approximation for diﬀerent values of the selectivity x. The symmetric bold lines correspond to a non-selective middle block (x = 0), the critical lines labeled by the numbers 1, 2, 3, 4 and 5 correspond to the values of the selectivity parameter x = 0.01, 0.1, 0.3, 0.5 and 0.8, respectively. The dashed lines cb and ab are the critical lines for x = ±1. Reprinted from [76] with permission.

Remarkably, the line fA = fC is not the only critical line even for x = 0. Another critical line is the curve ac, which is rather close to the straight line ac. For x = 0 the critical lines consist of two branches. In the limit x → 1 one of the branches, which corresponds to the case fC → 0.5, |χAC − χBC |

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χAC + χBC , approaches the diblock copolymer critical line ab whereas another branch, which corresponds to an AB copolymer with a short strongly interacting C block, [73] approaches Bc. WST Border Lines and the Phase Diagrams of the ABC Triblock Copolymers. As discussed in sections 2 and 3, taking into account the higher harmonics and other weakly ﬂuctuating ﬁelds results in a renormalization of the eﬀective 4th vertex β of the eﬀective free energy, and, eventually, in vanishing β moving oﬀ the critical lines. Therewith, the line β (fA , fC ) = 0 corresponds to the WST border line, i.e. a crossover line conﬁning the region beyond which the WST does not hold even qualitatively. In Fig. 18 the WST border lines calculated within the Hildebrand approximation for the linear and miktoarm ABC triblock copolymers are presented [76]. It is worth making two remarks here. First, as is seen from Fig. 18b, the WST border line can intersect the critical line. The intersection point is expected [76] to be the tricritical point where the line of the 2nd order phase transitions transforms into that of the 1st order ones. Thus, in general, a judgment on the validity of the RPA (and even critical point) analysis in each case requires the full WST analysis including ﬁnding of the 4th vertex. Second, the WST validity region in the composition triangle is far from being negligibly small, especially for linear ABCs. In particular, in the interval 0.42 ≤ fB ≤ 0.58 the WST phase diagram can be built for any asymmetry of the side blocks (see Fig. 19).

Fig. 18. The maps describing the WST application to the melts of the ABC a) linear and b) miktoarm triblock copolymers with the nonselective middle block (x = 0) within the Hildebrand approximation. The critical and WS border lines are shown by solid and short dashed lines, respectively. Note that the WST border line intersects the critical linefA = fC for miktoarm ABC triblock copolymers.

It is worth noting a rather broad (as compared to that for diblock copolymers) region (4) of the double gyroid (G) phase stability, which in this case can be predicted based on the WST only. We skip here many other phase

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361

Fig. 19. The WST phase diagram of the molten linear ABC block copolymers calculated fB = 0.55 within the Hildebrand approximation in the plane (σ, χ), ˜ where σ = fA /(fA + fC ) is the asymmetry parameter. The designations of the phases are the same as in Fig. 11. Reprinted from [76] with permission.

diagrams built and discussed in [76] the basic information of which is already given in Fig. 11. 4.1 Conclusion Even this brief survey of the current development of the weak segregation theory brings forward, hopefully, the idea that the potential of the theory is still far from being exhausted. As I tried to show in the review, the WST is capable of describing and predicting non-conventional phases in multicomponent block copolymer systems like gyroids (both single and double), face-centered and simple cubic phases as well as rather non-conventional new types of two-scale-length ordering. The WST easily incorporates various ideas of short-range thermodynamic interactions between the repeated units of polymer components. The main advantage of the WST is its capability to state analytically and solve both analytically and numerically the problems of optimization of various properties of ordered block copolymers and identify the most promising structures of multi-component block copolymers worth synthesizing by chemists, which could be too special to discover by chance. Still, the WST has some natural and rather important problems to be solved, in which case its predictive capacity could be much increased. First of all this includes extension of the WST to the level of the 3rd shell approx-

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imation. Such an extension seems to be a natural limit for the Landau-like theory, which would correspond to the Landau expansion up to the 6th order in powers of the order parameter and thus provide an opportunity to describe the ODT near the tricritical point and the WST border line discussed above. Thereupon it is worth mentioning here the work, [51] where the authors have shown that inclusion of the 6th order terms in the Landau expansion of the free energy of polydisperse copolymers results in considerable corrections to the previous results. The next problem is to improve our understanding of the ﬂuctuation eﬀects. In particular, the closely related problems of the interplay between the short- and long range ﬂuctuation corrections [128, 129] and theoretical explanation of the observed deviations (see [31] and Fig. 3) of the inverse structure factor from the RPA predictions are still open issues. Another important problem is to study coupling in the ordered phases between the density ﬂuctuations and transverse sound waves (shear ﬂuctuations), which until now has been considered [130, 131] only in the disordered state and is expected [132] to be especially important in the weak segregation regime where the shear modulus is small [133, 134]. But the most important problem is, of course, to establish a prompt and reliable feedback between theoreticians, experimentalists and industry since all good physics generates good technology, and there is no good technology which is not based (at least, implicitly) on good physics. Acknowledgement. To conclude, I thank my colleagues and friends Volker Abetz, Henk Angerman, Kurt Binder, Gerrit ten Brinke, Monica Olvera de la Cruz, Andrey Dobrynin, Alexander Grosberg, Jean-Francois Joanny, Albert Johner, Alexei Khokhlov, Ludwik Leibler, Marcus M¨ uller, Alexander Semenov, Friederika Schmidt and Reimund Stadler†, whose feedback helped me so much in my work.

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363

Appendix A. The Basic Weakly Segregated Morphologies The BCC family is determined by the set {qi } consisting of 12 vectors whose relative directions are given, e.g., by the six vectors listed below and the same vectors taken with the opposite sign:

( = (q

q 1 = q* qI

*

) 2 )(0, − 1, − 1),

2 (0,1, − 1),

( = (q

q 2 = q* q II

*

) 2 )(−1, 0,−1, ), 2 (−1, 0,1, ),

( = (q

q 3 = q* q III

*

) 2 )(−1, − 1, 0)

2 (1, − 1, 0),

Fig. A1. The planar mapping of the vectors characterizing the symmetry of the main harmonics a) for the BCC family and b) for the orthorhombic lattice Fddd (see explanations in the text). Reprinted from [76] with permission.

The vectors can be visualized as the edges of an octahedron [21] or a tetrahedron [26] or via their planar mapping as shown in Fig. A1. The arrow circuits in Fig. A1 correspond to the equalities qI = qII + q3 ,

qII = qIII + q1 ,

qIII = qI + q2 ,

q1 + q2 + q3 = 0.

1. For the conventional BCC all phases ϕ appearing in the deﬁnition (14) of the basic function Ψ are zero and the vertices read αBCC = 8γ/63/2 ,

βBCC = [λ0 (0) + 8λ0 (1) + 2λ0 (2) + 4λ (1, 2, 1)]/24. (A.1) 2. If the phases corresponding to the three vectors q1 , q2 and q3 , which form the base of the tetrahedron, and those of three non-coplanar vectors qI , qII and qIII (the thin and thick lines, respectively, in Fig. A1a) are equal to

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π/2 and 0, respectively, we get the lattice ﬁrst discussed in [12] and called (3) (3) (3) (3) there the BCC2 . For this lattice ΩABC = 3π/2, ΩABS = ΩACS = ΩBCS = (4) (4) (4) π/2, ΩABCSA = ΩBCASB = ΩCABSC = π and the vertices read αBCC2 = 0 ,

βBCC = [λ0 (0) + 8λ0 (1) + 2λ0 (2) − 4λ (1, 2)]/24. (A.2) 2

The BCC2 phase can be shown [76] to possess symmetry of the I41 32 space group (No. 214), which is non-centrosymmetric and closely related to the so-called single gyroid surface. Thus, it seems to be the simplest (and the only up to now) cubic non-centrosymmetric morphology that can be described (and for some cases predicted) within the WS theory. The BCC2 belongs to the class of morphologies like L, FCC and SC which we refer to as degenerate ones because for them the cubic vertex (20) identically equals zero due to symmetry reasons. For these degenerate morphologies the free energy (19) reads [21] ΔF = −τ 2 (4β ).

(A.3)

Thus, the most stable degenerate morphology is that having the least value of the quadric vertex. For reference, we also present the expressions for the vertices for FCC, L and SC morphologies: βFCC = [λ0 (0) + 6λ0 (4/3) − 2λ (4/3, 4/3)]/16, βL = λ0 (0)/4, βSC = [λ0 (0) + 4λ0 (2)]/12.

(A.4)

The G family is determined by 2 · 12 main harmonics given by the 12 vectors q01 =

q11 =

q21 =

q31 =

q∗ √ 6 q∗ √ 6 q∗ √ 6 q∗ √ 6

(−2, +1, +1), q02 = (−2, −1, −1), q12 = (+2, +1, −1), q22 = (+2, −1, +1), q32 =

q∗ √ 6 q∗ √ 6 q∗ √ 6 q∗ √ 6

(+1, −2, +1), q03 = (+1, +2, −1), q13 = (−1, −2, −1), q23 = (−1, +2, +1), q33 =

q∗ √ 6 q∗ √ 6 q∗ √ 6 q∗ √ 6

(+1, +1, −2), (+1, −1, +2), (−1, +1, +2), (−1, −1, −2). (A.5)

and 12 opposing ones. The planar mapping of the vectors is shown in Fig. A2a. For comparison, in Fig. A2b the planar mapping of the regular icosahedron is shown. Obviously, the set of vectors (A.5) (and the opposing ones) is obtained via a deformation of the regular icosahedron, which involves removing 6 of 30 edges of the icosahedron and proper rotation of the remaining edges, the resulting polyhedron being the Wigner–Zeitz cell of the corresponding crystal lattice. It is this relationship between the G family and the icosahedron symmetry which causes the famous 10 spot SAXS pattern observed in the gyroid phase.

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365

Fig. A2. The planar mappings of the vectors characterizing the symmetry of the main harmonics for the G family (left); and of the edges of the regular icosahedron (right). a) the vectors depicted by bold lines have zero phases for all three morphologies G, G2 and BCC3 , those depicted by dashed and thin solid lines have phases equal to π only for the double gyroid (G) and both for G and G2 , respectively (see the deﬁnitions of the corresponding phases in the text); b) the edges depicted by thin and bold lines correspond to the vectors to be removed and properly rotated to transform the icosahedron into the G cell. Reprinted from [76] with permission.

1. The morphology arising if all the phases ϕi are set to equal zero we call the BCC3 . It is just the ordinary BCC but for the fact that the dominant harmonics correspond here to the 3rd (rather than the 1st!) co-ordination sphere. αBCC3 = γ/33/2 βBCC = [Λ1 + 4[λ(1/3, 2/3) + λ(2/3, 5/3)] + 2λ(2/3, 2/3)] /48 (A.6) 3 Λ1 = λ0 (0) + 2 (λ0 (4/3) + 2 (λ0 (1/3) + λ0 (2/3) + λ0 (1) + 2λ0 (5/3))) 2. The trial function (14) with the main harmonics (A.5) and the phase choice ϕ12 = ϕ23 = ϕ31 = ϕ01 = ϕ02 = ϕ03 = 0, ϕ21 = ϕ32 = ϕ13 = ϕ11 = ϕ22 = ϕ33 = π. corresponds to the bi-continuous gyroid (G) or double gyroid morphology having the symmetry Ia¯ 3d. αG = γ/33/2 , βG = [Λ1 − 2Λ2 − 4λ(1/3, 2/3)] /48 Λ2 = 2λ(2/3, 5/3) − λ(2/3, 2/3).

(A.7)

It is seen from (A.6), (A.7) that due to the symmetry of the BCC3 and G lattices their cubic vertices are identical and, therefore, the BCC3 - G phase

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transition line (surface) is determined by the equation βG − βBCC3 = 0

(A.8)

3. The trial function (14) with the main harmonics (A.5) and the phase choice ϕ21 = ϕ32 = ϕ13 = π, ϕ12 = ϕ23 = ϕ31 = ϕ01 = ϕ02 = ϕ03 = ϕ11 = ϕ22 = ϕ33 = 0

(A.9)

¯ we refer to as the G2 . corresponds to the morphology of the symmetry I 43d For this morphology αG2 = γ/ 2 · 33/2 , βG2 = [Λ1 + 2Λ2 − 4λ (1/3, 2/3)]/3, 2/48. (A.10) For reference, we give here also the vertices for the HEX morphology [20] αHEX = 2γ/33/2 ,

βHEX = [λ0 (0) + 4λ0 (1)]/12.

(A.11)

The orthorhombic lattice (space symmetry group O70 or Fddd ) is generated by the 2 · 4 main harmonics given by the 4 vectors e1 = q˜∗ (a, b, c) ,

e2 = q˜∗ (a, −b, −c) ,

e3 = q˜∗ (−a, b, −c) , e4 = q˜∗ (−a, −b, c) (A.12)

#√ where q˜∗ = q∗ a2 + b2 + c2 , the values of the periods a, b, c of the reciprocal lattice are, in general, all diﬀerent and the choice of the phases is ϕ01 = π, ϕ02 = ϕ03 = ϕ04 = 0. Thus, in general, the lattice belongs to the class of the degenerate morphologies and the corresponding 3rd and 4th vertices read 3 # 0 0 16, (A.13) λ0 (hi ) − 2λ (h1 , h2 ) αortho = 0 , βortho = λ0 (0) + 2 i=1

where the designations hi = + a22 + a23 are introduced. In particular, for the case a = b = c the orthorhombic lattice just becomes the FCC one. If a2 = 2b2 = 2c2 then two second harmonics (0, ±2b, 0) and (0, 0, ±2c) also belong to the ﬁrst coordination sphere of the reciprocal lattice, which now contains (along with the main harmonics) 2 to 6 vectors and could be easily checked to correspond to the BCC lattice in the co-ordinate space. At last, if a2 = 4b2 = 12c2 then two second harmonics b1 = e1 − e2 = q˜∗ (0, 2b, 2c) and b2 = e4 − e3 = q˜∗ (0, −2b, 2c) as well as the fourth harmonic c = b1 + b2 = q˜∗ (0, 0, 4c) also belong to the ﬁrst coordination sphere with the radius q∗2 = a2 + b2 + c2 = 16c2 . The planar mapping of the vectors is shown in Fig. A1b. In this special case only those harmonics which have the same symmetry rather than belonging to the same coordination sphere of the reciprocal lattice, have equal amplitudes, so that the trial function (14) reads 4a2i /

a21

Theory of Microphase Separation in Block Copolymers

¯ (r) = AΨA (r) + BΨB (r) + CΨC (r) , Φ

367

ΨB = 2 (cos (b 2 r) − cos (b 1 r)) ,

ΨA = 2 (cos (e 2 r) + cos (e 3 r) + cos (e 4 r) − cos (e 1 r)) ,

ΨC = 2 cos (cr) . (A.14)

The choice of the signs in (A.14) is determined by equivalent requirements to satisfy the Fddd symmetry and provide the minimal free energy, which takes the form [92] F = V T min f (A, B, C) , where f (A, B, C) = τ 4A2 + 2B 2 + C 2 − 4γA2 B − 2γB 2 C + (λ (0) + 2 (λ (1/4) + λ (3/4) + λ (1) − λ (1/4, 3/4))) A4 + (λ (1) + (1/2) λ (0)) B 4 + (λ (0)/4) C 4 + 4λ (1, 3/2) A2 BC +4 (λ (1) + λ (3/2)) A2 B 2 + 4λ (3/2) A2 C 2 + 2λ (1) B 2 C 2. (A.15) Taking approximation (28) for the 4th vertices λ and minimizing the free energy (A.15) numerically with respect to all three amplitudes A, B, C results in the reduced phase diagram shown in Fig. 10b.

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99. Tarasenko S., Erukhimovich I. Polymer Science Series A 2006, 48, 207. 100. Smirnova, Yu., ten Brinke, G; Erukhimovich I. Polymer Science Series A 2005, 47, 430. 101. Smirnova, Yu., ten Brinke, G; Erukhimovich I. J. Chem. Phys. 2006, 124, 054907. 102. Pichugin V.E., Kuchanov S.I. J. Stat. Mech. P07009 (2005) 103. Pichugin V.E., Kuchanov S.I. ten Brinke G. e-polymers, 2006 104. Nap R., ten Brinke G. Macromolecules 2002, 35, 952. 105. Nap R., Erukhimovich I., ten Brinke G. Macromolecules 2004, 37, 4296 106. Nap R., Sushko N., Erukhimovich I., ten Brinke G. Macromolecules 2006, 39 (in press). 107. Kriksin, Yu.A., Khalatur P.G., Smirnova, Yu.G., Erukhimovich I., ten Brinke G. (in preparation). 108. Milner S.T., Olmsted P.T., J. Phys. II, 1997, 7, 249. 109. Marques C., Cates C.M. Europhys. Lett. 1990, 13, 267. 110. Olvera de la Cruz M., Mayes A.M., Swift B.W. Macromolecules 1992, 25, 944. 111. International Tables for Crystallography; Kluwer Publishing: Dortdrecht, The Netherlands, 1996, Vol.B. 112. Lifshitz I.M., Grosberg A.R., Khokhlov A.R. Rev. Mod. Phys.1978, 50, 683. 113. Dobrynin A.V., Erukhimovich I.Ya. Sov. Phys. JETP, 1991, 72, 751. 114. Yerukhimovich I. Ya. Polymer Science U.S.S.R. 1979, 21, 470. 115. Benoit H.C., Benmouna M. Polymer, 1984, 25, 1059. 116. Higgins J.S., Benoit H.C. Polymers and Neutron Scattering. Oxford: Clarendon Press, 1996. 117. Hong K.M., Noolandi J. Macromolecules 1983, 16, 1083. 118. Hansen J.P., McDonald I.R. Theory of Simple Liquids, Academic Press, London, 1986. 119. Khokhlov A.R. J. Physique, 1977, 38, 845. 120. Khokhlov A.R. Polymers, 1978, 19, 1387. 121. Schweizer K.S., Curro J.G. Adv. Polym. Sci. 1994, 116, 319. 122. Schweizer K.S., Curro J.G. Advances in Chemical Physics, edited by I. Prigogine and S. A. Rice, 1997, vol. XCVIII, Chap. 1. 123. Stepanov S. Macromolecules, 1995, 28, 8233. 124. Guenza M., Schweizer K. J. Chem. Phys., 1997, 106, 7391. 125. Hardy C. M., Bates F. S., Kim M.-H., Wignall G. D. Macromolecules 2002, 35, 3189-3197. 126. Holyst R., Schick M. J. Chem. Phys. 1992, 96, 7728. 127. Mogi Y., Nomura M., Kotsuji H., Ohnishi K., Matsushita,Y., Noda I. Macromolecules 1994, 27, 6755. 128. Maurer W.W., Bates F.S., Lodge T.P., Almdal K., Mortensen K., Fredrickson G.H. J. Chem. Phys. 1998, 108, 2989. 129. Kudlay A., Stepanow S. J. Chem. Phys. 2003, 118, 4272. 130. Doi M., Onuki A. J. Phys. II France 1992, 2, 1631. 131. Erukhimovich I.Ya., Kudryavtsev, Ya.V. Eur. Phys. J. E 2003, 11, 349. 132. Landau L.D., Lifshitz E.M. Electrodynamics of continuous media 1984 Pergamon Press; Oxford, New York, Beijing, Frankfurt. 133. Amundsen K., Helfand E. Macromolecules 1993, 26, 1324. 134. Kawasaki K., Ohta T. Physica 1986, 139A, 223.

Coarse-Grained Modeling of Mesophase Dynamics in Block Copolymers Zhi-Feng Huang and Jorge Vi˜ nals McGill Institute for Advanced Materials, and Department of Physics, McGill University, Montreal, QC H3A 2T8, Canada

1 Introduction Block copolymers are not only well suited for studying self-assembly and nanostructure formation in soft matter, but also provide a wide range of new applications in ﬁelds such as microelectronics, biomedicine, etc. Viewed as a simpliﬁed picture, a block copolymer is composed of two or more segments (blocks or sequences of monomers) that are chemically diﬀerent and mutually incompatible, joined by covalent bonds (see Fig. 1a for schematic of diblock and linear triblock). Of most interest in block copolymers is the emergence of mesophases below the order-disorder transition temperature TODT , showing as spatially ordered compositional patterns of diﬀerent types of symmetries, such as lamella (see Fig. 1b), cylinder, sphere, gyroid, etc. [1, 2].

Fig. 1. (a) Schematic of block copolymer molecules: AB diblock and ABC triblock. (b) Polycrystalline, defected conﬁguration for a diblock copolymer system in a lamellar phase. Also indicated are three types of topological defects: grain boundary, dislocation, and disclination.

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However, when a block copolymer melt sample is processed by temperature quench or solvent casting from a high temperature isotropic, disordered state, the resulting conﬁguration is usually polycrystalline with randomly oriented domains or grains, instead of a uniform, completely ordered structure as desired in most applications. The characteristic of such a mesoscopic, polycrystalline state is the presence of large amount of topological defects, such as grain boundaries, dislocations, and disclinations, as shown in Fig. 1b for a two-dimensional conﬁguration of a lamellar phase. Consequently, there is no long-range orientational order in the sample. These topological defects determine the longest relaxation times in a system which is outside of thermodynamic equilibrium, and hence their structure, interaction, and motion control the coarsening and evolution towards the equilibrium ordered state, as well as the response of the copolymer microstructure to external forces. Widespread applications of block copolymer materials, especially in nanotechnology [3] such as nanolithography [4–7], photonic crystals [8, 9], high density storage systems [10], etc., require well-ordered nanostructures free of topological defects that are usually detrimental to device performance. However, the full development of such ordered structures takes place over very long time scales. Furthermore, the essential mechanisms behind such evolution are still poorly understood. Therefore, increased emphasis has been put on the microstructural control of the copolymer conﬁguration, through e.g., external forces or conﬁnement, particularly for defect removal in order to achieve long range order. In this chapter we survey recent research on the theoretical description of the mechanisms controlling the dynamics of mesophases in extended block copolymer systems, including the motion of topological defects, coarsening of domains in partially ordered, nonequilibrium conﬁgurations, and the response of mesophases to external shear ﬂows for the control of long range order. Our focus here is on a mesoscopic modeling of block copolymer dynamics, based on the fact that the characteristic spatial and temporal scales of structural evolution in nanostructured copolymer materials are far beyond the individual molecular dimension, and hence a coarse-grained, reduced description can be adopted to account for the complex behavior of the system. The mesoscopic model equations as well as the associated amplitude/envelope equations derived from a multiple scale analysis will be introduced below in Sect. 2. The applications to various issues in block copolymer mesophase dynamics are given in Sect. 3 and Sect. 4.

2 Mesoscopic Modeling 2.1 Ginzburg–Landau Model Equations At a mesoscopic level over length scales much larger than the microscopic monomer scale and time scales long enough compared with the polymer chain

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relaxation time, the evolution of a block copolymer melt can be described by an order parameter ﬁeld ψ(r, t) which represents the local density diﬀerence of the constituent monomers, and a local velocity ﬁeld v(r, t). Given that relaxation of the concentration ﬁeld ψ is driven by local dissipation due to free energy minimization, we can adopt the following time-dependent Ginzburg– Landau equation ∂ψ/∂t + v · ∇ψ = −ΛδF/δψ, (1) where Λ is an Onsager kinetic operator (with Λ = −M ∇2 reﬂecting conserved dynamics for the order parameter). For a copolymer system under imposed shear ﬂow, the advection term (v · ∇ψ) in (1) cannot be neglected, and a hydrodynamic equation governing the velocity ﬁeld has to be incorporated into the description of copolymer dynamics. In (1), the coarse-grained free energy functional F is given by (the Ohta–Kawasaki energy [11]) τ u κ F[ψ] = dr − ψ 2 + ψ 4 + |∇ψ|2 2 4 2 B ¯ ¯ − r )(ψ(r ) − ψ), (2) + drdr (ψ(r) − ψ)G(r 2 with ψ¯ the spatial average of ψ. Here the ﬁrst part of the r.h.s. is identical to the Ginzburg–Landau free energy used for the description of phase separation in a binary mixture, and the second part (with the kernel G(r − r ) deﬁned by ∇2 G(r − r ) = −δ(r − r )) represents the long range interactions arising from the connectivity of diﬀerent blocks in a copolymer. In the weak segregation limit (i.e., near the order-disorder transition point), a block copolymer melt can be described by a simpler coarse-grained free energy (the Brazovskii or Leibler energy) [12, 13] 2 . τ 2 g˜ 3 u 4 ζ 2 ∗2 , (3) ∇ + q0 ψ F[ψ] = dr − ψ − ψ + ψ + 2 3 4 2 where the reduced temperature variable τ measures the distance from the order–disorder transition, with τ > 0 for T < TODT , and q0∗ is the wavenumber of the periodic structure. Note that here the order parameter has been ¯ the local deviation of the concentration ﬁeld from replaced by ψ → ψ − ψ, its spatial average. Substituting (3) into the Ginzburg–Landau equation (1), approximating Λ by M q0∗ 2 which is valid near TODT with negligible long range diﬀusion [14], and rescaling all quantities to be dimensionless, we obtain the so-called Brazovskii or Swift–Hohenberg model equation [15] 2 ψ + gψ 2 − ψ 3 , ∂ψ/∂t + v · ∇ψ = ψ − ∇2 + q02 (4) where = τ /ζq0∗ 4 with 0 < 1 corresponding to the weak segregation limit, and q0 = 1 after rescaling although we retain the symbol q0 in what follows for clarity of presentation. For lamellar phases which correspond to

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symmetric block copolymers (ψ¯ = 0) and are our major focus in this chapter, we have g = 0 in (4). Note that the free energy expressions given above are mainly applied to diblock copolymers, and that the generalization to multiblocks is much more complicated. Recent results of more general free energy expressions can be found in [16]. 2.2 Multiple Scale Approach and Amplitude Equation Formalism In the limit of weak segregation ( 1), a multiple scale approach [15,17] can be used to separate fast spatial/temporal scales of a base periodic or modulated structure and slow scales of evolution of its envelope (see Fig. 2), and then be used to derive the amplitude/envelope equations for slowly varying amplitudes of the base pattern. Within this amplitude equation formalism the properties and motion of a single topological defect in ordered phases have been well studied [15]. Here we present an example for a tilt grain boundary conﬁguration of a lamellar phase [18, 19] as shown in Fig. 2. Results for other types of defect conﬁguration can be obtained similarly (some of them will be given in Sect. 3.2 and Sect. 4.1).

Fig. 2. Envelope description of a tilt grain boundary conﬁguration of a lamellar phase in a multiple scale approach. The grain boundary is located at x = xgb , with velocity vgb and boundary thickness ξ ∼ λ0 −1/2 (λ0 = 2π/q0 ).

For a three-dimensional (3D) 90◦ conﬁguration comprising two lamellar domains of mutually perpendicular orientations (ˆ x orientation for domain A and zˆ (the vertical direction in Fig. 2) for domain B), we can expand the order parameter ﬁeld as the superposition of two base modes exp(iq0 x) and exp(iq0 z): 1 ψ = √ [A exp(iq0 x) + B exp(iq0 z) + c.c.] , (5) 3 with complex amplitudes A and B slowly varying in space and time: A = A(X = ¯= B = B(X

1/2

x, Y = x, Y¯ =

1/4

1/4

y, Z = y, Z¯ =

1/4

1/4

z, t),

1/2

z, t).

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Following a standard multiple scale expansion procedure (see e.g., [15, 18]), from the model equation (4) (with g = 0 for lamellar phases) we obtain the following amplitude equations (at O( 3/2 )) [19] 0 / ∂t A = − (2iq0 ∂x + ∂y2 + ∂z2 )2 A − |A|2 A − 2|B|2 A, (6) 0 / 2 2 2 2 2 (7) ∂t B = − (∂x + ∂y + 2iq0 ∂z ) B − |B| B − 2|A| B. The corresponding equations for a 90◦ grain boundary in two dimensions [18] are the same as (6) and (7) except for the absence of the terms proportional to ∂y2 simply due to the lack of the y direction in two-dimensional (2D) space.

3 Defected Structure and Dynamics The Ginzburg–Landau model equation introduced above has been widely applied to the study of mesophase dynamics in block copolymers, including domain evolution and the dynamics of topological defects. The discussion in this section is restricted to some recent research on 2D domain coarsening in lamellar and hexagonal phases controlled by grain boundary motion, as well as on grain boundary conﬁgurations and stability in 3D lamellar patterns. 3.1 Domain Coarsening and Pinning (2D) Lamellar/Stripe Phase Coarsening of defected, multidomain conﬁgurations in lamellar phases (or stripe phases in 2D space) is a long-lasting problem but still remains not well understood. The existence of statistical self-similarity in domain coarsening has been addressed both theoretically [20–24] and experimentally [4,25], with the characteristic length scale R(t) of a domain expected to scale as R(t) ∼ tα . However, no consensus has been reached on the value of the coarsening exponent α, which ranges from 1/5 to 1/2 in diﬀerent studies and appears to depend on the distance to TODT (i.e., the value of ), thermal noise, and the selected linear scale for analysis. A typical conﬁguration in the weak segregation regime obtained by numerical integration of the model equation (4) is shown in Fig. 1b. Large amounts of 90◦ grain boundaries exist, separating diﬀerently oriented domains with curved lamellae, as well as pointlike defects such as disclinations and dislocations. In the case of 1, the evolution is dominated by the motion of grain boundaries since bulk curved lamellae remain largely immobile as a result of the topological constraints set by the disclinations. Grain boundaries move over large distances driven by lamellar curvature and distortion within domains [23,26]. Given the characteristic curvature κ of the lamellae ahead of the grain boundary, the average boundary velocity can be obtained from the analysis of the 2D amplitude equations [26] as

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vgb ∼

−1/2 2

κ ∝ R−2

(8)

due to the fact that the distance between disclinations, which is proportional to the characteristic domain size R, determines the inverse curvature of lamellae constrained by these disclinations. Therefore, a coarsening law R(t) ∼ t1/3 is implied from dimensional analysis of (8). This coarsening behavior has been veriﬁed by the direct numerical solution of (4), which yields the same exponent 1/3 from the probability distribution of lamellar curvatures, a ﬁnite size analysis of the total grain boundary length, and the structure factor of the order parameter ψ [23]. However, when a system is far enough from the order–disorder threshold (i.e., with a larger, ﬁnite value of ), the above coarsening results no longer hold. In this case, the coupling between fast scales of base lamellar patterns and slow scales of the associated amplitudes/envelopes (as described in Sect. 2.2 and Fig. 2), referred to as the nonadiabatic eﬀects, have to be taken into account [24]. This results in an extra contribution to the grain boundary velocity (8), which is proportional to the amplitude of an eﬀective periodic potential induced by the underlying periodicity of the base pattern. Grain boundaries become pinned in this potential, with domains growing up to a pinning length scale given by (9) Rg ∼ λ0 −1/2 exp |α0 | −1/2 /2 , with the lamellar period λ0 = 2π/q0 and α0 a constant of order unity. For small enough (in the weak segregation limit) such that Rg is much larger than the system size, pinning eﬀects are negligible and domain coarsening obeys the t1/3 scaling law as discussed above. With increasing and hence decreasing Rg , defect pinning becomes more pronounced, leading to smaller eﬀective coarsening exponents (α < 1/3) in an intermediate time regime as well as a saturation of domain size and the appearance of glassy conﬁgurations at long times [23, 24]. Defect pinning eﬀects are believed to account for the range of coarsening exponents in previous studies. Hexagonal Phase For nonsymmetric block copolymer melts, mesophases with hexagonal symmetry have been observed and extensively studied in experiments, and can also be well modeled by the coarse-grained equations in Sect. 2. Similar to lamellar patterns, macroscopic samples do not have complete order; instead, polycrystalline conﬁgurations with the presence of topological defects are found, with a typical example given in Fig. 3 which shows three hexagonal domains separated by grain boundaries and is obtained by numerically integrating the model equation (4). The amplitude equation formalism has been used to analyze defect dynamics in hexagonal patterns. By considering nonadiabatic eﬀects arising from the

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Fig. 3. A defected, multidomain conﬁguration in a hexagonal phase obtained by numerical solution of the model equation (4).

coupling of slowly varying amplitudes and the phase of the defects, one can determine the grain boundary velocity to be [27] vgb = −phex sin [2q0 xgb sin(θ/2)] /D,

(10)

where xgb (t) is the grain boundary position, θ is the relative misorientation angle between the two uniform, hexagonal domains on either side of the boundary, and D is a friction coeﬃcient. The pinning force (or Peierls’ force) phex in (10) decays exponentially with the grain boundary thickness ξ, that is, phex ∼ exp [−2a∗ q0 sin(θ/2)ξ] ,

(11)

where a∗ is a dimensionless constant of order unity. In contrast to the case of lamellar phases, the pinning force here cannot be neglected in the√weak segregation limit → 0 due to the ﬁnite value of ξ( = 0) 15λ0 /(8 6πg) (with g given in (4)). For ﬁnite values of , phex is much larger than the pinning force in a lamellar phase (which scales as exp(− −1/2 ) due to ξlamellar ∼ −1/2 ; see also (9)). Therefore, defect pinning is unavoidable in the hexagonal phase, and the coarsening behavior in multidomain, partially ordered samples, as observed in both block copolymer experiments [28] and numerical studies of Ginzburg–Landau equations (1) and (2) [29] (with an eﬀective coarsening exponent around 1/4), likely occurs in an intermediate crossover regime before the system becomes pinned. 3.2 Grain Boundaries (3D) Due to the experimental observation [30, 31] that grain boundary defects predominate in block copolymer bulk samples, in the following we focus on grain boundary conﬁgurations in lamellar systems, which in 3D can be classiﬁed into two types: tilt and twist boundaries, as discussed respectively.

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Tilt Boundaries For a tilt grain boundary separating two diﬀerently oriented domains of lamellar symmetry, the structure can be viewed as the tilting of two grains with respect to an axis lying in the grain boundary plane, and the plane formed by lamellar normals of the two grains is perpendicular to the boundary plane. 90◦ tilt boundaries are commonly observed in 2D systems [17], and a stationary solution for the corresponding boundary conﬁguration (see Fig. 4a) is known to exist, with the same, unique value of wavenumber (q0 ) of the lamellae in both domains [18]. However, this is not the case in 3D patterns. In experiments of 3D lamellar block copolymers, 90◦ tilt boundaries are rarely observed [32], which we believe can be attributed to the ﬁnite wavenumber instability shown below.

Fig. 4. 90◦ tilt grain boundary conﬁgurations in (a) two-dimensional and (b) threedimensional lamellar phases, as calculated from either the order parameter model equation (4) or the amplitude equations (6) and (7). Note that a ﬁnite wavenumber instability develops along the grain boundary plane in the 3D conﬁguration (b).

The amplitude equations governing the evolution of a 90◦ tilt boundary are already given by (6) and (7). It is convenient to expand the complex amplitudes A and B in Fourier series (note that the grain boundary is on the y-z plane): ˆ y , qz , x, t)ei(qy y+qz z) , A(q A = A0 (x) + qy ,qz

B = B0 (x) +

ˆ y , qz , x, t)ei(qy y+qz z) , B(q

(12)

qy ,qz

with A0 (x) and B0 (x) the base state solutions corresponding to a planar and stationary grain boundary structure [18] and only dependent on the coordinate x that is normal to the boundary plane. Substituting the above expansion into (6) and (7) and retaining terms up to ﬁrst order of the perturbation ˆ which amplitudes, we can obtain the time evolution equations for Aˆ and B

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govern the stability of grain boundary conﬁguration. It can be found from the numerical study of these perturbed amplitude equations [19] that any initial ˆ will be ampliﬁed around the grain boundary plane, perturbations of Aˆ and B indicating an instability characterized by a positive perturbation growth rate σ. The maximum growth rate occurs at a ﬁnite wavevector (qy∗ , qz∗ ), and the corresponding wavelengths of maximum instability along the y and z direc∗ ) both decrease with the increase of , as shown in Fig. 5a. tions (λ∗y,z = 2π/qy,z Fig. 5a also indicates that wavelengths along both directions of the grain boundary plane are larger than the wavelength of the base lamellar pattern, and the mode of this ﬁnite wavelength instability is anisotropic on the boundary plane. This can be seen in Fig. 4b which presents a typical 3D grain boundary conﬁguration with ﬁnite wavenumber undulations of the boundary and is obtained from numerical solution of the amplitude equations (6) and (7) in a 2563 system for = 0.08. A similar instability (e.g., anisotropic instability wavelengths given in Fig. 5a and conﬁguration in Fig. 4b) can be derived from direct numerical solution of the order parameter model equation (4).

(a) λz

(b)

-8

4×10

*

-8

2×10

2D

100

λy,z

*

ΔF/V

0

-8

-2×10 *

λy

3D

-8

-4×10

-8

-6×10

10

0.01

ε

0.1

0

100

200

300

400

500

600

700

800

900 1000

t

Fig. 5. (a) Log-log plot of the most unstable wavelengths λ∗y and λ∗z along two directions y and z of the grain boundary plane as a function of . (b) Eﬀective free energy density diﬀerence ΔF/V as a function of time, for both 2D and 3D grain boundary conﬁgurations. See also [19].

Further insight into this instability can be gained from an eﬀective free energy analysis of the amplitude equations. By rewriting (6) and (7) in a potential form ∂t A = −δF/δA∗ ,

∂t B = −δF/δB ∗ ,

(13)

with ∗ denoting complex conjugation and F the eﬀective free energy given by 1 4 |A| + |B|4 + 2|A|2 |B|2 F = dr − |A|2 + |B|2 + 2

2 2

2 2 2

(14) + (2iq0 ∂x + ∂y + ∂z )A + (∂x + ∂y2 + 2iq0 ∂z )B ,

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we can examine the instability by the net energy change ΔF = F −F0 , with F0 the eﬀective energy of the base state. Time evolution of ΔF per unit volume V is plotted in Fig. 5b, for both 2D and 3D grain boundary structures. As expected, a stable 2D conﬁguration yields ΔF > 0, due to the energy penalty of any modulation of the stationary state. In contrast, we have ΔF < 0 for a 3D conﬁguration, indicating the instability as found in the stability analysis of the amplitude equations. Substituting the expansion (12) into (14), we ﬁnd that the negative contribution to ΔF arises from the coupling between A and B lamellar modes which is nonnegligible only in the region around the grain boundary plane. This cross mode coupling dominates in three dimensions, but is not strong enough compared to other positive, stabilizing contributions in the free energy of a 2D system. Further details can be found in [19]. Twist Boundaries A twist grain boundary can be constructed by rotating two lamellar domains with respect to an axis perpendicular to the grain boundary plane, with lamellar normals of the two joint domains staying in a plane that is parallel to the boundary plane. An example for a 90◦ twist boundary is shown in Fig. 6, with the boundary plane exhibiting a doubly periodic morphology. Despite the importance of twist boundaries in block copolymer samples [30], the related theoretical analyses are very limited [33–35]. Here we present a coarse-grained modeling of the twist boundary structure, based on the order parameter model equation and the amplitude description given above.

Fig. 6. A 90◦ twist grain boundary conﬁguration of lamellar phase, which is stable, in contrast to the unstable 3D tilt structure shown in Fig. 4b.

We ﬁrst apply the amplitude equation formalism to the description of twist boundaries. Similar to the tilt boundary case, we separate slowly varying amplitudes and fast spatial and temporal variables of the base lamellar pattern by expanding the order parameter ﬁeld 1 ψ = √ [A exp(iq0 y) + B exp(iq0 z) + c.c.] 3

(15)

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for a single boundary of the conﬁguration of Fig. 6, with the coordinate x normal to the y-z boundary plane. The amplitudes A and B scale as A = A( 1/4 x, 1/2 y, 1/4 z, t) and B = B( 1/4 x, 1/4 y, 1/2 z, t) respectively, given two diﬀerent orientations (ˆ y and zˆ) of constituent lamellar domains and the corresponding anisotropic spatial scalings. Thus, a multiple scale approach to the model equation (4) yields 0 / (16) ∂t A = − (∂x2 + 2iq0 ∂y + ∂z2 )2 A − |A|2 A − 2|B|2 A, 0 / 2 2 2 2 2 ∂t B = − (∂x + ∂y + 2iq0 ∂z ) B − |B| B − 2|A| B. (17) Note the diﬀerence between the above amplitude equations and those for tilt grain boundaries given in (6) and (7), which is due to the diﬀerent relationship between domain orientations and the boundary plane in these two types of grain boundaries. We perform the stability analysis on (16) and (17) by using the procedure described above for the tilt boundary, and obtain the perturbation growth rate σ ≤ 0 for all wavenumbers. This indicates stability of the twist conﬁguration, a result qualitatively diﬀerent from the tilt structure and consistent with the experimental ﬁndings in block copolymer melts. A simple dimensional analysis of (16) and (17) along the x direction (the boundary normal) yields a −1/4 dependence of the boundary thickness, in contrast to the −1/2 result of tilt boundaries. This is in agreement with both a numerical solution of the amplitude equations and of the model equation (4). Thus, in the limit of → 0, i.e., close to the order–disorder threshold, the width of the twist boundary region is much smaller than that of the tilt boundary. All the above results hold for twist boundaries of diﬀerent angles, as shown in [36].

4 Control: Shear Alignment As discussed in Sect. 1, polycrystalline copolymer samples are highly undesirable for most applications. Hence, external forces are usually imposed on defected samples to control microstructural evolution and achieve long range order. In this case, we must address not only the competition between differently oriented domains, but also the response of phases of diverse orientations to the external ﬁeld. Signiﬁcant eﬀects arise from the coupling between mesophase dynamics and hydrodynamic ﬂows in response to imposed ﬁelds. Here we are interested in the particular case of oscillatory shears, a widespread method to promote global order in bulk samples of block copolymers. The focus of the discussion that follows is on recent theoretical analyses within the framework of the coarse-grained modeling introduced in Sect. 2. A topic of special concern regarding shear alignment of lamellar block copolymers is the problem of orientation selection, that is, the selection of a particular lamellar orientation relative to the shear as a function of the parameters of the block copolymer and of the imposed shear. It is convenient to distinguish among three possible orientations of a lamellar phase:

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parallel (with lamellar layers parallel to the shearing surface), perpendicular (with lamellae normal parallel to the vorticity of the shear ﬂow), and transverse (with lamellae normal along the shear direction), as shown in Fig. 7. The mechanisms responsible for the response of the copolymer mesostructure and the orientation selection by shear, especially between parallel and perpendicular, still remain unknown despite intensive studies in recent years, both experimental (see [2] for a recent review, and [37] for recent progress in multiblocks) and theoretical [38–44].

Fig. 7. Shear alignment of a polycrystalline, defected conﬁguration, with three possible uniaxial orientations of lamellar phase: parallel, perpendicular, and transverse.

The coarse-grained model, (1) and (2), combined with the equation governing the local velocity ﬁeld, has been used to study secondary instabilities of the lamellar phase under oscillatory shears, in both 2D [45] and 3D [46]. It has been found that subjected to long wavelength perturbations, uniform lamellar conﬁgurations of all three orientations (parallel, perpendicular, and transverse) can be linearly stable under speciﬁc shearing conditions; however, the stability region turns out to be wider near the perpendicular orientation compared to the parallel one, with a much smaller projection in the transverse direction. Importantly, the viscosity contrast between the microphases of a copolymer is shown to play a negligible role in the stability of wellaligned lamellar structures, although this eﬀect has been proven crucial near the order–disorder transition point [39]. These stability results determine the parameter range over which speciﬁc orientations in the steady state can be in principle observed experimentally. However, the problem of orientation selection is of a dynamical nature, while the stability analysis does not provide for a selection mechanism among simultaneously stable states below the order–disorder transition. Attention below focuses on recent theoretical eﬀorts to model the competition between coexisting lamellar phases of diﬀerent orientations under oscillatory shear ﬂows of low enough shear amplitude γ and frequency ω (with ω < ωc , where ωc is the characteristic relaxation frequency of polymer chains). In the follow-

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ing we separately discuss parallel/transverse and parallel/perpendicular grain boundaries. 4.1 Parallel/Transverse Grain Boundary We consider a 90◦ grain boundary in a 2D x-z space (see Fig. 8), separating two lamellar domains A and B with the same wavenumber q0 but mutually perpendicular orientations. In the absence of shear this conﬁguration is known to be stationary, whereas boundary motion is expected upon the imposition of shear even though both bulk domains are linearly stable. As shown in Fig. 8, lamellae in domain A are transverse to the shear, hence the individual copolymers are compressed, leading to elastic response to the shear, while parallel lamellae in domain B respond like a ﬂuid to the shear. Consequently, extra elastic energy is stored in the compressed domain A, which can be relieved only through the motion of the grain boundary from the parallel (B) to the transverse (A) region.

Fig. 8. Schematic representation of a 2D grain boundary conﬁguration separating parallel and transverse lamellar domains subjected to an oscillatory shear. Also indicated are diﬀerent advection eﬀects of the shear on the two domains. Note that both domains are linearly stable in the corresponding bulk state.

Details of such grain boundary motion can be described at a mesoscopic level by the model equation (4), with the velocity ﬁeld v simply approximated by v0 , the imposed shear ﬂow, i.e., v v0 =

da ˆ = γω(cos ωt)z x ˆ, zx dt

(18)

with γ the shear strain amplitude, ω the shear frequency, and the shear strain a(t) = γ sin ωt. The shear ﬂow will advect the lamellae through the term v·∇ψ in (4), but with diﬀerent eﬀects on the two domains (see Fig. 8): no advection occurs in the parallel domain B because v0 · ∇ψ = 0, but transverse lamellae are distorted as a result of a nonzero advection term. The resulting motion of the grain boundary can then be obtained by numerical solution of (4) [47], with the introduction of a sheared frame of reference with coordinates

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Zhi-Feng Huang and Jorge Vi˜ nals

x = x − a(t)z

and z = z,

(19)

corresponding to an nonorthogonal basis set x ˆ = x ˆ and zˆ = a(t)ˆ x + zˆ. Typical results for the grain boundary location xgb and velocity vgb (both in the sheared frame) as a function of time are given in Fig. 9a. Note the oscillatory motion of the grain boundary, both forward (with positive vgb ) and backward (with negative vgb ). Boundary motion occurs through diﬀusive relaxation of the order parameter ﬁeld around the grain boundary, in addition to the rigid distortion of the transverse lamellae that accounts for the free energy imbalance between the two domains. This diﬀusive relaxation leads to the break-up and reconnection of the transverse lamellae at the boundary, and plays an important role in enabling dissipation of the excess energy stored in the transverse bulk region. Note also that the net motion of the grain boundary is still driven by free energy reduction, with positive value of time averaged boundary velocity vgb t (over a shearing period T0 ) as shown in Fig. 9b. This net average velocity monotonously increases with shear frequency ω and strain amplitude γ [47]. 2

0.01

(a)

(b) 0.008

0.006

t

x’gb / λ0

1

0.05

0 vgb

0.004 0

0.002

-0.05

-1

1

0

1

2

2

1.5

3

t/T0

model equation amplitude equations analytic approximation

3

2.5

4

0

0

0.02

0.04

ω

0.06

0.08

0.1

Fig. 9. (a) Time evolution of the grain boundary location in the sheared frame for = 0.04, γ = 0.3, and ω = 0.01. Here T0 = 2π/ω is the shear period. Inset: boundary velocity as a function of time; the results calculated from the model equation (4) are plotted as the solid curve, while those from the amplitude equations (20) and (21) are indicated as the open circles. (b) Plots of temporal average of grain boundary velocity vs. shear frequency ω, for = 0.04 and γ = 0.3. Solid line, results obtained from numerical solution of the model equation (4). Symbols, from the amplitude equations (20) and (21). Dashed line, from the analytic approximation (23). For more details see [47, 48].

Further information can be obtained from analysis of the corresponding amplitude equations, based on the fact that in the vicinity of the order– disorder transition, the characteristic length and time scales of defect motion are governed by slowly evolving amplitudes of the base patterns. Expressed

Coarse-Grained Modeling of Mesophase Dynamics in Block Copolymers

in the sheared frame (x , z ), up to O( equations [48]

3/2

385

) the amplitudes A and B obey the

∂t A = { − [2iq0 (∂x − a∂z ) + ∂z2 − q02 a2 ]2 }A − |A|2 A − 2|B|2 A, ∂t B = { − [2iq0 (∂z − a∂x ) + ∂x2 ]2 }B − |B|2 B − 2|A|2 B.

(20) (21)

Note that the term proportional to q02 a2 contributes to the equation for A in the entire transverse region, and represents the compression eﬀect described above. On the other hand, terms proportional to a∂x and q02 a2 ∂x are nonzero only around the grain boundary and incorporate local diﬀusive relaxation of the order parameter (or of the copolymer monomer concentration). Results calculated from (20) and (21) agree with those from the model equation (4), especially for small frequencies, as shown in Fig. 9. Diﬀusive relaxation of the monomer concentration ﬁeld incorporates the reorientation of copolymer molecules at the grain boundary, leading to diﬀusive monomer redistribution around the boundary so that the extent of the phase of transverse lamellae of higher energy can be reduced. We observe additional phase diﬀusion in the transverse phase corresponding to local wavenumber readjustment. It develops near the boundary and slowly propagates into the bulk, showing as a linear spatial dependence of the phase ϕA of the complex amplitude A, i.e., (22) ϕA ∝ −δq · x , with δq > 0, and hence a wavenumber decrease qx → q0 − δq. We have obtained ϕA from a one-dimensional (1D) approximation to (20) and (21) by assuming a planar grain boundary with only spatial dependence on x . The results are illustrated in Fig. 10, with the phase shift δq agreeing with that directly calculated from the model equation (4). At intermediate frequencies for which diﬀusive eﬀects are less important than elastic distortion eﬀects, we can derive an analytic expression for the boundary velocity [48], vgb =

2

d 0

F (x → +∞) − F (x → −∞) , +∞ dz −∞ dx (|∂x A |2 + |∂x B|2 )/d

(23)

with the eﬀective driving force F (x → +∞) − F (x → −∞) 1 = (−2q0 δq + q02 γ 2 sin2 ωt)2 [2 − (−2q0 δq + q02 γ 2 sin2 ωt)2 ] 2

(24)

arising from the free energy increase in the transverse region. The denominator of (23) represents an inverse mobility coeﬃcient that depends on the grain boundary proﬁle (A , B), with A deﬁned by A = A exp(−iδq · x ) as a result of wavenumber compression. The time average of (23) is plotted in Fig. 9b as a function of ω, agreeing well with the direct numerical integration of the

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Zhi-Feng Huang and Jorge Vi˜ nals 6 5

100 (γ=0.4)

50

4

ϕA

3

t=50 (γ=0.3)

2

100

1 0

(b) -1

3000

3500

4000

ix’

Fig. 10. Spatial dependence of phase ϕA of the complex amplitude A in terms of the grid index ix along the x direction, given by the 1D amplitude equations with Lx = 4096, = 0.04, ω = 0.04, and γ = 0.3 (solid lines) and 0.4 (dashed lines), and at times t = 50T0 and 100T0 . Also shown for comparison are dotted lines with slopes −δqΔx = −3Δx /128 (for γ = 0.3) and −5Δx /128 (for γ = 0.4), with Δx the grid spacing. Values of δq originate from the long time solution of the model equation (4).

amplitude equations at high enough frequencies. However, as frequency decreases the deviation between this analytic approximation and the numerical result increases, and (23) fails at very low frequencies, since diﬀusive eﬀects become more pronounced and lead to a smaller amplitude of the boundary velocity as compared to that given by the approximation (23). Note that the above results and discussion are not conﬁned to mesophase dynamics of block copolymers; instead, they should hold in other systems of the same symmetry in which solid-like or elastic response occurs on one side of a grain boundary, resulting in a structural distortion by the shear, and ﬂuid-like response is expected on the other side. 4.2 Parallel/Perpendicular Conﬁguration As shown above, we already have a fairly complete picture of the competition between coexisting parallel and transverse orientations, including the driving force for boundary motion that originates from the diﬀerent response to the shear ﬂow, as well as local phase diﬀusion eﬀects. On the other hand, ﬂuidlike response to the shear is expected for both parallel and perpendicular orientations; thus, the treatment given above for the parallel/transverse case cannot apply here. An alternative approach regarding mesoscopic hydrodynamics in block copolymers has been pursued [49,50]. The emphasis is put on the local velocity ﬁeld v = (vx , vy , vz ) for a block copolymer, which satisﬁes the equation ρ (∂v/∂t + v · ∇v) = −∇p + ∇ · σ D ,

(25)

Coarse-Grained Modeling of Mesophase Dynamics in Block Copolymers

387

as well as incompressibility ∇ · v = 0, with ρ the copolymer density, p the pressure ﬁeld, and σ D the dissipative stress tensor that still needs to be speciﬁed. Instead of addressing the velocity ﬁeld at spatial scales of the lamellar spacing, we adopt a long wavelength description of the dissipative part of the stress tensor σ D in (25) [51] D σij = ηDij + α1 n ˆin ˆj n ˆk n ˆ l Dkl + α56 (ˆ ni n ˆ k Djk + n ˆj n ˆ k Dik )

(26)

with i, j, k = x, y, z, which is the most general form appropriate for a phase of uniaxial symmetry, such as nematic liquid crystals [52, 53]. In (26), η is the Newtonian viscosity, α1 and α56 are two independent viscosity coeﬃcients for ˆ = (ˆ nx , n ˆy , n ˆ z ) represents the slowly uniaxial phases, Dij = ∂i vj + ∂j vi , and n varying normal to the lamellar planes. Note that the last two terms of (26) reﬂect the lowest order deviation from Newtonian behavior in the limits of low frequency and large scale ﬂow that is compatible with the lamellar phase symmetry. The assumption (26) leads to diﬀerent viscous responses in parallel and perpendicular regions: the dynamic viscosity η (= G /ω, with G the loss modulus) for a fully-ordered lamellar structure is a function of its orientation, = η + α56 ηparallel

and ηperpendicular = η,

(27)

consistent with experimental evidence [54, 55]. This proposed deviation from Newtonian response leads to a dynamical mechanism that favors one orientation over the other, even in regions in which both are equally stable in bulk. We consider the simple conﬁguration shown in Fig. 11 to address the competition between coexisting parallel and perpendicular domains under oscillatory shears. Both domains are linearly stable in the corresponding bulk state. To study the dynamics of competing lamellar domains, in principle the coupled equations (1), (25), and (26) should be solved, but this is complicated due to the ψ dependence of the dissipative stress tensor σ D , and nonzero ﬂuid inertia. Here we adopt the approximation that the dependence of σ D on ψ perturbations can be neglected. Therefore, we ﬁrst obtain the velocity ﬁeld from (25) and (26) without considering order parameter diﬀusion in (1), and then discuss the resulting advection eﬀects of the ﬂow ﬁeld obtained on the order parameter. The base state for the conﬁguration shown in Fig. 11 (with a planar interface at z = dA ) is the same as that of two superposed Newtonian ﬂuids with viscosities μA = η and μB = η + α56 . We examine the hydrodynamic stability of this conﬁguration and the eﬀects of any instability on domain competition. Note that the conﬁguration here is analogous to that of a two-layer Newtonian ﬂuid with viscosity stratiﬁcation which is known to be unstable under steady [56] and oscillatory [57] shears. We extend the analysis to the current system characterized by the additional viscosities given in (26). Expanding both velocity and pressure ﬁelds of A and B domains as ' A,B u ˆi (qx , qy , z, t) exp [i(qx x + qy y)] , viA,B = VA,B δiy + qx ,qy

pA,B = p0 + pA,B ,

(28)

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Zhi-Feng Huang and Jorge Vi˜ nals

Fig. 11. Schematic representation of a parallel/perpendicular conﬁguration under oscillatory shear. A hydrodynamic instability occurs at the interface, with the resulting secondary ﬂows (of ux = 0 and uy , uz = 0 corresponding to the critical mode of instability) also indicated.

with VA,B (periodic in time t) and p0 the base state solutions, and substituting into (25) and (26), we obtain the following linear stability equations governing and u ˆA,B : the perturbed velocity ﬁelds u ˆA,B x z / 0 2 2 2 2 A Re (∂t + iqy VA )(∂z2 − q 2 )ˆ uA uA ˆz z − iqy (∂z VA )ˆ z = (∂z − q ) u 2 2 2 −α56 qx2 (∂z2 − q 2 )ˆ uA ˆA z + iqx [2α1 qx − α56 (∂z − q )]∂z u x,

(29)

0 / 2 2 Re ∂t (∂z2 − q 2 )ˆ uA ˆA uA x + iqy (∂z − q )(VA u x ) + 2qx qy (∂z VA )ˆ z 2 2 2 A ˆA ux , = (1 + α56 )(∂z2 − q 2 )2 u x − 2α1 qx (∂z − qy )ˆ

for the perpendicular domain A (0 ≤ z ≤ dA ), and 0 / 2 uB uB Re (∂t + iqy VB )(∂z2 − q 2 )ˆ z z − iqy (∂z VB )ˆ 2 2 B = (1 + α56 )(∂z2 − q 2 )2 u ˆB ˆz , z − 2α1 q ∂z u

(30)

(31)

0 / 2 2 Re ∂t (∂z2 − q 2 )ˆ uB ˆB uB x + iqy (∂z − q )(VB u x ) + 2qx qy (∂z VB )ˆ z 2 B 2 2 3 B ˆB ˆx − iqx ∂z u ˆB ˆz , (32) = (∂z2 − q 2 )2 u x + α56 (∂z − q ) ∂z u z − 2iqx α1 ∂z u for the parallel domain B (dA ≤ z ≤ 1). Here Re = ρωd2 /η (with d the distance between shearing planes) is the Reynolds number, q 2 = qx2 + qy2 , and the above equations have been made dimensionless by introducing a length scale d, a time scale ω −1 , and by rescaling all viscosities by η (so that μA = 1, μB = 1 + α56 ). Boundary conditions include the continuity of velocity and stress and the kinematic condition at the interface z = dA + ζ(x, y, t), as well as rigid boundary conditions on the planes z = 0 and z = d (= 1 after rescaling) [50].

Coarse-Grained Modeling of Mesophase Dynamics in Block Copolymers

389

It is important to ﬁrst address the magnitude of two dimensionless parameters: the Reynolds number Re and the rescaled interfacial tension Γ = Γ/(ηωd) (with Γ the interfacial tension, as appearing in the boundary condition for the normal stress). For a typical block copolymer system of ρ ∼ 1 g cm−3 , η ∼ 104 − 106 P, Γ ∼ 1 dyn/cm, and d ∼ 1 cm, we have Re/ω ∼ 10−4 − 10−6 s and

Γ ω ∼ 10−4 − 10−6 s−1 .

(33)

Hence, Re 1 for the frequency range of interest, and the above stability equations (29)–(32) combined with boundary conditions can be solved analytically by an expansion of velocity and interfacial perturbations in orders of Re. Detailed solutions are given in [49, 50]; in the following we only discuss major results. The solutions of the perturbed velocity ﬁelds u ˆA,B z,x and interfacial proﬁle ˆ ζ (the Fourier transform of ζ) can be written as σt A,B u ˆA,B z,x (qx , qy , z, t) = e φz,x (qx , qy , z, t), ˆ x , qy , t) = eσt h(qx , qy , t), ζ(q

(34)

where σ is a Floquet exponent, representing the perturbation growth rate, and functions φA,B z,x and h are periodic in time when qy = 0 (according to Floquet’s theorem) or time independent for qy = 0. In the limit of Re → 0 with ﬁnite surface tension Γ = O(1), which might correspond to very low frequency ω according to (33), we ﬁnd σ ≤ 0 for all wavenumbers qx and qy due to the stabilizing eﬀect of surface tension, leading to a stable conﬁguration, or to coexistence of parallel and perpendicular domains under shear. However, diﬀerent results can be obtained for larger frequencies in the range ω ∼ 1 s−1 , as in most experiments, yielding Re 1, Γ 1, and Γ /Re = O(1). In this case, the perturbation growth rate σ is given by 1 B B (qx , qy ), (qx , qy )Γ /Re + δ 2 γ 2 fz1 σ/Re = fz0 2 1 1 = δ 2 f1 γ 2 q 2 − (θf0 ω −2 − δ 2 f2 γ 2 )q 4 + O(q 6 ), 2 2

(35) (36)

where δ = (μA /μB )/(dA + dB μA /μB ) with dB = 1 − dA , θ = Γ/(ρd3 ), and B B , fz1 , f0 , f1 , and f2 are complicated but known functions of α1 , α56 , and fz0 dA , but do not depend on the shear parameters γ and ω. Equation (36) is obtained from the long wave expansion of (35) along the y direction only, with qx = 0 and qy = q, based on the result that the maximum instability given by the numerical evaluation of (35) occurs at qx = 0. Equation (35) presents the stabilizing eﬀect of surface tension (given by the ﬁrst term of B the r.h.s., with fz0 ≤ 0 for all wavenumbers), and the destabilizing eﬀect of the imposed shear (given by the last term proportional to γ 2 , with maximum B being positive for certain range of parameters α1 , α56 , and dA ). value of fz1 A typical proﬁle of the perturbed velocity ﬁelds is given in Fig. 12. At the most unstable wavevector (qx = 0, qy = 0), the associated secondary ﬂow is

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Zhi-Feng Huang and Jorge Vi˜ nals

given by u ˆx = 0 and u ˆz = φz exp(σt), with the amplitude φz plotted in Fig. 12 as a function of spatial position z at t = T0 . We ﬁnd that ﬂow ﬂuctuations are directed only along the z and y directions (due to the incompressibility condition), develop around the interface, and relax respectively into the bulk regions of the perpendicular (A) and parallel (B) domains. 0.1

(Perpendicular A)

(Parallel B)

0.05

φz (t=T0)

Imag(φz), α56=9 0

Real(φz)<<1

Imag(φz), α56=−0.9 -0.05

φx = 0 -0.1 0

0.1

0.2

0.3

0.4

0.5

z

0.6

0.7

0.8

0.9

1

Fig. 12. Spatial dependence of the velocity amplitude φz for the most unstable mode, at time t = T0 (= 2π here) and with parameters α56 = −0.9 or 9 (i.e., μB = 1/10 or 10), α1 = 1, γ = 1, dA = 1/2, and Re = 5 × 10−4 .

The resulting stability diagram as a function of the eﬀective viscosity contrast μB and the domain thickness ratio dA /dB , as determined by (36), is shown in Fig. 13 (with σ > 0 indicating instability). The diagram is sym−1 , and shows a metric with respect to μB → μ−1 B and dA /dB → (dA /dB ) thin-layer eﬀect: instability occurs when the thinner domain has larger effective viscosity. This thin-layer eﬀect is analogous to the case of viscosity stratiﬁcation in Newtonian ﬂuids [58], but of diﬀerent origin: here the viscosity contrast is caused by the orientation dependence of the viscous response in uniaxial lamellar phases. Given the above interfacial instability and related perturbed ﬂows, we can infer the response of the lamellar domains, and argue about orientation selection. Since at instability ux = 0 and uz = 0, as shown schematically in Fig. 11, parallel lamellae will be compressed and expanded by the z-direction secondary ﬂow due to a nonzero advection term u · ∇ψ in (1), while no such eﬀect will occur for perpendicular lamellae. This situation is similar to the case discussed in Sect. 4.1 in which the ﬂow induced free energy imbalance leads to interface motion towards the distorted region (which is the parallel domain in the present case). Therefore, we argue that the hydrodynamic instability results in growth of the perpendicular lamellae at the expense of the parallel lamellae and hence the selection of the perpendicular orientation, whereas stability indicates the coexistence of parallel and perpendicular phases, as shown in the diagram of Fig. 13.

Coarse-Grained Modeling of Mesophase Dynamics in Block Copolymers 10

4

10 10

3

2

Perpendicular

Coexistence

(σ > 0)

μB (=1+α56)

Parallel / Perpendicular 10 10 10 10

1

0

-1

-2

Coexistence

Perpendicular

Parallel / Perpendicular

(σ > 0)

10

391

-3

-4

100.02

0.1

1

10

50

dA / dB

Fig. 13. Stability diagram of viscosity contrast μB vs. domain thickness ratio dA /dB , for phases of uniaxial symmetry. The selection between parallel and perpendicular orientations is determined by hydrodynamic stability, as addressed in [49,50].

Equation (36) can also be used to examine the dependence of the maximum instability growth rate σm on shear parameters. The line of constant σm is given by 2(θf0 ησm )1/4 γω 3/4 = 1/4 , (37) ρ δ(f1 d)1/2 which is equivalent to γω 3/4 = const. for a given block copolymer system. Since σm is proportional to Re 1, it is a very small quantity, perhaps of the order of the inverse observation time for typical experiments. In this case, this line can determine an eﬀective stability boundary. In this regard we note that the experimentally determined boundary separating perpendicular and parallel orientations in poly(styrene)-poly(isoprene) (PS-PI) copolymers [37, 59] is approximately given by γω = const. Finally, it is important to note that a polycrystalline conﬁguration always involves a variety of domain orientations and sizes. We would expect that lamellar domains with a component of the orientation along the transverse direction with respect to the shear will be eliminated quickly, according to Sect. 4.1, leaving only parallel and perpendicular domains. Their competition has been addressed in Sect. 4.2. Further eﬀorts beyond the stability analysis presented here are still necessary for a complete understanding of orientation selection in polycrystalline samples.

5 Perspectives One of the major challenges for widespread applications of nanostructured soft materials such as block copolymers is achieving precise microstructural control, and hence the study of the mechanisms underlying structure, nonlinear dynamics, and response of nanoscale phases needs to be undertaken. All three

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Zhi-Feng Huang and Jorge Vi˜ nals

aspects are strongly coupled to each other. In this chapter we have presented some recent research regarding these aspects in block copolymer mesophases, including structure and stability of topological defects, mesoscopic dynamics of domain evolution, as well as the eﬀects of external shears on response and orientation selection of mesophases. Although our primary focus of attention is on block copolymers, the methods and results are applicable to other types of mesophases of the same symmetry. However, our understanding in this ﬁeld is still far from complete, and future research would focus on further clarifying the interplay between structure, dynamics, and response. Examples include defect dynamics and domain coarsening, including hydrodynamic eﬀects that follow from the mesoscopic viscoelasticity of copolymers. Of particular interest are the nonlinear response of the copolymer system to external forces (e.g., shears, electric ﬁelds, etc.), defect removal and anisotropic domain coarsening processes, and the dynamics of block copolymer thin ﬁlms under surface conﬁning which serve as an important system for self-directed self-assembly of nanostructured copolymers.

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Eﬀective Interactions in Soft Materials Alan R. Denton Department of Physics, North Dakota State University, Fargo, North Dakota, 58105-5566, USA

1 Introduction Soft condensed matter systems are typically multicomponent mixtures of macromolecules and simpler components that form complex structures spanning wide ranges of length and time scales [1–6]. Common classes of soft materials are colloidal dispersions [7–10], polymer solutions and melts [11–14], amphiphilic systems [15,16], and liquid crystals [17–19]. Among these classes are many biologically important systems, such as DNA, proteins, and cell membranes. Most soft materials are intrinsically nanostructured, in that at least some components display signiﬁcant variation in structure on length scales of 1-10 nanometers. Many characteristic traits of soft condensed matter, e.g., mechanical fragility, sensitivity to external perturbation, and tunable thermal and optical properties, result naturally from the mingling of microscopic and mesoscopic constituents. The complexity of composition that underlies the rich physical properties of soft materials poses a formidable challenge to theoretical and computational modelling eﬀorts. Large size and charge asymmetries between macromolecules (e.g., colloidal or polyelectrolyte macroions) and microscopic components (e.g., counterions, monomers, solvent molecules) often render impractical the explicit modelling of all degrees of freedom over physically signiﬁcant length and time scales. Model complexity can be greatly reduced, however, by pre-averaging (coarse-graining) the degrees of freedom of some of the microscopic components, thus mapping the original model onto an eﬀective model, with a reduced number of components, governed by eﬀective interparticle interactions. The concept of eﬀective interactions has a long history in the statistical mechanics of liquids [20] and other condensed matter systems, dating back over a half-century to the McMillan–Mayer theory of solutions [21], the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory of charge-stabilised colloids [22, 23], and the pseudopotential theory of simple metals and alloys [24, 25]. In modelling materials properties of simple atomic or molecu-

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lar liquids and crystals, it is often justiﬁable to average over electronic degrees of freedom of the constituent molecules. The coarse-grained model then comprises a collection of structureless particles interacting via eﬀective intermolecular potentials whose parameters depend implicitly on the ﬁner electronic structure. In recent years, analogous methods have been carried over and adapted to the realm of macromolecular (soft) materials. Eﬀective interparticle interactions prove especially valuable in modelling materials properties of soft matter systems, which depend on the collective behaviour of many interacting particles. In studies of thermodynamic phase behaviour, for example, eﬀective interactions provide essential input to molecular (e.g., Monte Carlo and molecular dynamics) simulations and statistical mechanical theories. While such methods can be directly applied, in principle, to an explicit model of the system, brute force applications are, in practice, often simply beyond computational reach. Consider, for example, that a molecular simulation of only 1000 macroions, each accompanied by as few as 100 counterions, entails following the motions of 105 particles, computing at each step electrostatic and excluded-volume interactions among the particles. A far more practical strategy applies statistical mechanical methods to an eﬀective model of fewer components. This chapter reviews the statistical mechanical foundations underlying theories of eﬀective interparticle interactions in soft matter systems. After ﬁrst identifying and deﬁning the main systems of interest in Sect. 2, several of the more common theoretical methods are sketched in Sect. 3. Concise derivations are given, in particular, for response theory, density-functional theory, and distribution function theory. Although these methods are all well established, the interconnections among them are not widely recognized. An eﬀort is made, therefore, to demonstrate the underlying unity of these seemingly disparate approaches. Practical implementations are illustrated in Sect. 4, where recent applications to charged colloids, colloid-polymer mixtures, and polymer solutions are outlined. In the limited space available, little more than a sample of many methods and applications can be included. Complementary perspectives and details can be found in several excellent reviews [26–29]. Finally, in Sect. 5, a gaze into the (liquid) crystal ball portends an exciting outlook for the ﬁeld.

2 Systems of Interest The main focus of this chapter is eﬀective interactions among macromolecules dispersed in simple molecular solvents. The term “simple” here implies simplicity of molecular structure, not necessarily properties, thereby including water – the most ubiquitous, biologically relevant, and anomalous solvent. The macromolecules may be colloidal (or nano-) particles, polymers, or amphiphiles and may be electrically neutral or charged, as in the cases of charge-

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stabilised colloids, polyelectrolytes (including biopolymers), and ionic surfactants. Colloidal suspensions [7] consist of ultra-divided matter dispersed in a molecular solvent, and are often classiﬁed as lyophobic (“solvent hating”) or lyophilic (“solvent loving”), according to the ease with which the particles can be redispersed if dried out. Depending on density, colloidal particles are typically nanometers to microns in size – suﬃciently large to exhibit random Brownian motion, perpetuated by collisions with solvent molecules, yet small enough to remain indeﬁnitely suspended against sedimentation. The upper size limit can be estimated by comparing the change in gravitational energy as a colloid traverses one particle diameter to the typical thermal energy kB T at absolute temperature T , where kB is Boltzmann’s constant. To better appreciate these length scales, consider repeatedly dividing a cube of side length 1 cm until reaching ﬁrst the width of a human hair (10-100 μm), then the diameter of a colloid, and ﬁnally the diameter of an atom. How many cuts are required? Polymers [11, 12] are giant chainlike, branched, or networked molecules, consisting of covalently linked repeat units (monomers), which may be all alike (homopolymers) or of diﬀering types (heteropolymers). Polyelectrolytes [13, 14] are polymers that carry ionizable groups. Depending on the nature of intramolecular monomer-monomer interactions, polymer and polyelectrolyte chains may be stiﬀ or ﬂexible. Flexibility can be quantiﬁed by deﬁning a persistence length as the correlation length for bond orientations, i.e., the distance along the chain over which bond orientations become decorrelated. At one extreme, rigid rodlike polymers have a persistence length equal to the contour length of the chain. At the opposite extreme, freely-jointed polymers are random-walk coils with spatial extent best characterized by the radius of gyration, deﬁned as the root-mean-square displacement of monomers from the chain’s centre of mass. Amphiphilic molecules [15,16] consist of a hydrophilic head group joined to a hydrophobic tail group, usually a hydrocarbon chain. The head group may be charged (ionic) or neutral (nonionic). When suﬃciently concentrated in aqueous solution, surfactants and other amphiphiles organize (self-assemble) into regular structures that optimize exposure of head groups to the exterior water phase, while sequestering the hydrophobic tails within. The various structures include spherical and cylindrical micelles, bilayers, vesicles (bilayer capsules), and microemulsions. Common soap ﬁlms, for example, are bilayers of surfactants (surfactant-water-surfactant sandwiches) immersed in air, while biological membranes are bilayers of phospholipids immersed in water. Relative stabilities of competing structures are governed largely by concentration and geometric packing constraints, as determined by the relative sizes of head and tail groups [30]. Colloids, polyelectrolytes, and amphiphiles can acquire charge in solution through dissociation of ions from chemical groups on the colloidal surfaces, polymer backbones, or amphiphile head groups. For ions of suﬃciently low va-

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lence, the entropic gain upon dissociation exceeds the energetic cost of charge separation, resulting in a dispersion of charged macroions and an entourage of oppositely charged counterions. In an electrolyte solvent, charged macroions interact with one another, and with charged surfaces, via electrostatic interactions that are screened by intervening counterions and salt ions in solution. Equilibrium and nonequilibrium distributions of ions are determined by a competition between entropy and various microscopic interactions [30], including repulsive Coulomb and steric interactions and attractive van der Waals (e.g., dipole-induced-dipole) interactions. By changing system parameters, such as macroion properties (size, charge, composition), and solvent properties (salt concentration, pH, dielectric constant, temperature), the range and strength of interparticle interactions can be widely tuned. Rational control over the enormously rich equilibrium and dynamical properties of macromolecular materials relies on a fundamental understanding of the nature and interplay of microscopic interactions. In all of the systems of interest, the macromolecules possess some degree of internal structure. In charged colloids and polyelectrolytes, a multitude of microscopic degrees of freedom are associated with the distribution of charge over the macroion and the distribution of counterions throughout the solvent. In polymer and amphiphilic solutions, the polymer chains or amphiphilic assemblies have conformational freedom. Furthermore, the solvent itself contributes a vast number of molecular degrees of freedom. The daunting prospect of explicitly modelling multicomponent mixtures on a level so ﬁne as to include all molecular degrees of freedom motivates the introduction of eﬀective models governed by eﬀective interactions. The loss of structural information upon coarse graining necessitates an inevitable compromise in accuracy. The art of deriving and applying eﬀective interactions lies in crafting approximations that are computationally manageable yet capture the essential physics.

3 Eﬀective Interaction Methods 3.1 Statistical Mechanical Foundation Eﬀective interaction methods have a rigorous foundation in the statistical mechanics of mixtures [31]. These methods rest on the premise that, by averaging over the degrees of freedom of some of the components, a multicomponent mixture can be mapped onto an eﬀective model, with a reduced number of components. While the true mixture is subject to bare interparticle interactions, the reduced model is governed by coarse-grained, eﬀective interactions. In charged colloids, for example, averaging over coordinates of the solvent molecules and of the microions (counterions and salt ions) maps the suspension onto an eﬀective one-component model of mesoscopic “pseudomacroions” subject to microion-induced eﬀective interactions. The bare electrostatic (Coulomb) interactions between macroions in the true suspension are replaced by screenedCoulomb interactions in the eﬀective model. Similarly, in mixtures of colloids

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and non-adsorbing polymers, averaging over polymer degrees of freedom leads to polymer-induced eﬀective interactions between the colloids. Generic Two-Component Model We consider here a simple, pairwise-interacting, two-component mixture of Na particles of type a and Nb of type b, obeying classical statistics, conﬁned to volume V at temperature T . A given system is modelled by three bare interparticle pair potentials, vaa (r), vbb (r), and vab (r), assumed here to be isotropic, where r is the distance between the particle centres. The model system could represent, e.g., a charge-stabilised colloidal suspension, a polyelectrolyte solution, or various mixtures of colloids, nonadsorbing polymers, nanoparticles, or amphiphilic assemblies (micelles, vesicles, etc.). For simplicity, the discussion is conﬁned to binary mixtures, although the methods discussed below easily generalize to multicomponent mixtures. Throughout the derivations, it may help to visualize, for concreteness, the a particles as colloids and the b particles as counterions.

Fig. 1. Left: Generic model of a binary mixture of species labelled a and b. Right: Eﬀective one-component model, after coarse-graining of b species, and geometry for physical interpretation of response theory. Vectors r and r deﬁne centre-to-centre displacements of a particles. Vectors r1 , r2 , and r3 deﬁne points at which either the “external” potential of the a particles acts or a change is induced in the density of b particles (see Sect. 3.2). Reprinted with permission from [40]. Copyright (2004) by the American Physical Society.

Even before analysing the system in detail, it should be conceptually apparent that, since particles inevitably inﬂuence their environment, the presence of particles of one species can aﬀect the manner in which all other particles interact. Familiar analogies may be identiﬁed in any mixture of interacting entities – from inanimate particles to living cells, organisms, and ecosystems. In a simple binary mixture, for example, particles of type b can be regarded as inducing interactions between a particles. The induced interactions, which act in addition to bare aa interactions, depend on both the ab interactions and the

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distribution of b particles. The eﬀective interactions, which are simply sums of induced and bare interactions, may be many-body in character, even if all bare interactions are strictly pairwise, and may depend on the thermodynamic state (density, temperature, etc.) of the system. Reduction to Eﬀective One-Component Model These qualitative observations are now quantiﬁed by developing a statistical description of the system. We start from the Hamiltonian function H, which governs all equilibrium and dynamical properties of the system, and assume pairwise bare interactions. The Hamiltonian naturally separates, according to H = K + Haa + Hbb + Hab , into the kinetic energy K and three interaction terms: Nα 1 Hαα = vαα (rij ), α = a, b (1) 2 i =j=1

and Hab =

Nb Na

vab (rij ),

(2)

i=1 j=1

where rij = |ri − rj | denotes the separation between the centres of particles i and j, at positions ri and rj . Within the canonical ensemble (constant Na , Nb , V , T ), the thermodynamic behaviour of the system is governed by the canonical partition function Z =

exp(−βH)a b ,

(3)

where β = 1/kB T and · · ·α denotes a classical canonical trace over the coordinates of particles of type α: 1

exp(−βH)α = · · · drNα exp(−βH), (4) dr 1 α Nα !Λ3N α with Λα being the respective thermal de Broglie wavelength. The two-component mixture can be formally mapped onto an equivalent one-component system by performing a restricted trace over the coordinates of only the b particles, keeping the a particles ﬁxed. Thus, without approximation, Z = exp(−βHaa ) exp[−β(Hbb + Hab )]b a = exp(−βHeﬀ )a ,

(5)

Heﬀ = Haa + Fb

(6)

where is the eﬀective Hamiltonian of the equivalent one-component system and Fb = −kB T ln exp [−β(Hbb + Hab )]b

(7)

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can be physically interpreted as the Helmholtz free energy of the b particles in the presence of the ﬁxed a particles. Equations (5)-(7) provide a formally exact basis for calculating the eﬀective interactions. It remains, in practice, to explicitly determine the eﬀective Hamiltonian by approximating the ensemble average in (7). Next, we describe three general methods of attack – response theory, density-functional theory, and distribution function theory. While these methods ultimately give equivalent results, they have somewhat diﬀering origins and conceptual interpretations. 3.2 Response Theory The term “response theory” can have varying speciﬁc meanings, depending on discipline and context, but is used here to denote a collection of statistical mechanical methods that describe the response, in a multicomponent condensed matter mixture, of the density of one component to the potential generated and imposed by another component. Response theory has been systematically developed and widely applied, over the past four decades, in the theory of simple metals [20, 24, 25] to describe the quantum mechanical response of valence electron density to the electrostatic potential of metallic ions. More recently, similar methods have been carried over and adapted to classical soft matter systems, in particular, to charge-stabilised colloidal suspensions [32–34], polyelectrolytes [35–37], and colloid-polymer mixtures [38,39]. Although most applications have been restricted to the linear response approximation, which assumes a linear dependence between the imposed potential (cause) and the density response (eﬀect), increasing attention is being devoted to nonlinear response. Below, we outline the key elements of response theory, including both linear and nonlinear approximations, in the context of a classical binary mixture. Perturbation Theory To approximate the free energy (7) of one component (b particles) in the presence of another component (a particles) it is often constructive to regard the a particles as generating an “external” potential that perturbs the b particles and induces their response. This external potential, which depends on the ab interaction and on the number density ρa (r) of a particles, can be expressed as (8) vext (r) = dr vab (|r − r |)ρa (r ), and the ab interaction term in the Hamiltonian as Hab = dr ρb (r)vext (r), where

(9)

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ρα (r) =

Nα

δ(r − ri )

(10)

i=1

is the number density operator for particles of type α (α = a, b). If the a particles possess a property (e.g., electric charge or size) that can be continuously varied to tune the strength of vext (r), then Fb can be approximated via a perturbative response theory. A prerequisite for this approach is accurate knowledge of the free energy of a reference system of pure b particles (unperturbed by the a particles). Relative to this reference system, the free energy can be expressed as 1 1 ∂Fb (λ) = F0 + dλ dλ Hab λ , (11) Fb = F0 + ∂λ 0 0 where F0 = Fb (λ = 0) = −kB T ln exp(−βHbb )b

(12)

is the reference free energy, Fb (λ) = −kB T ln exp [−β(Hbb + λHab )]b

(13)

is the free energy of b particles in the presence of a particles “charged” to a fraction λ of their full strength, and

Hab λ =

Hab exp [−β(Hbb + λHab )]b ∂Fb (λ) =

exp [−β(Hbb + λHab )]b ∂λ

(14)

denotes an ensemble average of Hab over the coordinates of the b particles in this intermediate ensemble. (To simplify notation, we henceforth omit the subscript b from the trace over the coordinates of b particles: · · ·b ≡ · · ·.) Applying now a standard perturbative approximation, adapted from the theory of simple metals [20, 24, 25], the ensemble-averaged induced density of b particles may be expanded in a functional Taylor series around the reference system [vext (r) = 0] in powers of the dimensionless potential u(r) = −βvext (r): ∞ 1 dr1 · · · drn G(n+1) (r−r1 , . . . , r−rn )u(r1 ) · · · u(rn ), n! n=1 (15) where nb = Nb /V is the average density of b particles and the coeﬃcients δ n ρb (r) (n+1) (r − r1 , . . . , r − rn ) = lim G (16) u→0 δu(r1 ) · · · δu(rn )

ρb (r) = nb +

are the (n + 1)-particle density correlation functions [20] of the reference system. Equation (15) has a simple physical interpretation: the density of b particles induced at any point r results from the cumulative response to the external potentials at all points {r1 , . . . , rn }, propagated through the system via multiparticle density correlations.

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Further progress follows more rapidly in Fourier space, where the selfinteraction terms in the Hamiltonian (1) can be expressed using the identity Nα

vαα (|ri − rj |) =

i =j=1

=

dr ρα (r)ρα (r )vαα (|r − r |) − Nα vαα (0)

dr

1 vˆαα (k) [ˆ ρα (k)ˆ ρα (−k) − Nα ] , V

(17)

k

while the cross-interaction term (2) takes the form

Hab λ =

1 vˆab (k)ˆ ρa (k) ˆ ρb (−k)λ . V

(18)

k

Here vˆαβ (k) (α, β = a, b) is the Fourier transform of the pair potential vαβ (r) and (19) ρˆα (k) = dr ρα (r)e−ik·r is the Fourier transform of the number density operator (10), with inverse transform 1 ρα (r) = ρˆα (k)eik·r . (20) V k

The inverse transform is expressed as a summation, rather than as an integral, to allow the possibility of isolating the k = 0 component to preserve the constraint of ﬁxed average density in the canonical ensemble: ρˆα (k = 0) = dr ρα (r) = Nα . For charged systems, which interact via bare Coulomb pair potentials, special care must be taken to ensure that all longwavelength divergences formally cancel (see Sect. 4.1 below). Now Fourier transforming (15), we obtain 1 ˆ (3) ˆ (2) (k)ˆ G (k , k−k )ˆ u(k)+ u(k )ˆ u(k−k )+· · · ,

ˆ ρb (k) = G 2V

k = 0,

k

(21) ˆ (n) (Fourier transforms of G(n) ) are related to the where the coeﬃcients G ˆ (n) = nb S (n) , n-particle static structure factors of the reference system via G with the static structure factors being explicitly deﬁned by [20] S (2) (k) ≡ S(k) =

1

ˆ ρb (k)ˆ ρb (−k) Nb

(22)

and S (n) (k1 , · · · , kn−1 ) =

1

ˆ ρb (k1 ) · · · ρˆb (kn−1 )ˆ ρb (−k1 − . . . − kn−1 ) , Nb

n ≥ 3.

(23) ρa (k) [from (8)] into (21), the induced density Substituting u ˆ(k) = −βˆ vab (k)ˆ of b particles can be expressed in the equivalent form

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ˆ ρb (k) = χ(k)ˆ vab (k)ˆ ρa (k) +

1 χ (k , k − k )ˆ vab (k )ˆ vab (|k − k |) V k

ρa (k − k ) + · · · , × ρˆa (k )ˆ

k = 0,

(24)

where χ(k) = −βnb S(k)

(25)

is the linear response function and χ (k , k − k ) =

1 2 β nb S (3) (k , k − k ) 2

(26)

is the ﬁrst nonlinear response function of the reference system. Substituting the equilibrium density of b particles (24) into the ab interaction (18), the latter into the free energy (11), and integrating over λ, yields the desired free energy of the b particles to third order in the macroion density: Fb = F0 + Na nb lim vˆab (k) + k→0

1 2 χ(k) [ˆ vab (k)] ρˆa (k)ˆ ρa (−k) 2V k =0

1 χ (k , −k − k )ˆ vab (k)ˆ vab (k )ˆ vab (|k + k |) + 3V 2 ×

k =0 k ρa (k )ˆ ρa (−k ρˆa (k)ˆ

− k ).

(27)

Finally, this free energy may be substituted back into (6) to obtain the eﬀective Hamiltonian. Evidently, the term in Fb that is quadratic in ρˆa (k), arising from the term in ˆ ρb (k) that is linear in ρˆa (k), is connected to an eﬀective interaction between pairs of a particles. Similarly, the term in Fb that is cuρb (k), is connected to an bic in ρˆa (k), coming from the quadratic term in ˆ eﬀective interaction among triplets of a particles. Eﬀective Interparticle Interactions To explicitly demonstrate the connections between the free energy Fb and the eﬀective interactions, we ﬁrst identify (2)

vˆind (k) = χ(k)[ˆ vab (k)]2

(28)

as the interaction between pairs of a particles induced by surrounding b particles, in a linear response approximation [32–34]. As expected, the induced interaction depends on both the bare ab interaction and the response of the b particles to the external potential of the a particles. Combining the bare aa interaction with the induced interaction yields the linear response prediction for the eﬀective pair interaction: (2)

(2)

vˆlin (k) = vˆaa (k) + vˆind (k).

(29)

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The term on the right side of (27) that is second-order in ρˆa (k) can be manipulated using the identity [from (17)] Na

(2)

vind (rij ) =

1 (2) (2) (2) vˆind (k)ˆ ρa (k)ˆ ρa (−k) + Na na lim vˆind (k) − Na vind (0), k→0 V k =0

i =j=1

(30) where na = Na /V is the average density of a particles. Similarly, identifying vab (k)ˆ vab (k )ˆ vab (|k + k |) vˆeﬀ (k, k ) = 2χ (k , −k − k )ˆ (3)

(31)

in (27) as an eﬀective three-body interaction, arising from nonlinear response, and invoking the identity Na

(3)

veﬀ (rij , rik ) =

1 (3) vˆeﬀ (k, k )[ˆ ρa (k)ˆ ρa (k )ˆ ρa (−k − k ) V2 k

i =j =k=1

k

ρa (−k) + 2Na ], − 3ˆ ρa (k)ˆ

(32)

the eﬀective Hamiltonian acquires the following physically intuitive structure: Heﬀ = E +

Na 1 1 (2) veﬀ (rij ) + 2 3! i =j=1

Na

(3)

veﬀ (rij , rik ),

(33)

i =j =k=1

where E, veﬀ (r), and veﬀ (r, r ) are, respectively, a one-body “volume energy” and eﬀective pair and triplet interactions, induced by the b particles, between the a particles. A natural by-product of the reduction to an eﬀective one-component system, the volume energy is entirely independent of the a particle positions. Collecting coordinate-independent terms, the volume energy, expressed as E = Elin + ΔE, comprises a linear response approximation,

Na (2) 1 (2) v (0) + Na lim nb vˆab (k) − na vˆind (k) , (34) Elin = F0 + k→0 2 ind 2 (2)

(3)

and nonlinear corrections, the ﬁrst-order correction being na (3) Na (3) veﬀ (0, 0) − vˆeﬀ (k, 0) . ΔE = 6 V

(35)

k

(2)

The eﬀective pair interaction veﬀ (r) in (33) is the transform of (2)

(2)

(2)

vˆeﬀ (k) = vˆlin (k) + Δˆ veﬀ (k), where (2)

Δˆ veﬀ (k) =

na (3) 1 (3) vˆ (k, 0) vˆ (k, k ) − V eﬀ 3 eﬀ k

(36)

(37)

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is the ﬁrst nonlinear correction to the eﬀective pair potential, while the eﬀec(3) tive triplet interaction veﬀ (r, r ) is the Fourier transform of (31). Note that the ﬁnal term on the right sides of (34), (35), and (37) arise from the constraint of ﬁxed average density. Three observations are in order. First, since the volume energy depends, in general, on the mean densities of both a and b particles, it contributes to the total free energy and, therefore, thermodynamics of the system. This point has special signiﬁcance in applications to charged systems, as discussed in Sect. 4.1. Second, nonlinear response of the b particles generates not only eﬀective many-body interactions among the a particles, but also corrections to both the eﬀective pair interaction and the volume energy. In fact, as is clear (2) from (35) and (37), the nonlinear corrections to E and veﬀ (r) are intimately related to many-body interactions. Third, any inﬂuence of bb interactions on the eﬀective interactions enters through the free energy and response functions of the reference system. Thus, the quality of the eﬀective interactions is limited only by the accuracy to which the structure and thermodynamics of the pure b ﬂuid are known. Physical Interpretation While response theory is most easily formulated in Fourier space, its physical interpretation is perhaps more transparent in real space. The induced pair interaction in the linear response approximation (28) can be expressed in terms of real-space functions as (2) vind (r) = dr1 (38) dr2 χ(|r1 − r2 |)vab (r1 )vab (|r2 − r|). Here χ(|r1 − r2 |) is the real-space linear response function, which describes the change in the density of b particles induced at point r2 in response to an external potential applied at point r1 . Referring to Fig. 1, (38) can be interpreted as follows. A particle of type a, centred at the origin in Fig. 1, generates an external potential vab (r1 ), which acts on b particles at all points r1 . This potential induces at point r2 a change in the density of b particles given by dr1 χ(|r1 −r2 |)vab (r1 ). This induced density, which depends on pair correlations (via χ) in the intervening b ﬂuid, then interacts with a second a particle, at displacement r from the ﬁrst. The net result is an eﬀective interaction between the pair of a particles induced by the medium. The linear response contribution to the volume energy (per particle) associated with ab interactions (34) has a closely related form: (2) (39) dr2 χ(|r1 − r2 |)vab (r1 )vab (r2 ). vind (0) = dr1 The physical interpretation is similar, except that the induced density now interacts back with the ﬁrst a particle, generating a one-body (self) energy. An analogous interpretation applies to nonlinear response and induced manybody interactions [40].

Eﬀective Interactions in Soft Materials

407

3.3 Density-Functional Theory An alternative, yet ultimately equivalent, approach to deriving eﬀective interactions, follows from classical density-functional theory (DFT) [41,42]. Classical DFT has a long history, dating back a half-century to the earliest integralequation theories of simple liquids [20]. Following the establishment of formal foundations [43], DFT has been widely applied, in recent decades, to a variety of soft condensed matter systems, including colloids, polymers, and liquid crystals. Connections between density-functional theory and eﬀective interactions in charged colloids have been established by L¨ owen et al. [44, 45] and van Roij et al. [46, 47]. The essence of the theory is most easily grasped in the context of an ab mixture, where the challenge again is to approximate the free energy (7) of a ﬂuid of b particles in the presence of ﬁxed a particles. The basis of the density-functional approach is the existence [43] of a grand potential functional Ωb [ρb ] – the square brackets denoting a functional dependence – with two essential properties: Ωb [ρb ] is uniquely determined by the spatially-varying density ρb (r), for any given external potential vext (r), and is a minimum, at equilibrium, with respect to ρb (r). The grand potential functional is related to the Helmholtz free energy functional Fb [ρb ] via the Legendre transform relation Ωb [ρb ] = Fb [ρb ] − μb Nb ,

(40)

where μb is the chemical potential of b particles. The free energy functional naturally separates, according to Fb [ρb ] = Fid [ρb ] + Fext [ρb ] + Fex [ρb ],

(41)

into an ideal-gas term Fid , which is the free energy in the absence of any interactions, an “external” term Fext , which results from interactions with the external potential, and an “excess” term Fex , due entirely to interparticle interactions. The purely entropic ideal-gas free energy is given exactly by (42) Fid [ρb ] = kB T dr ρb (r)[ln(ρb (r)Λ3b ) − 1], where Λb is the thermal wavelength of the b particles, while the external free energy can be expressed as (43) Fext [ρb ] = dr ρb (r)vext (r), which is equivalent to Hab in (9). Inserting (42) and (43) into (41) yields Fb [ρb ] = kB T dr ρb (r)[ln(ρb (r)Λ3b ) − 1] + dr ρb (r)vext (r) + Fex [ρb ]. (44) The excess free energy can be expressed in the formally exact form,

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1 Fex [ρb ] = 2 =

1 2

dr

dr

dr

1

dλ ρbb [λvbb ; r, r ]vbb (|r − r |) (2)

0

dr ρb (r)ρb (r )

1

dλ gbb [λvbb ; r, r ]vbb (|r − r |), (45) (2)

0

where ρbb [λvbb ; r, r ] is the two-particle number density (a unique functional (2) of the pair potential), gbb [λvbb ; r, r ] is the corresponding pair distribution functional, and λ is a coupling (or charging) constant that “turns on” the interparticle correlations. In general, the pair distribution functional is not known exactly and must be approximated. For weakly correlated systems, it is often reasonable to adopt the mean-ﬁeld approximation 1 dr dr ρb (r)ρb (r )vbb (|r − r |), (46) Fex [ρb ] = 2 (2)

(2)

which amounts to entirely neglecting correlations and assuming gbb = 1. In a further approximation, valid for weakly inhomogeneous densities, the ideal-gas free energy functional is expanded in a functional Taylor series around the average density, which is truncated at quadratic order: δFid [ρb ] dr [ρb (r) − nb ] Fid [ρb ] Fid (nb ) + δρb (r) nb 2 δ Fid [ρb ] 1 + [ρb (r) − nb ][ρb (r ) − nb ]. (47) dr dr 2 δρb (r)δρb (r ) nb The ﬁrst term on the right, Fid (nb ) = Nb kB T [ln(nb Λ3b )−1], is the ideal-gas free energy of a uniform ﬂuid of b particles. The second (linear) term on the right vanishes identically by virtue of the constraint of constant average density. Evaluating the functional derivative in the coeﬃcient of the third (quadratic) term, kB T δ 2 Fid [ρb ] = δ(r − r ), (48) δρ(r)δρ(r ) ρb (r) and combining (41), (43), (46), and (47), the mean-ﬁeld free energy functional ﬁnally can be approximated by kB T Fb [ρb ] Nb kB T [ln(nb Λ3b ) − 1] + dr [ρb (r) − nb ]2 + dr ρb (r)vext (r) 2nb 1 dr dr ρb (r)ρb (r )vbb (|r − r |). (49) + 2 The equilibrium density is now determined by the minimization condition ρb (r) δΩb [ρb ] = ln(nb Λ3b )+ −1+βvext (r)+β dr ρb (r )vbb (|r−r |)−βμb = 0. β δρb (r) nb (50)

Eﬀective Interactions in Soft Materials

409

Fourier transforming and solving for the equilibrium density yields

ˆ ρb (k) =

−βnb vˆext (k) = χ(k)ˆ vext (k), 1 + βnb vˆbb (k)

where χ(k) =

k = 0,

−βnb 1 + βnb vˆbb (k)

(51)

(52)

is a mean-ﬁeld approximation to the linear response function introduced above in (25). Now expressing the free energy functional (49) in terms of Fourier components,

1 3 Nb vˆbb (k) + Na vˆab (k) Fb [ρb ] Nb kB T [ln(nb Λb ) − 1] + nb lim k→0 2 1 1 1 + ρˆb (k)ˆ vab (k)ˆ ρa (k) + ρb (−k), (53) vˆbb (k) + ρˆb (k)ˆ V 2V βnb k =0

k =0

and substituting for the equilibrium density from (51), we obtain – to second order in the a particle density – the equilibrium Helmholtz free energy of b particles in the presence of the ﬁxed a particles:

1 Nb vˆbb (k) + Na vˆab (k) Fb = Nb kB T [ln(nb Λ3b ) − 1] + nb lim k→0 2 1 2 χ(k) [ˆ vab (k)] ρˆa (k)ˆ ρa (−k). (54) + 2V k =0

After identifying 1 F0 = Nb kB T [ln(nb Λ3b ) − 1] + Nb nb lim vˆbb (k) k→0 2

(55)

as the free energy of the uniform b ﬂuid in the absence of a particles, (54) is seen to have exactly the same form as (27), to quadratic order in ρˆa (k). The same eﬀective interactions thus result from linearized classical densityfunctional theory as from linear response theory. Moreover, the same agreement is also found for the eﬀective triplet interactions derived from nonlinear response theory [40] and nonlinear DFT [48]. 3.4 Distribution Function Theory Still another statistical mechanical approach to calculating eﬀective interactions is based on approximating equilibrium distribution functions. This approach has been developed by many workers and applied to charged colloids in the forms of various integral-equation theories [49–55] and an extended Debye–H¨ uckel theory [56–59]. The basic elements of the method are sketched below, again in the context of a simple ab mixture.

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As shown above in Sect. 3.1, the key quantity in any theory of eﬀective interactions is the trace over degrees of freedom of the b particles of the ab interaction term in the Hamiltonian [see (2) and (11)]. This partial trace can be expressed in the general form

Hab = dr dr ρa (r)ρb (r ) vab (|r − r |), (56) where · · · denotes an ensemble average over the coordinates of the b particles with the a particles ﬁxed. The density of the ﬁxed a particles being unaﬀected by the partial trace, we can replace ρa (r)ρb (r ) by ρa (r) ρb (r ) in (56): (2)

Hab = nb dr dr ρa (r)gab (r − r )vab (|r − r |), (57) (2)

thus introducing the ab pair distribution function gab (r), which is deﬁned (2) via ρb (r) = nb gab (r). This pair distribution function is proportional to the probability of ﬁnding a b particle at displacement r from a central a particle. (2) More precisely, given an a particle at the origin, nb gab (r)dr represents the average number of b particles in a volume dr at displacement r. Distribution function theory evidently shifts the challenge to determining the cross-species (ab) pair distribution function. To this end, we consider an approximation scheme – rooted in the theory of simple liquids [20] – that illustrates connections to response theory and density-functional theory. The starting point is the fundamental relation, valid for any nonuniform ﬂuid, between the equilibrium density, an “external” applied potential, and the (internal) direct correlation functions. Minimization of the Helmholtz free energy functional (41) with respect to the density, at ﬁxed average density, yields the Euler–Lagrange relation [43]

ρb (r) =

eβμb (1) exp(−βvext (r) + cb [ρb ; r]), Λ3b

(58)

where the one-particle direct correlation functional (DCF), deﬁned as (1)

cb [ρb ; r] ≡ −β

δFex [ρb ] = −βμex [ρb ; r], δρb (r)

(59)

is a unique functional of the density that is proportional to the excess chemical potential of b particles μex (associated with bb interparticle interactions). Although it provides an exact implicit relation for the equilibrium density, (1) (58) can be solved, in practice, only by approximating cb [ρb ; r]. Approxima(1) tions are facilitated by expanding cb [ρb ; r] in a functional Taylor series about the average (bulk) density nb : (1) (1) (2) cb [ρb ; r] = cb (nb ) + dr cbb (r − r ; nb )[ ρb (r ) − nb ] + · · · , (60)

Eﬀective Interactions in Soft Materials

where (2) cbb (r

− r ; nb ) =

lim ρb (r)→nb

(1)

δcb [ρb ; r] δρb (r )

411

(61)

is the two-particle DCF of the (reference) uniform ﬂuid and, more generally, (1)

(n)

cb···b [ρb ; r1 , . . . , rn ] ≡

δ n−1 cb [ρb ; r1 ] δρb (r2 ) · · · δρb (rn )

(62)

is the n-particle DCF. Note that higher-order terms in the series, corresponding to multiparticle correlations, are nonlinear in the density. (1) Substituting (60) into (58), and identifying (eβμb /Λ3b ) exp[cb (nb )] as the bulk density nb , the nonuniform equilibrium density can be expressed as (2)

ρb (r) = nb exp −βvext (r) + dr cbb (r − r ; nb )[ ρb (r ) − nb ] + · · · . (63) Various practical approximations for the DCFs correspond, within the framework of integral-equation theory, to diﬀerent closures of the Ornstein–Zernike relation [20], (2) (2) (2) (2) (64) hbb (r) = cbb (r) + nb dr cbb (r − r )hbb (r ), which is an integral equation relating the two-particle DCF to the pair corre(2) (2) lation function, hbb (r) = gbb (r) − 1. Truncating the functional expansion in (63) and retaining only the term linear in density, thus neglecting multiparticle correlations in a mean-ﬁeld approximation, is equivalent to the hypernetted-chain (HNC) approximation in integral-equation theory. Making a second mean-ﬁeld approximation by (2) equating the two-particle DCF cbb (r; nb ) to its asymptotic limit [20], (2) lim c (r; nb ) r→∞ bb

= −βvbb (r),

thereby neglecting short-range correlations, (63) reduces to (2)

ρb (r) = nb exp −βvext (r) − β dr vbb (|r − r |)[ ρb (r ) − nb ] ,

(65)

(66)

corresponding to the mean-spherical approximation (MSA) in integral-equation theory. In passing, we note that (66) also provides a basis for the Poisson– Boltzmann theory of charged colloids and polyelectrolytes, if we identify (2) qb ψ(r) ≡ vext (r) + dr vbb (|r − r |)[ ρb (r ) − nb ] (67) as the mean-ﬁeld electrostatic potential energy of a b particle (charge qb ) brought from inﬁnity, where the potential ψ = 0, to a displacement r away from an a particle, and combine this with Poisson’s equation,

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Alan R. Denton

∇2 ψ(r) = −

1

qi ρi (r),

(68)

i

where is the dielectric constant of the medium and the sum is over all species with charges qi and number densities ρi (r). If we now make one further approximation by linearizing the exponential function, valid in the case of potential energies much lower than thermal energies, then (66) becomes (2)

ρb (r) = nb 1 − βvext (r) − β dr vbb (|r − r |)[ ρb (r ) − nb ] . (69) Fourier transforming (69) and solving for the equilibrium density ﬁnally yields

ˆ ρb (k) =

−βnb vˆext (k) = χ(k)ˆ vext (k), 1 + βnb vˆbb (k)

k = 0,

(70)

which is identical in form to the linear response and DFT predictions [see (24) and (51)] with the same linear response function χ(k) as before [see (52)]. Alternatively, we may ﬁrst linearize the exponential in (66) and then exploit the Ornstein–Zernike relation (64) to solve recursively for the equilibrium density, with the result (2)

ρb (r) = nb 1 − βvext (r) + dr cbb (r − r )[−βnb vext (r ) + · · ·] (2) = nb − βnb dr [δ(r ) + nb hbb (r )]vext (r ), (71) where (2)

−βnb [δ(r) + nb hbb (r)] = χ(r)

(72)

can be identiﬁed as the real-space linear response function. By Fourier transforming (71) and (72), we recover the linear response relation (70) with a linear response function, (2)

ˆ (k)] = −βnb S(k), χ(k) = −βnb [1 + nb h bb

(73)

deﬁned precisely as originally in (25). The connection to the linear response function in (52) is established via the Fourier transform of the Ornstein– Zernike relation (64), (2)

(2)

ˆ (2) (k) = h bb

cˆbb (k) (2)

1 − nb cˆbb (k)

−βˆ vbb (k) (2)

,

(74)

1 + βnb vˆbb (k)

where we have assumed a mean-ﬁeld (random phase) approximation for the (2) (2) two-particle DCF, cˆbb (k) −βˆ vbb (k) [cf (65)]. Distribution function theory therefore predicts the same linear response of the b-particle density to the

Eﬀective Interactions in Soft Materials

413

a-particle external potential – and thus the same eﬀective interactions – as do both response theory and density-functional theory. In closing this section, we emphasize that the eﬀective interactions derived here in the canonical ensemble apply to experimental situations in which the particle densities are ﬁxed. In many experiments, however, the system may be in chemical equilibrium with a reservoir of particles, allowing ﬂuctuations in particle densities. The appropriate ensemble then would be the semigrand or grand canonical ensemble for a reservoir containing, respectively, one or both species. It is left as an exercise to derive the eﬀective interactions in these other ensembles. (Hint: Consider carefully the appropriate reference system.)

4 Applications The preceding section describes several general, albeit formal, approaches to modelling eﬀective interparticle interactions in soft matter systems. To illustrate the practical utility of some of the methods, the following section brieﬂy outlines applications to three broad classes of system: (1) charge-stabilised colloidal suspensions, (2) colloid-polymer mixtures, and (3) polymer solutions. These systems exhibit two of the most common forms of microscopic interactions, namely, electrostatic and excluded-volume interactions. Further details can be found in [28, 33, 34, 38–40]. 4.1 Charged Colloids Primitive Model Charge-stabilised colloidal suspensions [7–10] are multicomponent mixtures of macroions, counterions, and salt ions dispersed in a molecular solvent and stabilised against coagulation by electrostatic interparticle interactions. For concreteness, we assume aqueous suspensions (water solvent). A reasonable model for these complex systems is a collection of charged hard spheres and point microions interacting via bare Coulomb pair potentials in a dielectric medium (Fig. 2). The point-microion approximation is valid for systems, such as colloidal suspensions, with large size asymmetries between macroions and microions. The macroions, of radius a (diameter σ = 2a), are assumed to carry a ﬁxed, uniformly distributed, surface charge −Ze (valence −Z), which may be physically interpreted as an eﬀective charge, renormalized by association of oppositely charged counterions (valence z Z). In a closed system, described by the canonical ensemble, global charge neutrality constrains the macroion and counterion numbers, Nm and Nc , via the relation ZNm = zNc . The salt-water mixture is modelled as an electrolyte solution of Ns dissociated pairs of point ions of valences ±z. The microions then number N+ = Nc + Ns positive and N− = Ns negative, totaling Nμ = Nc + 2Ns .

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Fig. 2. Left: Primitive model of a charge-stabilised colloidal suspension, consisting of hard spherical macroions (valence Z, diameter σ) and point monovalent microions (counterions, salt ions) dispersed in a dielectric continuum. Right: Eﬀective onecomponent model of pseudomacroions, after coarse-graining of microions. Reprinted with permission from [64]. Copyright (2006) by the American Physical Society.

Within the coarse-grained “primitive” model of charged colloids, the water is treated as a dielectric continuum, characterized entirely by a dielectric constant . This approximation amounts to preaveraging over the vast number of solvent degrees of freedom. For simplicity, we completely neglect charge-induced-dipole and other polarization interactions [60–62], which are shorter-ranged than charge-charge interactions and vanish if solvent and macroions are index-matched (i.e., have the same dielectric constant). The bare electrostatic interactions can be represented by Coulomb pair potentials: vmm (r) = Z 2 e2 / r (r > σ), vcc (r) = z 2 e2 / r, and vmc (r) = Zze2 / r (r > a). Note that in the primitive model, the solvent acts only to reduce the strength of Coulomb interactions by a factor 1/ . In addition, the macroion hard cores interact via a hard-sphere pair potential. Response Theory for Electrostatic Interactions Following the methods of response theory laid out in Sect. 3.2, eﬀective interactions now can be derived by integrating out the degrees of freedom of the microions, reducing the multicomponent mixture to an eﬀective one-component system of pseudo-macroions [32]. To simplify the derivation, we ﬁrst consider the rather idealized case of salt-free suspensions. The bare Hamiltonian of the two-component model then decomposes naturally, according to H = Hmm + Hcc + Hmc , into a macroion term Hmm

Nm 1 = HHS + vmm (rij ), 2

(75)

i =j=1

where HHS is the Hamiltonian for neutral hard spheres (macroion hard cores), a counterion term

Eﬀective Interactions in Soft Materials

Hcc

Nc 1 = Kc + vcc (rij ), 2

415

(76)

i =j=1

where Kc is the counterion kinetic energy, and a macroion-counterion interaction term Nm Nc Hmc = vmc (rij ). (77) i=1 j=1

By analogy with (11), the free energy of the counterions in the external potential of the macroions can be expressed as [20, 32]: 1 dλ Hmc λ , (78) Fc = F 0 + 0

where F0 = −kB T ln exp(−βHc ) is now the reference free energy of the counterions in the presence of neutral (hard-core) macroions, and the λ-integral adiabatically charges the macroions from neutral to fully charged. Neglecting counterion structure induced by the macroion hard cores, neutral macroions would be surrounded by a uniform “sea” of counterions. As the macroion charge is turned on, the counterions respond, redistributing themselves to form a double layer (surface charge plus neighbouring counterions) around each macroion. It is a special property of Coulomb-potential systems that, because volume integrals over long-ranged 1/r potentials diverge, each term on the right side of (78) is actually inﬁnite. Although the inﬁnities formally cancel, it proves convenient still to convert F0 to the free energy of a classical one-component plasma (OCP) by adding and subtracting the (inﬁnite) energy of a uniform compensating negative background 1 Ebg = − Nc nc lim vˆcc (0), k→0 2

(79)

where nc is the average density of counterions in the volume unoccupied by the macroion cores. Because the counterions are strictly excluded (with the background) from the hard macroion cores, the OCP has average density nc = Nc /[V (1 − η)], where η = π6 (Nm /V )σ 3 is the macroion volume fraction and V (1 − η) is the free volume. Thus, 1 Fc = FOCP + dλ Hmc λ − Ebg , (80) 0

where FOCP = F0 + Ebg is the free energy of the “Swiss cheese” OCP in the presence of neutral, but volume-excluding, hard spheres. All of the formal expressions derived in Sect. 3.2 for a generic twocomponent (ab) mixture now carry over directly, with the identiﬁcations a ↔ m and b ↔ c. In the linear response approximation [33, 34], the volume energy is given by

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Alan R. Denton

Nm (2) z (2) Z v (0) + Nm nc lim vˆmc (k) − vˆ (k) + vˆcc (k) , Elin = FOCP + k→0 2 ind 2Z ind 2z (81) the eﬀective (electrostatic) pair potential by (2)

(2)

vˆlin (k) = vˆmm (k) + vˆind (k),

(82)

with induced potential (2)

vmc (k)]2 , vˆind (k) = χ(k)[ˆ

(83)

and the eﬀective triplet potential by vˆeﬀ (k, k ) = 2χ (k , −k − k )ˆ vmc (k)ˆ vmc (k )ˆ vmc (|k + k |), (3)

(84)

where χ(k) and χ (k) are linear and ﬁrst-order nonlinear response functions of the uniform OCP. Similarly, the ﬁrst-order corrections for nonlinear response are given by ⎡ ⎤ Nm ⎣ (3) (3) vˆeﬀ (k, k ) − Nm vˆeﬀ (k, 0)⎦ , ΔE = (85) 6Vf2 k,k

and (2)

Δˆ veﬀ (k) =

k

1 (3) Nm (3) vˆ (k, k ) − vˆ (k, 0), Vf eﬀ 3Vf eﬀ

(86)

k

where Vf = V (1 − η) is the free volume. Random Phase Approximation Further progress towards practical expressions for eﬀective interactions requires specifying the OCP response functions. For charged colloids, the OCP is typically weakly correlated, characterized by relatively small coupling parameters: Γ = λB /ac 1, where λB = βz 2 e2 / is the Bjerrum length and ac = (3/4πnc )1/3 is the counterion-sphere radius. For example, for macroions of valence Z = 500, volume fraction η = 0.01, and monovalent counterions suspended in salt-free water at room temperature (λB = 0.714 nm), we ﬁnd Γ 0.02. For such weakly-correlated plasmas, it is reasonable – at least as regards long-range interactions – to neglect short-range correlations. We can thus adopt a random phase approximation (RPA) which equates the two-particle direct correlation function to its exact asymptotic limit: c(2) (r) = −βvcc (r) or cˆ(2) (k) = −4πβz 2 e2 / k 2 . Furthermore, we ignore the inﬂuence of the macroion hard cores on the OCP response functions, which is reasonable for suﬃciently dilute suspensions. Within the RPA, the OCP (two-particle) static structure factor and linear response function take the analytical forms

Eﬀective Interactions in Soft Materials

S(k) =

1 1 = 1 + κ2 /k 2 1 − nc cˆ(2) (k)

and

417

(87)

−βnc , 1 + κ2 /k 2

(88)

χ(r) = −βnc [δ(r) + nc hcc (r)] ,

(89)

χ(k) = −βnc S(k) =

where κ = 4πnc z 2 e2 / kB T is the Debye screening constant (inverse screening length), which governs the form of the counterion density proﬁle and of the screened eﬀective interactions. In the absence of salt, the counterions are the only screening ions. The macroions themselves, being singled out as sources of the external potential for the counterions, do not contribute to the density of screening ions. Fourier transforming (88), the real-space linear response function takes the form

where

βz 2 e2 e−κr (90) r is the counterion-counterion pair correlation function. Note the screenedCoulomb (Yukawa) form of hcc (r), with exponential screening length κ−1 . Equation (89) makes clear that there are two physically distinct types of counterion response: local response, associated with counterion self correlations, and nonlocal response, associated with counterion pair correlations. Proceeding to nonlinear response, we ﬁrst note that the three-particle structure factor obeys the identity S (3) (k, k ) = S(k)S(k )S(|k + k |) 1 + n2c cˆ(3) (k, k ) , (91) hcc (r) = −

where cˆ(3) (k, k ) is the Fourier transform of the three-particle DCF. Within the RPA, however, c(3) and all higher-order DCF’s vanish. Thus, from (25), (26), and (91), the ﬁrst nonlinear response function can be expressed in Fourier space as kB T (92) χ (k, k ) = − 2 χ(k)χ(k )χ(|k + k |) 2nc and in real space as χ (r1 − r2 , r1 − r3 ) = −

kB T 2n2c

dr χ(|r1 − r|)χ(|r2 − r|)χ(|r3 − r|).

(93)

Counterion Density Proﬁle An explicit expression for the ensemble-averaged counterion density is obtained by substituting the RPA linear response function (88) into the linear response relation

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Alan R. Denton

ˆ ρc (k) = χ(k)ˆ vext (k) = χ(k)ˆ vmc (k)ˆ ρm (k),

k = 0.

(94)

Inverse transforming (94) yields ρc (r) =

Nm

ρ0 (|r − Ri |) =

dr χ(|r − r |)

i=1

Nm

vmc (|r − Ri |),

(95)

i=1

which is the real-space linear response counterion density in the presence of macroions ﬁxed at positions Ri , expressed as a sum of single-macroion counvmc (k). terion density orbitals ρ0 (r) – the inverse transform of ρˆ0 (k) = χ(k)ˆ Substituting (89) and (90) for the real-space RPA linear response function into (95), the linear response counterion density proﬁle can be expressed as Nm −κ|r−r | κ2 e vmc (|r − Ri |) , (96) −vmc (|r − Ri |) + ρc (r) = βnc dr 4π |r − r | i=1 where the two terms on the right correspond again to local and nonlocal counterion response. For hard-core macroions, the form of the macroion-counterion interaction inside the core is arbitrary and can be speciﬁed so as to minimize counterion penetration inside the cores [46]. Thus, assuming 5 −Zze2 , r>a r (97) vmc (r) = −Zze2 α, r
and ρ0 (r) =

4πZze2 κ sin(ka) cos(ka) + (1 + κa)k 2 k

(98)

Z κ2 eκa e−κr , z 4π 1 + κa r 0,

(99)

r>a r < a,

which agrees precisely with the asymptotic (r → ∞) expression predicted by the DLVO theory of charged colloids [22, 23]. Eﬀective Electrostatic Interactions Practical expressions for the eﬀective electrostatic interactions are obtained by explicitly evaluating inverse Fourier transforms. By combining (81), (83), (84), (85), and (98), the volume energy can be expressed as the sum of the linear response approximation,

Eﬀective Interactions in Soft Materials

Elin = FOCP − Nm

Nc kB T Z 2 e2 κ − , 2 1 + κa 2

and the ﬁrst nonlinear correction, Nm kB T 3 2 ΔE = − (r)] − n (r)] dr [ρ . dr [ρ 0 0 c 6n2c

419

(100)

(101)

The ﬁrst term on the right side of (100) accounts for the counterion entropy and the second term for the macroion-counterion electrostatic interaction energy. The latter term happens to be identical to the energy that would result if each macroion’s counterions were all concentrated at a distance of one screening length (κ−1 ) away from the macroion surface. From (82), (83), (88) and (98), the linear response prediction for the effective pair interaction is given by κa 2 −κr e e Z 2 e2 (2) , r > σ, (102) vlin (r) = 1 + κa r which is identical to the familiar DLVO screened-Coulomb potential in the dilute limit of widely separated macroions [22, 23], while (86) yields the ﬁrst nonlinear correction kB T nc (2) Δveﬀ (r) = − 2 dr ρ0 (r )ρ0 (|r − r |) ρ0 (|r − r |) − . (103) nc 3 Finally, from (84) and (92), the eﬀective triplet interaction is kB T (3) dr ρ0 (|r1 − r|)ρ0 (|r2 − r|)ρ0 (|r3 − r|). veﬀ (r12 , r13 ) = − 2 nc

(104)

Note that the ﬁnal terms on the right sides of (100), (101), and (103) originate from the charge neutrality constraint. The above results generalize straightforwardly to nonzero salt concentration. Here we merely sketch the steps leading to the ﬁnal expressions, referring the reader to [34] and [40] for details. Assuming ﬁxed average number density (in the free volume) of salt ion pairs, ns = Ns /Vf , the total average microion density is nμ = n+ +n− = nc +2ns , where n± are the average number densities of positive/negative microions. Following [34], the Hamiltonian generalizes to H = Hmm + Hμ + Hm+ + Hm− , where Hμ is the Hamiltonian of all microions (counterions and salt ions) and Hm± are the electrostatic interaction energies between macroions and positive/negative microions. The perturbation theory proceeds as before, except that the reference system is now a two-component plasma. The presence of positive and negative microion species entails a proliferation of response functions, χij and χijk , i, j, k = ±, and a generalization of (89) to χ++ (r) = −βn+ [δ(r) + n+ h++ (r)] χ+− (r) = −βn+ n− h+− (r) χ−− (r) = −βn− [δ(r) + n− h−− (r)] ,

(105)

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Alan R. Denton

where hij (r), i, j = ±, are the two-particle pair correlation functions of the microion plasma. Substituting the ensemble-averaged microion number densities into the multi-component Hamiltonian yields expressions for the macroionmicroion interaction terms and, in turn, the eﬀective interactions. The ultimate eﬀect of salt is to modify the previous results as follows. First, the average counterion density that appears in the Debye screening constant and in the linear response function (88) is replaced by the total average microion density: κ = 4πnμ z 2 e2 / kB T and χ(k) = −βnμ S(k). The ﬁrst nonlinear response function retains its original form (92), but with the new deﬁnition of κ and with nc replaced by nμ . Second, the linear response volume energy becomes [34] Elin = Fplasma − Nm

(N+ − N− )2 kB T Z 2 e2 κ − , 2 1 + κa N+ + N− 2

(106)

where Fplasma = kB T {N+ [ln(n+ Λ3+ ) − 1] + N− [ln(n− Λ3− ) − 1]}

(107)

is the free energy of the unperturbed microion plasma (in the free volume) and Λ± denotes the thermal wavelengths of positive and negative microions. Third, the eﬀective triplet interaction and nonlinear corrections to the eﬀective pair interaction and volume energy are generalized as follows: Nm (n+ − n− ) 3 2 dr [ρ0 (r)] − nμ dr [ρ0 (r)] (108) βΔE = − 6 n3μ

nμ (109) dr ρ0 (r )ρ0 (|r − r |) ρ0 (|r − r |) − 3 (n+ − n− ) (3) dr ρ0 (|r1 − r|)ρ0 (|r2 − r|)ρ0 (|r3 − r|). (110) βveﬀ (r12 , r13 ) = − n3μ (2) βΔveﬀ (r)

(n+ − n− ) =− n3μ

These results imply that nonlinear eﬀects increase in strength with increasing charge and concentration of macroions and with decreasing salt concentration, and that eﬀective triplet interactions are consistently attractive. It is also clear that in the limit of zero macroion concentration (nc = n+ − n− → 0), or of high salt concentration (nμ → ∞), such that (n+ − n− )/nμ → 0, the leadingorder nonlinear corrections all vanish. This result – a consequence of charge neutrality – may partially explain the remarkably broad range of validity of DLVO theory for suspensions at high ionic strength. The wide tunability of the eﬀective electrostatic interactions leads to rich phase behaviour in charge-stabilised colloidal suspensions. Simulation studies [63] of one-component systems interacting via the screened-Coulomb pair potential have demonstrated that variation in the Debye screening constant – corresponding in experiments to varying salt concentration – can account for the observed cross-over in relative stability between stable fcc and bcc crystal

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421

structures. The density dependence of the volume energy, resulting from the constraints of ﬁxed density and charge neutrality, can have profound implications for thermodynamic properties (e.g., phase behaviour, osmotic pressure) of highly deionized suspensions. Speciﬁcally, the volume energy has been predicted by DFT [46, 47], extended Debye–H¨ uckel theory [59], and response theory [64] to drive an unusual counterion-induced phase separation between macroion-rich and macroion-poor phases at low (sub-mM) salt concentrations. Despite purely repulsive pair interactions, which oppose bulk phase separation, counterion entropy and macroion self-energy can, according to predictions, conspire to drive a spinodal instability. It remains unresolved, however, whether this predicted instability is related to experimental observations of anomalous phase behaviour in charged colloids, including stable voids [65] and metastable crystallites [66]. Predictions of response theory can be directly tested against simulations. As an example, Fig. 3 presents a comparison [40] with available data from ab initio simulations [67] for the total potential energy of interaction between a pair of macroions, of diameter σ = 106 nm and valence Z = 200, in a cubic box of length 530 nm (with periodic boundary conditions) in the absence of salt. The theory is in excellent agreement with simulation, although nonlinear eﬀects are relatively weak for these parameters. Figure 3 also illustrates the eﬀective triplet interaction between a trio of macroions arranged in an equilateral triangle for σ = 100 nm and two diﬀerent valences, Z = 500 and 700, computed from (110) [40]. The strength of the attractive interaction grows rapidly with increasing macroion valence and with decreasing separation between macroion cores. Other methods, including DFT [48] and Poisson– Boltzmann theory [68], predict qualitatively similar triplet interactions. In concentrated suspensions of highly-charged macroions, higher-order eﬀective interactions may become signiﬁcant. 4.2 Systems of Hard Particles Soft matter systems that contain hard (impenetrable) particles include colloidal and nanoparticle dispersions and colloid-nanoparticle mixtures. In such systems, the bare interparticle interactions depend, at least in part, on the geometric volume excluded by each particle to all other particles. In general, van der Waals, electrostatic, and other interactions may also be present. The simplest case is that of spherical particles, as in bidisperse or polydisperse mixtures of colloids and/or nanoparticles. In a binary (ab) hard-sphere mixture, the bare pair interactions have the form ∞, r < Rα + Rβ (111) vαβ (r) = 0, r ≥ R α + Rβ , where α, β = a, b and Rα denotes the radius of particles of type α. For other shapes, the pair interactions naturally depend also on the orientations of the

80

(3)

(2)

Linear Nonlinear Simulation

100

60 40 20 0

120

160

200

Triplet Potential v (r) / kBT

Alan R. Denton Pair Potential v (r) / kBT

422

240

280

Pair Separation (nm)

0

-200

Z=500 Z=700

-400

-600 1

1.5

2

2.5

Triplet Separation r / σ

Fig. 3. Left: Interaction energy of two macroions (diameter 106 nm, valence 200) in a cubic box of length 530 nm (with periodic boundary conditions) at zero salt concentration. The potentials are shifted to zero at maximum macroion separation. Dashed curve: linear response prediction. Solid curve: nonlinear response prediction [40]. Symbols: ab initio simulation data [67]. Right: Eﬀective triplet interaction between three macroions, arranged in an equilateral triangle of side length r, with macroion diameter σ = 100 nm, macroion valence Z = 500 (dashed curve) or Z = 700 (solid curve), volume fraction η = 0.01, and salt concentration cs = 1 μM. Computed from (110) [40].

particles. The methods outlined in Sect. 3 for modelling eﬀective interactions are easily adapted to hard-particle systems. In what follows, we ﬁrst present general formulae and then describe an application to a common model of colloid-polymer mixtures. Response Theory for Excluded-Volume Interactions As noted in Sect. 3.2, the perturbative response theory is most useful in cases where the mesoscopic particles possess a property that can be continuously varied to turn on the “external” potential for the other (microscopic) particles. In the case of electrostatic interactions, the obvious tunable property is the charge on the particles. Analogously, for excluded-volume interactions, the relevant property is the volume occupied by the particles. In an ab mixture, we can imagine the mesoscopic (a) particles to be inﬂated continuously from points to their full size. To grow in size, the a particles must push against the surrounding b particles. As they grow, the a particles sweep out spheres, of radius Ra + Rb , from which the centres of the b particles are excluded. The change in free energy during this process equals the reversible work performed by the expanding a particles against the osmotic pressure exerted by the b ﬂuid. Adapting (11) to quantify this conceptual image, the Helmholtz free energy of the b particles in the presence of the a particles can be expressed as 1 dλ Vexc [ρa ; λ] Πb [ρa , ρb ; λ], (112) Fb = F 0 + 0

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423

where F0 is again the free energy of the unperturbed (reference) b ﬂuid, the λ integral continuously scales the a particles from points to full size, and Vexc [ρa ; λ] and Πb [ρa , ρb ; λ] – both functionals of density – are, respectively, the total volume excluded to the b particles, and the osmotic pressure exerted by the b ﬂuid, in the presence of a particles expanded to a fraction λ of their full volume. Although (112) is formally exact for any shape of particle, the complicated dependence of the excluded volume and osmotic pressure on the scale parameter λ precludes an exact evaluation of Fb for an arbitrary conﬁguration of a particles. One approximation scheme is based on expanding the osmotic pressure in powers of λ. Expanding around λ = 0, for example, yields 1 ∂Πb (0) dλ λVexc [ρa ; λ] + · · · , (113) Fb = F0 + Πb (nb )Vexc [ρa ] + ∂λ 0 0 (0)

where Πb (nb ) is the osmotic pressure of the reference ﬂuid and Vexc [ρa ] is the total volume excluded to the b particles by full-sized a particles. In practical applications, such as colloid-polymer mixtures, the system is often coupled to an inﬁnite reservoir of b particles (e.g., a polymer solution) that ﬁxes the chemical potential μb of the b component. In this case, a more natural choice (0) (r) for Πb may be the osmotic pressure of the reservoir Πb , leading to the approximation (r) (114) Fb = F0 + Πb Vexc [ρa ]. The higher-order terms in (113), which depend on bb pair interactions, are difﬁcult to evaluate and commonly ignored. Nevertheless, (114) is often a reasonable approximation, especially in the dilute limit (na → 0), where Πb [ρa , ρb ; λ] depends weakly on na and λ. In the idealized case in which the b ﬂuid can be modelled as a noninteracting gas, the osmotic pressures of the system and reservoir are equal and (114) then becomes exact (see below). It remains a highly nontrivial problem, for an arbitrary conﬁguration of a particles, to approximate the excluded volume Vexc [ρa ], which requires determining the intersection volumes of the mutually overlapping exclusion spheres that surround the a particles. In the simplest case of non-overlapping spheres, the total excluded volume is merely the excluded volume of a single a particle times the number of particles. In the next simplest case, in which only pairs of exclusion spheres overlap, the intersection volumes of overlapping pairs must be subtracted to avoid double-counting. The next step corrects for the intersection volume of three mutually overlapping spheres. The geometrical problem is illustrated in Fig. 4 for a mixture of hard spherical colloids and coarse-grained spherical polymers. Expressing the total excluded volume as a sum of overlap terms, and assuming isotropic interactions (spherical particles), the free energy can be approximated by

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Alan R. Denton

⎛

Na 1 1 (r) (1) (2) Fb = F0 +Πb ⎝Na Vexc − Vov (rij ) + 2 3! i =j=1

Na

⎞ (3) Vov (rij , rik ) − · · ·⎠ ,

i =j =k=1

(115) is the excluded volume of a single a particle, is the where intersection volume of a pair of overlapping exclusion spheres surrounding (3) particles i and j, and Vov (rij , rik ) is the intersection volume of three mutually overlapping spheres surrounding particles i, j, and k. In the more general case of anisotropic interactions (nonspherical particles), the eﬀective interactions depend also on the relative orientations of the particles. (1) Vexc

(2) Vov (rij )

σc σp

σc+ σp

Fig. 4. Left: Colloid-polymer mixture with hard (excluded-volume) interactions. Depletion of polymer coils from spaces between colloids induces eﬀective interactions between colloids. Right: Asakura–Oosawa–Vrij model with colloids treated as hard spheres (diameter σc ) and polymers as coarse-grained spheres (diameter σp ). The eﬀective interactions are a function of the total volume excluded by the colloids to the polymer centres, which depends on the intersection volumes of mutually overlapping spheres of exclusion (diameter σc + σp ) surrounding each colloid.

From (115), the eﬀective Hamiltonian is seen to have the same general form as in (33), with a one-body volume term (r)

(1) E = F0 + Πb Na Vexc ,

(116)

an eﬀective pair potential (2)

(r)

(2) veﬀ (r) = vaa (r) − Πb Vov (r),

(117)

and an eﬀective triplet potential (3)

(r)

(3) veﬀ (rij , rik ) = Πb Vov (rij , rik ).

(118)

This simple low-density approximation exhibits several noteworthy features. The induced pair potential has the same sign as its electrostatic counterpart

Eﬀective Interactions in Soft Materials

425

(negative and attractive), while the eﬀective triplet potential has the opposite sign (positive and repulsive). The induced pair attraction originates from the depletion of b particles from the space between pairs of closely approaching a particles and the resulting imbalance in osmotic pressure. Furthermore, in contrast to systems of charged particles with electrostatic interactions, in systems of hard particles with excluded-volume interactions, the volume term depends only trivially on the density of the a particles and the eﬀective manybody interactions do not generate corrections to lower-order eﬀective interactions. One must take care, however, not to draw general conclusions, since the neglected higher-order terms in (113) can signiﬁcantly modify the eﬀective interactions – especially in concentrated systems – introducing densitydependence and even changing the sign. In binary hard-sphere mixtures, for example, packing of smaller spheres around larger spheres induces eﬀective pair interactions between the larger spheres that can exhibit a repulsive barrier and even long-range oscillations [39, 69]. A simple, but important, example of a binary mixture of spheres is the Asakura–Oosawa–Vrij (AOV) model [70, 71] of mixtures of colloids and free (nonadsorbing) polymers. The AOV model treats the colloids (a ↔ c) as hard spheres, interacting via an additive hard-sphere pair potential vcc (r) (111), and the polymers (b ↔ p) as eﬀective, coarse-grained spheres that have hard interactions with the colloids, ∞, r < Rc + Rp (119) vcp (r) = 0, r ≥ Rc + R p , but are mutually noninteracting (ideal): vpp (r) = 0 for all r. The radius Rp of the eﬀective polymer spheres is most naturally identiﬁed with the polymer radius of gyration. The neglect of polymer-polymer interactions is strictly valid only for theta solvents [11], wherein monomer-monomer excluded-volume interactions eﬀectively vanish. In this special case, the polymer in the system behaves as an ideal gas conﬁned to the free volume, Vf = αV , where the free volume fraction α is deﬁned as the ratio of the volume available to the polymer centres (i.e., not excluded by the hard colloids) to the total volume. At equilibrium, equality of the polymer chemical potentials in the system, (r) (r) μp = kB T ln(np Λ3p /α), and in the reservoir, μp = kB T ln(np Λ3p ), implies (r)

that the corresponding polymer densities must be related via np = αnp . This simple relation imposes equality also of the polymer osmotic pressures (r) (r) in the system and reservoir, Πp = Πp = kB T np , thus rendering (114) and (117) exact. Within the AOV model, not only are the eﬀective interactions exact, but the eﬀective one-body and pair interactions have analytical forms, since the (1) single-sphere excluded volume is simply Vexc = (4π/3)(Ra + Rb )3 , and the convex-lens-shaped pair intersection has a volume

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⎧

3 ⎨π 3 r 3r , σc < r < σ c + σ p (σ + σc ) 1 − + (2) 2(σc + σp ) 2(σc + σp )3 Vov (r) = 6 p ⎩ 0, r ≥ σc + σp , (120) where σc = 2Rc and σp = 2Rp are the particle diameters and q = Rp /Rc is the size ratio. The eﬀective one-body interaction (116), being linear in the colloid density, does not aﬀect phase behaviour, but does contribute to the total osmotic pressure. The eﬀective pair potential described by (117) and (120) consists of a repulsive hard-sphere core and an attractive well with a range equal to the sum of the particle diameters and depth proportional to the reservoir polymer osmotic pressure. For suﬃciently small size ratios (q ≤ 0.154), such that the spherical exclusion spheres surrounding three colloids never intersect, the eﬀective triplet (and higher-order) interactions are identically zero. For q > 0.154, the eﬀective triplet interaction is nonzero and repulsive. Although limited to ideal polymers, the AOV model gives qualitative insight even for interacting polymers. For suﬃciently large size ratios (q > 0.45) and high polymer concentrations, polymer-depletion-induced eﬀective pair attractions can drive demixing into colloid-rich (polymer-poor) and colloid-poor (polymer-rich) ﬂuid phases [8]. Simulations of the eﬀective Hamiltonian system [38] and of the full binary AO model [72] indicate that for smaller size ratios (q ≤ 0.45), ﬂuid-ﬂuid demixing is only metastable, being preempted by the ﬂuid-solid (freezing) transition. With the phase diagram of the AOV model now well understood, recent attention has turned to exploring the phase behaviour of mixtures of colloids and interacting polymers (Sect. 4.3). Cluster Expansion Approach An alternative, and elegant, approach to modelling eﬀective interactions in systems of hard particles has been developed recently by Dijkstra et al. [38,39]. This powerful statistical mechanical method is similarly based on integrating out the degrees of freedom of one species of particle in the presence of ﬁxed particles of another species. The essence of the method is perhaps most transparent in the context of the AOV model of colloid-polymer mixtures [38], deﬁned by the Hamiltonian H = K + Hcc + Hcp , where K is the kinetic energy and Hcc

Nc 1 = vcc (rij ), 2 i =j=1

Hcp =

Np Nc

vcp (rij )

(121)

i=1 j=1

describe the colloid-colloid and colloid-polymer interactions. Assuming an inﬁnite reservoir that exchanges polymer with the system, the natural choice of ensemble is the semigrand canonical ensemble, in which the number of colloids Nc , volume V , temperature T , and polymer chemical potential μp are ﬁxed. Within this ensemble, which treats the colloids canonically and the polymers

Eﬀective Interactions in Soft Materials

427

grand canonically, the appropriate thermodynamic potential is the semigrand potential Ω, given by: exp(−βΩ) =

exp[−β(Hcc + Hcp )]p c ∞ N zp p 1 Nc = dr exp(−βHcc ) drNp exp(−βHcp ) c Np ! Nc !Λ3N c Np =0 = exp(−βHeﬀ )c ,

(122)

where · · ·c and · · ·p denote semigrand canonical traces over colloid and polymer coordinates, zp = eβμp /Λ3p is the polymer fugacity, and Λc and Λp are the colloid and polymer thermal wavelengths. Here Heﬀ = Hcc + Ωp is the eﬀective one-component Hamiltonian, where Ωp represents the grand potential of the polymers in the presence of ﬁxed colloids, deﬁned by exp(−βΩp ) = exp(−βHcp )p Np Nc ∞ N zp p dr exp −β = vcp (|ri − r|) , Np ! i=1 Np =0 Nc = exp zp dr exp −β vcp (|ri − r|) . (123) i=1

Equating arguments of the exponential functions on the left and right sides, we have Nc vcp (|ri − r|) . (124) −βΩp = zp dr exp −β i=1

As shown by Dijkstra et al. [38, 39], the polymer grand potential can be systematically approximated by a cluster expansion technique drawn from the theory of simple liquids [20]. Deﬁning the Mayer functions −1, rij < Rc + Rp fij ≡ exp[−βvcp (rij )] − 1 = (125) 0, rij ≥ Rc + Rp , (124) can be expanded as follows: −βΩp = zp

drj

(1 + fij )

i=1

= zp

Nc &

drj

1+

Nc i=1

fij +

Nc

fij fkj + · · · .

(126)

i
Equation (126) is a form of cluster expansion well-suited to systematic approximation by diagrammatic techniques [20]. Successive terms in the summation, generated by increasing numbers of colloids interacting with the polymers,

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Alan R. Denton

correspond directly to the one-body volume term and eﬀective pair and manybody interactions. In fact, from (126), the semigrand potential can be written in a form that is precisely analogous to the free energy deﬁned in (114): ⎛ ⎞ Nc Nc 1 1 (1) (2) (3) − Vov (rij ) + Vov (rij , rik ) − · · ·⎠ , βΩp = βΩ0 +zp ⎝Nc Vexc 2 3! i =j=1

i =j =k=1

(127) where βΩ0 = −zp V is the grand potential of the reference system of pure polymer. In the case of ideal polymer, the polymer fugacity is simply related (r) to the reservoir osmotic pressure via zp = βΠp . Note that (114) and (127) are completely equivalent, diﬀering only with respect to the relevant ensemble, with (114) applying in the canonical ensemble and (127) in the semigrand canonical ensemble. For mixtures of colloids and polydisperse polymers, (126) generalizes to −βΩp =

zp(k)

k (k)

drj

Nc &

(k)

(1 + fij ),

(128)

i=1 (k)

where zp is the fugacity of polymer species k and fij is the corresponding Mayer function. From (128), the eﬀective pair potential is then simply a sum of depletion potentials, each induced by a polymer species of a diﬀerent size. On the other hand, in the case of interacting (nonideal) polymers [vpp (r) = 0], the eﬀective interactions – even for monodisperse polymers – are considerably more complex. Although (123) then can be formally generalized to exp(−βΩp ) = exp[−β(Hcp + Hpp )]p Np Np Nc & ∞ N & & zp p (c) (p) Np = (1 + fij ) (1 + fkl ), (129) dr Np ! i=1 j=1 Np =0

(c)

k
(p)

where fij and fij are the Mayer functions for colloids and polymers, respectively, practical expressions are less forthcoming. Using diagrammatic techniques, Dijkstra et al. [39] have further analysed (129) and demonstrated its application to the phase behaviour of binary hard-sphere mixtures. 4.3 Systems of “Soft” Particles Soft particles are here deﬁned as macromolecules having internal conformational degrees of freedom. Prime examples are ﬂexible polymer chains (linear or branched), whose multiple joints allow for many distinct conformations. While bare monomer-monomer interactions – either intrachain or interchain – can be modelled by simple combinations of excluded-volume, van der Waals, and Coulomb pair interactions, the total interactions between long, ﬂuctuating chains can be highly complex, rendering explicit molecular simulations of polymer solutions computationally challenging.

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Among several practical techniques developed for averaging over the internal structure of soft particles to derive eﬀective pair interactions, we discuss here two that are speciﬁcally suited to polymers in good solvents. A more extensive survey is given in the review by Likos [28]. One method is based on the general principles of polymer scaling theory [11], which describes the properties of polymers in the limit of inﬁnite chain length, i.e., segment number N → ∞. The starting point is a formal expression for the eﬀective pair interaction between the centres of mass, at positions R1 and R2 , of two isolated chains (labelled 1 and 2): 2 V ρ (R )ρ (R ) exp[−βH[{r }, {r }] , Dr Dr βveﬀ (R12 ) = − ln 1 2 cm 1 cm 2 1 2 Z12 (130) where Z1 is the partition function of a single isolated chain, Drα represents a functional integral over all conformational degrees of freedom of chain α, ρcm (Rα ) denotes the number density of the centre of mass of chain α, and the Hamiltonian H is a functional of the conformations, {r1 } and {r2 }, of the two chains. In the special case of two chains, with one end of each chain ﬁxed and the two ﬁxed ends separated by a distance r, (130) becomes Z2 (r) βveﬀ (r) = − ln , (131) Z2 (∞) where Z2 (r) is the partition function of the constrained two-polymer system and Z2 (∞) is the same in the limit of inﬁnite separation. Scaling arguments [73] suggest that in the limit N → ∞, in which the only relevant length scales are the separation distance r and the polymer radius of gyration Rg , Z2 (r)/Z2 (∞) ∝ (r/Rg )x , where x is a universal exponent. It follows that r (132) , r ≤ Rg , βveﬀ (r) ∝ − ln Rg which describes a gently repulsive eﬀective pair potential. Similar scaling arguments have been extended to star-branched polymers [28, 73], consisting of linear polymer chains (arms) all joined at one end to a common core. The same form of eﬀective pair potential results, but with an amplitude that depends on the number of arms and the solvent quality. A reﬁned analysis by Likos et al. [74] leads to an explicit expression for the eﬀective pair potential between star polymers, which extends (132) to r > Rg , speciﬁes the prefactor, and is consistent with experimentally measured static structure factors. An alternative coarse-graining approach, well-suited to dilute and semidilute solutions of polymer coils in good solvents, is based on the physically intuitive view of polymer coils as “soft colloids” [75], whose eﬀective interactions may be approximated using integral-equation methods from the theory of simple liquids [20]. The basis of this approach is the fundamental relation

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between the pair distribution function g(r), direct correlation function c(r), and pair potential v(r) of a simple liquid: g(r) = exp[−βv(r) + g(r) − c(r) − 1 − b(r)].

(133)

Equation (133) follows directly from the Euler–Lagrange relation (63) for the nonuniform density of particles ng(r) around a central particle, where the pair potential plays the role of the external potential and the bridge function b(r) subsumes all multiparticle correlation terms. Practical implementation begins with a numerical calculation of g(r) between the polymer centres of mass, e.g., by molecular simulation of explicit self-avoiding random-walk chains, and proceeds through inversion of (133) to determine the eﬀective centre-to-centre pair potential v(r). The inversion of g(r) combines (133) with the Ornstein–Zernike relation (64) and an approximate closure relation for the bridge function. Louis and Bolhuis et al. [75, 76] have demonstrated that the HNC closure, b(r) = 0, gives an accurate approximation for the eﬀective interactions. The resulting softly repulsive, Gaussian-like, eﬀective pair potential has a range comparable to the polymer radius of gyration, an amplitude 2kB T , and is only weakly dependent on concentration, implying the relative weakness of eﬀective manybody interactions [77]. In self-consistency checks, simulations of simple liquids interacting via the potential v(r) are found to reproduce, to within statistical errors, the same centre-to-centre g(r) as simulations of explicit chains. The same method has been applied also to calculate the eﬀective depletion-induced interaction between hard walls [75,76] and between colloids in colloid-polymer mixtures [78, 79]. The scaling and soft-colloids approaches both reach the common conclusion that eﬀective pair interactions between polymers in good solvents are ultrasoftly repulsive. As the centres of mass of two polymers approach complete overlap, the pair potential between star polymers diverges very slowly (logarithmically), while that between linear chains actually remains ﬁnite. These characteristically soft eﬀective interactions contrast sharply with the steeply repulsive short-ranged pair interactions between colloidal particles, going far in explaining the unique structural properties and phase behaviour of polymer solutions.

5 Summary and Outlook The key message of this chapter is that soft materials, comprising complex mixtures of mesoscopic macromolecules and other microscopic constituents, often can be eﬃciently modelled by preaveraging over some of the degrees of freedom to map the multicomponent mixture onto an eﬀective model, with fewer components, governed by eﬀective interparticle interactions. In general, the eﬀective interactions are many-body in nature and dependent on the

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thermodynamic state of the system. Brieﬂy surveyed were several recently developed statistical mechanical methods, including response theory, densityfunctional theory, and distribution function theory. These powerful methods provide systematic and mutually consistent approaches to approximating effective interactions, and have broad relevance to a variety of materials. Speciﬁc applications were illustrated for electrostatic interactions in charged colloids and excluded-volume interactions in colloid-polymer mixtures. Eﬀective interactions are often simply necessitated by the computational impasse presented by fully explicit models of complex systems, especially soft matter systems with large size and charge asymmetries. At the same time, however, eﬀective models can provide conceptual insight that may be diﬃcult or impossible to extract from explicit models. Consider, for example, the subtle interplay of entropy and electrostatic energy in charged colloids or the important role of polymer depletion in colloid-polymer mixtures, eﬀects that are elegantly and eﬃciently captured in eﬀective interaction models. While computational capacity will likely continue to grow exponentially in coming years, conceptual understanding of soft materials will also continue to beneﬁt from the theoretical framework of eﬀective interactions. Acknowledgement. Many colleagues and friends have helped to introduce me to the fascinating world of soft matter physics and the power of eﬀective interactions. It is a pleasure to thank, in particular, Neil Ashcroft, J¨ urgen Hafner, Gerhard Kahl, Christos Likos, Hartmut L¨ owen, Matthias Schmidt, and Alexander Wagner for many enjoyable and inspiring discussions. Parts of this work were supported by the National Science Foundation under Grant No. DMR-0204020 and the American Chemical Society Petroleum Research Fund.

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Ab-initio Coarse-Graining of Entangled Polymer Systems J.T. Padding and W.J. Briels Computational Biophysics, Faculty of Science and Technology, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands

1 Introduction 1.1 Viscoelasticity of Polymer Systems Ever since Richard Kuhn’s description of a polymer as a coiling ﬂexible thread [1], polymer systems have received continuous interest from both theorists and experimentalists. In semi-dilute and concentrated polymer solutions each polymer chain interacts with many other chains. The eﬀect of these intermolecular interactions is revealed by the peculiar ﬂow behaviour of these materials: they are very viscous and have surprising elastic properties. In uncrosslinked polymers these elastic properties manifest themselves temporarily, but still sometimes on time scales as long as seconds or hours. This peculiar viscoelastic behaviour is often rationalized by viewing polymer systems as temporary rubbery networks. Such a network arises as a result of mutual uncrossability of the polymer chains - they are entangled. Many attempts have already been made to fundamentally explain the entanglement phenomenon. The usual procedure is to propose a microscopic model, calculate the consequences for various dynamic properties, and compare the outcome with experiment, if available. Theoretical treatments of this sort include cooperative motion models, where the focus is on the increased friction experienced by a test chain because it drags other chains with it over ﬁnite distances [2]. A major diﬃculty in such an approach is the speciﬁcation of the location and duration of entanglements, because the exact nature of an entanglement is not known. With the advent of the reptation theory of Doi, Edwards, and de Gennes [3], a new concept was introduced in the theory of polymer dynamics. In reptation theory each polymer is supposed to move in a tube around a Gaussian path in space. The tube only serves one purpose, namely to roughly represent the uncrossability of the surrounding chains and to turn a diﬃcult multichain problem into a one-chain problem. The tube is clearly a mean ﬁeld concept, and to this day it is debated how much validity should be ascribed to it [2, 4].

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Fig. 1. Diﬀerent polymer simulation models. a) Atomistically detailed model. b) Coarse model, obtained by coarse-graining the atomistically detailed model (bottomup approach). When many atoms are combined into one new particle, the eﬀective interactions between the particles become so soft that bond crossings can no longer be prevented. c) Lattice model. Advantage is the great speed with which a computer can simulate such a model and the relative ease with which bond crossings can be prevented. d ) Coarse model, in which the interaction model has been chosen beforehand (generic polymer model; top-down approach). Usually these interactions are tuned such that bond crossings are energetically unfavourable.

Recent years have witnessed the rapid growth of another technique to gain fundamental understanding of the dynamics of polymer systems: computer simulations. By use of computers, a range of increasingly complex models can be solved. 1.2 Detailed Computer Simulations At a very detailed level, molecular dynamics (MD) simulations can be performed, in which each atom of a polymer chain is represented separately, see Fig. 1a. The atoms are modelled as interacting particles and they move according to Newton’s laws. Accurate force ﬁelds have been constructed to cater for anyone’s research interests, provided they exclude chemical reactions and other phenomena of a quantum mechanical nature. Bulk behaviour is simulated by applying periodic boundary conditions to the simulation box. Typical

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MD simulations cover the motion of a few tens of thousands of atoms over a period of a few nanoseconds; on current computers such a run would take a week to complete. There are limitations, however, to the length of polymers that can be simulated this √ way, for two reasons. First, the typical size of the polymer typically grows as n, where n is the number of monomers in a chain. A polymer should not interact with itself via the periodic boundaries, which means that the volume of the box, and hence the number of particles, should scale as n3/2 . Second, the longest relaxation time of a polymer chain scales very fast, usually as n2 or n3 . To obtain a well-equilibrated system, and also to measure certain long-time correlation functions, the simulation must be performed for at least as long as this time scale. It should come as no surprise that atomistically detailed MD simulations have only been performed for relatively short chains of up to 100 monomers or, if longer chains were studied, only for relatively short times. 1.3 Coarse-Grained Simulations In order to increase the time and length scales accessible in the simulation of polymers, detailed atomistic models are replaced by coarse-grained (CG) models in which each particle represents a collection of atomic particles. The coarse-graining and subsequent analysis of the dynamic data can be performed in two ways: bottom-up (ab-initio) or top-down. In the top-down approach a certain model for the polymer interactions is chosen beforehand. Examples include simulations on a lattice, see Fig. 1c, and simulations of chains of (relatively) hard beads, see Fig. 1d. These models are also referred to as generic models because they are thought to reproduce generic polymer behaviour. Usually the magnitude of the interactions are chosen such that bond crossings will be forbidden or at least energetically unfavourable. A well-studied example is the polymer model of Kremer, Grest and co-workers [6, 7], in which the polymer segments are modelled as relatively hard beads connected by ﬁnitely extensible non-linear springs. The simulations are performed in reduced units, i.e. length, mass, and energies are related to the size σ, mass m, and interaction energy of the beads. There exists a possibility to estimate the time and length scales occurring in the simulation a posteriori. However, when more than one length scale is relevant, the proportionalities between diﬀerent length scales may not be the same as those that occur in real polymer systems. Speciﬁcally, chemically realistic polymers are ﬂexible only at large length scales. At these length scales the beads will be almost empty and consequently very soft. Treating a polymer as a ﬂexible string of hard beads may therefore not give quantitative or even realistic results for (certain aspects of) the dynamics and rheology. In this chapter we will focus on the bottom-up (ab initio) approach where the interactions between the CG particles are derived from the underlying

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atomistic interactions by suitable averaging techniques [8–11]. The theory underlying this kind of coarse-graining is treated in Sect.2. Although the dynamics of the atomistic model is deterministic and conservative, the dynamics of the resulting coarse-grained model is stochastic and includes dissipation and possibly memory eﬀects. If the theory is followed to the rule, a correct description of the structure and dynamics follows automatically. One should be careful, however. For practical and computational reasons the eﬀective potentials, as well as frictions and random forces, are usually represented by single or pair and sometimes triplet terms (Sect. 3). The general observation is that when the degree of coarse-graining is suﬃciently large the eﬀective potentials become very soft, as represented in Fig. 1b. Employing a single or pair friction approximation, this usually means that the bonds between the CG particles can easily cross each other, leading to very unrealistic dynamics. Luckily this can be cured. In Sect. 4 we will introduce a constraint which, when introduced in a coarse-grained simulation, can re-establish the uncrossability of chains. At the end of this chapter we will give two examples of simulations of entangled polyethylene melts and entangled wormlike micelles.

2 Theory 2.1 Coarse-Grained and Bath Variables Coarse-graining is the process of removing those degrees of freedom that one is not going to use or measure in the ﬁnal analysis. An example will make things clear. Suppose that a full description of the system being studied needs N + B degrees of freedom R1 , R2 , . . . , RN , q1 , q2 , . . . qB ,

(1)

and that one is only interested in the behaviour of R1 , R2 , . . . RN . The latter may for example be coordinates describing the positions of N/3 colloidal particles while the q’s describe the conﬁguration of a solvent in which the colloids are dissolved. Usually in a situation like this, one is only interested in the dynamics and thermodynamics of the colloids. Another example more appropriate to this chapter is where the R’s provide a rough description of the conﬁgurations of polymers in a polymer melt, while the q’s are the remaining internal coordinates describing the details of the conﬁgurations. For instance, one may subdivide each of the chains in the melt into a number of subchains, and use R1 , . . . , RN/3 to describe the centres of mass of these subchains. Obviously, in order to describe the dynamics and thermodynamics of the R’s one cannot just ignore the q’s, but rather one has to perform a statistical average over all their possible values. We shall explain how this averaging should be done in case one wants to derive the appropriate equations of motion of the R’s [12–14]. We then automatically ﬁnd the appropriate averages needed to describe the thermodynamics of the R’s.

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˙ The probFig. 2. The ensemble of paths all starting at the same initial R and R. ability distribution for R after a time t is P (R, t). The dashed line is the average path of the ensemble. The generalized Langevin equation generates these paths.

So, suppose we are given an initial box described by the coordinates of (1) and the corresponding momenta P1 , . . . , PN , p1 , . . . , pB .

(2)

In general the momenta are deﬁned in terms of all coordinates and the corresponding velocities, i.e. time derivatives. For our purposes it is suﬃciently general if we assume Pi = M R˙ i , where M is the mass of Ri , and pi is some function of the q’s and their time derivatives. Now, given the initial conditions, we may solve the equations of motion and calculate R1 (t), . . . , RN (t) at all times of interest. Together these functions yield the path of the system in Rspace. This path depends on the initial coordinates and momenta. If we keep the initial R’s and P ’s constant, but vary the q’s and p’s, we get an ensemble of paths all starting at R = {R1 , . . . , RN } with velocities R˙ = R˙ 1 , . . . , R˙ N , gradually spreading with time as in Fig. 2. In this ﬁgure P (R, t) is the probability distribution of R(t), i.e. P (R, t)dR is the probability of ﬁnding R(t) in the interval [R, R + dR]. The aim of coarse-graining is to provide an equation of motion for the R’s, yielding the same ensemble of paths with the same distribution P (R, t), but without recourse to the ‘bath’ variables q and p. The generalised Langevin equation does exactly this. 2.2 An Exact Equation of Motion We will now give a derivation of the exact equation of motion for the R coordinates [12–14]. With some approximations this leads to the generalised Langevin equation. The dynamics of the full system is governed by the Hamiltonian H(P, R, p, q) = T (P ) + TB (p, q) + Φ(R, q),

(3)

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where T (P ) is the kinetic energy of the R-coordinates and TB (p, q) that of the q-coordinates; Φ(R, q) is the potential energy of the full system. The time derivative of any function F = F (P (t), R(t), p(t), q(t)) is found according to dp ∂F dq ∂F dP ∂F dR ∂F dF = + + + dt dt ∂P dt ∂R dt ∂p dt ∂q ∂H ∂F ∂H ∂F ∂H ∂F ∂H ∂F =− + − + ∂R ∂P ∂P ∂R ∂q ∂p ∂p ∂q ≡ iLR F + iLq F = iLF .

(4)

In the second line we have used Hamilton’s equations. In the last line we have deﬁned the Liouville operator L. For ease of notation we have omitted indices with all variables; absence of indices at coordinates and momenta indicates that they should be dressed with dummy indices which should then be summed over the appropriate ranges. Now let us make a Taylor expansion in time of the momentum of coordinate n,

∞ ∞ k

t tk dk k

(iL) Pn ≡ exp {iLt} Pn . P = (5) Pn (t) = n

k k! dt k! t=0 k=0

k=0

The last equality deﬁnes the exponential of an operator. Note that here and in the following all operators operate on a phase function F (P, R, p, q), which is a function of the phase point (P, R, p, q). The phase point should always be evaluated at the original time t = 0, i.e. we operate on F (P (0), R(0), p(0), q(0)), unless the phase function is explicitly followed by (t), in which case we operate on F (P (t), R(t), p(t), q(t)). For example, in the last equation exp {iLt} operates on the momentum Pn at the original time t = 0. With this deﬁnition the equation of motion that we are seeking reads dPn (t) = eiLt iLPn . dt

(6)

Obviously iLPn is the force that initially acts on coordinate Rn : iLPn = −

∂Φ . ∂Rn

(7)

The ‘propagator’ exp{iLt} turns this into the force at time t. Concomitantly with our wish to get rid of the q’s, we would like to replace the force −∂Φ/∂Rn with its average over the q’s and the p’s. Therefore we introduce with each function F (R, P, q, p) its partial average 1 (8) dpdq e−βHB (p,q;R) F (R, P, q, p) ≡ F B , PF (R, P ) = QB (9) HB (p, q; R) = TB (p, q) + Φ(R, q), QB (R) = dpdq e−βHB (p,q;R) . (10)

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Here β = 1/(kB T ) and P is a projection operator, with the property P 2 F = PF . Equation (8) is appropriate in those cases where the q’s are internal coordinates. In the colloidal example, i.e. in the case when Φ(R, q) remains ﬁnite when the q’s go to inﬁnity, it is often more useful to work in the semigrand ensemble, where the number of coarse-grained particles and the chemical potential of the q’s are ﬁxed [15]. We now rewrite (6): dPn (t) = eiLt (P + (1 − P)) iLPn dt = eiLt PiLPn + eiLt (1 − P) iLPn .

(11)

In the ﬁrst term we recognise the force on particle n averaged over the bath variables: 6 7 ∂Φ ∂A =− , (12) PiLPn = − ∂Rn B ∂Rn A(R) = −kB T ln QB (R). (13) Here A(R) is the free energy of the q’s at ﬁxed values of the R’s. On the basis of the second equality in (12) it is also called the potential of mean force. Again, note that QB (R) denotes the semigrand partition function in case of diﬀusive q’s. The ﬁrst term of (11) now reads eiLt PiLPn = −

∂A (t), ∂Rn

(14)

which is the mean force on coordinate Rn at time t obtained as the negative gradient of the potential of mean force when the R’s have values R1 (t), . . . , RM (t). The remaining term in (11) represents friction and random forces. We extract the latter by writing1 eiLt (1 − P)iLPn = e(1−P)iLt (1 − P)iLPn t + dτ eiL(t−τ ) PiLe(1−P)iLτ (1 − P)iLPn .

(15)

0

This looks rather formidable, so let’s give the ﬁrst term in (15) a simple name:

e 1

(1−P)iLt

(1 − P)iLPn ≡ FnR ,

(16)

R (1 − P)iLPn ≡ Fn,t .

(17)

Equation (15) follows from the following identity valid for any operator A and B: e(A+B)t = eAt +

t

dτ e(A+B)(t−τ ) BeAτ .

0

Substitute A = (1 − P)iL and B = PiL.

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R Fn,t is a random force because, when averaged over the bath variables, it yields zero: R (18) Fn,t B = Pe(1−P)iLt (1 − P)iLPn = 0.

The reader should beware that, contrary to the previous state functions, the R is not simply the function FnR evaluated at the phase point random force Fn,t R at time t. Rather, Fn,t depends on the initial state point, which is propagated by the operator exp{(1−P)iLt}. Since this is not the usual exp{iLt} operator, t now merely acts as a parameter. To avoid confusion we have purposely R , not as FnR (t). denoted the random force as Fn,t In the second term of (15) we recognise the same random force again, now with τ instead of t. The second term is simpliﬁed further by realising that2 R R R PiLFn,t = P (iLR + iLq ) Fn,t = PiLR Fn,t .

(19)

This may be worked out further using the deﬁnition of the Liouville operator: 8 9 . - Pm ∂ ∂ ∂H R R PiLR Fn,t = − Fn,t M ∂Rm ∂Rm ∂Pm m B 8 5 9 6 7 : R Pm ∂Fn,t ∂Φ R ∂ − F . (20) = M ∂Rm ∂Pm ∂Rm n,t B m B

The ﬁrst average can also be expressed as 9 8 R R ∂Fn,t ∂Fn,t 1 = dpdq e−βHB ∂Rm QB ∂Rm B 1 ∂ β ∂Φ R R = + F dpdq e−βHB Fn,t dpdq QB ∂Rm QB ∂Rm n,t 6 7 ∂Φ R =β F . (21) ∂Rm n,t B So for both averages in (20) we need to evaluate 7 6 ∂Φ R R F = − (iLPm ) Fn,t B ∂Rm n,t B R R = − (PiLPm ) Fn,t B − [(1 − P)iLPm ] Fn,t B R R R = − iLPm B Fn,t B − Fm Fn,t B . 2

This is true for any F (P (t), R(t), p(t), q(t)) because 1 PiLq F = QB =

1 QB

dpdq e

−βHB

dpdq

−

∂H ∂ ∂H ∂ + − ∂q ∂p ∂p ∂q

∂HB ∂ ∂HB ∂ + ∂q ∂p ∂p ∂q

where we have used that HB → ∞ as p → ∞ or q → ∞.

/

F

0

e−βHB F = 0,

(22)

Ab-initio Coarse-Graining of Entangled Polymer Systems

R Because Fn,t

B

445

= 0, the equation of motion now reads

∂A dPn R (t) = − (t) + Fn,t − dt ∂Rn m

-

t

dτ e

iL(t−τ )

0

∂ Pm − β M ∂Pm

.

R R Fm Fn,τ

B

.

R R (23) Note that this is still an exact equation of motion for Pn (t). Fm Fn,τ B is a function of τ for each point in phase space (R, P ). Sampling the random forces is terribly expensive, so this is just as diﬃcult as running the full system. 2.3 Generalized Langevin Equation R R Fn,τ is independent Let us gradually simplify (23). First we assume that Fm of the initial momenta P . We then get rid of the ∂/∂Pm term. The time propagator propagates the phase point to time t − τ : t dPn ∂A β R R R (t) = − Fm Fn,τ B (t − τ ). (24) (t) + Fn,t − dτ Pm (t − τ ) dt ∂Rn M 0 m This is the generalized Langevin equation alluded to before. Equation R R (24) should be interpreted with care. In particular, exp{iL(t − τ )} Fm Fn,τ B = R R R R Fm Fn,τ B (t − τ ) is the same function of τ as Fm Fn,τ B but now evaluated at the point R(t − τ ) in conﬁguration space. This is still very diﬃcult, so usually approximations are made for the τ -dependence of the correlation. The simplest approximation is to assume that the random force correlation diﬀers from zero only for very small values of τ , short enough for Pm to change only very little. In practice, such a complete separation of time scales is often hard to achieve, but can be approached if the mass of the CG particle is much larger than that of a bath particle. We can then bring the momenta outside the integral, t ∂A β R R dPn R =− Fm Fn,τ B (t − τ ). (t) + Fn,t − Pm (t) dτ (25) dt ∂Rn M 0 m We next assume that the coordinates R hardly change while the random forces decorrelate. We may then evaluate the random force correlation at the point R(t) instead of R(t − τ ): t dPn ∂A β R R R =− Fm Fn,τ B (t) (t) + Fn,t − Pm (t) dτ (26) dt ∂Rn M 0 m ∂A R (t) + Fn,t − Pm (t)ξmn (R(t)) . (27) ≡− ∂Rn m This is called the Langevin equation. In the last line we have deﬁned the friction ξnm (R(t)), which must be evaluated at the point R(t) in conﬁguration space. More precisely, we can interpret each term Pm (t)ξmn (R(t)) in

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J.T. Padding and W.J. Briels

(27) as the friction force on coordinate n caused by the momentum Pm (t) of coordinate m. As is obvious from these equations, the friction forces are not independent of the random forces. In fact, if we write the τ -dependence of the random force correlation as a delta function, we ﬁnd3 R R Fm Fn,τ B (t) = 2M kB T ξmn (R(t))δ(τ ). (28) This is a clear example of the so-called ﬂuctuation-dissipation theorem, which states that the random forces and friction forces are related. In summary, we have shown how to systematically coarse-grain a system by “projecting out” all irrelevant variables. This procedure is optimal if the memory of the random force correlation can be assumed to be short, that is if some separation of scales exists between the coarse-grained and bath variables.

3 Coarse-Graining in Practice 3.1 Representation of the Potential of Mean Force The obvious question now is: how does one coarse-grain a polymer system in practice? Unfortunately, there is no unique answer; the method of choice is often based on physical intuition. As suggested we may subdivide each polymer chain into subchains, partitioning the degrees of freedom into two sets: the coarse-grained coordinates R, which are the centres of mass of the subchains, and the coordinates q, which are the remaining internal coordinates describing the details of the conﬁgurations. If we now simulate a system that contains only CG particles, using the potential of mean force A(R) in (13) will ensure the correct distribution of R coordinates. However, the potential of mean force is generally a complicated function of all coordinates R, and possibly includes complicated multi-body interactions. For practical and computational reasons it is impossible to calculate and store the potential of mean force for all possible multibody conﬁgurations. Some approximations will need to be made. The most widely used approximation is that the potential of mean force can be assumed to be pairwise additive. For polymers a distinction may be made between nonbonded and bonded particles, with possibly an angular term for bonded triplets to account for chain stiﬀness. Schematically, the potential of mean force is expressed as ϕnb (Ri,j ) + ϕb (Ri,i+1 ) + ϕθ (θi ). (29) A(R) = i<j

i

i

The ﬁrst sum is over all non-bonded pairs at distance Ri,j , the second sum over all bonded particles at distance Ri,i+1 , and the third sum over all groups 3

The factor 2 in (28) arises because the integral in (26) is taken over half a delta peak.

Ab-initio Coarse-Graining of Entangled Polymer Systems

447

of three consecutive bonded particles with internal angle θi . The form and magnitude of the eﬀective potentials ϕnb and ϕb may be found by matching the structure at the pair-level in coarse-grained simulations with the structure measured in microscopic simulations [16], and similarly for ϕθ . 3.2 Polyethylene As an example, we will investigate polyethylene melts. To determine the target distribution functions, atomistically detailed simulations of C120 H242 chains were performed at T = 450 K [17,18], which is about the limit of what currentday computers can do for suﬃciently long time scales. There is much freedom to choose the number of monomers λ per CG particle. In our previous work, the following considerations were taken into account: (1) λ should be large enough to allow for a signiﬁcant increase in the time and length scales accessible to simulation, and (2) λ should not be so large that the “size” of a CG particle exceeds the typical diameter of the tube in the reptation picture, in other words the entanglement length. A suitable choice was λ = 20 CH2 units, which is still roughly one third of the smallest entanglement length reported in the literature [19]. With this choice each polyethylene chain of 120 CH2 units was represented by 6 CG particles. The structure could be matched reasonably well by expressing the nonbonded interaction as a Gaussian pair potential, and the bonded interactions as a sum of a repulsive interactions, described by two Gaussians, and an attractive term, described by a single power law. In formula: 2

ϕnb (R) = c0 e−(R/b0 ) , ϕb (R) = ϕrep (R) + ϕatt (R), ϕ

rep

ϕ

att

(R) = c1 e

−(R/b1 )2 μ

(R) = c3 R .

+ c2 e

−(R/b2 )2

(30) (31) ,

(32) (33)

The resulting interactions are shown in Fig. 3. Because the degree of coarsegraining is so high, the CG particles are rather empty and the interactions between the CG particles are very soft. By this we mean that there is a ﬁnite possibility that CG particles end up on top of each other. This is not an artefact. The centres-of-mass of two diﬀerent pieces of polymer may actually be at the same coordinates, without any of their constituent atoms overlapping. 3.3 Thermodynamic and Dynamic Consistency of the Potential of Mean Force At this point we would like to issue a warning. In the ideal case we would have available the exact potential of mean force A for all possible CG coordinates R, as well as all possible system densities ρ and temperatures T . This would guarantee a perfect description of the thermodynamic properties of our system. However, as explained before, for practical and computational reasons it

448

J.T. Padding and W.J. Briels

b

ϕ and ϕ

nb

[kJ/mol]

20

15

10

bonded

5

nonbonded

0 0

1

2

3

R [nm] Fig. 3. The potential of mean force between bonded (circles) and nonbonded (squares) coarse-grained pieces of polyethylene, each piece representing the centre-ofmass of 20 carbon groups. These potentials were obtained from distribution functions measured in atomistically detailed molecular dynamics simulations of C120 H242 . The solid lines are ﬁts with simple analytical functions (see text). Note that kT = 3.74 kJ/mol.

is impossible to calculate and store the potential of mean force for all possible conﬁgurations, so approximations will need to be made. The use of spatial correlations to construct the potentials is as arbitrary as any other object function, like for instance the compressibility of the system. From a thermodynamic point of view, the free energy may be a good starting point to judge the eﬀectiveness of a model. In particular, a coarse-grained model that has minimal free energy is believed to be the best representation of the ﬁne-grained, microscopic level [20]. Whatever choice is made for the object function, the reader should beware that this choice determines the applicability and non-applicability of the model for certain thermodynamic and dynamic properties. In practice it is only possible to correctly represent a few, but not all properties of the system. For example, in the previous subsection we have made the choice to approximate the free energy by eﬀective pair interactions which reproduce the bonded and nonbonded pair correlation functions as closely as possible. Often the thermodynamics of such a CG system will not correspond to the thermodynamics of the microscopic system. Indeed, it is intuitively clear that the softness of the interactions leads to a pressure which is generally too small and a system which is too compressible. This thermodynamic inconsistency

Ab-initio Coarse-Graining of Entangled Polymer Systems

449

is often worsened by ignoring the state point dependence of the potential of mean force. For a discussion on these matters, the reader is referred to [21–23]. 3.4 Representation of the Friction Once a choice is made for the representation of the potential of mean force, the dynamics of the CG coordinates can be optimized by a careful choice of the friction, which by the ﬂuctuation-dissipation theorem automatically determines the random forces. Note that the thermodynamics of the CG system is not inﬂuenced by the frictional and random forces, and is therefore already ﬁxed by our choice of the potential of mean force. As with the potential of mean force, the friction is generally a complicated function of all coordinates R, and possibly includes complicated memory and multi-body terms. Fortunately, if the degree of coarse-graining is very large and hydrodynamic retardation eﬀects may be neglected, the friction memory is short relative to the velocity decorrelation time of the CG particles. In that case, the Langevin equation (27) applies. As a further approximation we may assume that the friction ξmn depends only on the distance Rmn between particles m and n. Doing so, we arrive at the so-called dissipative particle dynamics method [24]. Although costly this method is rather popular since it conserves momentum, which is a prerequisite for hydrodynamic behaviour, and it is easy to use in non-equilibrium simulations. In the case of polymer melts, it is often even admissible to use a scalar friction with a static background, because the friction may be thought of as being caused by the motion of a (a part of a) chain relative to the rest of the material, which to a ﬁrst approximation may be taken to be at rest. Propagation of a velocity ﬁeld as in a normal liquid is highly improbable, meaning that hydrodynamic interactions are screened [3].

4 Twentanglement 4.1 Uncrossability of Chains The softness of the interactions between the CG particles poses another problem. At some point one may have found a realistic friction for the CG particles, a friction which correctly predicts the relaxation of a time correlation function sensitive to the dynamics at the scale of a few CG particles. One then usually ﬁnds the dynamics at the scale of an entire polymer chain is too fast. The reason, of course, is that the bonds are able to cross each other [18]. The entanglement eﬀect, leading to altered and much slower dynamics, is usually lost. In this section we will describe an algorithm designed to prevent unphysical bond crossings in simulations of polymer systems [18]. Employing this so-called Twentanglement algorithm to a simulation of highly coarse-grained polymer chains will reintroduce the entanglement eﬀect. The principle of the

450

J.T. Padding and W.J. Briels

Fig. 4. Principle of the Twentanglement algorithm. (a) Two line segments representing a bond are closing in on each other. (b) At a certain moment these bonds will touch. An ‘entanglement’ is created at the crossing point X. (c) After this, the bonds are viewed as slippery elastic bands. The elasticity will slow down the relative speed of the bonds. This sequence of events may also be reversed. Reprinted with permission from [18]. Copyright (2001) American Institute of Physics.

algorithm is depicted in Fig. 4. The bonds are considered to be elastic bands between the bonded particles. As soon as two of these elastic bands make contact, an ‘entanglement’ is created which prevents the elastic bands from crossing. To avoid any confusion, in the algorithm ‘entanglements’ are deﬁned as objects which prevent the crossing of chains. Usually only a few of these contribute to entanglements in the usual sense of longlasting obstacles, slowing down the chain movement. For instance, a C60 H122 chain is generally considered not to be entangled, yet many ‘entanglements’ occur in a coarse-grained simulation. 4.2 Algorithm At each simulation step, after updating the positions of the CG particles, the Twentanglement algorithm is called. It consists of three parts: 1. Given the new positions of the CG particles and the topology (order of CG particles and entanglements within each chain), move the entanglements to their new positions and calculate the resulting forces on the CG particles. 2. Detect new entanglements and disentanglements caused by movements of the CG particles and entanglements. 3. If possible, let entanglements slip across a CG particle or each other (topology-altering moves). Moving Entanglements Suppose entanglements already exist. In the algorithm entanglements have no volume and are fully characterised by their positions X. Each bond behaves as an elastic band with respect to the entanglements. This is accomplished as follows. The attractive potential ϕatt between bonded CG particles (33) is no longer a function of the direct distance Ri,i+1 , but of the path length Li,i+1 from CG particle i, via the entanglements, to CG particle i + 1. If there are p entanglements along its path, the path length is deﬁned as (Fig. 5)

Ab-initio Coarse-Graining of Entangled Polymer Systems

451

Fig. 5. Deﬁnition of the pathlength Li,i+1 between bonded coarse-grained entities at Ri and Ri+1 . Reprinted with permission from [18]. Copyright (2001) American Institute of Physics.

Li,i+1 = |Ri − X1 | + |X1 − X2 | + . . . + |Xp − Ri+1 | .

(34)

Since each CG particle represents a large collection of monomers, the heavy backbone of the polymer chain will generally move very sluggishly. This in contrast to an entanglement which at the atomic level includes only a few monomer units. Consequently, the time scale with which the entanglement position adjusts itself is much shorter than the time scale with which the polymer backbone is moving. Eﬀectively, on the coarse-grained time scale, there will be an equilibrium of forces at each entanglement. Such an equilibrium of forces is achieved by the following minimization: ϕatt (Li,i+1 (R, X)) , (35) min X

i

i.e. the entanglement positions X are determined by the requirement that the total attractive part of the energy is at its minimum. During this minimization the CG particles are kept at their respective positions R. Note that all other energy terms are still determined by the CG particle positions only. Detecting New (Dis)Entanglements In the algorithm, a polymer chain is viewed as a succession of objects, either coarse-grained particles or entanglements, connected by line segments. During the simulation the algorithm keeps track of all (unattached) pairs of line segments which are close together. For each pair of line segments and at each instant of time the following triple product is calculated: Vij = (ri − rj ) · [(ri+1 − ri ) × (rj+1 − rj )] ,

(36)

where we have used Fig. 4 as a reference for the indices. The absolute value of (36) is the volume of the parallelepiped deﬁned by the vectors ri+1,i , rj+1,j , and ri,j . Aside from some pathological cases [18], if Vij changes sign from one time step to the next, a bond crossing may have occurred. Additional checks are made to ensure that the crossing takes place along the physical part of the line segments (the above equation checks if two inﬁnite lines have

452

J.T. Padding and W.J. Briels

Fig. 6. Examples of topology-altering moves. (a) A slip past the end of a chain results in chain disentanglement. (b) If a chain is entangled with itself and the loop is shrinking as far as to pass through only one object apart from the entanglement, a self-disentanglement will occur. Reprinted with permission from [18]. Copyright (2001) American Institute of Physics.

crossed). If a real bond crossing has occurred, an entanglement is created at the crossing point. Subsequently, the associated volume Vij will serve to detect future disentanglements. If the volume Vij of the four objects surrounding an entanglement changes sign, a possible disentanglement has occurred, i.e. Fig. 4 may also be read backwards. Topology-Altering Moves While searching for its equilibrium position, an entanglement can move freely along the chain between two adjacent objects. In some cases, however, the attractive part of the energy might be lower if the entanglement could slip past a CG particle, or to the other side of another entanglement, or, in other words, if the topology would be altered. The algorithm detects if an entanglement has a tendency to get close to either one of its adjacent objects. If the distance is smaller than some prescibed value , where is suﬃciently small compared to the average bond length, a subalgorithm will check topology-altering moves. The list of possible moves is quite extensive. The reader is referred to [18] for a complete treatment. Two examples are given in Fig. 6: (a) If an entanglement slips past the last CG particle of a chain, the entanglement is lost, and (b) if a chain is entangled with itself, forming a loop, the loop will disappear if it shrinks far enough. We will now treat two applications of the above algorithm: polymer melts and semidilute solutions of wormlike micelles.

5 Application 1: Polyethylene Melts In Sect. 3 we have introduced the coarse-graining of a melt of C120 H242 chains into a melt of chains consisting of 6 coarse-grained particles each. The point

Ab-initio Coarse-Graining of Entangled Polymer Systems

10

453

-4 Experiment (Pearson et al.) MD simulation (Mondello et al.)

10

-5

MD simulation (Paul et al.)

2

D [cm /s]

CG simulation (Padding et al.)

10

10

-6

-2

-7

10

-8

10

2

10

3

10

4

10

5

MW [g/mol] Fig. 7. Self-diﬀusion coeﬃcient D vs molecular weight Mw for polyethylene at T = 450 K from diﬀerent simulation studies (symbols) and experiment (dashed line). All simulations were carried out at constant density.

of coarse-graining, of course, is to reach larger time and length scales. Having found the interactions between CG particles in a C120 H242 melt, we now simulate chains of higher molecular weight, assuming that the same interactions hold. We then measure various dynamic properties and compare with theoretical (scaling) predictions and experimental results. 5.1 Diﬀusion Results Pearson et al. have experimentally determined the self-diﬀusion coeﬃcient D in alkane and polyethylene melts at 450 K [25], and found that over the range from Mw = 600 up to 120000 it follows a power law, Dexp = 1.65/Mw2

(cm2 /s).

(37)

This result is plotted as a dashed line in Fig. 7. Atomistically detailed simulations [27, 28] generally ﬁnd good agreement (open symbols), but can only be performed in the lower part of the molecular weight range. The CG simulations can go to much larger Mw and are in excellent quantitative agreement (black squares) [26]. The uncrossability of chains is essential: without the Twentanglement constraint, the CG model yields diﬀusion coeﬃcients which are much too large and scale as D ∝ Mw−1 instead of the observed Mw−2 .

454

J.T. Padding and W.J. Briels

5.2 Rheology Results Much theoretical and simulation eﬀort has been spent to predict or reproduce the zero shear relaxation modulus G(t), which measures the relaxation of stress after applying a small step shear strain. Experimentally its Fourier transform is measured by applying small oscillatory shear, yielding the storage and loss moduli G (ω) and G (ω). A prominent feature of viscoelastic liquids is the (temporary) plateau that appears in G(t), signifying the elastic part of the relaxation behaviour that sets in after a liquid-like initial relaxation. The crossover time between the liquid-like and the plateau regime is identiﬁed as the entanglement time τe . Fig. 8 shows G(t) for four diﬀerent polyethylene melts. These were obtained from the ﬂuctuations of the stress tensor S in equilibrium simulations [3, 26]: G(t) =

V

Sxy (t)Sxy (0) , kB T

(38)

where V is the volume of the (periodic) simulation box. For a melt of C80 chains no plateau is observed and the shear relaxation corresponds well to predictions of the Rouse model (dashed lines) [3]. For a melt of C120 , deviations start to emerge for t > τe ≈ 6 ns. These deviations grow into plateau-like behaviour for melts of longer chain lengths. The deviations can be explained well if one assumes reptation relaxation for all modes larger than the tube size and Rouse relaxation for the other modes [26]. Pearson et al. have also measured the zero shear viscosity η0 in polyethylene melts [25]. They found that at low molecular weight Mw < Mc ≈ 5000 g/mol, the viscosity is well described by the power law η0 = 2.1 × 10−5 Mw1.8

(cP),

(39)

while at high molecular weight, Mw > Mc , the Mw dependence is much stronger: (cP). (40) η0 = 3.76 × 10−12 Mw3.64 The experimental ﬁts are plotted as dashed lines in Fig. 9. In simulations the zero shear viscosity can be obtained either in nonequilibrium, by shearing at suﬃciently low shear rates, or in equilibrium, by integrating the measured zero shear relaxation modulus: ∞ η0 = G(t)dt. (41) 0

The results of an atomistically detailed simulation (open symbol) [29] and the CG simulations (ﬁlled squares) [26] are shown in Fig. 9. Again the agreement is very good. The coarse-grained simulations have enabled us to study the dynamics of (in this case) polyethylene melt in great detail. We observed that the dynamics

Ab-initio Coarse-Graining of Entangled Polymer Systems

G(t) [MPa]

10 10 10 10

G(t) [MPa]

1

0

-1

C80

C120

C200

C800

-2

10 10 10 10

455

1

0

-1

-2

10

2

10

3

10

4

10

5

3

10

t [ps]

4

10

5

10

t [ps]

Fig. 8. Shear relaxation modulus of a polyethylene melt, calculated from equilibrium stress ﬂuctuations, for four diﬀerent chain lengths. Negative data are represented by black symbols in these logarithmic plots, to indicate when the (noisy) stress correlation decays to zero. Dashed lines are Rouse model predictions, solid lines are reptation model predictions. The arrows indicate estimates for the entanglement time τe .

is in approximate agreement with reptation theory, but one should be careful of literal interpretation of the tube concept. The chains are interacting with their neighbours on many diﬀerent length scales, even on scales smaller than the (interpreted) tube diameter. Proper care should also be taken of the fact that the chains are not Gaussian, caused by chain stiﬀness and other sources of non-harmonicity. For more details the reader is referred to [26].

6 Application 2: Wormlike Micelles 6.1 Wormlike Micelles as a Polymer System Let us brieﬂy consider another application. Under some circumstances surfactant molecules in aqueous salt solutions self-assemble reversibly into elongated structures. The average contour length of such a so-called wormlike micelle increases rapidly with increasing surfactant concentration. Above the overlap concentration entanglements form between the wormlike micelles, leading to a steep increase of the viscosity with surfactant concentration and a wealth of peculiar viscoelastic eﬀects [30, 31].

456

J.T. Padding and W.J. Briels

10

5

Experiment (Pearson et al.)

η0 [cP]

10

4

10

10

MD simulation (Moore et al.) CG simulation (Padding et al.)

3.6

3

2

1.8

10

1

Me

Mc

0

10 2 10

10

3

10

4

10

5

MW [g/mol] Fig. 9. The zero-shear viscosity η0 vs molecular weight Mw for polyethylene from diﬀerent simulation studies (symbols) and experiment (dashed line). All simulations were carried out at constant density.

In many respects a wormlike micellar network resembles a polymer network as treated in the previous section. The major diﬀerence is that strands in the micellar network can reversibly break. In the presence of oil, the wormlike micelles reassemble into spherical form, resulting in a low viscosity ﬂuid. This responsive behaviour makes them ideal for many industrial applications, such as hydraulic fracturing in the oilﬁeld [32]. From both a fundamental and practical point of view it is therefore important to understand the relation between structure and chemistry of the surfactants and the dynamics and rheology of the macroscopic ﬂuids. Simulations may provide this understanding. 6.2 Simulation Models As usual, atomistic models are limited to small length (10 nm) and time (10 ns) scales, whereas wormlike micelles have contour lengths in the range of micrometers and relaxation times of at least milliseconds. The only way forward is to coarse-grain. A few coarse-grained models have been proposed in the literature [34,35], but all of them may be classiﬁed as top-down models, where no reference is made to any speciﬁc wormlike micellar system. Again, in our opinion, great care must be taken if quantitative or even qualitatively realistic results for the dynamics and rheology are required. There are often multiple relevant length scales which must be taken into account.

Ab-initio Coarse-Graining of Entangled Polymer Systems

457

The philosophy behind the so-called Mesoworm model [33] is that the length scales and, more generally, the material properties of individual wormlike micelles can in principle be measured from detailed simulations or targeted experiments. Relevant properties include the solvent viscosity, worm diameter, bending rigidity (or persistence length), compressibility, and free energies associated with the breaking and fusion process of wormlike micelles. Allowing the model to have these parameters as input, the rheology can be predicted from realistic input, with as few assumptions as possible. In the Mesoworm model, a wormlike micelle is represented by a string of (rigid) rods, each rod representing one persistence length Lp of wormlike micelle. This degree of coarse-graining is as large as possible to permit a larger integration step and fewer particles, while it is still small enough to allow for an accurate description of the overall conformation of the wormlike micelle. Each rod has the possibility to fuse with other rods at its ends, forming breakable bonds each with a “scission” energy −Esc . The scission energy is a very important parameter because it determines the size of the wormlike micelles. In the simplest mean-ﬁeld theory, a wormlike micelle is treated as a random walk on a lattice, with an energy penalty Esc for each pair of chain ends [31]. By variationally minimising the free energy, under the constraint of ﬁxed volume fraction φ, the theory then predicts an exponential contour length distribution with an average contour length given by [31] ¯ p ≈ φ1/2 exp [Esc /(2kB T )] . L/L

(42)

This scaling has been conﬁrmed experimentally. We note that the details of the scission and fusion kinetics are very important for the resulting dynamics and rheology of the micellar solution [36,37]. In real wormlike micelles, before their ends can fuse, there are speciﬁc demands on the conformations of the surfactants in the spherical endcaps, giving rise to a considerable free energy barrier Eact associated with fusion. By increasing this activation barrier, not only the time-scales assocated with scission and fusion increase, but also the physics of the scission-fusion process changes from diﬀusion limited, where immediate self -fusion predominates, to reaction limited, where fusion is mostly with other chain ends [37]. Since most experimental wormlike micelles belong to the latter class [36], it is important to include such an activation barrier in the model. Finally, entanglements are very important for the rheology of a solution of wormlike micelles. In fact, the relaxation of stress in wormlike micellar solutions is believed to be determined for a large part by the balance between reptation and breaking of the micellar chains [31]. Entanglements emerge naturally when two wormlike micelles try to cross. In the Mesoworm model, the entanglements are handled by the Twentanglement algorithm. 6.3 Results A snapshot of a typical simulation box is shown in Fig. 10. This model repre-

458

J.T. Padding and W.J. Briels

Fig. 10. Snapshot of a Mesoworm simulation box.

sents wormlike micelles of the cationic surfactant erucyl bis-(hydroxyethyl)methylammonium chloride (EHAC) in aqueous solution of sodium chloride [38]. Most of the mechanical properties have been calculated from detailed atomistic molecular dynamics simulations of a small segment (about 10 nm) of wormlike micelle [33]. For example, the bending rigidity was determined, and corresponds to a persistence length of about 30 nm. A typical coarsegrained box contains 4000–32 000 persistence length units, at a box size of 300–600 nm. This allows for realistic micellar contour lengths in the range of micrometers. The computational speed is typically of the order of 0.1 - 1 ms per week on a single PC processor. The linear rheology can be measured similarly to the case of polymers from ﬂuctuations in the stress tensor, while the non-linear rheology can be measured by applying sheared periodic boundary conditions. Particularly, the transient rheology may be studied by suddenly applying shear to a sample initially at rest. If the shear rate is high enough, characteristic overshoots in the transient shear stress are observed [33]. After some time a steady state is established, when the shear viscosity is obtained from the ratio of shear stress to shear rate. An example is given in Fig. 11, where the shear-thinning behaviour of a wormlike micellar solution is studied as a function of surfactant concentration. These results are in good agreement with experimental viscosities obtained from rheometry [39].4 Supported by the quantitative agreement with experiment, the coarse-grained model now enables us to test various hypotheses about the dynamics and rheology of these wormlike micelles.

7 Conclusion Simulations are very helpful in understanding the link between microscopic interactions and processes on the one hand, and the macroscopic ﬂuid be4

It should be noted that, for computational reasons, the scission and activation energies were chosen lower than their realistic values, corresponding to a higher eﬀective temperature. It has been observed, however, that this does not inﬂuence the viscosity at high shear rates, but merely the critical shear rate where the transition from the Newtonian plateau to the shear thinning region takes place [38]

Ab-initio Coarse-Graining of Entangled Polymer Systems

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0

10

φ = 5.5% φ = 1.8% φ = 0.65%

shear viscosity [Pa s]

-1

10

-2

10

-3

10

o

T = 60 C

-4

10

Esc = 12 kT, Ea = 1.5 kT Lp = 30 nm, D = 4.8 nm

-5

10

1

10

2

10

3

10

4

10

5

10

6

10

7

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-1

shear rate [s ] Fig. 11. Viscosity vs shear rate for wormlike micellar solutions at various concentrations. Results were obtained by coarse-grained Mesoworm simulations (see text). Reprinted with permission from [33]. Copyright (2001) Institute of Physics.

haviour on the other hand. Sometimes they provide us with qualitative or quantitative predictions, but more importantly they enable us to verify or falsify assumptions that are made in approximate theories. Simulation studies of ﬂuids composed of relatively small molecules have traditionally focused on the Molecular Dynamics method. Such simulations are performed more or less routinely with accurate force ﬁelds that are readily available in the literature. For polymer systems, however, the time and length scales associated with collective dynamics and rheology are so large that an atomistic description is unfeasible and coarse-graining of the interactions becomes absolutely necessary. Unfortunately, this coarse-graining is neither unique nor trivial. The potential of mean force is, in principle, a complicated function of all coarsegrained coordinates. Additionally, the friction may have complicated memory terms. These problems may be overcome by a careful choice of approximations, always keeping an eye on the essential physics. In particular, we have shown that a correct treatment of the relative length and energy scales (persistence length, scale of bonded and nonbonded interactions) combined with the important uncrossability of bonds can lead to quantitative agreement of simulations with experimental results.

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References 1. W. Kuhn: Kolloid Z. 87, 3 (1939); Z. Physik. Chem. B42, 1 (1939) 2. T.P. Lodge, N.A. Rotstein, and S. Prager: Adv. Chem. Phys. 79, 1 (1990) 3. M. Doi and S.F. Edwards: The Theory of Polymer Dynamics (Clarendon, Oxford 1986) 4. R. Kimmich and N. Fatkullin: Adv. Polym. Sci. 170, 1 (2004) 5. R. Everaers et al.: Science 303, 823 (2004) 6. K. Kremer and G.S. Grest: J. Chem. Phys. 92, 5057 (1990) 7. M. P¨ utz, K. Kremer, and G.S. Grest: Europhys. Lett. 49, 735 (2000) 8. W. Tsch¨ op, K. Kremer, J. Batoulis, T. B¨ urger, and O. Hahn: Acta Polymer. 49, 61 (1998) 9. J. Baschnagel et al.: Adv. Polym. Sci. 152, 41 (2000) 10. R.L.C. Akkermans and W.J. Briels: J. Chem. Phys. 113, 6409 (2000) 11. F. Muller-Plathe: Soft Materials 1, 1 (2003) 12. J.M. Deutch and I. Oppenheim: J. Chem. Phys. 54, 3547 (1971); J. Albers, J.M. Deutch, and I. Oppenheim: J. Chem. Phys. 54, 3541 (1971) 13. D.L. Ermak and J.A. McCammon: J. Chem. Phys. 69, 1352 (1978) 14. T.J. Murphy and J.L. Aguirre: J. Chem. Phys. 57, 2098 (1972) 15. M. Dijkstra, R. van Roij, and R. Evans: Phys. Rev. E 59, 5744 (1999) 16. R.L.C. Akkermans and W.J. Briels: J. Chem. Phys. 114, 1020 (2001) 17. J.T. Padding and W.J. Briels: J. Chem. Phys. 114, 8685 (2001) 18. J.T. Padding and W.J. Briels: J. Chem. Phys. 115, 2846 (2001) 19. J.M. Carella, W.W. Graessley, and L.J. Fetters: Macromolecules 17, 2775 (1984) 20. R.L.C. Akkermans and W.J. Briels: J. Chem. Phys. 115, 6210 (2001) 21. W.J. Briels and R.L.C. Akkermans: Mol. Sim. 28, 145 (2002) 22. C.N. Likos: Phys. Rep. 348, 267 (2001) 23. A.A. Louis: J. Phys.: Condens. Matter 14, 9187 (2002) 24. P.J. Hoogerbrugge and J.M.V.A. Koelman: Europhys. Lett. 19, 155 (1992) 25. D.S. Pearson, G. Ver Strate, E. von Meerwall, and F.C. Schilling: Macromolecules 20, 1133 (1987) 26. J.T. Padding and W.J. Briels: J. Chem. Phys. 117, 925 (2002) 27. M. Mondello, G.S. Grest, E.B. Webb III, and P. Peczak: J. Chem. Phys. 109, 798 (1998) 28. W. Paul, G.D. Smith, and D.Y. Yoon: Macromolecules 30, 7772 (1997) 29. J.D. Moore, S.T. Cui, H.D. Cochran, and P.T. Cummings: J. Non-Newtonian Fluid Mech. 93, 83 (2000) 30. L.M. Walker: Curr. Opin. Coll. Int. Sci. 6, 451 (2001) 31. M.E. Cates and S.J. Candau: J. Phys.: Condens. Matter 2, 6869 (1990) 32. G.C. Maitland: Curr. Opin. Coll. Int. Sci. 5, 301 (2000) 33. J.T. Padding, E.S. Boek, and W.J. Briels: J. Phys.: Condens. Matter 17, S3347 (2005) 34. M. Kr¨ oger and R. Makhlouﬁ: Phys. Rev. E 53, 2531 (1996) 35. J.T. Padding and E.S. Boek: Phys. Rev. E 70, 031502 (2004) 36. B. O’Shaughnessy and J. Yu: Phys. Rev. Lett. 74, 4329 (1995) 37. J.T. Padding and E.S. Boek: Europhys. Lett. 66, 756 (2004) 38. S.R. Raghavan and E.W. Kaler: Langmuir 17, 300 (2001) 39. E.S. Boek, J.T. Padding, V.J. Andersson, W.J. Briels, and J.P. Crawshaw: to be published in J. Non-Newtonian Fluid Mech.

Computer Simulations of Nano-Scale Phenomena Based on the Dynamic Density Functional Theories Applications of SUSHI in the OCTA System Takashi Honda1 and Toshihiro Kawakatsu2 1

2

Japan Chemical Innovation Institute, and Department of Organic and Polymeric Materials, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152-8552, Japan Department of Physics, Tohoku University, Aoba, Aramaki, Aoba-ku, Sendai 980-8578, Japan

1 Introduction Multicomponent polymeric materials such as polymer blends, polymer melts, block copolymers, and polymer solutions, often show macro and micro phase separations that generate domains of the length scales of 1–100 nm. These polymeric materials with phase-separated domains are promising candidates for functional materials in nano-technologies [1–3]. The characteristic length scales of these domain structures are much larger than atomic length scales but are still smaller than hydrodynamic length scales. For phenomena on the micro and macroscopic length scales, there are well-established simulation techniques. For example, microscopic phenomena on atomic length scales can be dealt with using particle simulation techniques such as molecular dynamics (MD) simulations. On the other hand, macroscopic hydrodynamic phenomena are simulated with the ﬁnite element method (FEM). Compared to these extreme length scales, there have been very few simulation techniques for the intermediate length scales (the so-called mesoscopic scales) where the phaseseparated domains locate. To study the phase separated domains on mesoscopic scales, very useful tools are the density functional theories (DFTs) [4–7], where the phaseseparated domains are described in terms of the density distributions of monomers and solvents. One of the important features of DFT is that it can take into account the conformational entropy of polymer chains with any molecular architectures, i.e. the monomer sequence and the branching structures. Using this DFT, one can predict the equilibrium state of polymeric systems with mesoscopic structures, which is not easily accessible by the particle simulations or the ﬂuid dynamics simulations. Therefore the DFT plays

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an important role in bridging between microscopic particle simulations and macroscopic ﬂuid dynamics simulations. The DFTs for polymers are categorized into two major classes. One is based on the self-consistent ﬁeld (SCF) theory where the free energy of the system is evaluated using the path integrals of the chain conformations, and the other is based on the phenomenological power series expansion (the socalled Ginzburg–Landau (GL) expansion) of the free energy where the random phase approximation (RPA) is used in evaluating the expansion coeﬃcients. These DFTs have successfully reproduced the experimental results of polymeric systems, especially for block copolymers. The phase diagrams of diblock copolymers are predicted using the RPA [8] and the SCF theory [9,10], respectively. For the dynamical aspects, the physical responses of the micro domains in diblock copolymers to an external ﬁeld, such as a shear ﬂow or an electric ﬁeld, are simulated using a dynamical extension of the SCF theory [11, 12]. The order–order and disorder–order transitions in diblock copolymers are also investigated with the DFTs [13–17]. Despite the success of the DFTs on mesoscopic phenomena in polymeric systems, there have been very few general multi-purpose simulation tools for DFTs. We have been developing the “Open Computational Tool for Advanced material technology (OCTA)” system which is tailored for mesoscopic scale simulations. All source codes of the OCTA system are open to the public, and anyone can download these codes from the internet site and can use the OCTA system for nano-scale studies. In the OCTA system, there are several simulation engines. One of these engines is the “Simulation Utilities for Soft and Hard Interfaces (SUSHI)” which is designed to implement DFTs in general [18]. SUSHI can carry out static and dynamic calculations by using the SCF theory for any melts and blends of polymers with any types of architectures. With the static calculation, one can get the equilibrium state of the system. On the other hand, with the dynamic calculation, one can reproduce the time-dependent phase transition behavior such as the phase separation, the adsorption, and the order–order transition between microphase separated structures, and so on. The SCF theory presents a quantitatively accurate calculation method to deal with these phenomena because the conformational changes of polymers are taken into account in the calculations using the path integral technique. Moreover, with the SCF theory, one can obtain the physical quantities associated with the interfaces and domains such as the interfacial tension, critical micelle concentration, and so on. In return for the quantitative accuracy of the calculations, the SCF theory demands much computer power. To reduce the computational cost, it is worth introducing a DFT based on the series expansion of the free energy, such as GL theory combined with the RPA. We will abbreviate this method as GRPA [19]. Although the range of validity of GRPA is limited only to the weak segregation regime because of the truncation of the free energy expansion used in the GL theory, the computational eﬃciency is much higher than SCF theory.

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The basic assumption commonly used among the DFTs is the Gaussian chain approximation. In the next section, this Gaussian chain approximation is brieﬂy summarized. Then, we give an overview of the general features of DFTs (SCF theory and GRPA) in Sect. 3. The technical details of the SCF theory and the GRPA are given in Sects. 4 and 5, respectively. We also give several examples of the simulation results at the end of these sections. Finally, we discuss the perspective of the DFTs in Sect. 6.

2 Gaussian Chain Model The DFTs can treat mixtures of polymers with any molecular architectures and solvents. A polymer is a chain molecule composed of a sequence of repeating units. Here, the repeating unit is in general diﬀerent from the monomer. For example, an ethylene monomer becomes two methylene units in polyethylene upon polymerization. In the DFTs, the minimal structural unit of the polymer chain is called a segment, which consists of several repeating units. The number of repeating units in a segment is chosen so that the consecutive bond vectors connecting the segments are statistically independent. Such a chain is referred to as a Gaussian chain because, as will be shown below, the probability distribution of the chain conformation is described by a Gaussian distribution. The average length of the bond vector connecting the segments is called an eﬀective bond length. A Gaussian chain is also called an ideal chain, which means that the segment-segment interactions are negligible. Modeling a polymer chain on a coarse-grained scale with a Gaussian chain is called the Gaussian approximation. Due to the screening eﬀect, this Gaussian approximation becomes valid for highly concentrated polymer systems, i.e., polymer melts and blends, and dense polymer solutions in which the concentration is semi-dilute or more [7]. To explain the statistical nature of the Gaussian chain, we focus on a subchain which is a part of the whole ideal chain. As is shown in Fig. 1, it is convenient to introduce a lattice model of the polymer chain where the lattice constant corresponds to the eﬀective bond length b of the ideal chain. In Fig. 2, we show the possible positions of a nearest neighbor segment that is bonded to the segment located at the center of the cubic cell. The assumption of the ideal chain model is that the nearest neighbor segment can take any nearest neighbor sites freely irrespective of the occupation of the lattice sites by other segments. Let us calculate the statistical weight of a subchain whose end segments are speciﬁed by the indices i and i (i < i) and are ﬁxed at the sites r and r, respectively. We denote the statistical weight of this subchain by Q(i , r ; i, r). Due to the chain connectivity, the statistical weight Q(i , r ; i + 1, r) is related to Q(i , r ; i, r) through the following recurrence formula 1 Q(i , r ; i + 1, r) = Q(i , r ; i, r ), (1) z r

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Fig. 1. Schematic ﬁgures of lattice models of ideal chains

Fig. 2. Possible positions of the nearest neighbor segment in a lattice model of a polymer chain

where z is the number of nearest neighbor sites (z = 6 in the case shown in Fig. 2) and r is one of the possible positions of the nearest neighbor sites to r. Equation (1) means that Q(i , r ; i + 1, r) is the sum of the contributions from the statistical weight of the i-th segment placed on the nearest neighbor sites r . Since each nearest-neighbor position is statistically equivalent in the ideal chain model, each contribution is proportion to the probability 1/z. From the discretized expression of the recurrence formula (1), let us derive a continum formula. Subtracting Q(i , r ; i, r) from both sides of (1) leads to Q(i , r ; i + 1, r) − Q(i , r ; i, r) =

1 Q(i , r ; i, r ) − Q(i , r ; i, r). z r

(2)

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465

In the continuum limit, the left hand side becomes the derivative of Q(i , r ; i, r) with respect to the index i and the right hand side becomes the Laplacian of Q(i , r ; i, r). Thus (2) reduces to the following three dimensional (z = 6) continuum equation: ∂ b2 Q(s , r ; s, r) = ∇2 Q(s , r ; s, r), ∂s 6

(3)

where s is used as a continuous parameter instead of the discrete index i and the b2 factor appears when the ﬁnite diﬀerence is replaced by the Laplacian. Equation (3) can be regarded as a three dimensional diﬀusion equation of Q(s , r ; s, r) where the parameter s corresponds to time and the diﬀusion constant is given by D = b2 /6. Equation (3) can be solved when the initial condition at the end segment of the subchain is given. For example, if the end segment is ﬁxed at the position r , the initial condition is given by Q(s , r ; s , r) = δ(r − r ).

(4)

The solution of the diﬀusion equation (3) under this initial condition is given by Q(s , r ; s, r) =

3/2 3|r − r |2 3 . exp − 2 2π|s − s |b 2|s − s |b2

(5)

As this statistical weight distribution is a Gaussian distribution, the ideal chain obeys Gaussian statistics. Several physical properties of the Gaussian chain are obtained as follows. The probability distribution of the end-to-end vector R of a Gaussian chain composed of N segments is obtained from (5) as [7] P (R) = Q(0, 0; N, R) =

3R2 3 3/2 . exp − 2πN b2 2N b2

The average squared end-to-end vector < R2 > is given by ∞ 2 dR 4πR4 P (R) = N b2 , < R >=

(6)

(7)

0

where we assume that the chain is isotropic, i.e. the probability distribution function P (R) is a function only of R = |R|. The average squared end-to-end vector of a subchain whose end segments are s and s segments at r and r is also given by < (r − r )2 >= |s − s |b2 .

(8)

Using this relation, we can calculate the square of the radius of gyration RG that is deﬁned as the root mean square of the distance between each segment and the center of mass of the chain. The result leads to [20]

466

Takashi Honda and Toshihiro Kawakatsu 2 RG =

=

1 2N 2 1 2N 2

N

ds

0

0

N

ds < (r − r )2 >

0

N

ds

N

ds|s − s |b2 =

0

1 2 Nb . 6

(9)

The Fourier transformation of the distribution of the end-to-end vector of a subchain is given by −∞ 3/2 3 3r 2 dr exp ( iq · r) exp − 2π|s − s |b2 2|s − s |b2 −∞ = exp ( −

b2 |s − s ||q|2 ). 6

(10)

This expression gives the scattering intensity from a pair of segments s and s . Thus the total scattering function from the whole chain, i.e., the structure factor for a Gaussian chain, is given by [8]. 1 N N b2 ds ds exp(− |s − s ||q|2 ) S (q) = N 0 6 0 2N = 2 (e−x − 1 + x), x 0

(11)

2 where x ≡ (1/6)N b2 |q|2 = RG |q|2 and the function 2(e−x − 1 + x)/x2 in (11) is called the Debye function.

3 An Overview of the SCF Theory and GRPA A polymer chain in a polymer blend is not an ideal Gaussian chain if there are strong segment–segment interactions between diﬀerent types of segments or strong segment–object interactions exist, where “object” means impurity particles, such as the ﬁller particles, walls and surfaces. For example, in a system with a wall, the ideal Gaussian distribution of segments is distorted in the vicinity of the wall. The SCF theory is applicable to such systems under strong interactions. The distorted chain conformations are evaluated with the use of the path integral technique, where the segment-segment and segment-object interactions are treated as external ﬁelds acting on the ideal Gaussian chain. To obtain such external ﬁelds, an iterative reﬁnement procedure is necessary, which increases the computational cost considerably. On the other hand, there is no iteration procedure in the GRPA, and its computation is much faster than the SCF theory. However, in the GRPA, the free energy is expanded in power series around a uniform reference state, which means that the way of introducing external objects into GRPA is not trivial. Although the quantitative accuracy of the GRPA theory is poorer than that

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of the SCF theory, the GRPA is very useful in evaluating the phase behavior and the mesoscopic structures of polymer melts and blends qualitatively. In the subsequent sections, we explain the SCF theory and the GRPA in more detail. 3.1 Deﬁnition of the Molecular Architecture of a Polymer and the Characters of a Solvent A polymer is composed of subchain(s). For example, a homopolymer consists of a single subchain and a block copolymer consists of several subchains of diﬀerent kinds. Each segment has three important characteristics that are used in SCF calculations. These characteristics are the chemical species (segment type), eﬀective bond length, and the speciﬁc volume. The molecular structure of a polymer is deﬁned by the topology of the connectivity of the subchains, the type of the segments comprising each subchain, and the length of each subchain. We regard a solvent molecule as a special kind of segment without a bond. Thus, a solvent has two characteristics, i.e. the chemical type and the speciﬁc volume. We use superscript indices p and s (p, s = 1, 2, 3, · · ·) to identify the type of the polymer and the solvent, respectively, and we use a subscript index K to identify the chemical type of the segment (including the solvent as a special case). Subscript indices i and j (i, j = 1, 2, 3, · · ·) are used to identify all subchains in the system where we denote the number of subchains in a p-type ' polymer by n(p) and the total number of subchains in the system by n = p n(p) . We also deﬁne that the number of segments of the i-th subchain of the p-type (p) polymer is Ni and the total number of segments of the p-type polymer is ' (p) N (p) = i Ni . Figure 3 shows a typical topology of the subchain connectivity in a polymer chain (p-type polymer). Each subchain is numbered and the chemical type of the subchain is distinguished by the line style. A subchain is joined to other subchains by chemical bonds. We will deﬁne an end connected to other subchains as a junction point.

4 SCF Theory We will use the SCF theory to derive the expressions of density distributions of the segments and the solvents and the expression of the free energy. The SCF theory relies on the mean-ﬁeld approximation which assumes that the same types of segments feel an identical external mean ﬁeld. In other words, the SCF theory adopts a single body picture where a single chain is placed in an external mean ﬁeld. Under this approximation, the ﬂuctuations

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Takashi Honda and Toshihiro Kawakatsu

Fig. 3. Typical p-type polymer

among the chains are neglected and we only assume that the same types of polymers show the same statistical nature. 4.1 Solvents The mean-ﬁeld approximation can also be applied to solvents, which are molecules consisting of a single segment. The density of a K-type solvent at position r is in proportion to the Bolzmann factor as (s)

(s)

(s)

φK (r) = CK exp[−βρK VK (r)],

(12)

(s)

where β = 1/(kB T ), CK is a normalization constant, ρK is the speciﬁc vol(s) ume of the K-type solvent, and VK (r) is an external potential acting on (s) the K-type solvent. The external potential VK (r) is determined so that the constraints imposed on the system, such as the incompressibility condition, (s) are satisﬁed. The actual procedure to determine VK (r) is given in the next (s) section. The normalization constant CK depends on the statistical ensemble we use. There are two typical ensembles. One is the canonical ensemble which keeps the total volume fractions of each type of component in the system constant. The other is the grand canonical ensemble in which the system is assumed to be in equilibrium with the bulk reservoir, and therefore the total volume fractions of each type of component ﬂuctuate. Canonical Ensemble One can use the canonical ensemble for both ﬁnite systems surrounded by impenetrable walls and inﬁnite systems (surrounded by periodic boundaries or reﬂective boundaries in the actual simulations). This canonical ensemble is useful in calculating for example interfaces in an A/B polymer blend. (s) In this case, the normalization constant CK is given by

Computer Simulations of Nano-Scale Phenomena

M (s)

(s)

CK =

=

(s)

dr exp[−βρK VK (r)]

M (s) (s)

ZK

,

469

(13)

where M (s) is the total number of K-type solvent molecules in the system (s) (s) and ZK = dr exp[−βρK VK (r)] is the partition function of the K-type solvent. Grand Canonical Ensemble When one considers a system with a wall to which polymers are adsorbed from a bulk reservoir, it is convenient to use the grand canonical ensemble rather than the canonical ensemble. (s) In this case, CK is given by (s)

CK =

m(sb) (b)

exp[−βρK WK ]

=

m(sb) (sb)

ZK

,

(14)

where the superscript b means that the parameter is evaluated in the bulk uniform reservoir, i.e. m(sb) is the total number of K-type solvent molecules (b) per volume, WK is the segment-segment interaction potential acting on a K(sb) (b) type solvent molecule, and ZK = exp[−βρK WK ] is the partition function of the K-type solvent per volume. (b) The explicit expression of WK is given by (b) ¯(b) WK = (15) KK φK , K

where φ¯K is the total volume fraction of the K -type segment in the bulk reservoir, and KK is the interaction energy between K-type and K -type segments. (b)

4.2 Polymers The statistical nature of the distribution of the segments of polymers is different from that of solvents because there are extra contributions from the conformational degrees of freedom of the polymer chains. Path Integral We consider the i-th subchain composed of the K-type segments in the ptype polymer. As we have already deﬁned, the total number of segments in (p) this subchain is Ni . The K-type segment has two physical characteristics: the eﬀective bond length bK (the same as b in (3) ) and the speciﬁc volume ρK . As there is an arbitrariness in the deﬁnition of the eﬀective bond length

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bK (or equivalently the deﬁnition of the size of a segment), we should ﬁx this (p) parameter bK and the total number of segments in a subchain Ni simultaneously so that the radius of gyration of the i-th subchain RGi obeys the (p) 2 relation (9), i.e., RGi = (1/6)Ni b2K . The speciﬁc volume ρK is needed to relate the SCF calculation to the mass density of the repeating unit in the real system. We deﬁne the statistical weight of the i-th subchain as Qi (s , r ; s, r) and the external potential which acts on the segments of the i-th subchain as Vi (r). When the values of bK , ρK and Vi (r) are given, (3) is modiﬁed as b2 ∂ Qi (s , r ; s, r) = K ∇2 − βρK Vi (r) Qi (s , r ; s, r). ∂s 6

(16)

This equation is no longer a simple diﬀusion equation because the total amount of the diﬀusing material is not conserved due to the term containing Vi (r). Equation (16) is called the Edwards equation. The solution Qi (s , r ; s, r) deviates from the ideal Gaussian statistics when Vi (r) becomes large. The solution of (16) is the sum of Boltzmann weights for all possible chain conformations, which is identiﬁed as the path of a diﬀusing particle in (16). Therefore, Qi (s , r ; s, r) is called the path integral. Vi (r) in (16) can be divided into two contributions: a segment-segment interaction potential and an external potential due to the constraints on the segment density distributions, such as the incompressibility condition. These potentials are derived as follows. We denote the local segment density of the i-th subchain as φi (r) and assume that the segment-segment interactions are short range interactions that are proportional to the product of the pair of the local segment densities with the interaction energy KK introduced in (15). The contribution from the segment-segment interaction potential is then given by (17) WK (r) = KK φK (r), K

where φK (r) is the sum of the segment densities of the subchains composed of K -type segment and K -type solvent densities as (s) φj (r) + φK (r), (18) φK (r) = j∈K

where we assume that the size of a segment and the size of a solvent molecule are the same. The interaction parameter KK can be related to the Flory–Huggins interaction parameter χ. For an A/B binary mixture, we obtain χ ≡ zβ{

AB

1 − ( 2

AA

+

BB )}.

(19)

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471

The external potential for the constraint on the segment density distribution is a potential that keeps the proﬁle of the segment density distribution at a given proﬁle. Therefore, this constraining potential should balance with the chemical potential of the segment μi (r) so that the change in the proﬁles of the segment density do not occur. This leads to the result that the constraining potential is given by −μi (r). As a result, the self-consistent potential Vi (r) is given by Vi (r) = WK (r) − μi (r).

(20)

The same relation as (20) is also applicable to the K-type solvent as (s)

(s)

VK (r) = WK (r) − μK (r),

(21)

(s)

where μK (r) is the chemical potential of the K-type solvent. The potential Vi (r) and φi (r) are related to each other under the constraining condition and the total incompressibility condition [4, 7, 9, 21]. The total incompressibility condition is evaluated using the sum of the segment densities φi (r) over the index i. The segment density at the position (s, r) of the i-th subchain is calculated using two path integrals starting from both ends of the subchain. As was explained in (4) for the ideal homopolymer case, if the starting segment for the path integral calculation is a free end of the chain, the initial condition for (16) is given by Qi (0, r ; 0, r) = δ(r−r ). On the other hand, if the starting segment of the subchain is a junction point, the initial condition for (16) is a product of the path integrals of the subchains that meet at this junction point except for the subchain for which we are calculating the path integral. When we denote these initial statistical weights (p) (p) at both ends of the sumchain s = 0, Ni as qi0 (0, r ) and qi0 (Ni , r ), we can introduce the following integrated path integrals from both ends of the subchain as (22) qi (s, r) ≡ dr qi0 (0, r )Qi (0, r ; s, r) (p) q;i (s, r) ≡ dr qi0 (Ni , r )Qi (0, r ; s, r). (23) It is easy to conﬁrm that qi (s, r) and q;i (s, r) also satisfy (16). We will call these qi (s, r) and q;i (s, r) the normal direction integrated path integral and the reverse direction integrated path integral, respectively. When the end of a subchain is a free end, the initial value of the statistical weight of this free end is uniform everywhere, and we have qi0 (0, r ) = 1 or (p) qi0 (Ni , r ) = 1. Otherwise the end is a junction point as shown in Fig. 3, where the initial statistical weight of the end must be a product of all the statistical weights of the other subchains connected to this junction point. To calculate the statistical weight of a branching chain, the path integrals of all subchains should be evaluated sequentially starting from the free ends.

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We show, for example, a typical order of evaluation by the arrows with circled numbers on the subchains in Fig. 4. An arrow with a circled number along a subchain indicates the direction of the normal direction path integral and the opposing arrow on the same subchain indicates the direction of the reverse direction path integral. Having such an order of evaluation, one can avoid a loss of evaluation of the statistical weights of all subchains.

Fig. 4. Typical p-type polymer with order of evaluation of path integrals

Segment Density Calculation When the path integrals are evaluated, the segment density of the i-th subchain at position r is given by (p) φi (r) = C (p) ds qi (s, r); qi (Ni − s, r), (24) s∈i−th subchain (p)

where C is a normalization constant and the integral over s is taken over the whole i-th subchain. The integrand on the right-hand side is the statistical weight of the whole polymer chain under the condition that the segment s is located at position r because it is a product of the path integrals of the two parts of the whole chain meeting at this segment. To accumulate the contributions from all the segments of the i-th subchain, the integrand is integrated over s in the subchain. As in the case of the solvent, the expression for C (p) depends on the ensembles we use. Canonical Ensemble The normalization constant C (p) is given by C (p) =

M (p) (p)

dr qi (s, r); qi (Ni

− s, r)

=

M (p) Z (p)

,

(25)

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473

where M (p) is the total number of p-type polymer chains to which the i-th subchain belongs and Z (p) is the partition function of a p-type chain. Note that this partition function is independent of the subchain index i in the p-type chain. Grand Canonical Ensemble The normalization constant C (p) is given by C (p) =

m(pb) m(pb) = (pb) , (p) Z exp[−β ρK Ni WK (b)]

(26)

i∈p

where m(pb) is the total number of p-type polymers per volume in the bulk reservoir [4], and Z (pb) is the partition function of the p-type polymer per volume in the bulk reservoir. Note that the C (p) for the grand canonical ensemble is a constant determined only by the condition of the bulk reservoir while the C (p) for the canonical ensemble changes depending on the internal domain structure in order to keep the total volume fractions of polymers in the system. 4.3 Free Energy The Helmholtz free energy of the system can be given as a function of the (s) segment densities of subchains {φi } and the solvent densities {φK } as follows [4, 7, 9]. / 1 (p) 1 (p) (s) 0 F {φi }{φK } = − M ln Z (p) + M ln M (p) β p β p 1 (s) 1 (s) − M ln Z (s) + M ln M (s) β s β s 1 + dr KK φK (r)φK (r) 2 K K (s) (s) − drVi (r)φi (r) − drVK (r)φK (r). (27) i

K

In this expression, the ﬁrst and second terms correspond to the conformational entropy and the ideal mixing entropy of the polymers, respectively, and the third and fourth terms correspond to those of the solvents. The ﬁfth, sixth and seventh terms are the interaction energy among the segments and solvents, the interaction energy between the external ﬁelds and the segments, and the interaction energy between the external ﬁelds and the solvents, respectively. The most important factor is the ﬁrst term that comes from the

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Takashi Honda and Toshihiro Kawakatsu

conformational entropy of polymer chains. The advantage of SCF theory is that this conformational entropy can be quantitatively evaluated. A functional derivative of the free energy (27) with respect to φi (r) or (s) (s) φK (r) gives μi (r) or μK (r), which reduces to the relations (20) and (21). 4.4 Static Density Functional Theory In the equilibrium state of the system, the chemical potential diﬀerences between any pair of segment species and solvents are spatially uniform so that no mutual diﬀusion takes place. In this case, the total incompressibility condition is imposed on the segment densities φi (r) instead of constraining each seg(s) ment density proﬁle. Therefore, all the constraint potentials μi (r) and μK (r) are identical to the constraining potential due to the total incompressibility condition. (We assume that all the segments and the solvent molecules have the same volume.) Therefore we obtain (s)

μi (r) = μK (r) = μe (r),

(28)

where μe (r) is the chemical potential due to the total incompressible condition. 4.5 Dynamic Density Functional Theory For dynamical simulations, we introduce a simple diﬀusion process for each segment density. Here we assume Fick’s law of linear diﬀusion. Then, the time evolution equation for the segment density of the i-th subchain is given by [22, 23] ∂ φi (r, t) = ∇ · [Li (r, t)∇{μi (r, t) + λ(r, t)}] + ξi (r, t), ∂t

(29)

where Li (r, t) is the local mobility, λ(r, t) is a Lagrange multiplier for the local incompressibility condition, and ξi (r, t) is the random noise due to thermal ﬂuctuation. The Lagrange multiplier λ(r, t) is determined using the local incompressibility condition, and is expressed as ' Li (r, t)∇μi (r, t) . (30) ∇λ(r, t) = − i ' i Li (r, t) The thermal ﬂuctuation ξi (r, t) satisﬁes the following ﬂuctuation–dissipation relation

ξi (r, t)ξi (r , t ) = 2kB T Lij (r, t)∇2 δ(r − r )δ(t − t ).

(31)

Here, Lij (r, t) on the right-hand side is given by the local mobility Li (r, t) as

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Lij (r, t) = Li (r, t)δij +

Li (r, t)Lj (r, t) , Li (r, t)

475

(32)

i

where the second term corresponds to a Lagrange multiplier for the local incompressibility condition. The same type of equation as (29) can be used for the time evolution of (s) the solvent density φK (r, t). The local mobility Li (r, t) in general depends on the environment. In the simplest model, Li (r, t) is assumed to be a constant. This assumption is valid when the segment density ﬂuctuations are small compared to the average densities. Another simple model is to assume that Li (r, t) depends on the segment density as (33) Li (r, t) = L0 φi (r, t), where L0 is a constant. This model can be used for the Rouse dynamics or the reptation dynamics [20, 24]. In the dynamic simulations, each subchain has a diﬀerent chemical potential that depends on its instantaneous density proﬁle. For example, homopolymers with diﬀerent chain lengths have diﬀerent chemical potentials even if these are composed of the same type of segments. Thus we have to treat all μi (r, t) as independent variables. The total incompressibility condition imposes a constraint on these μi (r, t)’s. Therefore, the μi (r, t)’s are not independent of each other. 4.6 SCF Method In the SCF simulations, the value of Vi (r) in (16) is obtained by recursive calculations. As is shown in Fig. 5, the quantities φi (r), Vi (r) (including the (p) solvents in general), qi (s, r) and q˜i (Ni − s, r) are mutually related and form a self-consistent loop. To solve this self-consistent condition, one has to rely on a recursive calculation, where Vi (r) is adjusted so that the total incompressibility condition and the constraint on each segment density proﬁle φi (r) is fulﬁlled. When this recursive calculation converges, Vi (r) becomes the selfconsistent ﬁeld. This SCF scheme is basically a multi-dimensional optimization problem. The total number of parameters that should be adjusted is in proportion to the number of external ﬁelds at all mesh points in the system. (We use a spatial mesh to solve the Edwards equation.) Therefore, when the system size is large, a numerical method that uses second derivatives of the judge function with respect to the adjusting parameters, such as the conjugate gradient method, can not be used because the number of second derivatives is too large to be treated numerically. To avoid this diﬃculty, we propose a straightforward SCF method that is similar to the steepest descent method, where the second derivatives are not required. Such a straightforward SCF method falls into two categories.

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Fig. 5. A recursive calculation scheme in the SCF simulations

One is the static method explained in Sect. 4.4, by which one can generate the equilibrium state starting from an arbitrarily chosen initial state. For example, a convenient choice of the initial condition is to give a small random distribution of external potential and then introduce the segment-segment interactions gradually into the SCF calculations. In this case, the evolution of the system does not correspond to the actual time evolution. The other is the dynamic method explained in Sect. 4.5 where the equilibrium state is achieved by tracing the time evolution of the segment densities according to the diﬀusion process. Static SCF Method In the static calculation, all segments of the same type are subjected to the same self-consistent ﬁeld irrespective of the subchains they belong to. As was (s) discussed (28), we can assume that all μi (r)’s and μK (r)’s are equal to the same ﬁeld μe (r). In order to get the equilibrium state, a simultaneous updating of WK (r) in (17) and μK (r) should be performed starting from an appropriate initial distribution. This updating is realized in the following manner. φ (r) − W (r) (34) WK (r) −→ WK (r) + αW × KK K K K

μe (r) −→ μe (r) − αV × 1 − φK (r) ,

(35)

K

where αW and αV are appropriately chosen constants. Equation(34) introduces the segment-segment interactions gradually. Equation(35) guarantees that the total incompressibility condition is achieved when the iteration converges. Dynamic SCF Method As an initial condition for a dynamic calculation, segment density distributions (s) of subchains {φi (r)} and those for solvents {φK (r)} are given appropriately under the total incompressibility condition:

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φi (r) +

i

(s)

φK (r) = 1,

477

(36)

K

where, on the left-hand side, the ﬁrst term is the sum of the segment densities of subchains and the second term is the sum of the solvent densities. As in the case of the static SCF calculation (see (35)), the chemical potential μi (r) for the given segment density φi (r) is evaluated by recursive calculations as $ % μi (r) −→ μi (r) − αV × φi (r) − φi (r) , (37) where φi (r) is the segment density calculated by the SCF calculation, which should be equal to the given density distribution φi (r). As described in Sect. 4.5, individual chemical potentials are independent in the dynamic calculations, even if the subchains are composed of the same type of segments. On the other hand, recursive calculations for solvents are not needed to (s) obtain the chemical potentials. Solving (12) with respect to μK (r) for the (s) given solvent density φK (r) gives φ (r) 1 log K (s) + WK (r) βρK C (s)

(s)

μK (r) =

(s)

= −VK (r) + WK (r).

(38)

4.7 Features of SUSHI for the SCF Calculations In this section, we show the typical futures of SUSHI as a simulation tool for the SCF theory. Many functions are implemented in SUSHI [18]. Polymer Structure SUSHI can treat any types of polymers such as homopolymers, block polymers, comb polymers, star (multi-miktoarm) polymers and ring polymers. Any topological structures can be treated by SUSHI except for those that contain many loops such as cross-linking polymers, because an SCF calculation on many loop polymers requires extra recursive calculations. SUSHI can also simulate a tapered polymer that consists of several types of segments that are distributed along the chain with a gradient. By assigning a path integral to each type of segment, SUSHI evaluates the path integral of a tapered polymer [4]. Various Spatial Meshes SUSHI can treat various types of system such as an inﬁnite periodic system, a system that is in equilibrium with a bulk reservoir, an isolated micelle or a vesicle in a solvent, etc. For simulation of these various situations, the following spatial meshes are prepared in SUSHI.

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Takashi Honda and Toshihiro Kawakatsu

1. Regular and rectangular meshes are available to treat Cartesian coordinate systems with a rectangular shape. Depending on the spatial structures of the target system, the user can select one, two or three dimensional meshes. 2. A cylindrical mesh is available for simulating micelles, vesicles and so on. This cylindrical mesh is a two dimensional mesh system with radial and horizontal axes. 3. A spherical mesh is available for simulating micelles. This spherical mesh is a one dimensional mesh system with only radial coordinate. Boundary Conditions In SUSHI, a periodic boundary condition, a Dirichlet boundary condition and a Neumann boundary condition (reﬂective boundary condition) can be used. The periodic boundary condition is used when simulating an inﬁnite system where the eﬀects of the ﬁnite size of the simulation box should be eliminated (or reduced). The periodic boundary condition is also useful for simulating periodic domain structures in micro phase separations. The Dirichlet boundary condition is used for the calculation of the path integrals at the surface of a wall. On the other hand, the Neumann boundary condition is used in the dynamic SCF simulation for the segment density ﬁelds at the surface of a wall. This Neumann boundary condition is also used for the boundaries of the cylindrical and the spherical meshes where the boundary of the simulation box is not a realistic wall but a hypothetical boundary beyond which the system continues. By combining these Dirichlet and Neumann boundary conditions, one can simulate such a system with a wall and a system in equilibrium with a bulk reservoir. Other Features In SUSHI, one can perform simulations on the following situations. • • • • • •

a solid surface onto which polymers are grafted a polydisperse homopolymer system (static calculations only) chemical reactions polyelectrolyte and electrostatic interactions [4] a system onto which a shear ﬂow is imposed [11, 12] a system with hydrodynamic interactions.

4.8 Examples of the SCF Simulations on Block Copolymers As demonstrations of the usefulness of the SCF simulations, we show several simulation results of microphase separations of block copolymer systems.

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479

A block copolymer is composed of several blocks (subchains) whose ends are connected together by covalent bonds. When the blocks are mutually immiscible, a phase separation takes place. However, the connectivity of the diﬀerent blocks prevents a macro phase separation. As a result of these two eﬀects, a block copolymer shows a micro phase separation that has a periodic domain structure with a characteristic length scale of the order of 10–100 nm. AB type diblock copolymers are known to show typical microphaseseparated structures, and phase diagrams were obtained experimentally for various types of diblock copolymers. These phase diagrams have a similarity that does not depend on the chemical detail of the block copolymers. Such a universal nature is a target of the DFTs. Theoretical phase diagrams were actually obtained using RPA and SCF theories [8, 10]. Thin ﬁlms of diblock copolymers are expected to be used as nano-scale functional materials for data storage, photonic materials, etc [3]. The microphase-separated domains become much more complex when the numbers of blocks are increased. Making use of such complex nature, multiblock copolymers are used as industrial functional materials. For example, ABA type triblock copolymers form a network of micelles. One of these triblock copolymers is the SIS (S=styrene, I=isoprene) thermoplastic elastomer. Since the glass transition temperature of PS is higher than room temperature and that of PI is lower than room temperature, the PS micro domains perform as physical cross-linking points and the SIS shows elasticity [25, 26]. Star triblock polymers (three-miktoarm polymers) have also received considerable attention because of their fascinating microphase-separated structures [27–29]. Microphases of Diblock Copolymers The stable region of each microphase of a diblock copolymer is speciﬁed by two parameters, χN and f . Here, χ is the Flory–Huggins interaction parameter between the blocks, N is the total number of segments in the chain, and f is the block ratio deﬁned as the number of segments in one of the two blocks divided by N . Figure 6 shows snapshot pictures of typical microphase- separated domains of a diblock copolymer obtained by SCF simulations, where the isosurfaces of the volume fraction of the segment of the minor phase are shown. Figs. 6a-e correspond to body-centered cubic sphere (BCC), hexagonally packed cylinder (HEX), bicontinuous double gyroid (GYR), hexagonally perforated lamellar (HPL), and lamellar structures (LAM), respectively. Except for the HPL phase, all phases have their stable regions in the parameter space composed of χN and f . The HPL phase is a metastable phase and two types of stacking layers of hexagonally perforated lamellae are observed. The stacking layers are characterized by the relative positions of the hexagonally packed holes. One is an AB type and the other is an ABC type which is shown in Fig. 6d [30–32].

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Takashi Honda and Toshihiro Kawakatsu

Fig. 6. Typical microphase separated structures of diblock copolymers

Fig.7 shows the details of the GYR structures, where a super cell (2×2×2 unit cells) is shown in Fig. 7a. Figure 7b shows a cross section of the GYR structure, which shows a continuous “U” shaped structure that was actually observed in TEM observations [33].

Fig. 7. Super cell and a cross section of a bicontinuous double gyroid structure. (a) reprinted from [12]. Copyright (2006) American Chemical Society.

Epitaxial Transition of a Diblock Copolymer Upon a structural phase transition between diﬀerent types of micro phases, if the lattice constant and the positions of the periodic domains are preserved,

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481

the transition is called ”epitaxial transition”. For example, the GYR domain transforms into HEX domains aligned to the [111] direction of the GYR unit cell. This transition is a ﬁrst order transition where the nucleation and growth processes of HEX domains are expected. Figure 8 shows an epitaxial transition from GYR phase to HEX phase in a diblock copolymer (f = 0.35 and χN = 20.0) melt under a shear ﬂow [12]. Figure 8a is a cross section of a unit cell of a GYR phase, observed from the [111] direction. Under a shear ﬂow and a sudden temperature change to χN = 15.0, the GYR phase transforms into the HEX phase as shown in Fig. 8b. The black arrows indicate the direction of the shear ﬂow and the gray arrows show the time ﬂow. In this simulation, the expected nucleation of the HEX phase is observed.

Fig. 8. Epitaxial transition from a GYR phase to a HEX phase in a diblock copolymer melt. (b) reprinted from [12]. Copyright (2006) American Chemical Society.

Thin Films of Diblock Copolymers Figure 9 shows two dimensional cross sections of thin ﬁlms of an AB type diblock copolymer. The block ratio of the A block of this diblock copolymer is f = 0.34 and the interaction parameter is set to χN = 15.0. The upper black regions are the air phase which is simulated using a poor solvent, and the boundaries at the bottom are surfaces of solid substrates with which the minor A-type segments have more aﬃnity than the B-type segments. The

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Takashi Honda and Toshihiro Kawakatsu

value of the interaction parameters between the segments and the surface are assumed to be χsA = −0.5 and χsB = 0.0. The gray scales show the densities of the A-type segments of the diblock copolymer. We can clearly observe a HEX phase in these cross sections. The interaction parameters between the air phase and the polymers are changed in Fig. 9a-c. We call these interaction parameters χAV and χBV where V denotes the air (Void). At the free surfaces between the ﬁlm and the air, both A-type and B-type segments exist when χAV = χBV as shown in Fig. 9a. When χAV > χBV , the A-type segments are depressed at the interface as shown in Fig. 9b. Contrarily, when χAV < χBV , the A-type segments cover the surface as shown in Fig. 9c.

Fig. 9. Microphase separations in thin ﬁlms of an AB diblock copolymer with a block ratio f = 0.34. Figures a-c are for diﬀerent interaction parameters between the air and the segments of the diblock copolymer, i.e., (a) χAV = χBV = 5.0, (b) χAV = 5.0 and χBV = 2.5, and (c) χAV = 2.5 and χBV = 5.0, where A and B refer the two types of segments of the block copolymer and V refers to the air.

Local Segment Density Distributions in an ABA-type Triblock Copolymer in HEX Phase The degree of the physical cross-linking of SIS is related to the segment distributions at the nano-scale. The static SCF method is an appropriate method to investigate such segment distributions. We calculated a two dimensional HEX structure of an ABA triblock copolymer as shown in Fig. 10. Figure 10a is a HEX structure of an ABA triblock copolymer. To this structure, we add the same ABA triblock copolymer with a volume fraction of 0.0001 and conﬁne one end of it to one of the cylinder domains as shown in Fig. 10b by deﬁning the statistical weight of this end segment only in the speciﬁed cylinder domain. Figure 10b shows the spatial distribution of this end segment. Under this condition, we can obtain the distributions of the other segments of the added ABA triblock copolymer. The results are shown in Fig. 10. The

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483

schematic picture of the polymer at the center of Fig. 10 shows the speciﬁed segments and Figs. 10b-f show the distributions of these segments. As is shown in Fig. 10f, we can obtain the distribution of the other end segment. The fractions of the distribution of this end segment in the central cylinder domain and in the surrounding cylinder domains correspond to the fraction of the loop conformations and the bridge conformations, respectively. From this simulation data, one can predict the viscoelastic properties of the cylindrical domains due to physical cross-linking [26].

Fig. 10. Segment distributions of an A10 B40 A10 triblock copolymer in a HEX structure with χN = 30.0. (a) The HEX structure of the ABA triblock copolymer, and (b)-(f) segment distributions of the segment speciﬁed in the central schematic picture under the condition that one end segment is constrained in the central cylindrical domain. The details are described in the text.

Microphase Structures of Star Triblock Copolymers We calculated two-dimensional microphase structures of star triblock copolymers. The typical structures of these microphase structures are shown in Fig. 11. In these simulations, a matching between the natural periodicity of the domains and the side lengths of the simulation box has a crucial eﬀect on the simulation results. Thus, we optimized the side lengths of the simulation box so that the free energy density of the systems is minimized. The regions with diﬀerent gray colors in these ﬁgures correspond to the domains of individual types of segments. For this calculation, all the interaction parameters χij are ﬁxed to 0.6 and the numbers of the segments in the individual blocks are changed. Figs. 11a-b show LAM and HEX phases and Figs. 11c-d show complex structures.

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Takashi Honda and Toshihiro Kawakatsu

Fig. 11. Microphase structures of star triblock polymers. (a) A20 /B20 /C8 , (b) A20 /B20 /C20 , (c) A20 /B36 /C12 , (d) A20 /B40 /C40 with χij = 0.6.

5 A DFT for General Polymer Structures Using RPA In this section, we explain the GRPA applied to the dynamic density functional theory [19]. 5.1 Free Energy Model of GRPA The range of validity of the free energy obtained with the Ginzburg–Landau theory is in general restricted to the weak segregation regime because the expression of the Ginzburg–Landau free energy relies on a Taylor series expansion of the free energy with respect to the density ﬂuctuations of segments around the uniform mixed state [7] as 1 −1 F[{φi (r)}] = F0 [{φ¯i }] + (r − r )δφi (r)δφj (r ) + · · · dr dr Sij 2β ij 1 −1 ¯ = F0 [{φi }] + (q)δφi (q)δφj (−q) + · · · , dqSij 2β ij

(39) (40)

where φ¯i is the averaged segment density of the i-th subchain in the system and the ﬁrst term F0 [{φ¯i }] denotes the free energy of the reference uniformly mixed state. The local segment density ﬂuctuation of the i-th subchain at position r is deﬁned as

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δφi (r) ≡ φi (r) − φ¯i .

485

(41)

−1 The quantity Sij (r − r ) in (39) is the inverse of the density-density autocorrelation function between the segment density ﬂuctuations of the i-th and j-th subchains at positions r and r , respectively. It can be converted to the −1 (q) in Fourier space as in (40) where inverse of the scattering function Sij q means a scattering wave vector. The dots at the ends of (39) and (40) indicate the higher order terms in the Taylor series expansion in δφi (r) or δφi (q), respectively. Bohbot-Raviv and Wang have combined the second term of (40) with the Flory–Huggins free energy as follows [34]. φi (r) 1 1 1 dr dr F[{φi }] = ln φi (r) − δφi (r)δφi (r) β Ni 2β Ni φ¯i i i 1 −1 (q)δφi (q)δφj (−q), (42) dqSij + 2β ij

where the ﬁrst term on the right-hand side is the Flory–Huggins entropy term of the subchains and the second term is the second order term in the expansion of the entropy term in δφi (r). This second term is necessary to cancel the double counting of the second order terms in the expansion of the entropy term which is explicitly given by the third term. Equation (42) is regarded as a modiﬁcation of the right-hand side of (40). The ﬁrst term F0 [{φ¯i }] is eliminated because it is a constant and does not aﬀect the phase separation in the system. The higher order terms (dots) are replaced by the entropy term of the Flory–Huggins free energy. The third term on the right-hand side of (42) can be calculated using the auto-correlation function of the segment density ﬂuctuations. Thus it can account for the short range interactions between segments, such as the interaction energy of the Flory–Huggins free energy and the interfacial energy of a polymer blends [35], and can also account for the long-range eﬀects such as the interaction between domains composed of diﬀerent kinds of segments in a block copolymer [36]. Therefore the free energy (42) is a generalized free energy model for polymers with arbitrary topological structures. For this model to be used, one has −1 (q) for a given polymer. to ﬁnd the expression of the Sij 5.2 Subchain Scattering Function −1 We can derive an explicit expression for Sij (q) according to Leibler [8]. Vectors of the segment density ﬂuctuations of each subchain and the external potentials acting on each segment density with the wave vector q in the Fourier space are deﬁned, respectively, as

x(q) ≡ {δφi (q)} u(q) ≡ {ui (q)}.

(43) (44)

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Takashi Honda and Toshihiro Kawakatsu

The same vectors but in the real space are deﬁned as: x(r) ≡ {δφi (r)}

(45)

u(r) ≡ {ui (r)}.

(46)

The relation between x and u are given by 1 u(q) = − S−1 (q)x(q), β

(47)

where S−1 is the inverse of the scattering function matrix derived by the RPA under the incompressibility condition. The detail of the derivation of S−1 is described in Appendix A. The incompressibility condition sets the n-th elements of u and x as dependent variables on the other elements, and these elements are not used in the dynamic calculations. 5.3 Free Energy and Chemical Potential By using the inverse of the scattering function matrix, S−1 , the free energy of the system is given by n n 1 1 1 φi (r) F[{φi }] = ln φi (r) − dr ¯ δφi (r)δφi (r) dr β i Ni 2β i Ni φi (48) + dr xT (r)u(r), where xT (r) is the transposed vector of x(r). The vector u(r) in the third term of (48) is calculated by using the matrix S−1 (q) and the Fourier transformation of x. The last component, i.e. the n-th component of the segment densities is calculated as follows: φn (r) = 1 −

n−1

φi (r)

(49)

δφi (r).

(50)

i

δφn (r) = −

n−1 i

We can obtain the chemical potential of the i-th subchain under the incompressibility condition (49) and (50) as a functional derivative of (48) with respect to φi (r) as δF[{φi }] (51) δφi (r) 1 ln φn (r) 1 1 1 ln φi (r) 1 + − − − ¯ δφi (r) + ¯ δφn (r) = β Ni Ni Nn Nn φi Ni φn Nn +ui (r). (52)

μi (r) =

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5.4 Dynamic DFT for GRPA Similar to the dynamic SCF simulations, the chemical potentials obtained in (52) can be used in the dynamic DFT that was deﬁned in Sect. 4.5. Let us call this simulation a dynamic GRPA simulation. It should be noted that in the dynamic GRPA simulation, one should use the segment density dependent mobility given in (33). This is because the Taylor series expansion in terms of the density ﬂuctuation δφi (r, t) does not guarantee the positiveness of the value of φi (r, t). Moreover, in order to eliminate such a diﬃculty, the variable time mesh technique which varies the simulation time step width to keep the positiveness of the value of φi (r, t) is recommended for the dynamic GRPA. 5.5 Examples of the Dynamic GRPA The dynamic GRPA method is appropriate to study the dynamical transition of a large scale system. We will show the dynamical transition of the ABC layers of the HPL phase as shown in Fig. 6d under thermal noise for example. Dynamical Structure Change of HPL with ABC-Type Stacking The HPL phase is observed as an intermediate metastable phase from the HEX phase to the GYR phase under a deep quench [37]. To study the transition from HPL to GYR, we performed a dynamic GRPA simulation. The initial phase of the system is the ABCABC six layers as shown in Figs. 12a and d whose period is optimized with the static SCF method. The system is composed of diblock copolymers with block ratio f = 0.35 and χN = 20.0. For these parameters, the equilibrium phase is GYR phase [10]. We generate the thermal noise in (29) in the following way: ξi (r, t) = ∇v G (r, t),

(53)

where v G (r, t) is a vector whose elements are white Gaussian random numbers with a mean 0 and a standard deviation 0.001. We change the value of χN from 20.0 to 15.0 for the dynamic GRPA calculation because the stable phase for χN = 15.0 is the GYR phase. The initial ABC layers are shown in Fig. 12a. Upon the change in χN , this structure changes into the GYR as shown in Figs. 12b and c as time goes on. The lower ﬁgures Figs. 12d to f correspond to the same structures as those in the Figs. 12a to c, respectively, but observed from a diﬀerent view point. In the initial step, the HPL layers are undulating (Figs. 12b and e), and then one of the ABC three layers deforms largely (Figs. 12c and f) to form a three-fold junction (the circle in Fig. 12f). Such a three-fold junction is a characteristic partial structure of the GYR phase. Therefore, we expect that this simulation result shows a typical phase change from HPL to GYR.

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Fig. 12. A transition from ABC type HPL phase to GYR phase under a thermal noise, (a) and (d) are the initial structure at t = 0, (b) and (e) are the structure at t = 2537, and (c) and (f) are the one at t = 7965

6 Perspectives As shown in this chapter, the DFTs including the SCF theory and the GRPA are useful theories for the study of phase behaviors of polymeric materials because they cover the range from the small scale of polymer interfaces to the large scale of macro phase separations of polymer blends. One of the important factors in these DFTs is the quantitative evaluation of the conformational entropy of the polymer chains using the statistical mechanics of Gaussian chains. It is diﬃcult to reproduce the same simulation results as DFTs by using particle dynamics simulations such as the molecular dynamics simulations and the dissipative particle dynamics simulations [38] because these particle dynamics simulations demand a great deal of CPU power to obtain the full equilibrium structures of the system and to treat phenomena with large systems such as macro phase separations. However, multi-scale modeling combining coarse-grained molecular dynamics and the dynamic SCF theory has potential for the study of nano-scale materials as done for the ABA triblock copolymer [26]. On the other hand, the dynamic DFTs introduced in this chapter do not include the rheological eﬀects which are caused by the entanglements of polymers. Only recently, several trials have started to introduce rheological eﬀects to the SCF simulation [39–41].

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As the SCF theory demands much CPU power when dealing with large systems, a development of the technique of parallel computations is important, and such a trial has already been started in the MesoDyn project [11]. To overcome this diﬃculty in a diﬀerent manner, we proposed a new DFT by combining the SCF and GRPA theories to accelerate the dynamic SCF calculation without the loss of the quantitative accuracy of SCF theory [19]. The dynamic DFTs will be further extended and will be more useful by introducing new theoretical features such as rheological eﬀects and ﬂuid dynamics. At the same time, the graphic user interface (GUI) combined with the computer-aided design (CAD) systems for modeling meso-scale structures will be enhanced, which promotes the ease of meso-scale simulations. For the development of this ﬁeld, an accumulation of knowledge and experience is important. For such a purpose, open source code projects which have been developed in the ﬁeld of quantum chemistry will be required in polymer science. The OCTA project is one such attempt. Acknowledgement. The authors thank Dr. A. Zvelindovsky for giving us the opportunity to contribute this chapter. This study is executed under the national project on nanostructured polymeric materials, which has been entrusted to the Japan Chemical Innovation Institute by the New Energy and Industrial Technology Development Organization (NEDO) under METI’s Program for the Scientiﬁc Technology Development for Industries that Creates New Industries. This study is also supported by a grant-in-aid for science from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

Appendices A. Subchain Scattering Function by the RPA Hereafter, the expressions x and u without arguments are only used for those in the Fourier space. We deﬁne a matrix of the segment interaction energies and a matrix of the scattering functions between segments of subchains of ideal Gaussian chains as C ≡ {z ij } 0 S0 ≡ {Sij (q)},

(54) (55)

where ij denotes the interaction energy between the segments of the i-th and 0 (q) = 0 if the i-th and j-th subchains belong to the j-th subchains and Sij diﬀerent polymers. The detail of S0 is described in Appendix 6. We introduce a static pressure originating from the incompressibility condition in the Fourier space, and denote it as u∗ (q). This static pressure u∗ is the same for all subchains and can be derived as a function of u, S0 and C as

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Takashi Honda and Toshihiro Kawakatsu

u∗ = f (u, S0 , C).

(56)

Using the vectors and matrices mentioned above, the self-consistent equation is given by the RPA as [8] x = −βS0 (u + Cx + u∗ e),

(57)

where e is a vector whose elements are all unity. This equation is the fundamental equation which retains terms up to the second order in the expansion of the free energy. Solving (57) with respect to x gives x = −β(E + βS0 C)−1 S0 (u + u∗ e) = −β(Bu + u∗ Be),

(58)

where E is the identity matrix and B = (E + βS0 C)−1 S0 . Here the incompressibility condition must be satisﬁed for x as n

xi = 0.

(59)

i=1

From (58) and (59), u∗ is obtained as n n ∗

u =

Bij uj

i=1 j=1 − n n

.

(60)

Bij

i=1 j=1

Substituting (60) into (58) leads to x = −βSu,

(61)

where S = B + B and the element of the matrix B is deﬁned as n n Bij Bi j Bij =−

j =1 n

i =1 n

.

(62)

Bi j

i =1 j =1

If the inverse of S can be obtained in (61), we can get u = (−1/β)S−1 x and this u can be used for the coeﬃcient of the free energy. However, due to the incompressibility condition (59), the matrix S is a singular matrix, i.e, the rank of S is n − 1. This means that the segment densities of n subchains are not independent. Then, we take the segment densities of the subchains i ≤ n − 1 as the independent variables and the segment density of the n-th subchain as a dependent one. In this case, we should calculate the inverse matrix of the (n − 1) × (n − 1) block in the upper left part of S. Then, we deﬁne an n × n matrix S−1 , where the (n − 1) × (n − 1) block in the upper left part is ﬁlled with the inverse matrix obtained above and the n-th row and column are ﬁlled with 0 elements.

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B. Scattering Functions from an Ideal Gaussian Chain In a similar manner to which the Debye function in (11) is derived, the element 0 (q) in (55) can be obtained. We denote the indices of the subchains in the Sij p-type polymer by i and j . Then the scattering function from the p-type polymer which is assumed as an ideal Gaussian chain is given by (p)

2N 1 S 0 (q)|N =N (p) = (p)i 2 (e−x − 1 + x) (p) N N x i N (p) N (p) i j 1 q2 ds ds exp(− |s − s |b2K ) Si0p j (q) = (p) 6 N 0 0 Si0p i (q) =

Ni Nj e−z (p)

=

(p)

N (p) xy

(e−x − 1)(e−y − 1). (p)

(64) (p)

Here, S 0 (q)|N =N (p) is given by (11) with N = Ni , Ni i

(63)

(p)

and Nj being the

number of segments of the i -th and j -th subchains, respectively, and x, y, and z are given by 2 2 2 2 2 2 x ≡ RGi |q| , y ≡ RGj |q| , andz ≡ RGi j |q| ,

(65)

where RGi and RGj are the radii of gyration deﬁned by (9) of the i -th and j -th subchains, respectively, and RGi j is the radius of gyration of a sequence of subchains that connects the i -th and j -th subchains. Let us explain the meaning of this RGi j using the polymer shown in Fig. 13 as an 0p example. When we calculate S15 (q) (scattering function between segments of the 1st and 5th subchains) in Fig. 13a, it is enough to consider only the subchains shown in Fig. 13b, i.e. the 1st, 5th, and a sequence of the 3rd and 4th subchains. (The value of the eﬀective bond length is also averaged so that the average end to end distance is unchanged after the averaging.) This simpliﬁcation is due to the Gaussian chain approximation where each sunbchain is statistically independent.

Fig. 13. Ideal Gaussian chain model for the calculation of the scattering functions

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0 As a result, the explicit expression of the Sij (q) is obtained as a product of the average volume fraction of the p-type polymer in the system φ¯p and the Si0p j (q) as 0 (q) = φ¯p Si0p Sij j (q)

i=

p−1

n(p ) + i ,

(66) (67)

p 0 0 where S0 is a symmetric matrix that satisﬁes Sij (q) = Sji (q) and the diagonal 0 element Sii (q) is the self scattering function of the i-th subchain.

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11.

12. 13. 14. 15. 16.

17.

F. S. Bates, G. H. Fredrickson: Physics Today 52, 32 (1999). I. W. Hamley: Block Copolymers; Oxford University Press: Oxford, 1999. C. Park, J. Yoon, E. L. Thomas: Polymer 44, 6725 (2003). G. J. Fleer, M. A. Cohen Stuart, J. M. H. M. Scheutjens, T. Cosgrove, B. Vincent: Polymers at Interfaces; Chapman & Hall: London, 1993. M. W. Matsen, F. S. Bates: Macromolecules 29, 1091 (1996). M. W. Matsen: J. Phys. Cond. Matt. 14, R21 (2002). T. Kawakatsu: Statistical Physics of Polymers - An Introduction; SpringerVerlag, Berlin, 2004. L. Leibler: Macromolecules 13 1602 (1980). E. Helfand, Z. R. Wasserman: Macromolecules 9, 879 (1976). E. Helfand, Z. R. Wasserman: Macromolecules 11, 960 (1978). E. Helfand, Z. R. Wasserman: Macromolecules 11, 994 (1980). M. W. Matsen, M. Schick: Phys. Rev. Lett. 72, 2660 (1994). A. V. Zvelindovsky, G. J. A. Sevink, B. A. C. van Vlimmeren, N. M. Maurits, J. G. E. M. Fraaije: Phys. Rev. E 57, R4879 (1998). A. V. Zvelindovsky, B. A. C. van Vlimmeren, G. J. A. Sevink, N. M. Maurits, J. G. E. M. Fraaije: J. Chem. Phys. 109, 8751 (1998). A. V. Zvelindovsky, G. J. A. Sevink, J. G. E. M. Fraaije: Phys. Rev. E 62, R3063 (2000). A.V. Zvelindovsky, G. J. A. Sevink: Europhys. Lett. 62, 370 (2003). T. Honda, T. Kawakatsu: Macromolecules 39, 2340 (2006). M. Laradji, A.-C. Shi, J. Noolandi, C. R. Desai: Macromolecules 30, 3242 (1997). M. W. Matsen: Phys. Rev. Lett. 80, 4470 (1998). M. W. Matsen: J. Chem. Phys. 114, 8165 (2001). S. Qi, Z. G. Wang: Phys. Rev. Lett. 76, 1679 (1996). S. Qi, Z. G. Wang: Pys. Rev. E 55, 1682 (1997). S. Qi, Z. G. Wang: Polymer 39, 4639 (1998). M. Nonomura, T. Ohta: J. Phys. Soc. Jpn. 70, 927 (2001). M. Nonomura, T. Ohta: Physica A 304, 77 (2002). M. Nonomura, T. Ohta: J. Phys.: Condens. Matt. 15, L423 (2003). K. Yamada, M. Nonomura, T. Ohta: Macromolecules 37, 5762 (2004).

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18. (http://octa.jp) T. Honda,H. Kodama, J.-R. Roan, H. Morita, S. Urashita, R. Hasegawa, K. Yokomizo, T. Kawakatsu, M. Doi: SUSHI Users Manual ; OCTA: Nagoya, Japan, 2004. 19. T. Honda and T. Kawahatsu Macromolecules 40, 1227 (2007). 20. M. Doi, S. F. Edwards: The Theory of Polymer Dynamics; Oxford Science: Oxford, 1986. 21. K. M. Hong, J. Noolandi: Macromolecules 14, 727 (1981). 22. J. G. E. M. Fraaije: J. Chem. Phys. 99, 9202 (1993). 23. R. Hasegawa, M. Doi: Macromolecules 30, 5490 (1997). 24. P. G. de Gennes: Scaling Concepts in Polymer Physics Cornell University Press, Ithaca, 1979. 25. A. Hotta, S. M. Clarke, E. M. Terentjev: Macromolecules 35, 271 (2002). 26. T. Aoyagi, T. Honda, M. Doi: J. Chem. Phys. 117, 8153 (2002). 27. R. Stadler, C. Auschra, J. Beckmann, U. Krappe, I. Voigt-Martin, L. Leibler: Macromolecules 28, 3080 (1995). 28. W. Zheng, Z.-G. Wang: Macromolecules 28, 7215 (1995). 29. T. Gemma, A. Hatano, T. Dotera: Macromolecules 35, 3225 (2002). 30. I. W. Hamley, K. A. Koppi, J. H. Rosedale, F. S. Bates, K. Almdal, K. Mortensen: Macromolecules 26 5959 (1993). 31. S. Foerster, A. K. Khandpur, J. Zhao, F. S. Bates, I. W. Hamley, A. J. Ryan, W. Bras: Macromolecules 27, 6922 (1994). 32. M. E. Vigild, K. Almdal, K., Mortensen,I. W. Hamley, J. P. A Fairclough, A. J. Ryan: Macromolecules 31, 5702 (1998). 33. D. A. Hajduk, P. E. Harper, S. M. Gruner, C. C. Honeker, G. Kim, E. L. Thomas, L. J. Fetters: Macromolecules 27, 4063 (1994). 34. Y. Bohbot-Raviv, Z.-G. Wang: Phys. Rev. Lett. 85, 3428 (2000). 35. P. G. de Gennes: J. Phys. (Paris) 31, 235 (1970). 36. T. Ohta, K. Kawasaki: Macromolecules 19, 2621 (1986). 37. C-Y. Wang, T. P. Lodge: Macromol. Rapid. Commun. 23, 49 (2002). 38. R. D. Groot, T. J. Madden: J. Chem. Phys. 108, 8713 (1998). 39. T. Shima, H. Kuni, Y. Okabe, M. Doi, X.-F. Yuan, T. Kawakatsu: Macromolecules 36, 9199 (2003). 40. M. Mihajlovic, T. S. Lo, Y. Shnidman: Phys. Rev. E 72, 041801-1-26 (2005). 41. B. Narayanan, V. A. Pryamitsyn, V. Ganesan: Macromolecules 37, 10180 (2004).

Monte Carlo Simulations of Nano-Conﬁned Block Copolymers Qiang Wang Department of Chemical and Biological Engineering, Colorado State University, 1370 Campus Delivery, Fort Collins, CO 80523-1370, USA

1 Introduction Block copolymers consist of chemically distinct polymer chains (blocks) covalently bonded together. Unlike polymer blends exhibiting phase separation on a macroscopic scale, block copolymers spontaneously self-assemble into ordered microdomains on the length scale of tens of nanometers, a phenomenon known as microphase separation [1, 2]. Due to the uniformity and periodicity of these microdomains, block copolymers have great potential applications in nanotechnology (e.g., templates for nanolithography, nanowires, high-density storage devices, quantum dots, photonic crystals, nanostructured membranes, etc.) [3–5], where the size, shape and spatial arrangement of the microdomains (morphology) are utilized. Understanding, predicting and controlling the selfassembled morphology of block copolymers are therefore of paramount interest. For the simplest architecture of linear diblock copolymers AB, four morphologies have been determined to be thermodynamically stable in the bulk, depending on the temperature and the volume fractions of the two blocks: lamellae of alternating A-rich and B-rich layers, hexagonally packed cylinders of the minority component (A) in the matrix of the other component (B), A-spheres packed on a body-centered cubic lattice in the B-matrix, and bicontinuous gyroid phase [6,7]. For more complex molecular architectures such as linear triblock copolymers ABC, many other morphologies have been observed in experiments and their bulk phase behavior is not fully understood yet [2, 8]. In many applications, a solution of block copolymers is spin-coated on a supporting substrate (e.g., silicon wafer) to form a thin ﬁlm of tens to hundreds of nanometers thick, and the copolymers microphase separate in the ﬁlm upon solvent evaporation and/or annealing. Under such nano-conﬁnement, the tendency to resemble the bulk morphology with its characteristic period L0 , the surface-block interactions (surface preference) and the surface conﬁnement all have important eﬀects on the self-assembled morphology of block copolymers, thereby making it radically diﬀerent from its bulk counterpart.

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Qiang Wang

One can therefore use the conﬁning surface(s) to generate much more complex and fascinating morphologies, desirable for a broad array of applications [3–5]. The inﬂuence of conﬁnement on the microphase separation and morphology of block copolymers is also of fundamental interest in polymer science. The self-assembled morphology of block copolymers under nano-conﬁnement has been extensively studied by experiments, Monte Carlo (MC) simulations and various theories in the past decade. MC simulations are relatively easy to implement, and can give the exact solution (apart from statistical errors, which are controllable) to the model system studied. In addition to the selfassembled morphology, one can also access molecular details such as chain conformations and segmental distributions in MC. In this chapter, we focus on three-dimensional (3D) MC simulations of conﬁned block copolymers, and compare the simulation results with experiments and theories when available.

2 Lattice Models Full atomistic molecular simulations for concentrated polymer solutions or melts are out of the reach of current computation capability. Instead, all MC simulations of conﬁned block copolymers have been performed using a coarse-grained lattice model, where polymer chains are represented by segments connected by bonds. Such lattice simulations are generally much faster than continuum (oﬀ-lattice) MC simulations and molecular dynamics. Various lattice models used for conﬁned block copolymers are listed in Table 1. The simple cubic lattice (SCL) is the most basic model, where each lattice site has six nearest neighbors. Each polymer segment occupies one lattice site, and the bonds are restricted to the set of P(1, 0, 0), where P(x, y, z) represents all permutations and sign combinations of ±x, ±y and ±z (x, y and z are in units of lattice spacing). To represent the excluded volume eﬀects, segments cannot overlap. Unoccupied lattice sites can be considered as solvent molecules, and we use φC to denote the overall fraction of occupied lattice sites (referred to as the segmental density of block copolymers) in the system; systems of φC 0.5 are usually considered as concentrated polymer solutions or melts. For conﬁned systems, some lattice sites are used to represent the surface(s) and cannot be occupied by polymer segments; these surface sites (denoted by S) are not included in the calculation of φC . In lattice simulations, only nearest-neighbor interactions (denoted by ) are taken into account in the total energy E of the system1 , and periodic boundary conditions are applied in the unconﬁned direction(s). The process of MC simulations can be divided into three steps: First, an initial conﬁguration is generated by placing polymer chains on the lattice. Second, the initial conﬁguration is altered by trial moves that randomly 1

The “nearest-neighbor” refers to lattice sites that can be connected by one allowed bond vector in a model, except that√in BFM the nearest-neighbor interactions are counted only up to a distance of 6 [9].

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Table 1. Lattice models used in Monte Carlo simulations of conﬁned block copolymers. Here ns denotes the number of lattice sites occupied by each segment, and nb denotes the total number of allowed bond vectors. Model

ns

Allowed Bond Vectors

nb

SCL

1

P(1, 0, 0)

6

BFM

8 P(2, 0, 0) ∪ P(2, 1, 0) ∪ P(2, 1, 1) 108 ∪P(2, 2, 1) ∪ P(3, 0, 0) ∪ P(3, 1, 0)

BFM2

8 P(2, 0, 0) ∪ P(2, 1, 0) ∪ P(2, 1, 1) 54 P(1, 0, 0) ∪ P(1, 1, 0)

BFM1, BFM1× 1

18

displace polymer segments. In the simplest case of canonical-ensemble simulations (where φC , system volume and temperature T are ﬁxed), if a trial move does not violate the chain connectivity and excluded volume constraints (and sometimes the bond-crossing restriction), it is accepted according to the probability of min[1, exp(−ΔE/kB T )], where ΔE is the energy change due to the trial move, and kB the Boltzmann constant. This step needs to be repeated till the system is well equilibrated at the given temperature and its conﬁguration is completely uncorrelated with the initial one. Third, the conﬁgurational space of the system is sampled (again by trial moves) and data are collected at regular intervals. This step needs to be run long enough for the collected data to be statistically signiﬁcant. Typical trial moves in SCL include local moves shown in Fig. 1, and reptation moves where one detaches a chain end and re-attaches it to the other end of the same chain (for block copolymers segment types must be changed accordingly).

b

c

a Fig. 1. 2D illustration of the local moves typically used in SCL: (a) end-rotation, (b) kink-jump, and (c) crank-shaft. In 3D, there are ﬁve possible positions for end-rotation and three possible positions for crank-shaft (without considering the excluded-volume constraint).

For conﬁned block copolymers, the bond-ﬂuctuation model (BFM) [9, 10] and its variants have also been used. In BFM, each polymer segment occupies

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the eight √ lattice sites forming a unit cube, and the bond length can vary from 2 to 10 (whereas in SCL it is ﬁxed at 1) with 87 diﬀerent angles, in the interval of [0, π), between two consecutive bonds (whereas in SCL only 0 and π/2 are available). Its set of bond vectors is speciﬁcally chosen to prevent bond-crossing when only the trial moves of hopping are used, which displace a segment by one lattice spacing in any direction [9, 10]; this model is therefore also suitable to study polymer dynamics. It is, however, computationally much more expensive than SCL. Sommer et al. used a variant of BFM (denoted by BFM2) by reducing the number of bond vectors to 54 [11]. Another variant is the so-called single-site bond-ﬂuctuation model (denoted by BFM1 or BFM1×), where each segment occupies √only one lattice site as in SCL, but the bond length can vary between 1 and 2 [10]. The only diﬀerence between BFM1 and BFM1× is that bond-crossing is forbidden in the former while allowed in the latter.

3 Diblock Copolymer Thin Films Thin ﬁlms of tens to hundreds of nanometers in thickness are the most common form of nano-conﬁnement. In molecular simulations and theoretical studies, block copolymers are generally conﬁned between two hard (impenetrable) surfaces. In experiments, however, it is the most common case that the copolymers are spin-coated on a substrate to form a supported ﬁlm, with the top surface exposed to vacuum or air (free surface). In certain aspects, regions of the supported ﬁlm where the free surface is ﬂat (due to the surface tensions) can be viewed as conﬁned; there are, however, subtle diﬀerences between the free-surface conﬁnement and the hard-surface conﬁnement. For diblock copolymer thin ﬁlms, lamellae or cylinders both oriented perpendicular to the surfaces and having long-range order (over microns or longer) in the plane of the ﬁlm are desirable for applications such as nanolithography. The perpendicular orientation can be obtained by ﬁnely tuning the surface preference and ﬁlm thickness, but the in-plane ordering of such perpendicular structures on a homogeneous substrate is short-ranged (over only tens to a few hundreds of nanometers) [12–14]. Chemically nano-patterned substrates have therefore been used to induce the in-plane long-range order of perpendicular lamellae or cylinders. In both cases, a simulation box of sizes Lx × Ly × Lz (in units of lattice spacing) is used with periodic boundary conditions applied in both the x and y directions. The surfaces are represented by two extra lattice layers at z = 0 and Lz + 1, which cannot be occupied by polymers. In this Section we discuss ﬁrst the thin-ﬁlm morphology of lamellae-forming and cylinderforming diblock copolymers conﬁned between two homogeneous surfaces, and then the use of nano-patterned substrates to obtain both the perpendicular orientation and long-range in-plane order for these copolymers.

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499

3.1 Between Two Homogeneous Surfaces Table 2 lists the relevant MC simulations of diblock copolymer thin ﬁlms conﬁned between two homogeneous surfaces, which are generally set to be identical (referred to as “symmetric surfaces”). In most cases, only the A-B repulsion ( AB > 0) and surface-block interactions ( AS and BS ) are considered, with all other energetic interactions in the system set to 0. Symmetric diblock copolymers (where the volume fraction of the A block fA = 0.5) have been the most commonly studied. Due to the computation limitation, the box sizes and the copolymer chain length N are usually less than 100. Table 2. 3D lattice Monte Carlo simulations of diblock copolymer thin ﬁlms conﬁned between two homogeneous surfaces. Refer to the original papers for details. Ref.

Model

Lx × L y Lz

[15]

SCL

162 , 242 8 ∼ 32 0.5 16 0.8

[11]

BFM2

100 2

2

48

fA

N φC

Interactionsa AB = 2AS = 0.3 ∼ 0.6

0.5 24 0.6224 AB ≥ 0

2

[16, 17] BFM

93 , 96 30, 46, 0.5 32 0.5 56

AB = −AA = −BB = 0.177; BS = −AS = 0.1

[18]c

SCL

242

13 ∼ 37 0.5 24 0.8

AB = 0.435; AS = 0, 0.217, 0.870; BS = 0, 0.217, 0.870

[19]

BFM1× 162

8 ∼ 30 0.5 10 0.9

AB = 14.36; AS = 0, −2.872, −28.72

[20]d

SCL

b

28 × 34, 15 ∼ 85 0.25 36 0.7 56 × 67

AB = 0.667; AS = 0 ∼ 1.333; BS = 0 ∼ 1.333

In units of kB T . All other interactions were set to 0. b A box of sizes 642 × 256 was occasionally used. c Cases where the two surfaces are not identical were also studied. Lz = 58 was occasionally used. Cases of AS = 0.435 for one surface and BS = 0.174 for the other were occasionally studied. d In each case, either AS or BS was set to 0. a

Lamellae-Forming Symmetric Diblock Copolymers The simplest and most understood case of conﬁned block copolymers is that of symmetric diblock copolymers conﬁned between two ﬂat, hard, parallel and homogeneous surfaces. Over the past decade, such systems have been extensively studied by experiments [12–14, 21–23], MC simulations [11, 15–19] and various theories [15–17, 24–40]. The surfaces have several eﬀects on the thinﬁlm morphology: First, a surface generally exhibits an energetic preference for one of the two blocks, due to the diﬀerence in surface-block interactions. This

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results in lamellae oriented parallel to the surfaces [15, 18, 21–27, 31, 35–40]. Second, the two surfaces have a conﬁning eﬀect. When the surface separation D is incommensurate with the bulk lamellar period L0 (i.e., when D/L0 is not an integer in the case of two surfaces preferring the same type of segments, or not half an odd integer in the case of two surfaces preferring diﬀerent types of segments), such frustration forces the copolymers to change their lamellar period to be diﬀerent from L0 [15, 18, 21–23, 27, 31, 36, 39, 40], or even to adopt a diﬀerent (perpendicular) lamellar orientation to restore L0 when the frustration is large [12–18, 27–29, 31–34, 36, 39, 40]. The third eﬀect, referred to as the “hard-surface eﬀect” [18], is that a neutral and hard surface induces the formation of perpendicular lamellae [11, 14, 16, 18, 31–33, 36, 37, 39–43]. This eﬀect will be discussed in more detail below. By tuning the surfaceblock interactions and the thickness of the conﬁned ﬁlm, one can therefore obtain parallel lamellae, perpendicular lamellae, and even mixed lamellae of parallel orientation near one surface and perpendicular orientation near the other (in this case, both the parallel and perpendicular lamellae have a period L0 ) [13, 14, 18, 34, 40–45]. Inﬂuence of Periodic Boundary Conditions Kikuchi and Binder performed the ﬁrst MC simulations of symmetric diblock copolymers conﬁned between symmetric surfaces, where they observed parallel, perpendicular and mixed lamellae (at AB = 0.6) depending on the surface separation [15]. A critical issue in the study of periodic structures using a simulation box with periodic boundary conditions (PBC), however, is the matching between the intrinsic periodicity of the structure and PBC. Since the latter dictates the periodicity that can be obtained in the box, in the case of mismatch between the two it artiﬁcially changes the self-assembled structure of block copolymers. This is particularly important for lattice MC simulations using small boxes. It turns out that the mixed lamellae observed by Kikuchi and Binder were such an artifact caused by the mismatch between the intrinsic lamellar period (L0 ≈ 10) and their lateral box sizes (162 or 242 ); when appropriate box sizes (e.g., 202 ) were used, only perpendicular lamellae were observed at the same surface separation [46]. This agrees with self-consistent ﬁeld calculations showing that mixed lamellae are unstable between symmetric surfaces [31]. It is, however, diﬃcult to accurately determine L0 in lattice MC simulations. Although the copolymers tend to form the self-assembled structure with L0 , due to the ﬁxed box sizes the possible periods are quantized and diﬀerent periods correspond to diﬀerent orientations of the structure in the box. The free-energy barrier to be overcome in order for the whole structure to change its orientation can be rather large, and the simulated structure is very likely “trapped” in an orientation where its observed period diﬀers from L0 . A “brute-force” way to tackle this problem is to conduct many independent simulation runs and then take L0 as some average of the observed

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periods. To provide as many “options” for the period as possible, boxes with Lx = Ly = Lz (for bulk simulations, where PBC are applied also in the z direction), which avoid degeneracy between diﬀerent directions, are preferred over those with two or three equal sides. Using expanded grand-canonical ensemble simulations2 with judicious choice of box sizes, Wang et al. estimated L0 ≈ 12 for their model copolymers and chose the box sizes accordingly in their subsequent thin-ﬁlm simulations; the adverse inﬂuence of PBC on the self-assembled thin-ﬁlm morphology was therefore greatly reduced in their work [18]. Parallel and Perpendicular Lamellae With a good estimate of L0 , Wang et al. systematically studied the thin-ﬁlm morphology of symmetric diblock copolymers conﬁned between two homogeneous surfaces as a function of surface preference and separation [18]. For symmetric surfaces and antisymmetric surfaces (where the two surfaces prefer diﬀerent types of segments with the same strength), only parallel and perpendicular lamellae were observed, as summarized in Table 3. Similar results were also obtained by Yin et al. for a much more segregated system conﬁned between symmetric surfaces [19]. These results are well explained by the ﬁrst two surface eﬀects mentioned above. It is worth noting that the A-B interfaces in perpendicular lamellae between two preferential surfaces are not ﬂat but undulated due to the surface preference [15, 17, 18], as illustrated in Fig. 2(a). This agrees with self-consistent ﬁeld calculations [16, 17, 31, 32, 34, 40] and is recently shown to be closely related to the formation of mixed lamellae [40]. For symmetric diblock copolymers conﬁned between two neutral surfaces, only perpendicular lamellae (with ﬂat A-B interfaces) have been observed in MC simulations [11, 18, 19]. Both self-consistent ﬁeld calculations [31–33, 40] and density-functional calculations [39] have shown that perpendicular lamellae have a lower free energy than parallel lamellae between two neutral surfaces, regardless of the surface separation. While this has been mainly attributed to the surface-induced compatibilization between A and B segments (i.e., the A-B repulsion is eﬀectively reduced near the surfaces due to either the polymer segmental density decrease from that in the interior of the ﬁlm [16, 31, 33, 40] or the “missing-neighbor” eﬀect [16, 19]), the nematic ordering of copolymer segments near a hard surface is also responsible for this [32, 39]. MC simulations have also revealed that chain ends enrich near hard surfaces and that copolymer chains close to the surfaces orient parallel to them [11,18].

2

In such simulations one speciﬁes the chemical potential, instead of φC , of the copolymers, and includes trial moves to insert/remove copolymer chains piece by piece [18]. For long polymer chains at high densities, this simulation technique can relax and sample the system conﬁgurations much better than the commonly used canonical-ensemble simulations.

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Table 3. Morphology of symmetric diblock copolymers conﬁned between symmetric or antisymmetric surfaces. The symbol ||| denotes perpendicular lamellae, and ≡ [n] denotes parallel lamellae with n being the number of lamellar bilayers. α ≡ AS /AB (when AS > 0) or BS /AB (when BS > 0); see [18] for details. α = 0.5 represents a weak surface preference and 2 a strong preference. f ≡ D/(nL0 ) − 1 (assuming ≡ [n] form in the ﬁlm) measures the frustration imposed by the surface conﬁnement. Note that ||| were also observed in some independent runs for the cases marked by ∗, which may correspond to locally stable states. Adapted from [18].

D/L0 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3

(a)

Symmetric Surfaces f D=0.5 D=2 0 {[1] {[1] ||| 0.25 {[1] ||| {[2] 0.25 {[2] {[2] 0.125 0 {[2] {[2] 0.125 {[2]* {[2] ||| {[3] 0.167 {[3] {[3] 0.083 0 {[3] {[3]

(b)

Antisymmetric Surfaces f D=0.5 D=2 ||| {[1.5] 0.333 {[1.5] {[1.5] 0.167 0 {[1.5] {[1.5] ||| 0.167 {[1.5] ||| {[2.5] 0.2 {[2.5] {[2.5] 0.1 0 {[2.5] {[2.5] 0.1 {[2.5]* {[2.5] {[3.5]* {[3.5] 0.143

(c)

(d)

Fig. 2. Segmental density proﬁles for some stable morphologies obtained in 2D self-consistent ﬁeld calculations for symmetric diblock copolymers conﬁned between asymmetric surfaces preferring diﬀerent types of segments [40]; the light and dark regions correspond to A-rich and B-rich domains, respectively: (a) Perpendicular lamellae. (b) Mixed lamellae with one A-B interface (not perfectly ﬂat) parallel to the surfaces. (c) Mixed lamellae without the parallel A-B interface. (d) Doublemixed lamellae that can be considered as two mixed lamellae shown in (c) pieced together. In all cases, the two surfaces are at top and bottom, separated by L0 ; the upper surface prefers A segments and the lower surface prefers B segments with diﬀerent strengths. The black curves mark the A-B interfaces. The lateral period is L0 for all these morphologies. Reproduced from [40].

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503

z

z x

x

y y

Fig. 3. Representative conﬁguration (showing the six surfaces of the simulation box) of the mixed lamellae between an upper neutral surface and a lower strongly preferential surface (where AS /AB = 2) separated at D/L0 = 4.75. About two bilayers of parallel lamellae form near the lower surface preferring B segments (shown in dark). Reprinted with permission from [18]. Copyright (2000) American Institute of Physics.

Mixed Lamellae As shown in Fig. 3, Wang et al. also observed mixed lamellae in MC simulations when the two surfaces were asymmetric (i.e., neither symmetric nor antisymmetric) [18]. In this case the upper surface was neutral and the lower surface strongly preferred B segments; this morphology qualitatively agreed with the experimental observations of Huang et al. [14]. Another example of mixed lamellae was also given by Wang et al. [18], in accordance with the experiments of Koneripalli et al. [13]. Although Matsen concluded from his 2D self-consistent ﬁeld calculations that mixed lamellae were unstable for symmetric diblock copolymers conﬁned between two homogeneous surfaces, his statement was based on the assumption that mixed lamellae, if they were to occur, would be most stable between either symmetric or antisymmetric surfaces. Due to the asymmetry of mixed lamellae, this assumption is unfortunately not valid. Recent 2D self-consistent ﬁeld calculations identiﬁed three types of mixed lamellae, shown in Figs. 2(b)∼(d), as stable morphologies between asymmetric surfaces preferring diﬀerent types of segments [40]. Since in most experiments the two surfaces are diﬀerent, studies on thin-ﬁlm morphology between asymmetric surfaces are more relevant and desirable. Beyond Thin-Film Morphology In addition to the self-assembled morphology, MC simulations have also been used to reveal the details of thin-ﬁlm structures (e.g., density proﬁles, layer

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thickness, etc.) [15, 17, 18] and chain conformations (e.g., orientation, stretching, segmental distributions, etc.) [11, 17–19], and to examine the validity and accuracy of theory [16, 17]. In particular, Geisinger et al. compared in detail the phase stability and segmental density proﬁles predicted by the selfconsistent ﬁeld theory with MC simulations; qualitative agreement were obtained in general [16, 17]. The lack of ﬂuctuations in the self-consistent ﬁeld theory, however, results in a higher order-disorder transition temperature and a narrower A-B interfacial width. Using the convolution approximation with one ﬁtting parameter to take into account the capillary-wave broadening, they obtained almost quantitative agreement on segmental density proﬁles between the two [16, 17]. Cylinder-Forming Asymmetric Diblock Copolymers The thin-ﬁlm morphology of asymmetric diblock copolymers forming cylinders in the bulk is much more complex than that of symmetric ones (where only lamellae of diﬀerent orientations can form). Not only cylinders of diﬀerent orientations, but also various non-cylindrical morphologies (e.g., spheres, lamellae, perforated lamellae, etc.), can form in the ﬁlm. As shown in Table 2, the only MC study of asymmetric diblock copolymer thin ﬁlms was reported by Wang et al. using the expanded grand-canonical ensemble simulations. Similar to the lamellae case, they ﬁrst performed bulk simulations with different boxes of unequal sides and estimated the characteristic inter-cylinder distance L0 ≈ 16.8 for their model copolymers [20]. Since the hexagonal packing of cylinders is a 2D periodic structure, the inﬂuence of PBC is more severe than in the lamellae case and the box sizes must be carefully chosen in order to obtain well-ordered, hexagonally packed cylinders. Table 4 summarizes the thin-ﬁlm morphology of cylinder-forming asymmetric diblock copolymers conﬁned between two identical surfaces; depending on the ﬁlm thickness and surface preference, various morphologies as shown in Fig. 4 were obtained. In addition to visual inspection, ensemble-averaged segmental density proﬁles were also used to identify the morphologies due to their complexity and ﬂuctuations [20]. Because of the enrichment of both chain ends near a hard surface, an energetically neutral surface exhibits a slight entropic preference for the shorter (A) block [20], and one layer of half-cylinders parallel to the surfaces (denoted 1/2 by C≡ ) forms when the ﬁlm thickness is commensurate with the characteristic packing of cylinders (as shown in Fig. 4(a)). A slight energetic preference for the longer (B) block is therefore needed in order to obtain perpendicular cylinders (denoted by C||| ) at all surface separations (referred to as the “effectively neutral” condition). An energetic preference for A segments leads to one lamella with a ﬂat A-B interface parallel to the surface (denoted by L), whereas such a lamellar layer is absent near surfaces preferring B segments; this is consistent with experimental results [20]. The volume fraction of A in this lamellar layer is higher than fA due to the surface preference, which in

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505

Table 4. Morphology of asymmetric diblock √ copolymers conﬁned between two homogeneous and identical surfaces. L2 ≈ 3L0 was estimated to be 28 (in units of lattice spacing) [20]. The strength of surface preference α for each column is listed in the second row. The sA surface prefers the shorter (A) block (where α ≡ BS /AB ), the sB surface prefers the longer (B) block (where α ≡ AS /AB ), and the sN surface is energetically neutral. Cm ≡ denotes parallel cylinders with m being the number of 1/2 cylinder layers along the direction perpendicular to the surfaces; C≡ denotes one parallel layer of half-cylinders; C||| denotes perpendicular cylinders; L denotes one lamella having a planar A-B interface parallel to the surfaces; S denotes one parallel layer of A-spheres; P denotes one parallel layer of perforated A-lamella with B-rich holes. The mixed morphology of C≡ -C||| -C≡ , for example, represents C||| in the interior of the ﬁlm and one layer of parallel cylinders (C≡ and C≡ ) near each surface (formed by the interconnection of A domains at a small distance away from the surfaces; see [20] for details). Adapted from [20]. D/L2 0.5

5 P

2 P

sB-sB 1 P

0.75 CŁ2 1 1.25 1.5 1.75 2 3

P-P

P-P

P-P

0.2 C|||

sN-sN 0 CŁ½-CŁ½

0.5 L-L

C|||

C|||

C|||

L-S-L

L-S-L

L-S-L

CŁ2

C||| CŁ½-CŁ1-CŁ½ L-P-L

L-P-L

L-P-L

0.5 P

CŁ-C|||-CŁc CŁ-C|||-CŁc C||| CŁ3 CŁ3 CŁ-C|||-CŁc CŁ-C|||-CŁc C||| CŁ4 CŁ4 CŁ4

C||| C||| C||| C||| C|||

C||| CŁ½-CŁ2-CŁ½

L-CŁ2-L C||| L-CŁ2-L ½ 3 ½ CŁ -CŁ -CŁ L-CŁ3-L CŁ½-CŁ5-CŁ½

sA-sA 1 L-L

2 L-L

L-CŁ1-L L-CŁ1-L L-CŁ2-L L-CŁ2-L L-CŁ2-L L-CŁ2-L L-CŁ3-L

turn leads to the depletion of A segments in the interior of the ﬁlm. For small ﬁlm thicknesses, such depletion results in the formation of one layer of Aspheres (denoted by S) or A-lamella perforated by B-holes (denoted by P) in the interior of the ﬁlm (as shown in Figs. 4(c) and 4(d), respectively); the Aspheres or B-holes are hexagonally packed, as observed in experiments [47–50]. For larger ﬁlm thicknesses, parallel cylinders (denoted by Cm ≡ with the integer m being the number of cylinder layers) form in the interior of the ﬁlm (as shown in Fig. 4(b)). Similarly, surfaces preferring B segments lead to the formation of P at small ﬁlm thicknesses and cylinders for thicker ﬁlms. As in symmetric diblock copolymer ﬁlms, Cm ≡ form when the ﬁlm thickness is commensurate with the characteristic packing of cylinders, and C||| form in the interior of the ﬁlm otherwise. Interestingly, C||| have not been obtained between surfaces preferring A segments [20]. The MC simulations were in good agreement with experiments [20] and dynamic density-functional calculations [51,52]. Very recently, Yang et al. performed self-consistent ﬁeld calculations for diblock copolymers with fA = 0.3 conﬁned between two identical surfaces. In addition to the above morphologies, they also reported undulated parallel lamellae, undulated C1≡ , the asym-

506

Qiang Wang

(a)

(b)

x

z y

x y

(c)

x

z y

y

(d)

Fig. 4. Representative conﬁgurations of various morphologies of asymmetric diblock copolymers conﬁned between two identical surfaces; only A segments are shown as 1/2 1/2 dots: (a) C≡ -C1≡ -C≡ between neutral surfaces at D/L2 = 1, viewed along the cylinder axis. (b) L-C3≡ -L between sA-sA surfaces (α = 2) separated at D/L2 = 2, viewed along the cylinder axis. (c) L-S-L between sA-sA surfaces (α = 2) separated at D/L2 = 0.75; shown on the right is the interior of the ﬁlm (z = 7 ∼ 15). (d) P between sB-sB surfaces (α = 2) separated at D/L2 = 0.5. Refer to Table 4 and text for more details. Reprinted with permission from [20]. Copyright (2001) American Chemical Society. 1/2

metric morphology of C≡ -C1≡ , and some mixed morphologies of parallel cylinders with perforated lamellae [53]. Due to the low accuracy of their free-energy calculations, however, they were not able to determine the stability and detailed structures of some of these morphologies [53]. More studies are needed to further understand the complex morphology of asymmetric diblock copolymer thin ﬁlms.

Monte Carlo Simulations of Nano-Conﬁned Block Copolymers

507

3.2 On Nano-Patterned Substrates As discussed above, by ﬁnely tuning the surface preference and ﬁlm thickness, perpendicular lamellae or cylinders can be obtained on a homogeneous substrate. Transmission electron microscopy (TEM) or ﬁeld emission scanning electron microscopy (FESEM) images (e.g., Fig. 5), however, showed that the in-plane ordering of such perpendicular structures is short-ranged (over only tens to a few hundreds of nanometers) [12–14]. Chemically nano-patterned substrates are therefore used to induce the in-plane long-range (over microns or longer) order of perpendicular lamellae or cylinders. Such long-range, 3D well-ordered structures are of paramount interest in applications such as templates for nanolithography, high-density storage media, etc.

(a)

(b)

Fig. 5. Lack of in-plane long-range order of perpendicular structures on a homogeneous substrate: (a) Plane-view TEM image of perpendicular lamellae formed by poly(styrene-b-methyl methacrylate) (PS-PMMA) diblock copolymers; the light and dark regions are PMMA and PS domains, respectively. Adapted from [12]. (b) Plane-view FESEM image of perpendicular cylinders formed by PS-PMMA diblock copolymers after the removal of PMMA cylinders. Reproduced from [54].

Lamellae-Forming Symmetric Diblock Copolymers Between Patterned-Homogeneous Surfaces Wang et al. reported an MC study on the thin-ﬁlm morphology of symmetric diblock copolymers conﬁned between a lower stripe-patterned and an upper homogeneous surface, as shown in Fig. 6(a) [55]. The sA and sB stripes on the substrate had the same width, and strongly preferred A and B segments, respectively (i.e., sA−B = sB−A = 2 AB ). The upper sH surface either was neutral or preferred A segments (characterized by αH ≡ sH−B / AB ≥ 0). All other energetic interactions in the system were set to 0. To reduce the PBC inﬂuence, the simulation box sizes Lx and Ly were carefully chosen to be integer multiples of L0 . The substrate pattern period Ls , ﬁlm thickness D and upper surface preference αH were then varied to examine their eﬀects

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Qiang Wang

on the thin-ﬁlm morphology [55], and the simulation results were compared with the strong-stretching theory [56], as summarized in the phase diagrams shown in Fig. 7. z

z

z

Homogeneous Surface (sH)

Lz+1

Lz+1

Lz+1

x

o

x

sA sB

o

y Patterned Substrate

Ls

(a)

x

sA sB

o

y Ls

(b)

sA sB

y Ls

(c)

Fig. 6. Surface conﬁgurations of conﬁned symmetric diblock copolymers: (a) Patterned-homogeneous surfaces. Reproduced from [55]. (b) Symmetrically patterned surfaces. Reproduced from [57]. (c) Antisymmetrically patterned surfaces.

6

2 ~|||

d ≡ D/L0

|||

|||s

|||s−≡[m]

|||s−||| (a)

δH ≡ (σBH−σAH)/σAB

a

5

~||| +[1]−|||

4 3 2

1.5

4

1

2

3 p ≡ Ls/L0

(a)

+[2] / +[1] 4

5

3

4 b

+[1]−≡[3]

1

0.5

+[3] 1 0.5 0.2

3

≡[4] (b)

|||~ 0 0.2

|||

+[1]−|||,|||~

s

\ ||| −||| 1 s

|||s−||| p ≡ Ls/L0

2

3

(b)

Fig. 7. Phase diagrams of symmetric diblock copolymers conﬁned between (a) patterned-neutral surfaces (where αH = 0) and (b) patterned-preferential surfaces (where D/L0 = 2 and δH = αH ) calculated from the strong-stretching theory. For the |||s -≡ [m] morphology, the value of m is given in the ﬁgure. The open symbols represent the morphologies observed in Monte Carlo simulations: In both plots, denotes |||, denotes |||s , and ♦ denotes +[1]-|||; in (a), denotes |||s -||| and denotes +[2]; and in (b), denotes |||s -≡ [m], denotes |||s -|||, denotes ≡ [4], and denotes +[1]-≡ [3]. See text and [56] for more details. Reprinted with permission from [56]. Copyright (2000) American Institute of Physics.

Various morphologies were observed in MC simulations, which can be considered as composed of the following basic morphologies: (1) perpendicular lamellae registered with the substrate pattern, and thus having a period Ls and the long-range in-plane order (denoted by |||s ); (2) perpendicular lamellae ignoring the substrate pattern, and thus having a period L0 and only shortrange in-plane order (denoted by |||); (3) parallel lamellae having a period L0

Monte Carlo Simulations of Nano-Conﬁned Block Copolymers

509

(denoted by ≡ [m] with m being the number of chain layers in the direction perpendicular to the surfaces); and (4) checkboard morphology complying with the substrate pattern and having a period close to L0 in the direction perpendicular to the surfaces (denoted by +[m]). The more complex, mixed morphologies were denoted by two or more basic morphologies (from the lower substrate to the upper surface) connected by a “-”. Due to their complexity and ﬂuctuations, ensemble-averaged proﬁles of segmental distributions and chain orientation were used to identify the morphologies, in addition to visual inspection [55]. The morphologies obtained in MC simulations were used to construct the phase diagrams using the strong-stretching theory, since the latter requires a priori knowledge about the possible morphologies in order to calculate their free energies [56]. As shown in Fig. 7, good agreement was obtained between the two. This combined simulation-theoretical study revealed that two conditions are essential for obtaining the long-range ordered perpendicular lamellae (|||s ): a stripe-patterned substrate with Ls comparable to L0 , which directs the in-plane ordering of perpendicular lamellae over a macroscopic scale, and a neutral or weakly preferential surface on the top, which stabilizes the perpendicular lamellae [55, 56]. This guided the experimental design for symmetric diblock copolymers on chemically nano-patterned substrates, where stripe-patterned substrates were created using extreme ultraviolet interferometric lithography [58]. When Ls = 47.5nm matched L0 ≈ 48nm of the poly(styrene-b-methyl methacrylate) (PS-PMMA) diblock copolymers used in the experiments, long-range ordered perpendicular lamellae were obtained after depositing the copolymers on the patterned substrate, as shown in Fig. 8 [58]. Symmetric diblock copolymers on stripe-patterned substrates have also been studied by other experiments [59–63] and theoretical calculations [64– 68]. In particular, both the self-consistent ﬁeld calculations by Petera and Muthukumar for patterned-neutral surfaces [66] and the Ginzburg-Landau free-energy functional calculations by Tsori and Andelman for patternedpreferential surfaces [68] have predicted the morphology of tilted lamellae when Ls > L0 ; the A-B interfaces in the tilted lamellae form an angle of sin−1 (L0 /Ls ) with the surfaces to comply with the substrate pattern while retaining the bulk lamellar period. Such a morphology was not found in the MC simulations, probably because the theoretical calculations were performed in the weak segregation regime where the eﬀects of the substrate pattern can propagate far into the ﬁlm, while the simulations were performed in the intermediate segregation regime [69] where the substrate eﬀects were less pronounced. This led to the idea of using two stripe-patterned surfaces to enhance the surface eﬀects; tilted lamellae were indeed observed in MC simulations under certain conditions, as discussed next.

510

Qiang Wang

Fig. 8. SEM top-view of the perfect epitaxial ordering of the PS-PMMA ﬁlm (L0 ≈ 48nm, D = 60nm) on a chemically nano-patterned substrate (Ls = 47.5nm). The light and dark regions are PS and PMMA domains, respectively. The lamellae orient perpendicular to the substrate and are macroscopically registered with the underlying substrate pattern. There is no defect in this 5μm×5μm image, and this is just a small part of the actual sample. These straight lines can run over 400μm, which is the size of the substrate pattern. Reproduced from [58].

Between Two Stripe-Patterned Surfaces In experiments it would be diﬃcult to conﬁne copolymers between two parallel surfaces that are both chemically nano-patterned and separated by only tens or hundreds of nanometers. This, however, can be readily studied in simulations and theory. The study of such systems is complementary to those of thin ﬁlms between patterned-homogeneous surfaces, and can further our understanding on the eﬀects of nano-patterned surfaces on the thin-ﬁlm morphology. As an extension of the simulation work discussed above, Wang performed MC simulations for thin ﬁlms of symmetric diblock copolymers conﬁned between two stripe-patterned surfaces shown in Figs. 6(b) (referred to as “symmetrically patterned surfaces”) and 6(c) (referred to as “antisymmetrically patterned surfaces”) [57]. For a given Ls > L0 and surface conﬁguration, the period L of the tilted lamellae complying with both surface patterns is dictated by the ﬁlm thickness D; simulations were therefore performed at the ﬁlm thicknesses most suitable for the formation of tilted lamellae (i.e., L ≈ L0 ). In such cases, tilted lamellae were indeed observed when D 2L0 , as shown in Fig. 9(a); the A-B interfaces were basically perpendicular to the surfaces in the immediate vicinity of the surfaces (due to the parallel chain orientation induced by the hard-

Monte Carlo Simulations of Nano-Conﬁned Block Copolymers

511

surface eﬀect in these regions), and exhibited some undulations further away from the surfaces. For smaller D, a checkerboard morphology was observed instead, as shown in Fig. 9(b). At the ﬁlm thicknesses most unsuitable for the formation of tilted lamellae, a mixed morphology of |||s (when Ls /L0 = 1.5) or +[1] (when Ls /L0 = 2) near each surface and ||| in the interior of the ﬁlm was generally observed. The severe frustration imposed by the two patterned surfaces, however, led in some cases to highly irregular or unexpected morphologies, which corresponded to locally stable states. More than one locally stable morphologies and the ﬂipping between them were sometimes observed during a single simulation run, demonstrating the eﬃcient sampling of the expanded grand-canonical MC simulations [57].

30 20 z

z

15

10

10 5 10

10

20 y (a)

20 y

30

30

(b)

Fig. 9. Ensemble-averaged 2D proﬁles showing the density diﬀerence between A and B segments in (a) the tilted lamellae (D = 32) and (b) the +[2] morphology (D = 16) between symmetrically patterned surfaces at Ls /L0 = 1.5. The light and dark regions are A-rich and B-rich domains, respectively. The black curves in (a) mark the A-B interfaces. Adapted from [57].

Cylinder-Forming Asymmetric Diblock Copolymers With judicious choice of the simulation box sizes to minimize the PBC inﬂuence, Wang et al. also studied the thin-ﬁlm morphology of their model asymmetric diblock copolymers conﬁned between a lower nano-patterned and an upper homogeneous (sH) surface [70]. For such systems, knowing the intrinsic packing period of the cylinders is critical to obtain meaningful results. Here the preference of the sH surface is characterized by αH ≡ ( sH−B − sH−A )/ AB ; for αH > 0 (where sH−A = 0) it prefers A segments, and for αH < 0 (where sH−B = 0) it prefers B segments. Three substrate patterns are examined, as shown in Fig. 10. In addition to the sA (sB) sites strongly preferring A (B) segments (i.e., sA−B = sB−A = 2 AB ), the sC sites also populate the

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Qiang Wang

substrate with their preference characterized by αC ≡ ( similar to that of the sH surface [70].

sC−B

−

sC−A )/ AB ,

wsB

wsx x

wsx

θ wsy

wsy

Ls

(a)

(b)

(c)

y Fig. 10. Chemically nano-patterned substrates used for cylinder-forming asymmetric diblock copolymers: (a) Hexagonally patterned substrate, where the light regions consist of sA sites strongly preferring the shorter (A) block, the dark regions consist of sB sites strongly preferring the longer (B) block, and the rest are sC sites. This pattern is ﬁxed with wsx = 7 and wsy = 8 (in units of lattice spacing) in the ﬁgure, and θ = tan−1 (7/4) ≈ 60.26◦ ; the hexagonal pattern is therefore commensurate with the perpendicular cylinders having bulk characteristic dimensions and packing (in the framework of the simple cubic lattice). (b) Stripe-patterned substrate, where the dark regions are sB sites and the rest are sC sites. (c) Square-patterned substrate, where the light regions are sA sites and the rest are sB sites. This pattern is ﬁxed with wsx = wsy = 8. Periodic boundary conditions are applied along the x and y directions in all cases. Reproduced from [70]. Copyright (2003) American Chemical Society.

On Hexagonally Patterned Substrates Since the dimensions of the hexagonally patterned substrate are commensurate with perpendicular cylinders having bulk characteristic dimensions and packing, such a substrate with sC sites preferring B segments (e.g., αC = −2) would be suitable for inducing the long-range ordered perpendicular cylinders registered with the substrate pattern throughout the ﬁlm. This is indeed the case if the upper surface is “eﬀectively neutral” (i.e., αH = −0.2; see Sect. 3.1), as shown in Fig. 11(a). Note that, although the sA regions on the substrate are rhombic, the A-rich domains in the ﬁlm are still circular. This suggests that the detailed shape of the sA regions is not crucial; instead, their hexagonal arrangement is more important for inducing the long-range order of perpendicular cylinders. If the regions preferring A segments are not hexagonally packed (e.g., by changing αC to 0), parallel structures are obtained in

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the simulation. On the other hand, if the upper surface deviates too much from the “eﬀectively neutral” condition, its preference can lead to parallel structures (cylinders or lamellae) in the ﬁlm [70].

x

50

30 z

20 10

10 20

40

60

10

30 y

y (a)

50

(b)

Fig. 11. Ensemble-averaged 2D proﬁles showing the density diﬀerence between A and B segments in perpendicular cylinders on (a) a hexagonally patterned substrate (αC = −2) and (b) a stripe-patterned substrate (Ls = 14, wsB = 5 and αC = −0.2). An upper “eﬀectively neutral” surface (αH = −0.2) is used in both cases. The light and dark regions correspond to A-cylinders and B-matrix, respectively. Diﬀerent gray scales are used in (a) and (b) to highlight the cylinders. The box sizes are 56 × 64 × 85 in (a) and 67 × 56 × 29 in (b). One can clearly see the registration of cylinders with the substrate pattern shown in Figs. 10(a) and 10(b), respectively. Adapted from [70]. Copyright (2003) American Chemical Society.

On Stripe-Patterned Substrates A stripe-patterned substrate may also be used to induce the long-range ordered perpendicular cylinders, when the sC stripes are close to “eﬀectively neutral”; all cylinders in this case stand on the sC stripes, as shown in Fig. 11(b). The stripe width is not critical, as long as the substrate pat√ tern period Ls ≈ 3L0 /2, but an upper “eﬀectively neutral” surface is again needed. Due to the translational symmetry of the stripe pattern along the x direction shown in Fig. 10(b), long-range order of perpendicular cylinders may not be obtained in this direction. On the other hand, a stripe-patterned substrate with sC stripes preferring A segments can induce parallel cylinders registered with the substrate pattern [70]. On Square-Patterned Substrates A square-patterned substrate commensurate with L0 is used, together with an upper “eﬀectively neutral” surface, to examine its eﬀects on the packing of

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perpendicular cylinders. As shown in Fig. 12(a), the cylinders comply with the square pattern shown in Fig. 10(c) only over a short distance from the substrate. Further away from the substrate, the cylinders re-arrange themselves to recover their bulk hexagonal packing. One way of doing this is for adjacent rows of cylinders to slide along the y direction to develop a staggered packing; as shown in Fig. 12(a), the square packing of cylinders is thereby gradually transformed into a hexagonal packing as z increases. Note that the hexagon formed in this way is not a regular hexagon but is stretched along the x direction. The cylinders therefore also bend in the x direction to reduce this stretching, as shown in Fig. 12(b). In the vicinity of the upper surface, the cylinders orient perpendicular to it. Fig. 12(b) is reminiscent of the tilted lamellae discussed above. This example demonstrates the importance of using a nano-patterned substrate having a pattern commensurate with the bulk morphology of the copolymers, if the substrate pattern is to be propagated throughout the ﬁlm [70].

z

x

x

y

(z=1~5)

y

(a)

(z=6~29)

z

20 10 20

40

60

x (b) Fig. 12. Perpendicular cylinders between a square-patterned substrate and an upper “eﬀectively” neutral surface (αH = −0.2): (a) Representative conﬁguration with only A segments shown as dots. (b) Ensemble-averaged 2D proﬁles showing the density diﬀerence between A and B segments. The light and dark regions correspond to A-cylinders and B-matrix, respectively. The box sizes are 64 × 64 × 29. Adapted from [70]. Copyright (2003) American Chemical Society.

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Spatial Fluctuations of Perpendicular Cylinders Wang et al. also examined the spatial ﬂuctuations of perpendicular cylinders on diﬀerent substrates, namely, a homogeneous substrate (same as the upper sH surface), a stripe-patterned substrate (Ls = 14, wsB = 5 and αC = −0.2) with the stripes now along the y direction, and a hexagonally patterned substrate (αC = −2) [70]. In all cases, the upper homogeneous surface was “effectively neutral” (αH = −0.2), and a large simulation box of 112 × 128 × 85 (containing 64 perpendicular cylinders) was used to reduce the PBC suppression of the spatial ﬂuctuations. The ﬁlm was evenly divided into 5 slices along the z direction, and the ensemble average of the deviation of cylinder positions in each slice from their “ideal” hexagonal packing positions (referred to as the “spatial ﬂuctuations”) was calculated [70]. The results given in Fig. 13 show that the spatial ﬂuctuations of perpendicular cylinders in the ﬁrst slice (closest to the substrate) are clearly reduced by the substrate pattern, which explains why nano-patterned substrates can induce long-range order of perpendicular cylinders. The results also show that this reduction of ﬂuctuations unfortunately does not propagate far into the ﬁlm; the spatial ﬂuctuations in the interior of the ﬁlm are about the same, regardless of the substrate used. This is due to the short-range interactions between the substrates and copolymers. Based on these results, it was proposed that combining nano-patterned substrates (to induce the in-plane long-range order) with electric ﬁelds applied perpendicular to the ﬁlm [54, 71] (to propagate the substrate eﬀects further into the ﬁlm) could achieve the ultimate goal of producing long-range threedimensional order in self-assembled block copolymer ﬁlms, useful as templates for nanolithography or high-density storage media [70]. To demonstrate the PBC suppression of the spatial ﬂuctuations, the results for the hexagonally patterned substrate obtained using a box of 224×256×57 (containing 256 perpendicular cylinders) are also shown in Fig. 13. This is so far the largest box reported in lattice MC simulations of block copolymers. If we map it to an experimental system with L0 ≈ 30nm, this box corresponds to a ﬁlm about 100nm thick with a lateral area of about 0.4μm×0.5μm. This ﬁlm was evenly divided into 3 slices along the z direction, and the spatial ﬂuctuations of perpendicular cylinders in each slice were found to be systematically larger than those obtained for the same system but in the smaller box (the slice thicknesses in these two boxes were comparable) [70]. Larger box sizes and longer simulations will give more accurate results on the spatial ﬂuctuations, but the above qualitative conclusions may not change.

4 Triblock Copolymer Thin Films Table 5 summarizes MC simulations for triblock copolymer thin ﬁlms conﬁned between two homogeneous and identical surfaces. Both linear ABA and ABC copolymers have been studied. Although the bulk morphologies of ABA

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Fig. 13. Spatial ﬂuctuations of perpendicular cylinders δ in each slice on diﬀerent substrates: “Pattern 0” refers to an “eﬀectively neutral” homogeneous substrate, “Pattern 1” refers to a stripe-patterned substrate, and “Pattern 2” refers to a hexagonally patterned substrate. See text for details. In all cases the upper homogeneous surface is “eﬀectively neutral”. The simulation box sizes are listed in the legend. The standard deviation of δ obtained in the smaller box was estimated to be 0.05 [70]. Reproduced from [70]. Copyright (2003) American Chemical Society.

copolymers are the same as those of AB [72, 73], they could have improved mechanical properties due to the bridging of the mid-block between the two end-blocks not in the same microdomain [74]. On the other hand, ABC copolymers exhibit much more fascinating morphologies than AB [2,8]. To date, even their bulk morphology has not been well explored and understood yet, due to the large parameter space involved [2,8]. More parameters are needed for conﬁned ABC copolymers. Although this large parameter space imposes a great challenge for us to explore and understand the phase behavior of conﬁned ABC copolymers, it also provides us with great opportunity and ﬂexibility in the control and design of self-assembled nano-structures for various applications. 4.1 ABA Copolymers Feng and Ruckenstein simulated both symmetric (Am B20−2m Am ) and asymmetric (Am B20−m−n An with m = n) triblock copolymers of chain length N = 20, where the copolymer composition (m and n), surface separation and preference were varied to examined their eﬀects on the thin-ﬁlm morphology and segmental distributions [75, 76]. Although various morphologies were obtained, they are unfortunately diﬃcult to understand and rationalize because no eﬀort was made to estimate the bulk morphology and packing period for any of the copolymers used. In most cases, the same (small) lateral box sizes of 212 , which seem to be arbitrarily chosen, were used for copolymers over a wide

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Table 5. 3D lattice Monte Carlo simulations of triblock copolymer thin ﬁlms conﬁned between two identical surfaces. Refer to the original papers for details. Ref. Model Lx × Ly Lz

Copolymer

φC Interactionsa

[75]b BFM1 212

Am B20−2m Am

0.95 AB = 0.3

21

[76]c BFM1 212 , 422 3, 10 ∼ 21 Am B20−m−n An Am B15−2m Cm

0.95 AB = BC = AC = 0.3

[77] BFM 962

12 ∼ 40

A8 B48 A8

0.5 AB = −AA = −BB = 0.2495, 0.3742

[78] BFM1 482

4 ∼ 26

A8 B16 A8

0.8 AB ≥ 0

2

3 ∼ 32

A5 B5 A5 , A5 B5 C5 0.94 AB = BC = AC = 0.3

d

[79] BFM1 32 a

In units of kB T . All other interactions, except those with the surfaces, were set to 0. b Box sizes of 422 × 21 and 212 × 11 were occasionally used. c AB = BC = AC = 1 were occasionally used. Copolymers of A10 B20 A10 , A10 B10 C10 , A7 B3 C7 (with AB = BC = 0.5 and AC = 0.1), and A2 B2 C9 (with AB = BC = 0.1 and AC = 0.5) were occasionally used. d Boxes of lateral sizes 482 and 642 were occasionally used for A5 B5 A5 . Copolymers of A6 B5 A4 , A7 B5 A3 and A8 B5 A2 were occasionally used with a box of 322 × 16, and Am B21−2m Cm were occasionally used with a box of 442 × 16.

range of compositions (in some cases one block had only one segment) [75,76]. This is not appropriate. As shown in Sect. 3.1, even for the simplest case of fA = 0.5, where lamellae are expected, an arbitrarily chosen box could signiﬁcantly inﬂuence the thin-ﬁlm morphology. The same problem persisted in the simulations of Huang et al. on symmetric triblock copolymers with fA = 2/3, where the (arbitrarily chosen) lateral box sizes of 322 were used in most cases without knowing the bulk morphology and packing period [79]. Szamel and M¨ uller studied the symmetric triblock copolymers forming cylinders in the bulk at two diﬀerent segregation strengths: AB = 0.3742 and 0.2495 [77]. In the former case, they roughly estimated the inter-cylinder distance L0 ≈ 1.56Re , where Re = 25 (in units of lattice spacing), in bulk simulations starting with a prepared cylindrical structure3 ; in the latter case, the initial cylindrical structure evolved to another morphology in the bulk simulation, probably due to the mismatch between their box sizes (963 for the bulk simulations) and the characteristic hexagonal packing of cylinders. Using lateral box sizes of 962 , they then studied the thin-ﬁlm morphology as a function of surface separation and preference (denoted by H and w ≡ AS = − BS , respectively, in their work); the results are shown in Fig. 14 [77]. Compared to asymmetric diblock copolymers having the same fA = 0.25, the top two rows in their phase diagram had thicknesses (D/L2 ≈ 0.58 and 0.46, 3

Cylinders did not form in bulk simulations starting from random initial conﬁgurations [77].

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respectively) comparable to the ﬁrst row in Table 4, and good qualitative agreement was obtained between the two.

Fig. 14. Phase diagram of symmetric triblock copolymers (with AB = 0.3742) conﬁned between two identical surfaces. Symbols are placed (approximately) where the simulations were performed, and the phase boundaries are to guide the eyes only [77]. Notations diﬀerent from Table 4 are used for the thin-ﬁlm morphologies: 1/2 1/2 1/2 W corresponds to L-L, C⊥ corresponds to C||| , C corresponds to C≡ -C≡ , C corresponds to C1≡ , and PL corresponds to P in Table 4. In addition, Wa denotes the “asymmetric wetting” morphology observed in [77], where A segments cover only one of the surfaces due to the small ﬁlm thickness. In some cases, two coexisting 1/2 morphologies were observed in the simulation, e.g., C /C⊥ . The phase diagram for AB = 0.2495 is the same, except that all the PL morphologies in the top row are replaced by C /PL and the phase boundary between these morphologies and 1/2 C /C⊥ is removed [77]. Provided by Grzegorz Szamel.

Nie et al. investigated the chain conformations and bridging fractions of symmetric triblock copolymers of fA = 0.5 conﬁned between two neutral surfaces as a function of the A-B segregation strength (inversely proportional to the system temperature) and ﬁlm thickness; only perpendicular lamellae were obtained at temperatures below the order-disorder transition temperature in the ﬁlm (TODT ) [78]. They identiﬁed TODT by the peak position of the heat capacity (calculated from the ﬂuctuations of the system energy E) in a cooling run4 , at which the copolymer radius of gyration also exhibited a change in its slope with the system temperature. A critical ﬁlm thickness Dc = 3 ∼ 4Rg,0 , where Rg,0 denotes the copolymer radius of gyration in the athermal state ( AB = 0), was found, below which the surface conﬁnement 4

Note that several factors aﬀect the accurate determination of TODT : First, although the ODT is a ﬂuctuation-induced ﬁrst-order transition, the peak in the heat capacity is rounded in simulations due to the ﬁnite-size eﬀects. Second, the mismatch between PBC and L0 stabilizes the disordered phase and thus decreases the observed TODT . Third, copolymers exhibit hysteresis, i.e., TODT determined in a heating run is diﬀerent from that determined in a cooling run, presumably due to the slow chain relaxation in simulations.

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notably suppressed TODT due to the surface-induced compatibilization between A and B segments (caused by the “missing-neighbor” eﬀect). Diﬀerent deformations of the mid-block were also found for ﬁlm thicknesses above and below Dc [78]. 4.2 ABC Copolymers Only two MC studies have so far been reported on the thin-ﬁlm morphology of linear ABC triblock copolymers conﬁned between two identical surfaces, where the surfaces interacted with only one block [76, 79]. Unfortunately, the bulk morphology and packing period were not estimated for any of the copolymers used. Some understanding can only be gained for the most studied case of fA = fB = fC = 1/3 and ≡ AB = BC = AC = 0.3, where one may expect that the bulk morphology would be lamellae of ABCCBA with the B layer thickness being about half of those of the A and C layers: When the surfaces preferred an end (A or C) block, parallel or perpendicular lamellae were obtained depending on the ﬁlm thickness and surface preference [79]. When the surfaces preferred the middle (B) block, perpendicular lamellae throughout the ﬁlm were obtained over a wide range of ﬁlm thicknesses if the surface preference was not too strong (e.g., BS / = −1) [79]; for stronger surface preferences ( BS / = −3.33 or −6.67) a layer of B segments covered the surfaces while the perpendicular lamellae remained in the interior of the ﬁlm [76]. Finally, when the surfaces were neutral, perpendicular lamellae were obtained [76]. The last two cases agreed with the 2D self-consistent ﬁeld calculations by Pickett and Balazs [80].

5 Nano-Conﬁnement in Two and Three Dimensions For block copolymers under nano-conﬁnement, the geometry of the conﬁning surface(s) plays an important role in determining their self-assembled morphology. For example, as shown in Sect. 3.1, various non-cylindrical morphologies can be obtained in thin ﬁlms of cylinder-forming asymmetric diblock copolymers. Thin ﬁlms represent 1D conﬁnement. Recently, experimental eﬀorts have been devoted to exploring the self-assembled morphology of block copolymers under 2D and 3D conﬁnement. Concentric cylindrical [81–83]/spherical [84] structures have been reported for symmetric diblock copolymers conﬁned in pores/spheres; in the former case, an unexpected stacked-disc or toroidal-type structure [82, 85] was also reported. Even more interesting is the morphology of cylinder-forming diblock copolymers conﬁned to cylindrical pores, recently studied by Russell’s group with the pore wall preferring the shorter (A) block [81, 85, 86]; when the pore diameter was comparable to the bulk inter-cylinder distance, the morphology of multiple helices formed by A was observed [85,86]. Helical and other exotic morphologies (e.g., stacked donuts, helix around a rod, etc.) were also found by Wu et al. in

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silica-surfactant composite mesostructures conﬁned to cylindrical pores [87]. It seems that these exotic morphologies are general features of cylindrical structures under 2D conﬁnement. Very recently, sphere-forming diblock copolymers were also conﬁned to cylindrical pores and helical strings of spheres were observed experimentally [85]. Table 6 lists the MC simulations of block copolymers conﬁned to cylindrical pores reported so far, where Lz denotes the box length in the direction along the pore axis (where PBC are applied), and D denotes the pore diameter5 . Table 6. 3D lattice Monte Carlo simulations of block copolymers conﬁned to cylindrical pores. Refer to the original papers for details. Ref. Model

Lz D

Copolymer

φC Interactionsa

[88]b BFM1

100 92

A10 B10

0.7 AB = −AS = BS = 0.5

[89] BFM1

100 32 (AB), A30 B30 , 34 (ABC) A20 B30 C10

0.7 AB = BC = AC = −AS = BS = CS = 0.5

[90]c BFM1

80 16, 22, 26 A10 B10

0.73 AB = 0.5; AS = −0.2 ∼ 0

[91] BFM1×

5 ∼ 43

A3 B9 , A2 B10 0.85 AB = 10.51; AS = −BS = −10.51, 0, 10.51

a

In units of kB T . All other interactions were set to 0. b The copolymers were conﬁned to both cylindrical pores and spheres; in the latter case, D denotes the diameter of the sphere. c Lz = 60 ∼ 100 were used for D = 22 and AS = 0.

He et al. ﬁrst reported MC simulations of symmetric diblock copolymers conﬁned to cylindrical pores; due to the cylindrical geometry of the conﬁning surface and its preference for one of the blocks, concentric cylinders having the same axis as the pore were obtained [88, 89], in agreement with experiments [81–83] and dynamic density-functional calculations [92]. The symmetric diblock copolymers were also conﬁned to a sphere preferring one of the blocks, and concentric spheres were similarly obtained [88], again consistent with recent experiments [84]. He et al. also simulated in one case linear triblock copolymers A20 B30 C10 conﬁned to a cylindrical pore preferring A segments, and obtained concentric cylinders [89]. A more systematic study was recently reported by the same group, where symmetric diblock copolymers were conﬁned to cylindrical pores; the pore diameter and preference were varied and several morphologies were reported, as summarized in Table 7 [90]. When the pore wall was neutral, lamellae oriented perpendicular to the pore axis (the “stacked disk” in Table 7) were obtained, in agreement with dynamic density-functional calculations [92]. From 5

Since the simulations are performed on a lattice, the conﬁning pore is actually a polygon.

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this morphology they roughly estimated L0 ≈ 16. Interestingly, single-helix structures were also observed in neutral pores, probably due to the mismatch between Lz = 80 and the true L0 of their copolymers6 . Concentric cylinders (the “barrel” in Table 7) were obtained at the strongest surface preferences, where the number of cylindrical shells varied with the pore diameter. Other more complex morphologies were also obtained, as shown in Fig. 15 [90]. Table 7. Morphology of symmetric diblock copolymers conﬁned to cylindrical pores. The surface preference is characterized by α ≡ −AS /AB . Diﬀerent morphologies at the same pore diameter D and surface preference are obtained in diﬀerent simulation runs. Adapted from [90]. α

D = 16

D = 22

D = 26

0

Stacked disk

Stacked disk Single helix

Stacked disk Single helix

0.05 Stacked disk Single helix

Single helix Gyroidal

Stacked disk Single helix

0.1 Stacked disk Single helix Catenoid cylinder

Catenoid cylinder Stacked disk Single helix

0.15 Catenoid cylinder

BA-barrela

Disordered

0.2 Defective BA-barrel BA-barrel

BA-barrel Defective BA-barrel

0.25 BA-barrel

BA-barrel

Stacked circle

0.3 BA-barrel

BA-barrel

Stacked circle

0.35 BA-barrel

BA-barrel

Stacked circle ABBA-barrelb

0.4 BA-barrel

BA-barrel

ABBA-barrel

a

This morphology consists of a B-cylinder coated by an A-shell, denoted as “monocylinder barrel” in [90]. b This morphology consists of an A-cylinder coated by an inner B-shell and an outer A-shell, denoted as “two-layer concentric cylinder barrel” in [90].

Using the simulated annealing technique, Yu et al. studied cylinderforming asymmetric diblock copolymers conﬁned to cylindrical pores at strong segregation between A and B segments [91]. They estimated the characteristic inter-cylinder distance L0 for their copolymers in bulk simulations using rectangular boxes of Lx = Ly = Lz . The pore diameter and preference were 6

Their simulations showed that varying Lz aﬀects the probability of observing the single-helix structure [90], which may be a “trapped” state according to [92].

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(a)

(b)

(c) Fig. 15. Representative conﬁgurations of some complex morphologies of symmetric diblock copolymers conﬁned to cylindrical pores preferring A segments: (a) Single helix at D = 16 and α = 0.05. The middle shows only A segments, and the right shows only B segments. (b) Catenoid cylinder at D = 22 and α = 0.1. The middle shows only A segments, and the right shows only B segments. (c) Stacked circle at D = 26 and α = 0.3. The right shows the cut-view along the pore axis. Reproduced from [90]. Copyright (2006) American Institute of Physics.

then varied to examine their eﬀects on the conﬁned morphology; the results are shown in Fig. 16. Various complex morphologies (including spheres, cylinders, helices, toroids, and their hybrid structures) were obtained; their appearance can be correlated to D/L0 , indicating the conﬁnement-induced structural frustration [91]. The MC results were in qualitative agreement with available experiments [81,85–87], 3D self-consistent ﬁeld calculations of a blend of asymmetric diblock copolymers AB with homopolymers C conﬁned to cylindrical pores preferring C [87], and 2D self-consistent ﬁeld calculations of diblock copolymers AB conﬁned to a circle preferring B [93]. More studies are needed to further quantify and understand these complex morphologies.

6 Perspectives Independent of, yet complementary to, experiment and theory, Monte Carlo simulation is an essential tool for us to explore and understand the selfassembly of block copolymers under nano-conﬁnement. Although its basic principles are straightforward to understand and implement, care must be taken to reduce or eliminate possible artifacts in the simulation, including the mismatch between the periodic boundary conditions and the intrinsic periodicity of the self-assembled structure, inappropriate choice of trial moves, insuﬃcient equilibration and/or sampling of the system conﬁgurations, etc. When properly conducted, Monte Carlo simulation gives the exact solution to the model system studied. It can therefore be used to examine the validity and/or accuracy of the theory that provides approximate solution to the same (or a similar) model system. By comparing between the simulation and

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Fig. 16. Morphology of asymmetric diblock copolymers conﬁned to cylindrical pores. Both a top-view (along the pore axis) and a side-view are given for each morphology. In the top-view, the outmost circle represents the pore wall, and the inner ring formed by the copolymers is shown separately in some cases. D/L0 is given underneath each morphology. The surface preference is characterized by α ≡ AS /AB = −BS /AB . The top row shows the case of fA = 1/6 (L0 ≈ 9.93) and α = 1, the middle row shows the case of fA = 1/6 and α = −1, and the bottom row shows the case of fA = 1/4 (L0 ≈ 10.67) and α = 0. Reproduced from [91]. Copyright (2006) by the American Physical Society.

experimental results, one can also examine the appropriateness of the model system extracted. As discussed in this chapter, Monte Carlo simulations have been extensively used in the past decade to explore and understand the self-assembled morphologies of block copolymers under nano-conﬁnement. In the near future, it would be of great interest to address the following issues: • •

Eﬃcient methods to estimate the intrinsic periodicity of the self-assembled structures. A promising direction is to develop fast oﬀ-lattice simulation techniques in a constant-pressure ensemble. Free-energy calculations for a given morphology. So far, no free-energy data have been directly calculated from Monte Carlo simulations of conﬁned block copolymers. Such data are needed in order to determine which morphology observed in simulations is the most stable phase. For this

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purpose, combining Monte Carlo simulations with some theory (e.g., the self-consistent ﬁeld theory) to estimate the free energy is also desirable. Further simulations of asymmetric diblock copolymer thin ﬁlms, 2D and 3D conﬁnement of diblock copolymers, as well as the self-assembly of linear and star ABC triblock copolymers under nano-conﬁnement. Due to the complexity of the self-assembled morphologies and their ﬂuctuations, ensemble-averaged proﬁles (e.g., segmental distributions, chain orientation, etc.) would be very helpful in identifying various morphologies, in addition to the commonly used visual inspection of system conﬁgurations.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

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Understanding Vesicles and Bio-Inspired Systems with Dissipative Particle Dynamics Julian C. Shillcock Max Planck Institute of Colloids and Interfaces, Theory Department, 14424 Potsdam, Germany

1 Introduction 1.1 Cells, Membranes and Amphiphiles Biological organisms have always been a source of inspiration for the designers of artiﬁcial materials and machines. At the boundaries of Biology, Chemistry and Physics, this process of borrowing from Nature is becoming increasingly important for several reasons. Industrial sectors, such as pharmaceutical companies and the chemicals industry [1], are keen to understand and rationally modify the design of natural materials so as to minimise manufacturing costs and create new materials whose properties are better suited to their purpose. Biologists and biophysicists would like to understand how organisms perform crucial life functions, such as protecting themselves against infectious agents, so that they can mimic these functions in the ﬁelds of medical treatment and bioweapons detection. Novel treatments for genetic diseases require the controlled replacement of defective genes with healthy ones, and will only fulﬁll their promise if their delivery can be made routine and reliable. The construction of artiﬁcial viruses [2] to deliver genetic material is a promising route, but understanding the interactions of their components, and optimising their structure, are challenging tasks. Progress in the experimental techniques of electron microscopy, ﬂuorescence imaging and micromanipulation is revealing more and more details of cellular structures and molecular interactions in natural and artiﬁcial cells and organelles. Simultaneously, the exponential increase in computing speeds, and concomitant drop in price, of the last two decades, provides researchers with the tools to construct ever more comprehensive computer models of biological systems. These models are becoming essential aids in helping to make sense out of the vast quantities of data that are generated by experiments. They allow dynamic processes occurring in biological systems to be visualized and probed on length and time scales that are experimentally inaccessible, and reveal how molecular-scale interactions propagate to the macroscopic level. This

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allows them to be used for hypothesis testing prior to performing experiments, and to direct experimental resources to investigating the most promising areas of a system’s behaviour. Computer simulations have been used for many decades to study model systems. However, traditional Molecular Dynamics simulations are limited to systems with linear dimensions in the nanometre range, and can only reach time-scales of perhaps a hundred nanoseconds. In recent years, a variety of mesoscale simulation techniques have been developed to go beyond these limitations. It is now possible to simulate models of biological processes, such as the fusion of a 28 nm vesicle to a 50 × 50 nm2 planar lipid membrane, containing tens of thousands of lipid molecules surrounded by millions of water particles using a single-processor desktop computer [3]. The applications of mesoscale simulation techniques to soft matter are only beginning to be explored. They include visualizing the self-assembly of complex macromolecular assemblies from rationally-designed polymers [4, 5], predicting the material properties of soft matter, such as phospholipid membranes and polymersomes [6], and the design of drug delivery vehicles [7, 8] and artiﬁcial viruses [2]. Liposomes and polymersomes are closed, spherical sacs containing solvent that form spontaneously when amphiphilic molecules, phospholipids and diblock copolymers respectively, are immersed in aqueous solvent [9]. Figure 1 shows a typical equilibrium state of a polymersome and lipid vesicle in aqueous solvent. Membranes such as these are examples of soft surfaces: ﬂuid interfaces that are highly ﬂexible and readily undergo shape changes [10]. Amphiphiles are molecules that contain two parts: a hydrophilic, or water-loving, part chemically bonded to a hydrophobic, or water-hating, part. The hydrophobic part is often a linearly-connected hydrocarbon chain. The drive for the hydrophobic portion of the molecules to be shielded from the aqueous solvent by the hydrophilic part, referred to as the hydrophobic eﬀect, leads amphiphiles to spontaneously assemble into a variety of aggregates. These include micelles (spherical droplets of amphiphiles with their hydrophobic parts in the interior of the droplet shielded from the surrounding solvent by their hydrophilic parts covering the surface), bilayers (two planar layers of amphiphiles with the hydrophobic ends of the molecules in each layer adjacent), and vesicles (spherical sacs in which the amphiphiles organize into a thin, closed, bilayer membrane surrounding a volume of solvent). The type of aggregate formed depends on the molecular details of the components [11]. Phospholipid membranes are ubiquitous in nature, and essential for cellular life. Nature has created a bewildering variety of phospholipid molecules (more simply referred to as lipids) for use in constructing the cell and organelle membranes that occur in biological organisms. Lipids diﬀer in the number, length and degree of unsaturation of their hydrophobic chains, and the size, structure, and presence or absence of charge, of their hydrophilic headgroup. The complexity of lipid membranes is still being appreciated. In the 30 years since it was realised that the cellular plasma membrane is a ﬂuid bilayer [12]

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Fig. 1. Cut through a simulation snapshot of a polymersome containing 2616 diblock copolymers with molecular architecture H29 I T47 representing a typical molecule such as poly(ethylene-oxide)-polyethylethylene as described in [6], and a lipid vesicle containing 5887 lipid molecules with architecture A3 (B4 )2 representing a typical lipid such as dimyristoylphosphatidylcholine as modelled in [3]. Bead types H and A represent the diblock’s hydrophilic monomers and lipid headgroup constituents respectively, and T and B represent the hydrophobic monomers and lipid tail segments respectively. Solvent particles are invisible for clarity. The bead-bead interaction parameters, and bond potentials, are chosen independently for the two aggregates so as to reproduce their equilibrium structural properties. Both aggregates have an outer diameter of approximately 30 nm. The simulation box contains more than 2.3 million particles in total. Whereas the lipids are anchored ﬁrmly at the hydrophobic-water interface, the membrane being compressed by the hydrophobic eﬀect of the lipid tails in water, the hydrophilic parts of the diblock copolymers are well hydrated and extend into the bulk water phase. The hydrophobic region in the polymersome is approximately twice as thick as that in the lipid vesicle, and the chains from the two leaﬂets are signiﬁcantly more interdigitated than the lipid tails in the vesicle (although this is not obvious from the ﬁgure). Equilibration of this system requires about 23 cpu-days of processing time on a single Intel Xeon 2.0 GHz processor.

our knowledge of membranes has increased enormously. The original picture of a membrane as a roughly-homogeneous, relatively-empty, lipid “sea” that provides a medium for sparsely-distributed proteins and structural support for the cell, has been replaced by a more heterogeneous, dynamic one [13]. In this new picture, diﬀerent lipid types cluster preferentially around embedded proteins, segregating the membrane into distinct regions; and oligomeric transmembrane proteins may occlude a large fraction of the lipidic surface from the surrounding environment. Understanding the organization and interactions of such a complex system is beyond the reach of traditional mathematical models in which the membrane is treated as an inﬁnitely-thin elastic sheet [14–16], although such methods can still yield insight in idealised situations [17]. Vesi-

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cles constructed out of diblock copolymers [18, 19] have properties that can be tuned more widely than those composed of phospholipids [20], and this makes them attractive for biotechnological applications such as drug delivery vehicles [7]. The optimization of the physico-chemical properties of such novel materials requires understanding of the inﬂuence of their molecular structure on their dynamical behaviour. Varying a large number of independent molecular parameters and measuring the subsequent changes in the material properties is time consuming and costly if carried out in experiments, but much faster and simpler by creating and modifying suitable computer models of the materials. Computational models of membranes that capture just those molecular details that are relevant for the macroscopic processes of interest can yield insight into how molecular features propagate from the microscopic scale to the macroscopic: the so-called Twilight Zone [21]. This helps in developing a greater understanding of the dynamic role of membranes in biophysical and biochemical processes in the cell, and their rational modiﬁcation for biomedical and other applications. Hence, the quantitative estimation of physical properties of membranes and the visualization of their dynamical behaviour are important steps towards increasing our understanding of soft materials. 1.2 Visualization and the Search for Answers The material properties of membranes are crucial for their function, both in natural biological structures such as the cell’s plasma membrane, and in artiﬁcial vesicles. The membrane thickness, ﬂexibility, ﬂuidity and porosity to the passage of small and large molecules, are major properties that determine the utility of membranes for medical and other applications. The resistance of a membrane to rupturing under tension, or as a result of attack by membranemodifying agents such as phospholipases, clearly limits its usefulness. But predicting the dependence of such properties on the membrane’s molecular constituents using continuum theories is restricted to highly-simpliﬁed cases, and the results of experiments are often obscured by the complexity of the molecular interactions. Mesoscale or coarse-grained computer simulations have emerged in the last 15 years as important tools for studying amphiphilic membranes. Computational models of vesicles containing tens of thousands of amphiphiles have been followed on a 50 nm length-scale for many microseconds [3, 22]. Coarsegrained simulation techniques reach these length and time scales by a combination of collecting groups of atoms, or molecular groups, into particles or beads, providing a spatial coarse-graining; and replacing the complex atomistic force ﬁelds, which typically involve Lennard-Jones potentials, with softer eﬀective forces, allowing a temporal coarse-graining. Because the soft forces allow a larger integration time-step in solving the Newtonian equations of motion, coarse-grained simulation techniques are being extensively developed

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Fig. 2. Transient shape deformation of a lipid vesicle in shear ﬂow. The upper image shows the side view, and the lower image an oblique view of the same state. The vesicle is similar to that in Fig. 1 and contains 5887 lipid molecules with architecture A3 (B4 )2 representing a typical lipid such as dimyristoylphosphatidylcholine. Initially the vesicle is relaxed and spherical, a shear ﬂow is created in the surrounding solvent by applying a constant external force to solvent particles outside the vesicle such that some move upwards in the simulation box and others move downwards. The resulting drop in pressure around the circumference of the vesicle causes it to contract at the poles (top and bottom of the image) and expand laterally. This shape is transient, and the vesicle continues to deform in the ﬂow ﬁeld until the ﬂow is switched oﬀ. The similarity to the erythrocyte membrane’s equilibrium shape is obvious, although the erythrocyte shape is determined by the minimum energy conﬁguration of the ﬂuid lipid membrane and proteinaceous cytoskeleton, whereas the purely lipidic vesicle shown here is only in a transient state. The simulation box contains approximately 1.2 million particles, and the state shown here requires 17 cpu-hours of simulation time on the same processor as in Fig. 1.

for studying biological systems, including membranes and protein-driven biophysical processes, whose natural length and time scales exceed those that are achievable using atomistic Molecular Dynamics simulations. Figure 2 shows a transient shape deformation of a 28 nm diameter vesicle deforming in a shear ﬂow. The shape transformation is driven by hydrodynamic forces that would be prohibitively expensive to simulate using Molecular Dynamics simulations, and impossible in solvent-free models. Coarse-grained simulations that include solvent particles, such as Dissipative Particle Dynamics, allow the simulation of very large systems in which hydrodynamic forces are included, and their eﬀects on soft matter systems can be visualized. A consequence of the coarse-graining is that eﬀective parameters appear whose values are not unambiguously related to the microscopic properties of the molecules, and must be input into the technique. However, the almost unlimited control over these parameters contrasts with experiments where many parameters are poorly controlled or inextricably coupled. Such control allows the eﬀects of modiﬁcations of molecular details on large-scale material properties to be systematically explored. For many applications it may be sufﬁcient to obtain qualitative results for a wide range of parameters rather than more accurate results for a restricted region of parameter space. Mesoscale

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models can be used to obtain approximate answers to important questions in the design of novel bio-inspired materials such as vesicles and nanostructured materials before applying the full machinery of atomistic Molecular Dynamics. The remainder of this chapter is organized as follow. In the next section, we survey the simulation techniques that have been developed to study simpliﬁed models of membranes. Atomistic Molecular Dynamics (MD) simulations have been used for many years to study small systems containing perhaps a few hundred molecules. Although MD simulations provide the most accurate atomic details, their computational demands limit them to small systems and short time scales. Coarse-grained methods, such as Brownian Dynamics (BD), Dissipative Particle Dynamics (DPD), and specialised solvent-free simulations have been developed to go beyond the limitations of MD by ignoring many molecular details thereby allowing the simulation of systems containing tens of thousands of lipids. In Sect. 3, we focus on the application of coarse-grained simulations to three problems: the self-assembly of amphiphilic molecules into aggregates; the material properties of planar bilayer membranes composed of lipids or diblock copolymers; and the fusion of a vesicle to a membrane. Finally, in Sect. 4 we present our perspective on the developments in the ﬁeld of mesoscale simulations that we believe will make them increasingly important in the rational design of soft biomaterials.

2 Computational Membrane Models 2.1 Starting at the Bottom: Atomistic Molecular Dynamics Molecular Dynamics simulations have been used for 50 years to explore the behaviour of small amounts of matter [23]. Atomistic MD simulations use atoms as the fundamental constituents with complex potentials tuned to reproduce the structural properties of the systems of interest. Once the molecules and their interactions are speciﬁed, the algorithm integrates Newton’s equations of motion to generate the system’s dynamical behaviour, which should apply as long as the size of the particles is large compared to their de Broglie wavelength. The great advantage of MD simulations is that the dynamics of the system are (presumably) the same as those obeyed by the physical system, and the maximum information is gained about all processes taking place within the length and time scales of the simulation. But this advantage also means that atomistic MD is constrained to small systems, a few hundred molecules in a membrane is typical, simulated for short times, usually less than a hundred nanoseconds. Although MD simulations are able to reach the microsecond scale in special cases [24], such runs require huge computational resources beyond the means of most groups. Biological processes often occur on millisecond time-scales or longer, and involve millions of molecules, and the routine modelling of such processes using atomistic MD is not yet feasible. Spatially-restricted questions of membrane behaviour involving a few hundred lipids are however intensively studied using atomistic MD. Fundamental

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properties of membranes can be precisely measured, such as the ﬂuidity and in-plane diﬀusion of lipid and lipid-cholesterol mixtures [25, 26]; and the variation of membrane structural properties followed as the constituent lipid tail length and unsaturation are modiﬁed [27]. The perturbative eﬀects of incorporating antimicrobial peptides [28] or bundles of alpha-helical peptides [29] into lipid membranes have also been explored. Phase transitions between diﬀerent lipid phases [30] have been simulated, as has the eﬀects of dehydration on the lipid headgroups in closely-apposed membranes [31]. The latter work showed that redistribution of lipids occurs when the membranes are dehydrated, but the membranes were restricted to 56 lipid molecules each. These systems have in common that their dominant length scale is not many times larger than the width of the membrane, so that the process can be captured within the length and time span of the MD simulations. The desire to capture phenomena whose natural scales are microns and milliseconds or larger, has driven the development of new simulation techniques in which atoms are lumped together into so-called coarse-grained particles, and the complex, atomistic force ﬁelds are replaced by simpler, eﬀective forces. Coarse-grained MD simulations achieve their speed-up in two ways: ﬁrst, several atoms or atomic groups are replaced by a single, usually-spherical particle. This reduces the number of degrees of freedom that must be integrated to solve the system’s equations of motion, and is especially useful in reducing the number of water molecules that are present in almost all biological problems of interest. Second, the complex atomistic force ﬁelds are replaced by simpler sets of Lennard-Jones interactions between the particles, and these features extend the temporal range of MD simulations to tens of microseconds. The price one pays for these gains is that the coarse-graining introduces ambiguity in the assignment of mass, length and time scales, which results in many parameters needing to be calibrated for accurate predictions from the simulations. However, this cost is outweighed by the enormous increase in system size, from tens of lipid molecules in atomistic MD to thousands in coarse-grained MD simulations [32,33]. But even this substantial gain is insufﬁcient to bridge the gap between the atomistic world and the mesoscopic world of blood, vesicles and cells. This requires a further degree of coarse-graining. 2.2 A Problem of Length-Scale: Coarse-Grained Models In a typical MD membrane simulation, most of the computational eﬀort goes into calculating the motion of the solvent molecules simply because these outnumber the lipid molecules by more than 10 to 1. One approach to increasing the achievable system sizes is to eliminate the solvent particles. Because this eliminates the hydrophobic eﬀect that drives the formation of amphiphilic membranes, solvent-free models are obliged to include more complex intermolecular forces to restore its eﬀects. Various simulation schemes have been constructed to retain the self-assembly properties of amphiphiles after removing the solvent molecules.

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A recent “water-free” model of a ﬂuid membrane [34] uses non-additive, pairwise potentials between amphiphiles to cause them to assemble into, and remain in, a planar bilayer structure. The amphiphiles in this model are rigid rods containing a single hydrophilic head particle and two hydrophobic tail particles connected in a linear chain. The model possesses both ﬂuid and crystalline phases as the area per molecule is varied, and allows the extraction of the membrane’s elastic properties. Pores form in the membrane at low area densities that promote inter-monolayer ﬂip-ﬂop of molecules. This model has recently been extended to allow simulations of rod-like DNA molecules adhering to planar, cationic lipid membranes [35]. A second solvent-free model is the Espresso code [36]. This has been used to study the self-assembly and material properties of amphiphilic membranes [37]. An amphiphile in this work is a three-particle, linear chain containing one hydrophilic head particle and two hydrophobic tail particles. The chains are ﬂexible, however, and possess a long-range attractive potential between their tail particles that serves to drive self-assembly of the molecules into a membrane aggregate. The range of this attractive potential is a key parameter, and can be used to tune the membrane’s elastic properties. Realistic values of 5–20kB T for the membrane’s bending modulus are found that make the model suitable for lipid membrane simulations. This makes the scheme very attractive for studying membrane processes such as domain formation in multi-component membranes, budding in vesicles, and also dynamic processes such as the engulfment of a colloidal particle by a membrane. Using this method it is possible to simulate domain formation and budding in a 60 nm diameter, two-component vesicle containing about 16,000 molecules. Domain formation is driven by the preference of the amphiphiles to pack next to their own kind instead of mixing. Another example of a solvent-free model has been introduced recently [38]. An amphiphile is again represented by a three-bead linear molecule containing a single hydrophilic bead and two hydrophobic beads. The beads are connected by ﬁnitely-extensible-nonlinear elastic (FENE) springs, and the molecule is not rigid but possesses a bending stiﬀness potential. The absence of the solvent is compensated for by using a multibody density-dependent potential. Oﬀlattice Monte Carlo simulations are used to generate the states of the system, and ensemble averages provide the mean values of equilibrium observables. The model produces lateral diﬀusion coeﬃcients and elastic moduli in good agreement with experimental data on lipid vesicles [39]. Brownian Dynamics (BD) simulations also eliminate the solvent molecules, and have been used, for example, to follow the self-assembly of amphiphilic bilayer vesicles containing around 1000 molecules [40]. The amphiphiles are represented by rigid rods containing one hydrophilic head particle and two hydrophobic tail particles. In BD simulations, each particle experiences forces resulting from its interactions with neighbouring particles together with a viscous drag and random force that mimic (some of) the eﬀects of the missing solvent. It should be noted that the random force is added independently to

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each particle’s equation of motion so that the dynamics of the system is diffusive and hydrodynamic modes are not included. Other forces that depend on the presence of a solvent also have to be incorporated in an ad hoc way. As mentioned earlier, the hydrophobic eﬀect of amphiphiles in water drives their self-assembly into ordered aggregates such as micelles, membranes and vesicles. The removal of the solvent in BD simulations must therefore be compensated for if the amphiphiles are to aggregate as expected. In the work of Noguchi and Takasu [40], this is done by adding a density-dependent force between the hydrophobic particles in the amphiphiles. This allowed the authors to simulate the fusion of two vesicles containing about 1000 amphiphiles each, which were approximately 20 nm in diameters [41], and the adhesion of solid nanoparticles to vesicles [42]. Fusion of the vesicles appeared to follow diﬀerent pathways at diﬀerent temperatures, and the nanoparticles were observed to induce budding of a single vesicle into two and the opposite process of two vesicles fusing into one. Solvent-free models gain a computational advantage through not using processor cycles to simulate the bulk solvent, and have been shown to be useful in extracting equilibrium material properties of membranes [37]. But their continued development for studying problems for which hydrodynamic modes are potentially important is limited. Such problems include a wide range of biologically-interesting processes, such as the diﬀusion of signalling molecules in cytoplasm, the hydrodynamic ﬂow of a vesicle’s internal solvent being expelled under pressure, and the controlled release of the contents of many vesicles held near the pre-synaptic plasma membrane in a more realistic model of synaptic vesicle fusion. Secondly, while providing an excellent framework for studying equilibrium membrane properties, their dynamics do not always correspond to the Newtonian equations of motion. A coarse-grained simulation technique invented in the early 1990’s attempts to address these deﬁciencies. 2.3 Dissipative Particle Dynamics: History and Algorithm Dissipative Particle Dynamics is a particle-based, explicit-solvent simulation technique that was created for the simulation of ﬂuids at larger length and time scales than is possible using atomistic Molecular Dynamics, whilst retaining the hydrodynamic modes that are missing in techniques such as Monte Carlo and Brownian Dynamics. The original DPD algorithm of Hoogerbrugge and Koelman [43] had some deﬁciencies that were removed by later workers [44, 45], and it is the GrootWarren version of the DPD integration scheme that is now most commonly used. An early review of the technique was published by Warren [46], and a comparison of various methods of simulating surfactant solutions followed [47]. A derivation of the DPD scheme from a set of underlying MD particles allows the simulation of dissipative particles whose size can change with time, improving the scheme’s performance when multiple length scales appear in a

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simulation, such as for hard colloidal particles in a solvent [48]. Early applications of the DPD technique included microphase separation of polymeric mixtures [49], the dynamics of an oil droplet near a hard surface in shear ﬂow [50], aggregation of surfactants onto a polymer in a bulk surfactant solution [51], colloidal motion in a solvent [52], and rupture of a planar membrane patch by incorporation of nonionic surfactants [53]. It was also applied to the packing of surfactants at an oil-water interface and their eﬃciency at reducing the surface tension [54], following the evolution of the interface between pure surfactant and water [55] and the behaviour of grafted polymer brushes subject to shear ﬂow [56]. As DPD has become more widely used, attempts have been made to improve the scheme’s thermostat, which has been shown [57, 58, 60] to lead to spurious behaviour if too large a time step is used in the integration scheme. These include changing the type of thermostat [61, 62] to improve the temperature control [63], combining two diﬀerent thermostats, and randomly selecting one or the other for each interacting particle pair, that allows the viscosity of a DPD ﬂuid to be varied by orders of magnitude [64]. The latter thermostat can also be applied to other particle-based simulation techniques, such as MD [65]. Other changes have been suggested to allow simulations in new ensembles such as constant pressure and constant surface tension ensembles [59], and to replace the original potentials with density-dependent ones that include an attractive part thereby allowing liquid-gas interfaces to appear in the simulations [66], a process that is forbidden in the original algorithm by the quadratic nature of the DPD ﬂuid’s equation of state. We now describe the Hoogerbrugge–Koelman/Groot–Warren DPD scheme that is most commonly used. The macroscopic equations of ﬂuid dynamics treat a ﬂuid as a continuous medium with well-deﬁned values at all points in space of density, velocity, pressure (and, if it is relevant, temperature). By invoking the conservation laws of mass, momentum and energy, one may derive the fundamental equations of ﬂuid ﬂow, the so-called Navier-Stokes equations. In this procedure, it is assumed that the macroscopic properties of the ﬂuid at any point, for example, the velocity or pressure, represent averages over a number of atoms or molecules located near that point. A ﬂuid element is therefore an idealized small volume that contains many molecules but is smaller than any relevant macroscopic length scale. The Dissipative Particle Dynamics simulation algorithm also deﬁnes small volume elements as its fundamental units [43]. This makes it distinct from classical Molecular Dynamics which uses atoms and molecules. The volume elements, which are often referred to as beads, are assumed to contain a number of molecules or molecular groups. But unlike the ﬂuid dynamical approach, the size of the beads is not assumed to be much larger than the molecular scale. A bead in a typical DPD simulation is often assumed to contain between 3 to 10 molecules [53]. The exact mapping from one bead to a number of molecules depends on the size of the molecules the bead is to represent. The

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physical masses of all beads (m) in a DPD simulation are usually assumed to be identical, as are their sizes (a0 ). Note that this length a0 is the distance at which all non-bonded, bead-bead forces vanish, and it corresponds to the diameter of a bead if one pictures two beads at the extreme of their interaction range as two spheres whose surfaces just touch. The remaining physical unit required to convert dimensionless simulation quantities to physical quantitites is a time-scale. This can be extracted from the time-scale of a relevant process in the simulated system. For the simulation of ﬂuid amphiphilic membranes and vesicles, it is common to use the amphiphiles’ in-plane diﬀusion coeﬃcient to set the time-scale. This is obtained by calculating the average of the mean-square displacement of all the molecules in a membrane, and taking the ratio of its long-time limiting value to the elapsed time as a measure of the diﬀusion coeﬃcient. This is not the only choice, however, and the timescale associated with a given simulated process may even depend on details of the DPD implementation. For further details the reader is referred to the literature [53, 64]. Once the beads are deﬁned, they need to interact. The forces between beads in a DPD simulation are eﬀective forces that are chosen so that they locally conserve mass and momentum and are pairwise additive. These conservation laws, together with the fact that the total force on a bead is the sum of all the forces due to its neighbours within a ﬁxed distance, ensure that hydrodynamic interactions emerge in a DPD ﬂuid for much smaller particle numbers and on shorter time-scales than would be possible in MD simulations. A DPD simulation of a pure ﬂuid contains a single bead type. The total force between two beads in the ﬂuid is the sum of three contributions. Each contribution has a parameter that sets the relative magnitude of the interaction and a functional form that determines how the force varies with increasing separation of the beads. Beyond a ﬁxed distance, the so-called cutoﬀ distance a0 , all the forces vanish. The ﬁrst type of force is a conservative interaction that corresponds in purpose, although not in its functional form, to the Lennard-Jones interaction between two atoms in an MD simulation. It allows beads to be given an identity, so that a water bead and an oil bead feel a mutual repulsion that leads to phase separation. The magnitude of the conservative force is related to the compressibility of the ﬂuid being simulated [45]. The other two forces are a random and a dissipative force that together constitute a thermostat that adds energy to the ﬂuid and extracts energy from the ﬂuid respectively. Unlike the thermostats commonly used in MD simulations (for a brief review of their properties see [64]), the DPD thermostat conserves momentum locally, and it is this that allows the hydrodynamic interactions in the ﬂuid to propagate. To maintain the ﬂuid at a pre-set temperature, the magnitudes and functional forms of the thermostat forces have to be chosen appropriately; when this is done, they satisfy a ﬂuctuation-dissipation theorem, and the equilibrium states of the ﬂuid are generated with a probability that obeys the Boltzmann distribution.

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Simulating a mixture of N ﬂuids using DPD involves a straightforward generalisation of the scheme in the previous paragraph. Instead of a single parameter deﬁning the conservative force between two like beads, there is an N × N matrix of such parameters. The diagonal elements in this matrix are directly related to the compressibilities of the pure ﬂuid components, while the oﬀ-diagonal elements are related to the ﬂuids’ relative solubilities. Note that the interaction matrix is symmetric by deﬁnition. There are also two further symmetric matrices for the random and dissipative forces whose elements are connected by the ﬂuctuation-dissipation relation (see the description of the random force below (3). In principle, the random and dissipative force parameters between bead pairs of diﬀerent types can be chosen independently as long as the temperature deﬁned by the ratio of the force parameters is the same. In practise, this ﬂexibility has not been explored in great detail in the DPD literature. This completes the set of non-bonded forces between bead types in a multi-component DPD ﬂuid simulation. In order to include molecules in such simulations, bond forces must be introduced to tie beads together. Hookean spring forces are typically used for this purpose. The spring constant and unstretched length are determined by comparison of physical properties of the molecules, such as their end-to-end length and bond angle distributions, to the corresponding quantities in the simulation. Once the forces between beads have been speciﬁed, the state of the ﬂuid is evolved in time by integrating Newton’s equations of motion for all the beads in the system. Because the thermostat involves a stochastic term (in the random force) and a velocity-dependent term (in the dissipative force), the choice of integrator is not as simple as for MD simulations that contain only (velocity-independent) Lennard-Jones interactions. A variety of integration schemes have been proposed [61,63,67] to reduce artifacts due to the step size and to handle the thermostat force appropriately, but the most commonlyimplemented one, which has been shown to be as good as more complicated schemes if the integration time-step is chosen small enough [58], is a modiﬁed velocity-Verlet scheme introduced by [45]. For more details of the scheme, the reader is referred to their original paper. We note here that temperature enters into the DPD formulation in two ways. First, the average kinetic energy of the beads deﬁnes the kinetic temperature; and, second, the ratio of the random force parameter to the dissipative force parameter deﬁnes the thermostat temperature. These should be the same for any simulation, but if the integration scheme is not chosen carefully, or the integration step size is chosen too large, the kinetic and thermostat temperatures may diverge. This leads to the somewhat surprising result that the kinetic temperatures of diﬀerent bead types may be diﬀerent in the same simulation. These artifacts have recently been shown [60], [68] to be directly attributable to the use of too large an integration time step. For concreteness, we have found that an integration step of 0.02 (in DPD units of ma20 kB T ) is usually appropriate for lipid membrane simulations, although occasionally

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an even smaller value may be necessary, especially if stiﬀ Hookean springs are used within molecules. In this chapter we are mainly interested in molecules that have some biological signiﬁcance, such as phospholipid molecules or diblock copolymers. As was described in the introduction, these are amphiphiles that contain a hydrophilic part chemically bonded to a hydrophobic part. Phospholipids come in many shapes and sizes, and may possess one, two or more hydrocarbon tails, and have a charged, polar or neutral headgroup with a diﬀerent degree of bulkiness. A simple molecular architecture that reﬂects these properties consists of three hydrophilic beads (designated H) to which are attached two linear hydrophobic tails each containing four chain beads (C). Such an amphiphile is represented, using an obvious symbolism, as H3 (C4 )2 . The amphiphiles are contained within bulk solvent (W). Each solvent bead represents a volume of bulk water consisting of several molecules. Because a solvent bead represents several molecules of solvent, and has a length scale of the order of 1 nm, there is no explicit modelling of hydrogen bonds. We now summarise the functional form of the various non-bonded and bonded forces between all bead types. The conservative force between two beads i, j separated by a distance rij is (1) FijC = aij (1 − rij /a0 ) rˆij for rij < a0 , and zero otherwise. The range of the force is set by a0 , and aij is the maximum force between beads of types i, j; rij is the distance between the centres of beads i, j, and rˆij is the unit vector pointing from bead j to bead i. Note that the conservative force is always ﬁnite, taking its maximum value, aij , at zero separation. The dissipative force between two beads is linear in their relative momenta and takes the form 2

FijD = −γij (1 − rij /a0 ) (ˆ rij .vij ) rˆij

(2)

where γij is the strength of the dissipation between beads i, j, and vij = vi −vj is their relative velocity (which is the same as their momentum as m = 1 in our simulations). Finally, the random force between a bead pair is FijR = 2 γij kB T (1 − rij /a0 ) ζij rˆij (3) where values of the random force are generated by sampling a uniformlydistributed random variable, ζij (t), that satisﬁes ζij (t) = 0 and

ζij (t) ζi j (t ) = (δii δjj + δij δji ) δ (t − t ). The random force has the symmetry property ζij (t) = ζji (t) that ensures local momentum conservation, and hence the correct hydrodynamic behaviour of the simulated ﬂuid on long length scales. Note that we have replaced the (nominally-independent) random force parameter σij by its value as determined from the ﬂuctuation-dissipation relation σ 2 = 2 γij kB T as shown by [45].

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Molecules are constructed by tying beads together using Hookean springs with the potential U2 (i, i + 1) = 1/2k2 (rii+1 − l0 )

2

(4)

where i, i + 1 label adjacent beads in the molecule. The spring constant, k2 , and unstretched length, l0 , are chosen so as to ﬁx the average bond length to a desired value. Both parameters may be speciﬁed independently for each bead type pair allowing a linear chain’s bond strength to vary along its length. Chain stiﬀness is modelled by a three-body potential acting between adjacent bead triples in a chain, U3 (i − 1, i, i + 1) = k3 (1 − cos (φ − φ0 ))

(5)

where the angle φ is deﬁned by the scalar product of the two bonds connecting the pairs of adjacent beads i−1, i and i, i+1. In general, the bending constant, k3 , and preferred angle, φ0 , may be speciﬁed independently for diﬀerent bead type triples allowing the chain stiﬀness to vary along a molecule’s length. A preferred angle of zero means that the potential minimum occurs for parallel bonds in a chain.

3 Simulations of Soft Biomaterials 3.1 Self-Assembly of Amphiphilic Molecules Early attempts at modelling aggregates of amphiphilic molecules focussed on their self-assembly and equilibrium properties. Surfactant micelle selfassembly using coarse-grained MD simulations was ﬁrst presented in 1990 [69], and later models explored the dependence of micelle properties on surfactant architecture [70]. These models were limited to small systems, containing of the order of tens of amphiphiles, because of the limitations of the available computer hardware. The small system sizes meant that the simulations were unable to explore the thermodynamic equilibrium state in which aggregates of diﬀerent sizes are in equilibrium with each other. This distribution is expected to be exponential for linear aggregates, such as cylindrical micelles, and Gaussian for spherical micelles [11]. The subsequent increase in computer processing speeds over the next few years allowed the self-assembly of micellar systems [71, 72], planar lipid bilayers [73], and even small vesicles [74] to be followed using atomistic MD. During this time, coarse-grained MD simulations were developed to study vesicle systems [75] for longer times than was possible using all-atom MD. But even coarse-grained MD simulations struggle to capture the dynamics of an ensemble of such aggregates. It is only recently that such systems have been explored using dynamical self-consistent ﬁeld theory (SCFT) [76], and the results show a rich variety of aggregate shapes and sizes. The process of

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vesicle formation, appearance of metastable, multilamellar nano-containers, and even vesicle fusion were observed in these simulations. Recent advances in experimental techniques have revealed that many more systems than phospholipids can self-assemble into vesicles [20]. Some of these systems are quite exotic, including vesicles composed of diﬀerent polymers in the inner and outer leaﬂets that can turn inside out in response to changes in the external solvent or speciﬁc binding to the outer leaﬂet. Such systems have potential as drug delivery vehicles as their response to their environment can be used to induce them to release their contents under controlled conditions. Simulations have a long way to go before such systems can be said be properly modelled, but progress has been made in estimating the material properties and stability of planar bilayers and vesicles made of lipids and diblock copolymers. We summarise these results in the next section. 3.2 Bilayers, Vesicles and Polymersomes The equilibrium properties of planar lipid bilayers, including the lateral stress proﬁle and surface tension were ﬁrst extracted from coarse-grained MD simulations in 1998 [77], a work that also introduced additional degrees of freedom into coarse-grained lipid models such as the bending stiﬀness of the hydrocarbon chains. A variety of coarse-grained MD models have since been used to measure the equilibrium properties of planar lipid membranes containing a few hundred lipids [32,33,78]. These models have been extended to measuring the in-plane diﬀusion of the lipids [79], shape ﬂuctuations of two-component membranes [80], the formation of pores in a membrane in response to electrical and mechanical stress [81], the formation of non-lamellar phases, such as hexagonal phases [82], and even the fusion of small vesicles [22, 83]. It has recently been realised that self-assembly into membranes and vesicles is a generic behaviour of many diﬀerent types of molecule [20], and some of the resulting agregates have exciting properties for medicinal and pharmaceutical applications. Several groups have used DPD to simulate the formation [84] and equilibrium properties of amphiphilic bilayer membranes, including the lateral stress proﬁle [85]. The advantage of DPD over MD for these systems is that membranes containing tens of thousands of lipid molecules can be simulated within a few days of CPU time. This allows good statistics to be collected on membrane observables. These simulations have been extended to more complex systems, such as the appearance of diﬀerent phases of a membrane as the temperature and amphiphile interactions are varied [86], the eﬀects of a cosurfactant, such as alcohol, on the membrane’s structure [87], and the dependence of a membrane’s material properties on the symmetry and length of its tails for two-tailed amphiphiles [88]. Electrostatic forces between a polyelectrolyte and charged surfactants in bulk solution have been included in a recent extension of DPD to charged systems [89], but to our knowledge this extension has not been used in other studies. Planar amphiphilic membranes

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in aqueous solvent tend to close into vesicles, and DPD has been used to follow vesicle formation [90], domain growth in vesicles [91] and budding and ﬁssion of two-component vesicles [92, 93]. The range of systems now being studied by DPD simulations is expanding fast. It includes vesicles formed of diblock copolymers, called polymersomes [6], the aggregation of copolymer analogues of the exon1 fragment of Huntington’s disease proteins driven by the relative hydrophobicity of diﬀerent regions of the fragment [94], the tension-induced fusion of a vesicle to a planar membrane [3], the behaviour of a worm-like chain model of DNA polymers [95], and the inﬂuence of model proteins embedded in a ﬂuid membrane [96]. Diblock copolymers form an exciting class [18] of bilayer and vesicle forming molecules. Each molecule consists of a hydrophilic block, containing watersoluble monomers, chemically bonded to a hydrophobic block, containing water-insoluble monomers. The properties of the polymers, and the aggregates they form, depend on the relative lengths of the two blocks, the molecular weight of the polymer, and the presence of functional groups attached to side chains. A typical example of this class of polymer is poly(ethylene-oxide)polyethylethylene (PEO-PEE). The size of each block can be varied from a few monomers up to hundreds of monomers. Polymersomes have physical properties, such as the membrane thickness and elastic moduli, that span a wider range of values than lipid vesicles [7,97]. The interior structure of the membrane of a polymersome diﬀers substantially from that of a phospholipid membrane. The hydrophobic block is sequestered between the well-hydrated hydrophilic blocks, and the aqueous solvent penetrates to the edge of the hydrophobic region. The entanglement of the individual molecules leads to much slower inplane diﬀusion, and a greater resistance to rupture under lateral stress. Recent experiments have explored the interactions of short, amphipathic peptides, such as alamethicin, with (uncharged) diblock copolymer membranes [98] and found that even though the peptides are less than one half of the diblock membrane width they permeabilize it quite eﬀectively. Other experiments using 50 micron diameter polymer vesicles have shown that they undergo fusion when subject to ultrasound [99], which makes them attractive as drug delivery vehicles. Molecularly-detailed simulations of the fusion of two polymersomes would be valuable for exploring the molecular architectural space of polymersomes, and for optimizing their physico-chemical properties to make them more suitable for clinical applications. Unfortunately, the high molecular weight of some diblock copolymers, and the large diameter of polymersomes, currently restricts atomistic MD simulations to a few tens of molecules, and even coarse-grained MD are limited to patches of a hundred or so molecules [100]. These limitations, and the soft nature of polymer vesicles, makes them a prime target for DPD modelling. The variation in polymersome physical properties with the molecular weight of the constituent molecules forms an important link between experimental results, analytical theories and simulations. Ortiz et al. [6] have performed DPD

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simulations of membrane patches of diblock copolymers of various molecular weights, calibrating the DPD parameters using data obtained from atomistic MD simulations of the same system. Typically, every bead of each molecular species in a multi-component DPD simulation is considered to contain the same amount of matter, and the selfinteraction parameters are chosen so that the compressibility of each pure species matches that of water at room temperature. The cross-terms are then matched to the relative solubility of each species in the others. For species that are mutually soluble, such as the PEO block and water relevant for polymersome simulations, a diﬀerent property has to be chosen. Ortiz et al. [6] choose the radial distribution function of PEO in water for this purpose. Using the conventional DPD mapping for simulations of a planar membrane patch of diblocks, they ﬁnd that the geometry of the hydrophobic block is incorrect and the hydrophobic density is too high. This leads to unphysical values for the membrane area stretch modulus. A revised mapping was developed in which the beads of each species are considered to contain an amount of matter that depends on the species, and is chosen so as to reproduce the experimental bulk density. For the PEO-PEE diblock considered this leads to the densitybased mapping of 1.392 PEO monomers/bead, 0.774 PEE monomer/bead and 3.01 water molecules/bead. This is in contrast to the conventional mapping in which the ﬁrst two ratios are unity. Using the new mapping, the membrane area stretch modulus is found to be 137 mN/m, which is in good agreement with the experimental value of 120+/-20 mN/m [97]. Additionally, the scaling of the membrane hydrophobic block thickness with polymer molecular weight was found to obey the experimentally-observed scaling law d ∼M1/2 . Neither of these results were obtained using the conventional DPD mapping. The new DPD mapping was used by Ortiz et al. [6] to simulate the rupture of a 28 nm diameter polymersome containing 1569 diblock copolymers after being inﬂated with excess internal solvent. The ﬁrst step in the rupture pathway appears to be micellization of the inner leaﬂet which subsequently weakens the outer leaﬂet allowing the solvent to escape to the external volume via multiple pores. 3.3 Vesicle Fusion Vesicle fusion is essential for cell viability, and takes place in processes as diverse as signal transmission at neuronal synapses, fertilization of an egg by a sperm, and viral entry into cells. Fusion appears to conﬂict with the primary requirement of biological membranes, which is to provide a barrier between intracellular compartments, and between a cell and its external environment. Understanding how the stability of lipid membranes is overcome by the cellular protein machinery when required, is a major topic of research, and several reviews have appeared in the last few years [101–105] including two recent ones [106, 107]. Although fusion of giant (1 - 20 micron diameter) vesicles can be observed using fast optical video microscopy [126], ﬂuorescence microscopy

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[108] and optical dark-ﬁeld microscopy [109], and SNARE-mediated fusion of liposomes to a supported planar membrane has been followed using total internal reﬂection ﬂuorescence microscopy [110], the molecular rearrangements that take place during the ﬁnal stage of the fusion process, when the two initially-distinct membranes join and produce a fusion pore, cannot yet be resolved by these experimental techniques. Current in vitro vesicle fusion assays indicate a time scale of tens of milliseconds for fusion [111, 112], and in vivo experiments suggest it may require hundreds of microseconds [113]. The relevant length scales range from less than a nanometre for the initial fusion pore width up to tens of microns for the vesicle diameter. This renders the use of atomistic MD simulations computationally prohibitive, and even coarse-grained MD simulations are restricted to small systems. Coarse-grained MD simulations of the fusion of two 15 nm vesicles, containing about 1000 lipids each, found that constraining the vesicles to be close together for tens of nanoseconds was suﬃcient to cause them to fuse [83]. Successful fusion depended on the lipid species in the vesicles: mixtures of dipalmitoylphosphatidylcholine (DPPC) and palmitoyloleoyl phosphatidylethanolamine (POPE) fused most easily at separations up to 1.5 nm; vesicles composed of pure DPPC only fused when held together closer than 1 nm for more than 50 ns; and vesicles containing DPPC and 25% lysoPC were not seen to fuse at all within 200 ns. The initial contact between the vesicles was provided by a few lipids whose protrusion ﬂuctuations caused them to merge into the apposed monolayer. The small size of the vesicles, and consequent high curvature, provides the driving force for the observed fusion. The fusion pathway observed in these simulations proceeds as follows. First, a contact zone, or stalk is formed by the cis monolayers. Next, the stalk converts into a transmembrane contact in which the trans monolayers touch. Finally, the contact zone ruptures and the inner compartments of the vesicles become contiguous. An unexpected ﬁnding in some of the fusion events was the mixing of lipids from both the outer and inner monolayers. This appears to result from a reduced line tension for pore formation in the vicinity of the stalk, and an asymmetrical expansion of the contact zone into a “bananashaped” region. Stevens et al. [22] simulated the fusion of two vesicles containing about 1000 lipids each for hundreds of microseconds. In this protocol, a transient force was applied to all membrane molecules in the vesicles to push them together. The force was removed after a few lipids had exchanged between the vesicles. Fusion appeared to start at the edge of the ﬂattened contact zone between the two vesicles, where the curvature of the surface is greatest. Interestingly, this results in the fusion pore forming at points quite distant from the point of closest approach of the vesicles. The stalk in the fusion events of both [22] and [83] appears to expand asymmetrically around the strained edge of the contact zone leading to a partially-conﬁned solvent cavity between the two liposomes.

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The vesicles in the work of [22] and [83] had diameters around 15 nm. The high curvature of such vesicles may inﬂuence their fusion pathway, and larger vesicles would be a better model for experimental systems such as synaptic vesicle fusion which involves vesicles of 40 nm diameter. The fusion of two closely-apposed, tense planar membrane patches composed of amphiphilic copolymers has been studied using canonical ensemble, lattice Monte Carlo simulations in three dimensions [114]. The size ratio of the hydrophilic to hydrophobic sections (11 segments and 21 segments respectively) is chosen to be close to that appropriate to biological lipids. The solvent is a homopolymer. The molecules in these simulations do not obey Newtonian dynamics, but evolve according to a Markov process (using the Metropolis algorithm) that ensures the correct statistical weight for states of the system in equilibrium. Ensemble averages then provide the connection with physical properties. Because the total densities of each segment type are conserved, the motion of the polymers is diﬀusive. Contacts between the molecules in the membranes arise naturally in this model as a result of thermal shape ﬂuctuations. Most of these contacts rapidly disappear, but some lead to formation of a “stalk” or merging of regions of the closest (cis) monolayers. Once a stalk has formed, the probability of a hole appearing in one or other bilayer near the stalk increases markedly. The presence of the nearby hole then appears to cause the stalk to traverse around it and form a ring-like connection between the membranes. The authors explain the increased probability of hole formation close to a stalk as the result of a lowering of the line tension around such a hole caused by the reduction in the curvature of the piece of membrane between the hole and the stalk. The ﬁnal stage in the observed fusion process is the appearance of a second hole in the other membrane and the movement of the stalk to surround both holes. This results in the full fusion pore connecting the distal sides of the membranes. Similar results were observed in the SCFT simulations of [76] on large systems of symmetric, amphiphilic diblock copolymers, including the observation of pore formation close to the stalk connecting two fusing membranes. Recent work on a similar model using SCFT methods emphasizes the key role of the line tension in this interpretation of membrane fusion [115, 116]. The key steps leading to membrane fusion are: close proximity of the two membranes; initial contact and inter-penetration of the outer leaﬂets; opening of a pore connecting both membranes; and release of the vesicle contents. Here, we focus our attention on the steps subsequent to the initial membrane contact. The simplest driving force for membrane fusion is tension. Increasing the tension in a membrane eventually results in its rupture. If an alternative pathway is possible, such as merging with a closely-apposed, less tense membrane, the rupture end-point can be avoided. Fusion of a vesicle to a planar membrane therefore provides a model of synaptic vesicle fusion in which a small, 40 nm diameter vesicle fuses to the much larger pre-synaptic membrane of the nerve cell.

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Fig. 3. Sequence of snapshots of the tension-induced fusion of a 28 nm diameter vesicle to a 50 x 50 nm2 planar membrane patch. The whole sequence shown here represents 240 nanoseconds of real time, and the time between snapshots is 32 ns (snapshots proceed across each row in turn). The vesicle contains 5684 and the planar membrane 5455 lipid molecules respectively, and the planar membrane is under a higher tension than the vesicle. Solvent particles outside the vesicle are invisible for clarity, while those initially inside the vesicle are shown. Note that very few solvent particles diﬀuse out of the vesicle before the fusion pore appears compared to the more than 50,000 inside it. The initial contact of the vesicle to the planar membrane occurs by its shape ﬂuctuations and lipid protrusions. Once the vesicle outer leaﬂet has contacted that of the planar membrane, its lipids are drawn across by the higher planar membrane tension. The perturbation of the contact zone caused by the merging of the two membranes transiently disrupts its stability and a pore appears that allows the (tense) vesicle to discharge its contents. Reprinted from [3] with permission.

In a drive to get closer to the experimental length scale for synaptic vesicle fusion using particle-based simulations, we have recently studied the tensioninduced fusion of a 28 nm diameter vesicle to a 50 × 50 nm2 planar membrane patch using DPD simulations [3]. When the vesicle is relaxed it contains approximately 6,500 amphiphiles, and the relaxed planar membrane contains around 8,200 amphiphiles. Such system sizes are an order of magnitude larger than any published atomistic, or coarse-grained, Molecular Dynamics fusion study. The membranes used in this study are signiﬁcantly more stretchable than typical lipid membranes, and are more similar in this respect to those formed of diblock copolymers. This may inﬂuence the time-scale over which the fusion pore opens up, and partially explain the diﬀerence between the hundreds of nanoseconds required for fusion in the simulations and the hundreds of microseconds estimated from experiments [113]. However, the model system captures the features that we believe are important for understanding the molecular rearrangements that occur during the fusion of tense membranes, and the predicted fusion times are consistent with extrapolation of the experimental data of [126]. The global tensions in the vesicle and planar membrane, which are are created by appropriately choosing the number of molecules

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in each membrane, are used as control parameters. Repeating the protocol for each tension pair, using thermodynamically equivalent, but molecularly distinct, initial states, allows the probability of the various outcomes to be obtained. A typical fusion event is presented as a series of snapshots in Fig. 3. After the initial contact there is a rapid transfer of molecules from the vesicle to the cis leaﬂet of the (more tense) planar membrane which destabilizes the merged contact zone, and leads to the fusion pore. Once the pore has been created, it grows rapidly due to the large tension present in the planar membrane. These results predict that fusion of tense membranes only typically occurs when they are already so tense as to be close to spontaneously rupturing. In order to relax their tensions, the vesicle and the planar membrane can explore several pathways: they can rupture independently, merge into a stable hemifused state, or fuse and transfer molecules between them to relax the initial tensions. A second observation from this work is that all successful fusion events occurred between 150 and 350 ns after initial contact of the two membranes even though the simulations were run out to 2 microseconds. The fusion time is deﬁned here as the simulation time between the ﬁrst contact of the two membranes, and the time when the pore has expanded approximately to the diameter of the vesicle. The upper cut-oﬀ of the fusion time distribution arises from the stabilization of the hemifused state in the membrane geometry used here. Because the hemifused state is metastable for relatively large initial tensions, fusion can occur only at even higher tensions for which the fusion pathway exhibits no activation barrier. The stabilization of the hemifused state depends on the membrane areas that are initially stretched: if the vesicle and planar membrane areas are comparable, the planar membrane can relax its (higher) tension by incorporating vesicle amphiphiles before a fusion pore can appear. If the planar membrane area is much larger than the vesicle area, the hemifused states are only stabilized for smaller initial tensions, and the region of successful fusion is shifted towards smaller tensions. These results show that modulating the global tensions in a vesicle-planar membrane system is an unreliable means of inducing their fusion because the required tensions, which need to be large so as to raise the probability of fusion, also allow the system to explore alternative mechanisms for releasing this tension, such as premature rupture of the vesicle or planar membrane. A more realistic fusion protocol is to embed model proteins in relaxed membranes, and to explore their possible actions in driving the membranes to fuse. Several groups are exploring how to incorporate model peptides or proteins into simulated membranes using atomistic MD [28, 29], coarse-grained MD [100] and DPD [96]. The challenge for these groups is to integrate the results of diﬀerent simulation techniques into an understanding of the complete fusion process from the molecular rearrangements occurring on the microscopic scale to the time course of the fusion pore conductivity on physiological length and time scales.

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4 Perspectives In this section, we present our expectations for the development of particlebased simulations of soft biomaterials, and list some key areas in which we believe signiﬁcant advances can be made. Given that the largest particle-based simulations of vesicle fusion published so far involve aggregates with sizes some tens of nanometers in diameter [3, 22, 93], the ﬁrst priority is to extend the simulations to larger length and time scales. Dissipative Particle Dynamics and coarse-grained Molecular Dynamics hold out the most promise in this regard. Using a parallel simulation code running on a cluster of a 1000 processors, it ought to be possible to simulate a micron-sized cube of matter for milliseconds using DPD. In order to calibrate the parameters involved in the simulation, atomistic MD simulations can be performed, albeit on smaller systems, and key physical properties of the molecular species involved compared between the two techniques, as described in [6]. If preliminary DPD simulations reveal interesting phenomena in a region of the model’s parameter space, it is important that the same processes be simulated using more quantitative methods, such as coarse-grained Molecular Dynamics simulations. This ensures that the results are robust against details of the simulation method. Other methods of simulating across many scales, so-called multi-scale modelling, are being developed by several groups [117–120], and may lead to models that span length and time scales from the microscopic, via particle-based simulations, to the macroscopic, using simulation results as input to chemical reaction-diﬀusion equations. Once micron-sized simulations are common, new areas of nanostructured material simulations become possible. Microﬂuidics, or lab-on-a-chip, involves fabricating tiny channels, reservoirs, pumps and syringes on microchips in order to manipulate volumes of ﬂuid less than a microlitre [121]. Because their surface area to volume ratio increases as their linear dimension decreases, microreactors achieve larger conversion rates in chemical reactions. They also allow manipulation of cell-sized objects with a precision unobtainable in macroscale reactors. It is possible to subject living cells, even Drosophila embryos, to a temperature gradient such that one end of the embryo is at a diﬀerent temperature than the other [121]. The simulation of ﬂuid ﬂow within such channels is feasible using DPD, and other techniques not discussed here such as Lattice Boltzmann [122]. The non-slip boundary conditions that are necessary for simulating ﬂow problems within solid tubes have recently been implemented within DPD by Pivkin and Karniadikis [123]. Another promising area in which DPD can contribute is the modelling and design of liposomal [8] and polymersomal [124] drug delivery vehicles. These have sizes ranging from 100 nm diameter vesicles to micron-sized polymersomes, and even millimeter-sized artiﬁcial cells. Simulations of vesicles with sizes near the lower end of this range are already possible with DPD, and we expect such systems to be routinely modelled within the next few years. This will allow the stability and properties of the drug carriers to be explored

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as their constituent molecular architecture is varied, and the external conditions changed. It is straight-forward to incorporate multiple diﬀerent types of molecule into a DPD vesicle, and this makes it possible to add functionality to the vesicles. This includes shielding them from molecular binding by attaching long PEO chains to a fraction of the molecules in the vesicle’s outer surface; inducing degradation of the vesicle’s surface, or lysis, by making some of the molecules break up in response to a change in the external solvent conditions; and incorporating transmembrane receptors into the vesicle that can be made to coalesce into receptor-rich domains with improved ligand-binding properties. Extrapolating over the next half-decade, we expect the natural progression of coarse-grained simulations to make possible the modelling of the physicochemical properties of self-assembled nanoparticles for drug delivery, dubbed artiﬁcial viruses by Mastrobattista et al. [2]. The optimization of the structural features of nanoparticles that are critical for the safe and eﬃcient delivery of target material (drug molecules or genes for example) to cells is a key application area for simulations. Further into the future, the simulation of a bacterial cell membrane that incorporates the lipids, transmembrane channels, receptors and their ligands would provide a testing ground for hypotheses about signal transduction through membranes and the response of actively-controlled membrane processes to external perturbations. Work on such a model has already begun [125], and we predict that experiments guided by modelling will be the route of choice for many biotechnological materials design problems in the future. Acknowledgement. It is a great pleasure for me to thank Professor Reinhard Lipowsky for his insight and encouragement during the years in which I have worked at the Max Planck Institute of Colloids and Interfaces, and to acknowledge the funding received from the Max Planck Gesellschaft.

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Theoretical Study of Nanostructured Biopolymers Using Molecular Dynamics Simulations A Practical Introduction Danilo Roccatano School of Engineering and Science, Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany

Lex. I. Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nili quatenus a viribus impressis cogitur statum illum mutare. Lex. II. Muationem motus proportionalem esse vi motrici impressae, et ﬁeri fecundum lineam rectam qua vis illa imprimitur. Lex. III. Actioni contrariam semper et equalem esse reactionem: sive corporum duorum actiones in se mutuo semper esse aequales et in partes contrarias dirigi. Isaac Newton. Philosophiae Naturalis Principia Mathematica. London, 1686

1 Introduction In 1686, Isaac Newton published the ﬁrst edition of his Philosophiae Naturalis Principia Mathematica, one of the greatest scientiﬁc masterpieces of all time. Page 12 of this magnum opus reported the famous three laws that carried his name and from which classical physical mathematics began. 350 years after his publication, the same laws, formulated to explain the motion of stars and planets, turned out to be useful in creating mathematical models that describe the dynamics of the atomic world. In this chapter, I will show how this task is accomplished by introducing molecular dynamics (MD) techniques. This computational chemistry method, based on the classical mechanics description of atomic motion, is now a powerful tool for the study of complex molecular systems. Its success is strongly connected with the continuous and exponential increase in computer power that allows molecular systems of expanding complexity to be simulated on longer time scales. The study of the properties of nanostructured biopolymers (namely natural and synthetic polymers constituted of amino acids, carbohydrates and

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nucleic acids) can beneﬁt from theoretical investigations of their mechanical properties at the atomic scale. In fact, this information can be used to understand the properties of these substances and to provide the foundation for engineering new materials with enhanced properties. For this purpose, MD simulation techniques oﬀer a very powerful method to analyze these materials at diﬀerent time and space scales. Several comprehensive reviews [1–4] and textbooks [5–8] have been published on this topic. Therefore, the aim of this chapter is to provide a short and concise guide to molecular dynamics simulation. It is intended for those just entering the ﬁeld to acquire this technique. The chapter is organized as follows. In the ﬁrst part, a basic introduction to the force ﬁelds is reported; together with an overview of currently available libraries of parameters. This is followed by a short description of the basic algorithms and techniques used to model molecular systems. A list of the most commonly used packages for MD simulation is provided with information concerning the techniques that are included. In the second part of the chapter, some examples of application to molecular systems will be given to familiarize the readers with the results that can be obtained from the simulation of complex molecular systems. Finally, in the Perspective section, an outline of the future development of this technique is reported. 1.1 Modeling the Atomic World In the ﬁrst decade of the last century, the dawn of the quantum mechanics marked the beginning of the mathematical modeling of the atomic world. The equation of Schr¨ odinger, like Newton’s equations, allowed the formulation in a mathematically elegant and concise way, building on shining intuitions and experimental results accumulated in previous decades. Immediately, it was evident that although this equation could be used to describe the physicochemical properties of any molecular system, it turned out to be impossible to resolve it analytically, when more than two particles are involved. The advent of electronic computers has, however, allowed us to overcome this problem by solving the many-electron problem numerically. Despite the continuous and rapid development of computer performance, the use of a quantum-mechanical approach to describe the physico-chemical properties of large macromolecules is still far from being a routine tool of investigation. The challenge lies in the number of calculations required by this approach which is proportional to third or higher power of the total number of electrons present in the system. This makes it computationally intractable when the number of atoms in the system is more than one hundred. Fortunately, it is possible to introduce ad hoc approximations to model the atomic interactions by reducing them to a classical description where the electrons in atoms are not explicitly considered but their mean-ﬁeld eﬀect is taken into account. In fact, the dynamical behavior of a molecular system can be decomposed into the motion of the nuclei and that of the electrons. Using the Born–Oppenheimer approximation, the two

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motions can be treated separately. In this way, a molecular system can be described as a system of simple particles interacting in a mean-ﬁeld produced by electrons. This mean-ﬁeld can be approximated as an analytic function. Furthermore, when reactions, excited states, and low temperature eﬀects are not considered, it is reasonable to assume that the motion of atoms of the system can be described by the laws of the classical mechanics. With these approximations, the computational demands of the model are strongly reduced and, therefore, it is possible to simulate molecular systems of large sizes (thousand of atoms) on time scales of nanoseconds or more. The ﬁrst simulation of atomic ﬂuid using this approximation was performed about 40 years ago by Alder and Wrainwright [9]. They developed and used the method in order to study simple ﬂuid models where atoms were represented as rigid spheres. These ﬁrst pioneering studies marked the birth of the classical molecular dynamics simulation techniques. A very important landmark in biomolecular application of MD was the simulation of the ﬁrst protein (the bovine pancreatic trypsin inhibitor) by McCammon and Karplus in 1977 [10]. This initial attempt was followed by several important contributions that deﬁnitively collocated the MD technique among the theoretical tools for understanding the dynamics of biomolecules. The successes obtained from reproducing structural properties of proteins have been enormous in the following years, and we have witnessed the growth of MD as a tool within the studies of structural biology and material science. The continuous increase of computer power and programming languages has concurred with the further reﬁnement of the technique and its application to larger molecular systems. MD represents a powerful tool to investigate the structural and dynamical behavior of complex molecular systems in conditions not accessible by experimental techniques. Nowadays, MD simulation techniques are used in diﬀerent ﬁelds of chemistry, physics and biology for studying the dynamical and structural property of proteins, or in general nanostructured biomaterial. In the following sections, I will introduce the techniques in a practically oriented manner by brieﬂy describing the “principal ingredients” that are necessary to set up and analyze simulations of biomolecular systems.

2 Force Fields As mentioned in the previous section, the atomic interactions in the classical MD are treated using an analytic function. This function, usually referred to as the force ﬁeld function or eﬀective potential, depends on the atomic coordinates of the system and on a certain number of parameters (force constants, partial charges, reference distances and angles) obtained from experimental data. A typical example of a force ﬁeld, used in diﬀerent programs for MD simulations, is the following:

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V' (r1 , r2 , r3 , . . . , rN ) = ' ⎫ 1 1 2 2 ⎪ 2 Kb (b − b0 ) + 2 Kθ (θ − θ0 ) ⎬ bonds ' angles ' Bonded 1 2 + Kφ [1 + cos(nφ − δ)] ⎪ interactions 2 Kξ (ξ − ζ0 ) + ⎭ proper improper dihedral (1) dihedral

⎫ 12 6 '' ⎪ σij σij ⎪ 4εij − rij ⎬ rij j>i + i' Non-bonded interactions ' qi qj ⎪ ⎪ + ⎭ 4πε0 εr rij i j>i

The function is divided into two parts: the bonded and non-bonded interaction functions. The bonded interactions are usually described using functions depending on the coordinates of 2 to 4 atoms. These interaction functions are used to describe harmonic oscillations with force constants obtained from experimental crystallographic and spectroscopic data [7,11]. The ﬁrst term in the (1) represents the energy of bond vibrations. It is usually modeled as a harmonic function where b0 , represents the equilibrium bond length, and kb the force constant with values depending on the bond type. As we will see later, the integration of the Newton equation is performed using time steps that depend on the faster vibration mode in the system. Since bond vibration (in particular bonds containing hydrogen atoms) [7] does not aﬀect the large scale motions this term, for large biomolecules, is not calculated and the bond lengths are kept constant, and equal to the crystallographic values, using holonomic constraints. These time-independent constraints are applied at each simulation step using numerical algorithms such as SHAKE [7] or LINCS [12] for large molecules and SETTLE [13] for smaller compounds. The second term in (1) describes the interaction energy for the bond angle vibration. It is a three-body interaction function since the angle (θ) depends on the positions of three atoms. In this case, the function describes a harmonic motion where θ0 represents the reference value of the bond angle and kθ the vibration force constant. The next two terms in the equation are used to describe the four-atom interaction potential. The ﬁrst function describes the so-called improper dihedral angle vibration. This term is used in order to avoid chirality inversion of enantiomeric carbon atoms and deformation of planar geometries of, for example, aromatic rings. The chirality inversion of enantiomeric carbon atoms is a problem when united carbon atom types are used in the model. These atom types are used to describe CH3 , CH2 and CH groups as a single particle with optimized Lennard–Jones and charge parameters. In this way, enantiomeric carbons could have a chirality inversion during the simulation by umbrella ﬂipping of the three bonded non-hydrogen atoms. The improper dihedral function avoids this problem by introducing an extra potential that keeps the right stereochemistry of the chiral atom in place. The second four-atom interaction potential term describes the energy involved in the rotation around bonds. It is usually modeled as a sinusoidal function where kφ is the force constant, n the number of multiplicity of the function and δ its phase shift. The last two terms in the (1) describe non-

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bonded atomic interactions. These potential terms are usually modeled using pair interaction functions describing van der Waals and electrostatic interactions. For the van der Waals interactions, the Lennard–Jones (LJ) function is generally used where σij and εij values represent the distance at which the function √ is zero and the value of the function at its minimum (located at rm = σij / 6 2), respectively. The twelve-power term in the function describes the repulsive interactions (London forces). The nature of these forces is better described by an exponential term as in the case of the Buckingham or the Hill potential [5]. The second term in the LJ function describes the attractive interactions, a result of the mutual atomic induced dipoles. The LJ constants are obtained from experimental data (crystallographic data, compressibility data) or quantum mechanical calculations of simple compounds. The parameters describing the interaction between diﬀerent atomic species are derived using the Lorentz–Berthelot mixing rules based on geometric or arithmetic average of the homo-atomic parameters [7] : σij =

σii + σjj √ ; εij = εii εjj 2

(2)

The electrostatic interactions are modeled using a simple coulomb term. Fixed partial atomic charges, calculated using a quantum mechanical method, are assigned to each atom of the system. The relative dielectric constant, εr , is generally considered equal to one. In some cases, this approximation provides a poor description of the phenomena where the polarizability of the atoms plays an important role. It is possible to account for this polarizability by optimizing the value of the partial charge to reproduce experimental data (e.g. in water models [14]). However, the best solution to this problem is the explicit inclusion of the atomic polarizability that allows a correct calculation of the electric ﬁeld eﬀect at every point of the simulated system. Unfortunately, the use of a polarizable force ﬁeld is still inconvenient for large molecular systems because it is computationally expensive and it also requires new parameterization of current force ﬁelds [7, 15, 16]. Despite these diﬃculties, there are eﬀorts in developing new force ﬁelds and MD programs that implement this feature [14,15]. The non-bonded interactions between atoms separated by two bonds are not calculated. For atoms separated by three bonds, the calculation is performed using reduced parameters in order to avoid the strong repulsions caused by the short distance. During a MD simulation, the largest part of the calculation time is spent on evaluating the non-bonded interactions and in particular the long-range electrostatic interactions. In fact, the number of these interactions is proportional to the square of the number of atoms whereas for bonded interactions, it is a linear function of the number of atoms. Hence, the performance of MD simulations is strictly connected to the use of eﬃcient algorithms to speed up the evaluation of these interactions. Several methods have been proposed to improve the eﬃciency of the non-bonded force calculation [6]. Among those, the most commonly used are based on the distance truncation of interactions

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using cutoﬀ criteria, and on the Ewald summation based algorithms. These two approaches will be shortly reviewed in the next section. 2.1 Force Field Parameterization The parameters present in the force ﬁeld potential are evaluated from experimental data or from quantum mechanical (QM) calculation on small model molecules. Force constants for the harmonic potential of bonds and bond angles are modeled to reproduce the properties of reference compounds obtained from crystallographic structures, IR/Raman spectroscopy and from QM calculations in vacuum [17]. In the case of partial charges and LJ parameters, the following procedure is normally followed. The starting partial charges used for the optimization procedure of a new set of parameters are obtained from Hartree–Fock calculations (basis set 6-31G* or 6-31+G* for anions) while the starting parameters for the LJ function are the same as those used for the corresponding atom types present in similar molecules. The ﬁrst attempt in order to improve the partial charges and the LJ parameters is made by modeling these functions to reproduce energetic and structural properties of the molecule obtained from QM calculations of the isolated molecule and in small cluster of waters. For neutral molecules, the parameters are further improved by simulations of the liquid phase. The density, the vaporization enthalpy, the thermal compressibility at constant pressure and the diﬀusion constant are calculated and compared with the experimental values. The values of the partial charges and the LJ parameters are modiﬁed until they reproduce both the data in gaseous and liquid phases. In the case of charged molecules (i.e. ions), the same procedure can be performed by calculating the solvation free energies. Diﬀerent types of force ﬁelds have been developed from various research groups all over the world. These force ﬁelds constitute a library of parameters that have been optimized for use in several classes of macromolecules (usually proteins or nucleic acids). The mostly commonly used force ﬁelds are the following: • GROMOS: The force ﬁeld GROningen MOlecular Simulations was developed at the University of Groningen (The Netherlands) and at the ETH in Zurich (Switzerland) by H.J.C. Berendsen and W.F. van Gunsteren. The latest version (GROMOS96) [18, 19] is the main force ﬁeld used in the packages for MD simulations GROMOS96 and Gromacs. It is a united atom force ﬁeld. The parameters have been optimized using the twin-range cutoﬀ method and the reaction ﬁeld approach. The atomic charges on the molecule are distributed to form charge groups. • OPLS: The Optimized Potential for Liquid Simulations was developed by W. L. Jorgensen [20, 21]. It is one of the most complete and detailed force ﬁelds for simulation of organic molecules. Hundreds of diﬀerent atomic types optimized for diﬀerent organic molecule classes are present. The

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original force ﬁeld was developed for Monte Carlo calculation of small molecules. Subsequently, the force ﬁeld was adapted for use in MD calculation adopting the bond parameters from the AMBER force ﬁeld. • CHARMM: The Chemistry at Harvard, Macromolecular mechanics was developed by M. Karplus and B.R. Brooks [22, 23] at Harvard University. In this force ﬁeld, the partial charges are modeled to reproduce the heat of condensation, the dipole moments of the compound and the interaction energies and geometries in gaseous phase of a water molecules cluster. The interaction energies were obtained from QM calculations using the 6-31G* basis set, multiplied by a factor of 1.16, the same factor that exists between the energies of interaction of water molecules in the TIP3P model and those obtained from the QM calculation. • AMBER: The Assisted Model Building using Energy Reﬁnement [24–26] is a force ﬁeld developed by P. Kollman, at the UCSF. The partial charges are calculated by electrostatic ﬁtting to the electron density obtained by Hartree-Fock calculations using the 6-31G* basis set. The Lennard–Jones parameters are taken from the OPLS force ﬁeld.

3 Cutting the Energy Oﬀ The simplest procedure for calculating the non-bonded interactions is the use of the so-called cutoﬀ criterion. It is applied by calculating, for each atom of the system, the interactions with atoms located within a distance Rc (cutoﬀ distance). The assumption is that beyond this distance the atomic contribution to the interaction energy is negligible. The interactions within Rc are collected in a list (the Verlet’s list [7]) which is used to calculate some interactions in the next m steps. To compensate for errors in the estimation of interactions due to the diﬀusion of atoms inside and outside Rc during the m steps, a larger cutoﬀ radius is used to calculate the interactions at the m-th step. The use of two cutoﬀ radii is usually referred as the twin-range cutoﬀ method. By using the cutoﬀ and the Verlet’s list method, the number of non-bonded interactions calculated at each step becomes proportional to the number of atoms within the cutoﬀ radius. The use of the cutoﬀ criterion is a good approximation for the calculation of short-range interactions, but it can give rise to artifacts with electrostatic interactions. In fact, the discontinuity of energy at the cutoﬀ distance introduces an artiﬁcial increase of kinetic energy (i.e. temperature) of the system. This excess is normally eliminated by coupling the system with an external bath. Moreover, in the case of highly charged systems (i.e. ionic liquids, ions, nucleic acids), anomalous accumulations of charges in proximity of the cutoﬀ radius caused by the presence of residual dipolar interactions can occur. These artifacts can be partially reduced by damping to zero the value of the long-range energy function at the cutoﬀ radius. This can be done by multiplying (switching functions) or adding (shifting functions) a term that

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smoothes the interaction energy function uniformly to zero at the cutoﬀ distance [6]. The use of these functions has unfortunately no physical basis and their use can introduce other artifacts to the simulated system. A more convenient way to reduce the range of electrostatic interactions is achieved by grouping partial charges of the molecule to form groups of charge having net zero charge. If the long-range interactions are calculated between charge group pairs, the resulting interaction will be dipolar. This means that the interaction energy will decay with the distance as r−3 but not as r−1 . In this way, the perturbation due to the cutoﬀ truncation will be considerably reduced. A more accurate method for the treatment of the electrostatic interactions is based on the sums of Ewald [27]. This method, in its initial formulation, was used to study periodic systems like ionic crystals, but it appeared to be adequate also in simulation of polar solutions. The advantages of the Ewald sums include a substantial reduction of the calculation demand from the power square of the number of atoms (N ) to ∼ N log N and at the same time maintaining a good accuracy in the calculation of long range interactions [27]. This method is replacing the use of the cutoﬀ in MD simulations and several algorithms have been proposed to further speed up the calculation and implement the method on parallel computers [27].

4 Atomic Starting Positions The initial positions of the atoms in the simulated systems are normally obtained from experimental X-ray crystallographic or Nuclear Magnetic Resonance structural data. When experimental structures are not available, geometrically modeled structures can also be convenient. In the latter case (but also for low resolution experimental structures), careful attention should be paid in performing MD simulations since these structures can give rise to artifacts. It is also very important to always incorporate the crystallographic water molecules. In most cases, these molecules are important for preserving the structural integrity of the model. Some of the repositories of molecular structures are reported in Table I. Table 1. List of the most common repositories of molecular structures. Crystal structure of small organic and inorganic molecules: CSD, Cambridge Structural Database — www.ccdc.cam.ac.uk OCD,Open Crystallographic Database — sdpd.univ-lemans.fr/cod/index.html KEGG ligand database — www.genome.jp/ligand/ Crystal structure of protein and nucleic acids: PDB, Protein Data Bank — www.rcsb.org NDB, Nucleic Acid Bank database — ndbserver.rutgers.edu

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5 Chiare, Fresche et Dolci Acque . . . [28] The solvent plays an important role in the structural and dynamical properties of biopolymers in solutions. For this reason, it is crucial to perform simulations in the same solvent condition in which the experimental data was collected. The most common solvent used for MD simulation is water. Several water models have been proposed in the last three decades for computer simulations [29]. Only a few of them are commonly applied to study biopolymers in solution [14]. The most common models are the SPC [30] (or SPC/E [31]) and TIP4P [32] water models. The several variants of these two models reproduce diﬀerent physico-chemical properties of real bulk waters with good accuracy [14]. Most of the force ﬁeld parameters of biomolecules have been optimized in combination with these two solvent models. In the simulation, solutes are usually surrounded by several layers of water molecules. Bulk conditions are obtained using the so-called periodic boundary conditions (PBC) [7]. These conditions are analogous to those normally used in 2D videogames, in which space ships or characters leave from one side of the screen and reenter from the opposite site. In the last case, this topology corresponds to a torus surface, and in the 3D case to a hypertorus surface. In this manner, each molecule is uniformly surrounded by other solvent molecules, as in bulk conditions, and boundary eﬀects caused by rigid conﬁnement walls are avoided. Simulation boxes can have diﬀerent shapes. To apply the PBC, these shapes need to be space-ﬁlling solids. Cubic, truncated octahedral, or truncated dodecahedral are the geometries normally adopted for box shapes and they are ordered in descending volume occupancy and ascending order concerning spherical symmetry properties. Once the shape of the box is deﬁned, the solute is centered in the box and the remaining empty volume is ﬁlled uniformly with solvent molecules. The initial coordinates of the solvent molecules are generated geometrically by superposition of an equilibrated solvent box and removal of the solvent molecules overlapping with the solute. This procedure avoids generating solvent molecules that are too close to the solute. Close contacts would produce strained conﬁgurations that need to be relaxed by a further minimization, a procedure to move the coordinates of the molecule to a conformation having lower energy. In this way, unfavorable atomic contacts that might cause integration problems at the start of the MD run are removed. The energy minimization is usually performed using the steepest descent and the conjugate gradient algorithm [2].

6 Fiat Vis! Once the simulation box is set up, we can proceed with the simulation. A long time ago, Sir Isaac told us that the forces (Fi ) acting on the ith atom of our system are related to the mass (mi ) and acceleration (ai ) of the particle and to the gradient of the potential energy by the equations:

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Fi = mi ai = −

∂V ∂ri

(3)

This is a second order diﬀerential equation. Once the initial positions and velocities of the atoms in the system are known, it can be integrated to obtain the new position and new velocity of the particle at time t+dt. The algorithms that perform such integration are well-documented in diﬀerent textbooks and reviews [7]. A variation of Verlet’s method (known as “leap frog” [7]) is normally used for its simplicity, stability and eﬃciency. In the case of the leap-frog method, the ﬁnal equations of motion are given by the formulae: r(t + Δt) = r(t) + v(t + Δt 2 )Δt v(t + 12 Δt) = v(t − Δt )Δt + aΔt 2

(4)

It is important to note that the velocity and the position of the atoms in this algorithm are calculated at half integration time steps (Δt/2). The initial velocities of every atom are randomly assigned by sampling a Maxwell’s velocity distribution at the temperature of the simulation. The temperature of the system is calculated from the kinetic energy by the relation M '

T =

mi vi2

i=1

3N kB

(5)

where N is the number of particles of mass mi and kb is the Boltzmann constant.

7 Keeping Molecules Warm and Under Pressure The physical conditions at which experiments are conducted are characterized by two important thermodynamic properties: temperature and pressure. Other thermodynamic variables can be taken into account. However, under normal circumstances, the previous two factors are most important. In a conservative system, the total energy of the system is constant; but large ﬂuctuations in the kinetic energy occur during simulations. From statistical mechanics, this condition is known to provide a microcanonical ensemble. A more realistic physical condition is a canonical ensemble where temperature, pressure, or both are constant [7]. In MD simulations, the canonical ensemble can be obtained in various ways. The most common algorithms for a thermostat are the Berendsen’s weak coupling approach [33] and the Nose–Hoover thermostat [34]. For pressure, the weak coupling [33] or the Parrinello–Rahman [7] method are the most commonly used approaches. These methods can be either approximate or exact, accordingly to the mathematically proved capability to reproduce an exact statistical canonical ensemble. The Berendsen’s methods

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are approximate methods. The Berendsen’s thermostat is based on the correction at each time step of the atomic velocities by a factor that accounts for the actual temperature and the reference temperature [6, 8]. The Berendsen’s barostat is based on the correction of the box sizes by a factor that takes into account the current pressure and the reference pressure [6, 8]. When an accurate determination of the thermodynamics properties is not required, it is the most commonly used and the simplest approach to keep temperature and pressure constant.

8 Starting the Simulation We now have all the ingredients to start a MD simulation. In order to adjust the system density to the correct value, a preliminary equilibration of the system is necessary. The length of the equilibration depends on the size of the system. Normally, a few hundred picoseconds are suﬃcient for most cases. To avoid possible deformation of the solute during the equilibration procedure, its atomic coordinates are restrained to a reference position (the starting conformation) by positional restraints, e.g. harmonic potentials that keep the atomic positions vibrating around the reference ones. In this way, the solvent molecules can redistribute all around the solute. The second step of the equilibration procedure is the relaxation of the solute in the solvent. This task is achieved by progressively heating the system from a low temperature up to a target temperature at which the simulation is carried out. This procedure can be performed either by starting several discontinuous simulations of ﬁxed length at increasing temperatures, or a single continuous simulation in which the temperature of the thermal bath is raised progressively during the simulation. Once the system has been equilibrated, it is possible to start the production runs. The length of a production run depends on the size of the system and the nature of the phenomenon to be studied. Small biomolecular systems (up to 20 000 atoms) can be simulated for hundreds of nanoseconds or even microseconds in a relatively short time and with inexpensive PC clusters. Larger systems require more computational eﬀort and the simulation length is normally limited to several tens of nanoseconds.

9 Advanced MD Methods Several advanced methods have been invented either to improve the performance of the MD simulation in accomplishing speciﬁc studies that are beyond the capability of standard MD simulations, or to extract relevant information from the trajectory analysis. In this section, some of the most common methods are summarized, to give readers an idea of their use and capability. Several other advanced techniques (for example thermodynamic calculation,

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non-equilibrium calculation, replica exchange methods) are excluded from this chapter; they are explained in the comprehensive review and textbooks mentioned in the introduction. 9.1 Steered Molecular Dynamics The Steered Molecular Dynamics (SMD) method is a non-equilibrium method. An external time-dependent force is applied to the system in order to steer it along a particular degree of freedom. This method can be used to investigate diﬀerent processes, for example ligand binding to and unbinding from proteins, domain motion and protein folding [35, 36]. SMD provides a means to accelerate processes by applying external forces that lower the energy barrier and thus allow the study of long time scale processes on a nanosecond timescale. The methodology is analogous to single molecule manipulation techniques such as atomic force microscopy and optical tweezing [37]. A quantitative comparison with the experimental data is however limited due to the fact that the velocity applied to the pulling force is too fast to be studied experimentally. Nevertheless, for a qualitative description of the mechanism involved, the agreement with experimental data is still remarkably good. Recent theoretical reformulation of non-equilibrium thermodynamics [38, 39] has opened a new way to extract free energy data from the non-equilibrium SMD simulations. This direction seems promising but the method is still technically demanding compared to more standard and assessed techniques. 9.2 Investigating the Collective Motions Biopolymers are characterized by dynamical processes that span several orders of magnitude in space and in time. This behavior is due to the rugged free-energy surface that describes the conformational space of these complex molecules. As mentioned previously, MD simulation can only partially cover the dynamic range of a protein’s motion. One interesting feature of this motion is the spatial and temporal correlation. Small-amplitude, fast and locally uncoupled atomic motions, as well as large scale collective motions are both present in biomolecules. In equilibrium conditions, the biomolecules diffuse among free-energy basins (large collective motions) with diﬀerent level of roughness (small local motions). The analysis of the conﬁguration space spanned by the protein during its equilibrium dynamics can be performed by the essential dynamics (ED) analysis [40] of the MD trajectories. This method allows characterization of both the conﬁguration subspace (the “essential subspace”), in which the largest collective motions occur, and the subspaces of the faster small amplitude motions. The analysis is done by building, from the MD trajectories, the covariance matrix (C) of the atomic positional ﬂuctuations deﬁned as: Cij = (xi − xi ) (xj − xj ) .

(6)

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This matrix can be diagonalized giving C = Z−1 ΛZ where Z and Λ are the matrices that contain the orthonormal set of eigenvectors deﬁning a new set of generalized coordinates and the corresponding eigenvalues, respectively. The eigenvectors represent the orthonormal direction in which the overall motion of the molecule can be decomposed, while the eigenvalues provide the extent of variation along that direction. The essential subspace is deﬁned by the set of eigenvectors with the largest eigenvalues of the atomic positional ﬂuctuations covariance matrix [41]. In the second part of this chapter, example of an application of this analysis is given. 9.3 Essential Dynamics Sampling When the time scale of a biological process is beyond the capability of conventional MD simulation, the use of new computational approaches can enhance the exploration of the protein conformational space. One of these methods is the Essential Dynamics Sampling [42]. It is based on the use of the information obtained using the ED from a normal MD simulation to bias the system to move along the directions of the ”essential space”. In this way, the system avoids wasting simulation time to sample local minima and it is biased to diﬀuse quickly along basins that characterize conformations accessible by slow collective modes. This method has been successfully applied to protein dynamics [43], domain motion [44] and even to study protein folding [45, 46]. In the case of the domain motion study, the essential dynamics sampling was combined with the so-called rigid body essential dynamics RBED analysis [44, 47]. 9.4 Molecular Dynamics Docking Understanding the basis of molecular interaction is very important for understanding the processes of molecular recognition. The enormous interest in industrial application has boosted the development of simpliﬁed and fast method for screening large compound libraries and protein-protein complexes [48, 49]. The interaction of two molecular systems can, in principle, be performed using MD. This allows us to include the full ﬂexibility of the substrate and protein and to use an explicit solvent environment. However, the demanding calculation time requires the use of tricks to accelerate the search for docking conﬁgurations within the time scale aﬀordable by MD. A direct approach is using SMD by applying a pulling force on the ligand to drive it into the active site [50]. However, this approach requires pre-knowledge of the docking site(s). When this prerequisite is missing, a convenient way to accelerate the docking process is to artiﬁcially increase the diﬀusion of the ligand around the receptor. This is the approach used in the MDD (Molecular Dynamics Docking) method [51, 52]. The velocity of the mass centre of the ligand is separated from the internal degrees of freedom; the velocity is coupled to an external bath at a higher temperature while the internal motions are kept at

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a lower temperature by coupling to another bath. By doing this, the system can rapidly diﬀuse within the simulation box and explore more eﬃciently the surface of the protein. The combination of this simple approach with essential dynamics sampling could allow us to speed up docking processes where conformational changes in the receptor (or in the ligand) play a major role in recognition.

10 Molecular Modeling Programs Several packages for MD simulation of large biomolecular systems are available. In Table II, some of the most common packages are listed. The basic features of them are very similar. All of them use diﬀerent methods for treating long-range interactions, the temperature and pressure coupling. Algorithms for thermodynamics calculations are available in all of them. There are two types of structure in MD packages: 1) separate programs for diﬀerent purposes (i.e. GROMACS) and 2) a single monolithic program that can be used to perform most of the MD task (i.e. CHARMM). Every package oﬀers the possibility to run simulations on parallel computers but they have varying performance due to the diﬀerent parallelization approaches adopted. For simulating relatively small systems on PC clusters, the best choice is GROMACS. This package oﬀers terriﬁc performance on PC clusters since the core routines for the non-bonded calculations are mostly written in assembly language by squeezing the parallel pipeline feature of the processor. For large systems that require massive parallel computing, the best choice is NAMD or the PMEMD module of AMBER. In both cases, the scalability is assured up to hundreds of processors. Also the programming languages used in developing various packages are diﬀerent. FORTRAN (77 or 90) is used for AMBER, CHARMM and GROMOS, C for GROMACS and C++ for NAMD (and the new GROMOS05). Normally all the MD programs have compression algorithms to reduce the size of trajectory ﬁles. In the case of GROMACS, a very powerful compression method is used to considerably reduce the space occupied by the trajectories. A good overview of the diﬀerent MD programs can be found in the December 2005 special issue (Volume 26, Issue 16) of Journal of Computational Chemistry. Table 2. List of the most common packages for molecular dynamics simulations. Package AMBER CHARMM GROMACS GROMOS NAMD

Homepage amber.scripps.edu www.charmm.org www.gromacs.org www.igc.ethz.ch/gromos www.ks.uiuc.edu/Research/namd

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11 Analysis of the Simulation Trajectory The analysis of an MD simulation trajectory comprises both extracting diﬀerent structural, thermodynamic and kinetic properties, and comparing them with experimental data. In this way, it is possible to verify the quality of simulation models and to predict properties that are diﬃcult to access by experimental techniques. The physico-chemical properties are extracted from MD trajectories using statistical mechanics relations that are functions of average positions, velocities, or energies of the system. According to the so-called ergodic hypothesis [7], these averages are representatives of a statistical mechanics ensemble if the length of the trajectory is long enough (in principle inﬁnite) to extensively explore the conformational space of the molecular system. An extensive sampling of the conformational space of a large molecular system is usually very demanding in terms of computational time. Therefore, it is important to bear in mind that MD simulations provide, with their limited trajectories, an approximate estimation of the statistical mechanical properties of the molecular system under investigation. The accuracy of this estimation can be improved by producing several short trajectories starting from various positions of the conformational space of the system. This latter approach is becoming more frequently used thanks to the availability of new high performance supercomputers as in the case of the IBM Blue Gene project (researchweb.watson.ibm.com/bluegene), distributed computing facilities such as the Folding@home project (folding.stanford.edu), and the lowcost parallel PC cluster. Physico-chemical properties calculated from MD simulations are classiﬁed in the following three groups: 1. Structural. Derived from atomic coordinates of a single structure and of the averages along the trajectory. 2. Dynamic. Derived from atomic velocities and coordinates. 3. Thermodynamic. Derived from averages and ﬂuctuations of diﬀerent mechanical properties (energy, coordinates, forces). These properties can be compared with experimental data obtained from various experimental techniques (see Table 3 for some examples). In the following paragraphs, I will give an overview of some properties that can be obtained from the analysis of the trajectory of biomolecular systems. In these examples, I will use data obtained from the analysis of the simulation of three molecular systems: 1. TEM-1 β-lactamase. This is a bacterial enzyme responsible for the hydrolysis of β-lactam and cefalosporine antibiotics. 2. β-hairpin 41-56 from protein G B1 domain. This is a 16-mer β-hairpin peptide, present in the B1 domain of the bacterial protein G. It was the ﬁrst example of β-hairpin forming peptides in solution.

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3. Capped 10-mer polyalanine peptide. This is a 10 alanine neutral peptide with the N- and C-terminal residues acetylated and amidated, respectively. The simulations were performed using the Gromacs program suit. For system 1, the OPLS force ﬁeld was used. For the other two systems, the GROMOS96 force ﬁeld was employed. All the simulations were performed in SPC water. More detail concerning the MD simulation of these systems can be found in the literature [53–55]. 11.1 Structural Properties Molecular Visualization

Table 3. Some examples of possible comparisons between experimental data and properties obtained from MD simulations. Physical properties Structure - Positions, Size - Distance Mobility - Debye–Waller factors

Experimental methods

Gas Solution Membrane Crystal

X-ray diﬀraction, neutrons, electrons X NMR, FRET, EXAFS

X-ray diﬀraction, neutrons, electrons X NMR,FRET

- Relaxation, contact Dynamic - Vibration frequencies IR Spectroscopies X - Spin relaxation, diﬀusion NMR - Contacts Time-resolved laser spectroscopies Thermodynamics - Density Densitometry - Enthalpy of evaporation, heat capacity Calorimetry - Viscosity Viscosimetry

X

X X

X X

X

X X

X X

X

X

X

X

X

X

X

X

X

X

X X

X

X

The visualization of molecular structure is a powerful tool to analyze the complex dynamics of biomolecular systems. The quality of the available graphical programs adds a touch of artistic beauty to the scientiﬁc content of the

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molecular structure representation. This artistic impression can, however, be misleading. In fact, the wrong molecular model can look as beautiful as the correct one on the screen. Hence, caution should be taken when treating these graphical programs as black boxes for molecular modeling. Nevertheless, the visualization of molecular dynamics trajectories is a quick and eﬃcient way to analyze the structural behavior of the system. There are plenty of programs for molecular visualization (for examples see molvis.sdsc.edu/visres). A popular program is VMD (www.ks.uiuc.edu/Research/vmd). This program oﬀers both a user-friendly interface for routine applications and the possibility to use a script-oriented python based language for more complex analyses. The Root Mean Square Deviation The root mean square deviation (RMSD) is a measure of the deformation of the macromolecular structure with respect to a reference one. Figure 1A shows the superimposition of one frame of an MD simulation (in gray) and the reference structure (in black). The arrows indicate the distances between pairs of corresponding Cα atoms used to calculate the RMSD. The average value of these deviations is calculated at each time frame of the MD trajectory as total RMSD using the following formula: 1 2 rmsd(t) = [ri (t) − rref (7) i ] N i

Fig. 1. A) Example of superimposition of an MD frame structure (in gray) on the reference one (in black). Some distances between corresponding Cα atoms are indicated. B) Example of backbone RMSD from the simulation of the protein TEM1 at two diﬀerent temperatures (300K, black; 450K gray). Each frame from the simulation has been rotationally and translationally ﬁtted on the crystallographic starting structure.

In Fig. 1B, an example of RMSD curve obtained from the simulation of protein TEM-1 at two diﬀerent temperatures is shown. It is evident that high

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temperature destabilizes the structure, thus increasing the value of RMSD. It is possible to quantify the deformation of each residue by averaging the RMSD for a single atom (or residue) in the structure, calculated using the following expression: 1 2 [ri (t) − rref (8) rmsd(i) = i ] Frames t In Fig. 2 two examples of rmsd(i) plot for the protein TEM-1 are reported. The rmsd(i) plot provides an easy way to identify residues having the largest deviation during the course of the simulation.

Fig. 2. Example of backbone RMSD per residue output for the protein TEM-1 at two diﬀerent temperatures (300K: black and 450K: gray).

Solvent Accessible Surface Area The principal driving force for biopolymer folding is the burying of hydrophobic residues in the core of the 3D structure. The residues exposed on the surface are mainly hydrophilic in nature and they tend to oﬀer a minimum surface. The deﬁnition of molecular surface is based on the approximation of the atomic shape as spheres of van der Waals radius. Using this approximation, it is possible to calculate the solvent accessible surface (SASA) of a molecule by rolling a sphere (having a radius equal to that of a solvent molecule), over the van der Waals surface (see Fig. 3A). If the surface is colored according to the polarity of the amino acid, it becomes easier to recognize the presence of large hydrophobic and hydrophilic patches on the surface of the protein and hence to identify possible binding sites. The SASA is also used to evaluate the stability of the protein during the course of the simulation (see Fig. 3B).

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Fig. 3. A) Top: Cartoon showing the calculation of SASA. The probe sphere is rolled over the van der Waals surface of the atoms in the molecule. The trajectory of the center of the probe is used to deﬁne the surface. Bottom: Example of SASA for the TEM-1 β-lactamase. B) Time dependant curves of the diﬀerent SASA components from the 300 K simulation of the TEM-1 β-lactamase.

The Radius of Gyration The mass distribution of a large molecular system can be evaluated using its radius of gyration that corresponds to the radius of the sphere having a uniform mass distribution equal to the total protein mass (see Fig. 4A). The Rg is calculated using the following formula: ' [ri − rcom ]2 i mi' (9) Rg = i mi where r i is the position of the ith atom and r com the position of the centre of mass of the molecule. As for the RMSD and the SASA, the radius of gyration provides information of the structure deformation during the course of the simulation. Fig. 4B is an example of the time course of the radius of gyration. The increase in this value indicates an expansion of the protein volume. The Protein Secondary Structure Secondary structure elements in a protein are α-helices, β-strands and loops. An example of a 3D representation of TEM-1 secondary structure elements is shown in Fig. 5A. The diﬀerent elements are conventionally represented as cylinders (α-helices), ribbons (β-strands) and tubes (coil regions). Secondary structure can be identiﬁed using programs such as DSSP [56] and STRIDE [57]. These programs use hydrogen bond patterns and/or Ramachandran plot

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Fig. 4. A) The radius of gyration is deﬁned as the average sphere having a uniform mass distribution equal to the total protein mass. B) Example of the time course of radius of gyration from the simulation of TEM-1 at two diﬀerent temperatures (300 K black, 450 K gray).

propensity of the φ/ψ angles to classify each residue. The analysis can be performed along a simulation trajectory and the result obtained is shown in Fig. 5B, in which each residue along the trajectory is color-coded according to its secondary structure. From the graph, it is possible to identify the change of secondary structure elements during the course of the simulation. The Cluster Analysis The ensemble of structures obtained from an MD simulation can be classiﬁed and compared in order to identify clusters of structures with similar features. This analysis is particularly useful when the structure under investigation is ﬂexible and several conformational states are accessible (for example small peptides). The comparison of the molecular structures is accomplished by performing RMSD of each frame structure in the simulation. The cluster analysis is subsequently performed using diﬀerent algorithms. Two commonly applied algorithms are the Jarvis–Patrick [5] and the gromos [58]. In the ﬁrst method, a structure is added to a cluster when this structure and a structure in the cluster are neighbors of each other or at least both have a ﬁxed number of neighbors in common. In the gromos method, neighbors of each structure are evaluated by selecting those conformations having RMSD below a certain cutoﬀ. The neighbors of a structure are a ﬁxed number of structures or all the structures within a certain cutoﬀ. The structure having the largest number of neighbors together with all its neighbors is considered as a cluster and thus eliminating it from the pool of clusters. The same procedure is repeated for other structures within the pool. From such cluster analysis, one can retrieve diﬀerent information. Figure 6 is an example of cluster analysis of the G B1 peptide using the gromos method. The analysis was performed on the non-hydrogen

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Fig. 5. A) Example of secondary structure representation in proteins. α-helices and β-strands are conventionally represented as cylinders and ribbons, respectively. B) Secondary structure change during the 300 K simulation of TEM-1. The diﬀerent secondary structure elements are deﬁned as in the legend.

atoms of the peptide using a cutoﬀ of 0.2 nm. In panel A, the classiﬁcation of each residue to the diﬀerent clusters is reported. It is obvious that only a few clusters representing most of the structures are present in the trajectory. Figure 6B shows the cumulative number of clusters along the trajectory. The convergence of this curve to an asymptotic number of clusters can be used to assess the equilibration of the molecular system [59]. For each cluster, it is possible to identify a representative structure. In the case of G B1 peptide, the representative structures for the ﬁrst four clusters are illustrated in Fig. 6C. Solvent Structure The ﬂuid state is characterized by the absence of a permanent structure. This means that every molecule (or atom) of the ﬂuid has a continuous rearrangement with respect to other molecules. The solvent reorganization results in deﬁned structural correlations that can be observed or measured using different experimental techniques. In MD, structural correlations are obtained from the atomic coordinate using a canonical distribution of general form [7]: U (r,...rN a ) − kB T Nb (Nb − 1) dr3 . . . rN a e gab (r1 , r2 ) = U (r,...rN a ) − kB T ρ2 dr3 e

(10)

where the integral in the denominator is the partition function, and the one in the nominator excludes the r 1 and r 2 coordinates of the two particles (a, b) from the integration. In the case of a homogenous system, the relative

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Fig. 6. Cluster analysis of the 30 ns simulation of the system (2). A) The cluster map along the trajectory. Points indicate which cluster each simulation structure belongs to. B) Cumulative sum of the number of clusters along the trajectory. C) Representative structures of the ﬁrst 4 clusters.

positions between molecules is meaningful, and in the case of an isotropic system the function can be mediated over the angular distribution without loss of information, thus reducing (10) to: gab (r) =

nb (dr) ρb (r + dr) 3 = 0 ρ 4 ρ0 Nb [(r + dr)3 − r3 ]

(11)

where dr is the thickness of a spherical sector centered on the reference particle, N the total number of particles, Na and Nb the numbers of particles for components a and b, (ndr) the number of particles in the spherical sector, and ρ0 the total density. The resulting distribution function gab (r) describes the local organization around every atom in the system. The g(r) plays a central role in the physics of the liquid state and all the functions that depend on the distance of separation (e.g. the interaction function and the pressure) can be used to derive pair distribution functions. The deﬁnition of g(r) implies that 2πg(r)dr is proportional to the probability of ﬁnding an atom in the volume element dr at a distance r from a given atom, and, in the case of a homogenous and isotropic ﬂuid, n(r) = 4πρg(r)r2 dr represents the average number of atoms within a shell of radius r and thickness dr that surrounds the atom. The g(r) functions can be obtained from experimental techniques, like small angle scattering of neutrons or X-ray absorption

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spectroscopy. In fact, the g(r) is related to the structure factor S(k ) by the Fourier transform [60].

Fig. 7. A) Examples of g(r) from the simulation of a box of water. The solid thick line and the thick dashed line are the oxygen-oxygen and the oxygen-hydrogen radial distribution functions, respectively. The corresponding n(r) functions are represented by thin lines. B) Calculated structure factor from the water box simulation.

The structure factor can be obtained from measurements of X-ray diﬀraction or neutron diﬀusion experiments. The relation that joins S(k ) together with g(r) is given by: S(k) = 1 + ρ g(r)e−k·r dr (12) For an isotropic ﬂuid, this equation can be simpliﬁed to: sin kr g(r)r2 dr S(k) = 1 + ρ4π kr

(13)

In Fig.6B, an example of structure factor for the water molecule obtained from 100 ps simulation of a 5×5×5 nm3 water box is reported. A good review of the experimental and simulation data on the water structure is given in [60]. In the case of biomolecules, the g(r) is used to analyze the solvent distribution around the molecule. This information is interesting when one would like to investigate the eﬀect of cosolvents. For industrial applications, the use of non-aqueous environments is important for many processes. The stability or activity of biomolecules in these non-natural environments is of fundamental importance for their putative applications. In fact, the preferential solvation by cosolvent can aﬀect either the structural stability or the catalytic activity of the biopolymer. Molecular dynamics simulations of the biopolymer in such conditions can be readily performed and the qualitative result provides a mechanism for understanding these processes [53, 61].

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11.2 Dynamic Properties Calculation of the Diﬀusion Constant The diﬀusion of a molecule in a ﬂuid is deﬁned by the diﬀusion constant D that determines the velocity at which the process takes place. In 1905, Albert Einstein proved that D was related to the average distance covered by the particle in time t with respect to its starting position (mean square displacement, msd). This relationship is described as follows: 6Dt =

N 1 2 [ri (t) − ri (0)] N i=1

(14)

where r i (t) is the coordinate vector of the ith particle at time t, r i (0) the coordinate vector of the ith particle at time t = 0 and N the total number of particles [7]. In Fig. 7, an example of msd curve calculated from the simulation of a box of SPC water molecules at 300 K is provided. The slope obtained from the ﬁtting curve is 3.7 × 10−5 cm2 /s, 1.5 times the experimental value at the same temperature.

Fig. 8. Mean square displacement (msd) curve from a water box simulation. The straight line is the linear ﬁtting of the msd curve.

Another way to calculate D is the Green–Kubo formula: TM D

0

where

N 1 Ci (t) dt = 3D N 1

(15)

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Ci (t) =

1 TM D − t

TM D

vi (t )vi (t + t)dt

579

(16)

0

is the autocorrelation function of the velocities and TM D is the total simulation time. However, this method is comparatively less accurate [7]. The calculation of diﬀusion constant is normally used to evaluate the mobility of solvent around the protein or the ligand in the active site. This information can then be used to compare with NMR measurements. The Root Mean Square Fluctuation The root mean square ﬂuctuation (RMSF) is a measure of the dynamics of each atom or residue in the protein structure. It is calculated from the second momentum of the atomic position ﬂuctuation along the MD trajectory using the formula: 1 rmsf (i) = [ri (t) − ri ]2 (17) Frames t

Fig. 9. Example of RMSF output for the protein TEM-1 at two diﬀerent temperatures (300K: thick line and 450K: thin line).

In Fig. 9, an example of backbone RMSF from the TEM-1 simulation is reported. As for the residue RMSD (see Fig. 2), the secondary structure of the protein is indicated at the bottom of the graph. The correspondence between the larger ﬂuctuations and the loop regions of the structure is obvious from such a plot. α-helices and β-sheets show a much lower ﬂuctuation due to the internal hydrogen bonds.

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Essential Dynamics Analysis The essential dynamics analysis of the trajectory provides plenty of information which can be used for diﬀerent purposes. From the diagonalization of the covariance matrix, a set of eigenvectors and corresponding eigenvalues are obtained. Fig. 9 shows the ED analysis of a 30 ns simulation of the 16 residue long β-hairpin peptide from the protein G B1 domain. The analysis was conducted on the mainchain atoms. In this case, the number of degrees of freedom is 16 × 4 × 3 = 264. The total number of non-zero eigenvalues is 264-6 since we have removed 6 degrees of freedom by ﬁtting the structure to the same reference structure. In Fig. 10A, the eigenvalues in descending order are reported in the inset. As mentioned before, the sum of eigenvalues gives the overall ﬂuctuation of the atoms used in the ED analysis along the trajectory whereas the single eigenvalues provide an indication of the extent of motion along each eigenvector. Each contribution can be better represented using a cumulative relative contribution plot as shown in Fig. 10A. The number of eigenvalues deﬁning the large amplitude motion (essential motions) is related to the size of the protein and is always a small fraction (between 5 and 10) of the total number of degrees of freedom. The collective motions describe the diﬀusion of the coordinates of the protein on a multi-minima energy surface. The projection of the protein trajectory coordinates on the eigenvector allows us to ﬁgure out the nature of the eigenvector. In the case of the essential eigenvector, the histogram of the projection is often given by a multi-modal distribution curve, where the maxima points describe the diﬀerent conformational states of the system. The shape of the distribution tends to become more and more gaussian with the increase of the eigenvector index. The higher eigenvectors describe small amplitude protein motions that are quasi-harmonic. The characterization of the nature of anharmonic motions in protein has been addressed by diﬀerent authors [55, 62]. An important aspect in this analysis is the convergence of the simulation that is connected with the conformational space sampling. As mentioned at the beginning of this section, one limitation of MD is the limited exploration of the conformational space for large macromolecular systems. During a single MD trajectory, the system has low probability of overcoming high barriers between diﬀerent conformational energy minima of the molecule. In such conditions, the trajectory represents a non equilibrium state of the protein and the eigenvectors are not representative of the equilibrium collective motion of the system. For a simulation starting from a crystal structure, the trajectory would represent the relaxation process from the crystallographic enviroment. Several descriptors can be used to evaluate the convergence of eigenvectors in a simulation. Hess has provided a good criterion based on the inner product of trajectory segments [63]. In the case of the β-hairpin in our example, the Hess method provides a convergence indicator close to unity that indicates a rather good equilibration of the trajectory.

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Steered Molecular Dynamics Steered molecular dynamics is normally used to study processes where the application of external forces can speeds things up. This process is physically equivalent to connecting a spring to the molecule and pulling it at constant velocity in a certain direction. The external perturbation is a time-dependant force: f (x) = k(x − vt) (18) where k is the force constant of the spring, v the pulling velocity and x the pulling direction. The elongation of the spring is proportional to the barrier encountered during the pulling process. A typical application example is the study of protein or peptide unfolding. Figure 11 demonstrates the unfolding of a 10-mer capped alanine peptide. The N -terminal of the peptide is ﬁxed using a position restraint while the pulling spring is applied to the mass center of NH2 group at the C-terminal along the Z direction. Two force proﬁles are reported: the pulling force versus time (panel A) and versus the end-to-end distance (panel B). In Fig. 11C, some structures obtained during the pulling process are shown. SMD experiments are capable of providing a qualitative picture of the process. The time scale is however too fast to be compared with real AFM experiments. Nevertheless, extrapolation to low velocity regimes provides remarkably good agreement between MD and experimental data. The unfolding free energy barrier is in principle the integral of the pulling force in the case of inﬁnitely slow extension process. It is worth mentioning that the integral of several repeated experiments provides only a gross estimation of this value that can later be used for comparison. SMD experiments are also applied in driving more complicated processes of large, complex systems such as molecular motors, channel diﬀusion, docking or undocking processes [35, 36].

12 Perspectives At the beginning of this chapter, I mentioned that MD is subject to diﬀerent approximations that reduce its accuracy. The removal of the electronic degrees of freedom and the lack of suﬃcient conformational sampling are the most signiﬁcant limitations. To tackle the ﬁrst limitation, new developments are proceeding along two directions. The ﬁrst direction is the improvement of the long-range interaction forces by accounting for the polarizability eﬀect. Polarizable force ﬁelds can improve the treatment of protonation state of biomolecules in solution by properly reproducing the dielectric conditions in their interior [15]. The other direction is more fundamental, an attempt to ﬁnd a theory that combines both quantum mechanics and the treatment of the enzymatic reactions centers. The conformational sampling problem is a more general issue related to the statistical mechanics foundation of the MD methods.

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Fig. 10. A) Relative cumulative ﬂuctuation as a function of the eigenvector index. In the inset the corresponding eigenvalues are reported. B) Projection of the trajectory along the ﬁrst two eigenvectors. On the axis, the histograms of the projections are reported. C) Histogram of the trajectory projection along the 200th eigenvector. D) Graphical representation of the peptide motion along the ﬁrst three eigenvectors. The initial and ﬁnal conformations are in thick lines.

The simulation of a molecular system at temperatures larger than absolute zero generates representative conﬁgurations of a particular statistical ensemble. The thermodynamics properties of such systems are deﬁned from their medium values, estimated by the generated conﬁgurations. From the ergodic hypothesis, the average of such trajectories over an inﬁnite time is equivalent to the average carried out on the statistical ensemble [7]. The application of this hypothesis in a rigorous way is practically impossible because of the need of a trajectory of inﬁnite length. However, if the conﬁguration space of the system is not too large, the time required to explore a large part of it becomes reasonable for MD simulation. In order to explore eﬀectively the conﬁguration space within a reasonable simulation time, it is possible to introduce further approximations that reduce the number of degrees of freedom of the system. These approximations are valid as long as the excluded degree of freedom does not inﬂuence the properties under investigation. The solution to both the aforementioned problems relies on the development of faster parallel computers. In fact, the availability of low cost parallel computers (PC clusters) has opened new horizons for MD, allowing us to afford, even at low budget level facility, the study of large complex molecular systems and/or for simulation of long time scale. The exponential development

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Fig. 11. Steered Molecular Dynamics analysis of the capped decaalanine unfolding. A) Pulling force curve as function of time. B) Pulling force versus peptide end-toend distance. The average is reported as smooth black line. C) Peptide structures sampled along the pulling trajectory.

of computer power (an order of magnitude in the calculation speed every 5 years) will improve not only the size and the length of the simulations but also the accuracy. A positive outcome would be to use more reﬁned atomic models in the near future that would improve the quality of the properties predicted using MD simulation. Furthermore, it would be possible to study many complex processes that are involved in material science and in structural biology. Currently, we can aﬀord to simulate biological systems containing up to 106 atoms for time scales of tens of nanoseconds [64,65]. We can expect that in 510 years, the same system will be studied on a microsecond timescale, getting MD simulations closer to our perception of time. Acknowledgement. The author would like to thank Steven Hayward and Tuck Seng Wong for reading and commenting on the manuscript.

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Understanding Liquid/Colloids Composites with Mesoscopic Simulations Ignacio Pagonabarraga Departament de F´ısica Fonamental, Universitat de Barcelona Carrer Mart´ı i Franqu´es 1, 08028-Barcelona, Spain

1 Introduction The study of complex ﬂuids and soft materials contains a number of basic conceptual and practical challenges. The most basic diﬃculties are related to the fact that diﬀerent length and time scales compete, making it impractical to carry out ﬁrst principles modeling of these materials [1]. Colloidal particles are rigid molecular aggregates of sizes varying between a few nanometers and a micron. As a result, they are at least one order of magnitude larger than the molecules that characterize the solvent they are suspended in. Colloids move at smaller velocities than solvent molecules as a result of their size mismatch, and hence their characteristic time scales are also orders of magnitude larger than solvent ones. Moreover, colloid interactions also diﬀer qualitatively from their molecular counterparts. Although they derive from the speciﬁc atomic interactions among the atoms that constitute the colloids, the fact that colloids are made of thousands of atoms implies that the eﬀective strength is typically of a few times the characteristic thermal energy, kB T , at room temperature, with kB referring to the Boltzmann constant [2]. The functional forms of these interactions diﬀer from the usual Lennard– Jonnes form, and are a combination of an excluded volume associated to their ﬁnite extension together with an attractive and/or repulsive contribution with a range of the order of the particle size. The standard theory that accounts for dispersion and electrostatic interactions is DLVO, a potential that has been successfully applied to understand the stability and phase diagrams of colloidal suspensions [3]. The fact that interactions are weaker for colloids than for molecules has deep implications for the collective properties of soft materials. They give rise to very soft solids (hence the generic name “soft matter”), and their response is very sensitive to thermal ﬂuctuations. In the extreme case where the potential range is negligible with respect to the colloid size, colloids are regarded as interacting through excluded volume interactions only. Hard sphere colloids have in fact become as a reference system for soft

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materials, and it exposes the central role entropy plays in determining the collective behavior of these systems [4]. The excluded volume interaction associated to particles’ ﬁnite size also has deep implications for the dynamics of these materials. At medium and high concentrations, geometric restrictions hinder the conﬁgurational relaxation. As a result, even the approach towards equilibrium may become extremely slow. Hence, although knowledge of the equilibrium behavior and the corresponding phase diagram can be derived from the eﬀective colloid-colloid interactions, materials’ properties will be mostly determined from the systems’ kinetics. This fact opens the possibility to control and tune the properties of these materials in ways that diﬀer qualitatively from the strategies followed for molecular materials, in which the thermal structures play a central role. Gels, jammed structures, glasses, and their response to applied stresses among others can be exploited to control these materials and produce new structures [5]. The solvent plays an additional role in soft material dynamics. The fact that colloids are at least an order of magnitude larger than solvent molecules implies that a colloid interacts with such a large number of solvent molecules at a time that it is sensitive to the solvent’s collective modes, and in particular to local hydrodynamic ﬂows. Hence, if a colloid moves due to the action e.g. of an applied force, it will induce a ﬂow ﬁeld that will decay algebraically slowly (because for colloids the Reynolds numbers are always small). A second colloid will be dragged by such a ﬁeld; as a result colloids interact dissipatively through the solvent. Such interactions, called hydrodynamic interactions, induce long range dynamic correlations in colloidal suspensions and signiﬁcantly aﬀect their collective dynamic properties. These interactions modify suspensions’ diﬀusivity and viscosities and generate long range dissipative structures under the action of external ﬁelds; e.g large vortices are observed during the sedimentation of colloidal particles [6]. Such external ﬁelds and the shears they induce also profoundly aﬀect the aggregate structures and generate structures that diﬀer qualitatively from their equilibrium counterparts. 1.1 Mesoscopic Methods The peculiar speciﬁties of soft materials pose serious challenges to their numerical simulation. Basic methods, such as molecular dynamics (MD), are ruled out because they cannot reach the relevant time and length scales in which colloids evolve. MD can be used only for very small nanocolloids to explore their behavior at short times (usually at the expense of not resolving appropriately the separation in time scales between solvent and colloidal relaxation) [7]. Nonetheless, this approach is useful because it takes the solvent molecular details into account and can address basic issues, as for example the relation between friction and diﬀusion, or the validity of eﬀective boundary conditions used on eﬀective models and theories [8].

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In order to reach longer time scales and to enforce the appropriate scale separation between colloids and the embedding molecular solvent, it is necessary to reduce the degrees of freedom of the solvent and introduce coarse grained descriptions that focus on a set of prescribed solvent variables. Such a choice requires physical insight, and diﬀerent approaches have been followed during recent years. The traditional method, based on the Langevin picture of colloid/solvent interaction, considers the solvent as a passive medium that exerts a friction on the colloids. In this method, Brownian dynamics (BD), a friction force proportional to the particles velocity is added to the conservative colloid/colloid interactions, and a random force is also included to ensure that the ﬂuctuation-dissipation theorem is fulﬁlled and that proper thermal equilibrium can be reached [9]. Even if this method has proved very useful in the exploration of the structures that colloids may form, and in the dynamics on time scales in which conﬁguration relaxes, it misses the fact that momentum conservation gives rise to induced solvent ﬂows in response to colloidal motion. In an attempt to ensure a more realistic description of the solvent, in recent years there has been a growing interest in the development and use of mesoscopic simulation methods which capture at a coarse grained level the collective modes of the solvent. The goal is to eliminate degrees of freedom of the solvent, so that the computational eﬀort can be focused on the study of colloidal particles, while ensuring that momentum is conserved, a basic feature missing in BD. One approach consists in the use of particle-based methods, in the same spirit as molecular methods such as MD, but introducing eﬀective forces. It is well known that the reduction of degrees of freedom introduces friction forces. One can accordingly extend MD schemes by including friction and random forces such that equilibrium is recovered. The use of momentum conserving thermostats has opened the possibility to enforce the development of collective hydrodynamic modes on shorter scales than in standard MD. Dissipative particle dynamics (DPD) proposed a thermostat in which there is a friction force that depends on the relative velocity between particles, combined with pairwise additive random forces [10]. The amplitudes of these two forces should be carefully tuned to ensured that ﬂuctuation-dissipation theorem is satisﬁed in equilibrium. An alternative momentum-conserving thermostat has been proposed by Lowe [11], which is a local version of the traditional Anderson thermostat, and which captures proper hydrodynamics. For the DPD thermostat theoretical analysis has proved how the Navier–Stokes equation follows from these non-Hamiltonian forces [12]. However, in order to reach meaningful time scales, these thermostats should be combined with eﬀective conservative forces. At the simplest level, these are regarded as coming from soft potentials, although there is no derivation from ﬁrst principle that allows us to link the parameters appearing in such potentials with their atomic counterparts [13]. Such forces allow the use of longer time steps, hence favouring the reaching of meaningful time scales.

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An alternative approach is based on the use of kinetic theories. The idea is to develop simpliﬁed models for the solvent that capture the collective solvent modes while disregarding any information related to its molecular character. Such approaches were introduced with the idea of producing minimalist dynamic models, analogous to the use of Ising models in critical phenomena, where the microscopic details are irrelevant. The Lattice gas cellular automata (LGCA) [14] or the lattice Boltzmann (LB) method [15], have opened the door for kinetically-based models for ﬂuids that can be coupled to colloids and other soft materials. These models are very competitive computationally because they are deﬁned on a lattice and are based on local rules. In this chapter I will describe these diﬀerent approaches to the modeling of suspensions. As an example of a kinetic model I will concentrate on LB. After describing the fundamentals and how colloids can be included, I will discuss how to deal with complex mixtures and analyze some examples of collective properties obtained with this method. In subsequent sections I will discuss some of the particle based mesoscopic models that have been proposed recently and how eﬀective soft forces can be introduced and the potential use of many body interactions.

2 Kinetic Approaches: Lattice Boltzmann In order to overcome the computational limitations in the study of ﬂuids at high Reynolds numbers, in the early nineties a lattice gas cellular automaton (LGCA) was introduced to study ﬂuid ﬂow [16]. It was deﬁned on a lattice, moves were restricted to a subset of neighbouring nodes, and single node occupation was enforced. Even if they were unconditionally stable and easily parallelizable, it was not possible to reach Reynolds numbers as large as expected. Moreover, the need to average ﬂuctuations made them computationally intensive and they suﬀered from a number of physical limitations, such as the fact that Galilean invariance is lost. In the context of LGCA, the lattice Boltzmann method (LB) was introduced as a preaveraged version of LGCA. It keeps both space and time discretization, but considers as the fundamental dynamical variable the one particle distribution function, fj (ri , t). It can be regarded as the number of particles at node ri at time t with velocity cj . The set of allowed velocities, cj , joins a given node with a prescribed set on neighbours. Hence, space and time still remain discretized. The symmetry of the lattice and the minimum allowed set of velocities should conform to a minimum set of symmetry properties that ensure that the underlying anisotropy of the lattice does not aﬀect the response of the system at the Navier–Stokes level [17]. The collision matrix determining the relaxation of fj (ri , t) toward equilibrium was introduced at ﬁrst as a result of averaging the collision rules of the underlying LGCA. However, in a second stage, this collision mechanism was substituted by a lin-

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earized collision matrix [18]. The simplicity of the dynamics when considered from this second perspective has made LB a powerful tool. 2.1 Hydrodynamic Fields The densities fj (ri , t) are the elementary dynamical variables in LB. The connection with the collective dynamics of the ﬂuid is done taking moments of these elementary densities, in the same way that hydrodynamic variables are obtained as moments of the one-particle distribution function in kinetic theory. Hence, the local density ρ(ri , t) momentum ρv(ri , t) and momentum ﬂux P(ri , t) are obtained from ρ(ri , t) = fk (ri , t) k

ρ(ri , t)v(ri , t) =

ck fk (ri , t)

k

P(ri , t) =

ck ck fk (ri , t)

(1)

k

where the index k runs over the subset of allowed velocities. 2.2 LB Dynamics The time evolution of the distribution function is composed of discrete time steps. Two dynamic events compose each of these time steps: collision and propagation. At the collision, the distribution function is relaxed toward the corresponding equilibrium distribution. This is a purely local movement. During the propagation step, the densities move to a neighboring node according to their velocity. We can express it as fj (ri + cj , t + 1) = fj (ri , t) + Λjk [fk (ri , t) − fk (ri , t)eq ] (2) k

where the index k spans the velocity subspace, fk (ri , teq ) is the equilibrium distribution function, and Λjk is a matrix that mixes the densities with different velocities at the corresponding node and ensures that both mass and momentum are conserved. This collision matrix is intimately related to the viscosities of the ﬂuid. In the simplest model (referred to as exponential relaxation time (ERT)), Λjk = τ −1 δjk is diagonal; the resulting LB is the lattice equivalent of BGK [19]. 2.3 The Equilibrium Distribution The equilibrium distribution is chosen to ensure that the collision operator conserves mass and momentum, which requires

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fk (ri , t)eq = ρ(ri , t)

k

ck fk (ri , t)eq = ρ(ri , t)v(ri , t)

(3)

k

ck ck fk (ri , t)eq = ρ(ri , t)v(ri , t)v(ri , t) + P(ri , t).

k

There exist further constraints on the acceptable forms of the equilibrium distribution function. They only become clear when analyzing the macroscopic equations that the dynamics deﬁned by (2) gives rise to on long length scales [19]. A suitable form for the equilibrium distribution is a quadratic polynomial in the velocities (which is equivalent to a low velocity expansion of the velocity distribution). Speciﬁcally,

fieq = ρ a0 + a1 v · ci + a2 vv : ci ci + a3 v2 (4) where the coeﬃcients {ai } depend on the lattice geometry and velocity subset. In the absence of additional physical requirements, the equilibrium distribution is characterized by the equation of state of an ideal gas, P = ρc2s 1, with cs being the speed of sound. Hence such an LB model behaves as a dissipative ideal gas. This simpliﬁed model is already useful for a number of colloidal hydrodynamic problems. I will describe later diﬀerent approaches to account for non-ideal ﬂuids within LB schemes. 2.4 Collective Dynamics Starting from the evolution equation (2), it is possible to perform an expansion in spatial gradients of the conserved hydrodynamic variables, analogous to the Chapman–Enskog expansion that is carried out on the Boltzmann equation. Such analysis allows us to derive the hydrodynamic equations that describe the evolution of the hydrodynamic ﬁelds. One can show that to lowest order in gradients, the LB equation recovers the expected hydrodynamic equations, ∂ρ + ∇ · (ρv) = 0 ∂t

(5)

which expresses mass conservation (the continuity equation), and 3η ∂ρv + ∇ · (ρvv) = −∇(ρc2s ) − ∇ · Π − ∇∇ : (ρvvv) ∂t ρ

(6)

where the viscous contribution to the stress tensor is given by ¯ + ξ∇ · 1) Π = η ∇v

(7)

¯ meaning the symmetric and traceless contribution from the tensor with ∇v ∇v while 1 is the identity tensor. The shear and bulk viscosities, η and ξ

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respectively, are given in terms of the eigenvalues of the collision matrix Λ [17, 19]. For the ERT model, the eigenvalue τ can be conveniently expressed as τ = 1/λ − 1/2. The ﬂuid viscosities in turn read 1 2 η= −1 6 λ 2 (8) ξ= η 3 The balance equation (6), together with the constitutive relation (7) recover essentially the Navier–Stokes equation, except for the last term in the balance equation. Such a term is of higher order in velocities, and hence can be neglected under usual conditions. Note that (6) has been written down for an ideal ﬂuid (the generic expression for the equilibrium pressure tensor Peq has been particularized to Peq = ρc2s 1 with cs being the speed of sound. The corresponding Navier–Stokes equation for a generic equation of state is discussed in [20]. It is possible to go beyond the hydrodynamic regime, analyzing the hydrodynamic modes of LB at ﬁnite wave vectors. In the case of an ideal gas equation of state it has been shown that such dependence is analytic in the wave vector [21]. This fact shows one of the useful aspects of working with a kinetic equation, rather than with the Navier–Stokes equations themselves. Even if LB is tailored to reproduce hydrodynamics, the underlying kinetic model ensures a physically meaningful behavior in regimes that go beyond the hydrodynamic limit, avoiding an uncontrolled response in those regimes. Another important aspect related to the kinetic-equation structure of LB modeling is the fact that the collision matrix can be also used to model the decay of the fast (non-conserved) modes. Although in general models this feature is not used (and such eigenvalues are set to their maximum value to ensure a proper separation of time scale that ensure proper hydrodynamic behavior), tuning them appropriately can be used at one’s advantage to model e.g viscoelasticity [22].

3 Non-Ideal Fluids: A Binary Mixture There exist a number of proposals to deal with non-ideal solvents. Some of them are based on non-local interaction rules [23], and others introduce two sets of distribution functions that follow LB kinetics like the one described previously, although only conserving the barycentric velocity, rather than each species’ momentum [24]. An alternative, fruitful approach uses the equilibrium distribution to enforce an appropriate equilibrium stress tensor consistent with a prescribed free energy functional. The equilibrium distribution, (4), has to be generalized to

fieq = ρ a0 + a1 v · ci + a2 vv : ci ci + a3 v2 + G : ci ci (9)

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introducing a second term quadratic in the microscopic velocities to account for the energetic cost of generating spatial gradients of the density ﬁelds. The coeﬃcients of fieq are now ﬁxed by imposing again (1), where now P derives from a free energy functional. This approach does not require any modiﬁcation of the LB dynamics, and it has been applied to model liquid/gas kinetics within a van der Waals model [25]. In order to model a binary ﬂuid mixture, it is necessary to introduce a second distribution, gi (r, t), that accounts for the local relative concentration of one of the two ﬂuids. One can assume that such a distribution also obeys a dynamics analogous to (2), namely, g gj (ri + cj , t + 1) = gj (ri , t) + Ljk [gk (ri , t) − gk (ri , t)eq ] (10) k

In this case, to recover the proper hydrodynamic behavior, it is enough to impose the conservation of the local concentration eq gk (ri , t) = gk (ri , t) (11) k

k

As it was the case for the distribution fj , the moments of this distribution are linked to the macroscopic physical quantities gk (ri , t) = φ(ri , t)

k

ck gk (ri , t) = φ(ri , t)v(ri , t)

k

ck ck gk (ri , t) = φ(ri , t)v(ri , t)v(ri , t) + M μ(ri , t)1

(12)

k

where now, at rest, the second moment is related to the chemical potential diﬀerence between the two species, μ. This quantity also derives from the same free energy functional from which the pressure tensor is obtained (used to determine (9)), keeping thermodynamic consistency. M is a proportionality constant related to the concentration diﬀusivity and the equilibrium density gieq has the same functional form as fieq . The speciﬁc values of the coeﬃcients appearing in the expansion are determined by requiring that in equilibrium (12) are satisﬁed, together with the appropriate expression for the chemical potential. A standard model for binary ﬂuid dynamics derives from the φ4 free energy functional, 1 A 2 B 4 κ 2 F [φ, ρ] = dr φ + φ + |∇φ(r)| + ρ(r)(ln(ρ(r) − 1) (13) 2 4 2 3 where there is no interaction associated with density variations. Hence, only a demixing transition can take place as temperature decreases (the coeﬃcients A and B depend implicitly on temperature). It can be viewed as a minimalist

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free energy model that has a second order phase transition sustaining ﬁnite width interfaces. A, B and κ are parameters that deﬁne the thermodynamics of the model. B needs to be positive to ensure thermodynamic stability. A ≥ 0 corresponds to the high temperature phase where the two species are miscible. For A ≤ 0 there is a phase separation in which two phases with concentrations φeq = ± −A/B coexist. κ determines both the interfacial width, ξ, and surface tension, σ, between the two ﬂuid phases at low temperatures, ξ =

−κ/(2A) and σ = −8κA3 /(8B 2 ). The gradient term in (13) can also be viewed as a low wave vector expansion of an underlying, more microscopic functional [26]. The dependence of the second moment of the gj on the chemical potential, together with the conservation of the local concentration, φ, is enough to ensure that φ follows a convection-diﬀusion equation at long scales. It is not necessary to impose momentum conservation because the barycentric velocity is the conserved variable for a ﬂuid mixture. For example, if we assume a BGK model for the dynamics of gi with Lgij = γδij , and we perform a Chapman– Enskog expansion of (10), along the lines of that described in Sect. 2.4, we arrive at ∂φ φ 1 1 2 ∇ (M μ) − ∇ · + ∇ · (φv) = − ∇·P (14) ∂t γ 2 ρ which is a convection-diﬀusion equation for the concentration, where the relaxation coeﬃcient γ gives us essentially the mobility. The last term in the previous equation is again small whenever compressibility does not play a relevant role. Nonetheless, appropriate minor modiﬁcations in the equilibrium distribution may be enough to remove such a spurious contribution [27]. The evolution equation for the barycentric velocity and the mean density are the same as the ones obtained for an ideal ﬂuid in Sect. 2.4, with the only modiﬁcation that one has to incorporate the appropriate pressure tensor (see [20]). This means that this simple LB with two densities gives rise to the appropriate hydrodynamics, with viscosities that are the same for both species (the relaxation of fi towards equilibrium does not depend on φ). This approach has been extended to simulate other types of complex ﬂuids. In particular, diﬀerent nematic and cholesteric phases have been addressed in detail [28]. The φ4 free energy has been generalized by including appropriate gradient terms to model ternary suspensions and lamellae [29]. There have been also extensions to model ternary mixtures [30]. An alternative approach to non-ideal ﬂuids has been developed recently in the framework of electrohydrodynamics. In this approach one does not regard the concentration as an eﬀective second one-particle distribution function. Rather, one couples LB to the convection-diﬀusion of a scalar ﬁeld. It is important to introduce the dynamics accounting appropriately for the underlying lattice symmetries to ensure that the conservation laws associated to convection-diﬀusion are satisﬁed to numerical accuracy. Such a scheme

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has proved fruitful to analyze electrokinetics of charged colloids of various shapes [31] 3.1 Spinodal Decomposition If two miscible ﬂuids are subject to a temperature quench below their critical temperature, they will start to phase separate. The initial stages of phase separation are controlled by the formation of domains where locally the concentration reaches the value of one of the coexisting phases. After this ﬁrst stage, once local domains are separated by well deﬁned interfaces, the further process of phase separation toward thermodynamic equilibrium (where a single interface will separate the two ﬂuids) will depend on the relative amount of each species. For symmetric mixtures where one has similar volume of the two ﬂuids, interconnected domains will coarsen, and the basic theoretical assumption is that a dynamic universal scenario is achieved, so that domains depend on time only through their average size L(T ) [32]. Domain growth is determined by the competition between energy release by minimizing the surface area, ∇ · P ∼ σ/L2 , where σ is the interfacial tension [20] and the dominant dissipation mechanism.; either viscous dissipa˙ 2 ) or the convection down to small scales tion (that scales as ν∇2 v ∼ ν L/L where dissipation takes place (that scales as v · ∇v ∼ L˙ 2 /L). Dynamic scaling implies that if lengths are made dimensionless by an intrinsic length scale L0 = η 2 /(ρσ), and the same is done with time, T0 = η 3 /(ρσ 2 ), then the dimensionless length ˜l ≡ L/L0 and time t˜ ≡ t/T0 must follow a universal curve. This universal curve will have two diﬀerent regimes; a viscous regime (when viscous dissipation dominates) where ˜l = b1 t˜, and an inertial regime (when convective transport to short scales dominates) where ˜l = b2/3 t˜2/3 . The amplitudes b1 and b2/3 will be also universal (once a procedure to measure the length is prescribed). These two regimes will be separated by a crossover region, which should also be universal. LB has proved an invaluable tool to establish the existence of such a universal curve [33]. While there existed good evidence of the viscous regime for symmetric quenches, a proper characterization of the inertial regime was missing [20]. By modifying the ﬂuid parameters (viscosity, surface tension), and hence L0 and T0 , it has been possible to cover 5 orders of magnitude in reduced length and 7 orders of magnitude in dimensionless times, and by superimposing a number of diﬀerent runs reconstruct the universal curve on all achieved dynamical regimes [20, 34, 35].

4 Colloidal Suspensions One of the basic strategies to simulate complex ﬂuids in LB consists in assuming a simple solvent where the solute (mesoscopic) objects are suspended.

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Colloidal particles are then characterized as rigid objects that are in thermal equilibrium because they interact with the thermal ﬂuctuations of the solvent. One can account for the colloids through their shape and combine the colloid rigid body dynamics with the ﬂuid densities on the lattice nodes [36]. To be speciﬁc, a spherical colloid can be deﬁned as the set of links that join ﬂuid to solid nodes. This set of links is uniquely deﬁned if we know the colloid’s center of mass position; hence, this set should be recomputed when the colloid displaces. The propagation step for ﬂuid densities moving along a boundary link diﬀers from the rest of the ﬂuid densities to ensure that the ﬂuid follows the solid velocity (stick boundary conditions). To this end, these ﬂuid densities reverse their velocity (bounce back) in the frame of reference comoving with the colloidal particle. Let us consider a ﬂuid density fi (r, t) which moves from node r to r + ci at time t. If after the collision (and prior to the propagation step) the density moving along the boundary link is fi (r, t)∗ (and fi (r+ci , t)∗ the corresponding density moving in the opposite direction, where I have introduced the notation ci = −ci ), then bounce-back can be written as fi (r + ci , t + 1) = fi (r + ci , t)∗ + 6ti ρvw · ci fi (r, t + 1) = fi (r, t)∗ − 6ti ρvw · ci

(15)

for a colloid interface moving with velocity vw , where the coeﬃcient ti depends on the geometry of the underlying lattice and is related to the weights of the diﬀerent velocity subsets in the equilibrium distribution. These coeﬃcients ti can be found by imposing that bounce-back does not modify the equilibrium distribution. For moving colloids, a fraction of the ﬂuid density can leak into the colloid. For simple solvents the closed, solid interface characterizing the particle decouples the interior from the exterior ﬂuid on relevant time scales for colloidal dynamics; hence one can ﬁll the colloidal particles with ﬂuid. The eﬀect of the inner ﬂuid will only show up at very short times (on the time scale in which the speed of sound propagates a distance of the order of the colloidal size) [37]. Such a diﬀerence has been quantiﬁed, and therefore it can be easily corrected for if very short time eﬀects are of interest. Alternatively, one can model colloids with no inner ﬂuid densities. In this case additional rules must be considered to account for the momentum exchange at the solid/ﬂuid interface, and to manage density updates as ﬂuid nodes are created and destroyed due to colloidal motion. The density modiﬁcation at the interface implies momentum exchange. The overall momentum exchange due to the ﬂuid on the boundary adds up to give us the force exerted by the ﬂuid on the colloid (and analogously the torque). This overall force is now used to update the position of the colloid, following a simple MD step. To be speciﬁc, the force at the boundary link due to the bounce-back is given by

1 F(rb , t + ) = 2 fi (r, t)∗ − fi (r + ci , t)∗ − 6ti ρvw · ci ci 2

(16)

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where rb = r + ci /2 is the approximate position of the boundary between nodes r and r + ci . Since the reversed densities depend on the colloid velocity (through the link velocity vw ), care should be taken in the consistency of the MD step. The implementation of an implicit, self-consistent scheme, ensures that the code is stable for buoyant colloids [38]. The overall scheme is a mixed algorithm in which the LB dynamics controlling the solvent dynamics is coupled to an MD-like scheme for the colloidal particles in the suspension. The description, even if put in the perspective of colloids, is completely general for any ﬂuid/solid interface. In fact, the ﬂexibility in deﬁning interfaces is one of the great advantages of LB, and it has been used to study suspension and ﬂow problems in porous media and generic geometries [39–41]. A similar scheme has also been used to model polymer solutions. In this case the beads of the polymer are treated as solid spheres, and are connected with springs. In principle, one could deal with every bead in the same way I have described for the colloids. However, due to its computational cost, in [42], they consider the bead to have a radius smaller than the lattice spacing. The lack of spatial resolution is solved by assuming a friction coeﬃcient for the beads that must be ﬁxed numerically. This method has been recently extended to develop an alternative way to simulate colloids in LB [43,44], and has been applied to study colloid electrophoresis. In order to simulate a suspension in equilibrium, thermal ﬂuctuations of the solvent must be included. A consistent treatment of such ﬂuctuations requires that we take into account that in equilibrium energy is equipartitioned among all the degrees of freedom allowed for the distribution functions (and hence that energy is injected appropriately for all possible velocities of the distribution functions). Such a procedure is described in detail in [45] and it has shown to reproduce correctly the equilibrium distribution of the ﬂuctuating densities. Wetting For binary mixtures it is necessary to specify the interaction between the concentration ﬁeld and the colloidal particles (and generically at any solid interface). In this case it is particularly relevant that no concentration is deﬁned inside the particles, and hence the bounce-back rules for the distributionfi must be modiﬁed appropiately [46]. One can enforce stick boundary conditions by performing the same kinetic rules for gi as introduced for fi along boundary links and appropriate modiﬁcations for a moving colloid [47]. This rule leads to colloids that neutrally wet the two phases. It is possible to tune the wettability of a solid by introducing an additional boundary condition on φ (see e.g. [48]). In this case the contact angle may be ﬁxed and tuned from complete wetting to complete dewetting.

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4.1 Colloidal Hydrodynamics The LB method to deal with suspensions described in the previous sections has the advantage that it incorporates all the relevant dynamical couplings between the colloids and the collective modes of the solvent. All the (dissipative) interactions are properly accounted for through local rules. Long-range hydrodynamic interactions ( colloid interactions mediated by the ﬂow they induce in the solvent) build up dynamically as a result of momentum conservation. This approach is then distinct from Stokesian Dynamics (SD) [49], where the solvent is integrated out and it works directly with the friction matrices which depend on colloid conﬁgurations, which assumes zero Reynolds number. SD incorporates lubrication, while in LB it has to be added on top of the basic scheme (lubrication is lacking as a result of the existence of an underlying lattice which sets a minimum resolution length scale) [19]. However, SD has to deal with the inversion of large matrices, and unless appropriate tricks are included, the method becomes numerically expensive. LB works by resolving the dynamics starting from the time scale at which momentum propagates in the solvent. This has an additional computational cost when compared to SD, because it is necessary to bridge the time scales up to the scale at which colloids diﬀuse. These two may be separated by ﬁve orders of magnitude for micron size colloids in water. How much can be achieved depends on the possibility of narrowing such a gap. In practice, it is enough to keep the proper separation of time scales by typically one order of magnitude to ensure proper physical behavior [50]. Extensive studies have quantiﬁed the performance of LB to simulate colloidal hydrodynamics. Mobility coeﬃcients and viscosities are recovered and shown to get excellent agreement with theoretical predictions and experiments in a variety of situations [51,52]. As concentration is increased, a better resolution of colloidal particles is required to account for near ﬂow ﬁeld eﬀects [41]. The diﬀerent dynamical regimes of suspensions have then be explored with this method, ranging from short [53] to long times [54]. 4.2 Colloids Under High Conﬁnement One advantage of LB is its ﬂexibility to deal with general geometries. This is useful when considering systems conﬁned to small scales, as can be the case in nanotubes or in microﬂuidic devices. The latter beneﬁt from the technology developed in recent decades in chip miniaturization to design small circuits to control the ﬂow of small quantities of ﬂuids. This is thought to be a growing area which will allow the delivery of small and well controlled amounts of materials as well as the development of electro/biological interfaces. The understanding of the ﬂow of complex ﬂuids under such conditions poses new challenges related to ﬂow control. In this respect, the use of colloids becomes natural, given that their size is smaller than the typical width of these circuits [55]. Hence, the interest in understanding colloidal dynamics under high conﬁnement.

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The eﬀect of conﬁnement on colloidal diﬀusion constitutes a fundamental question in this context. One can address this issue analyzing the velocity autocorrelation function (vacf). It will provide us with the relevant dynamical information that characterizes the hydrodynamical couplings between suspended objects and the solvent. For an unbounded suspension such a coupling is well understood. The initial velocity of a colloid of size R and mass M will aﬀect the surrounding solvent inducing both a density and a vorticity perturbation. The former will propagate away from the colloid by sound waves, aﬀecting colloid dynamics on a time scale τs ∼ R/cs (of the order of 10−8 s for an ordinary colloidal suspension). Due to momentum conservation, vorticity can only diﬀuse away from the colloid, and it does so on a time scale τν ∼ R2 /ν (of the order of 10−6 s). This turns out to be of the same order of magnitude as the time scale in which the colloid loses memory of its velocity (the inertial time scale, τξ ∼ M/(νRρ), related to the inertial response of the colloid to the ﬂuid friction). The fact that τν and τξ are of the same order implies that the dynamics of a colloid on this time scale will be controlled by a mode- coupling between these two dynamic variables. These are the relevant time scales that control the vacf. There exists still a third scale, related to the diﬀusion of the colloid itself, τD ∼ kB T /(ρνR) (of the order of 10−3 s). In the absence of mode-coupling eﬀects, the colloid’s vacf will decay exponentially due to the solvent friction. The implications of the dynamic coupling described above can be understood on simple physical grounds. For simplicity’s sake, let us assume that the velocity is a scalar. Momentum conservation leads to velocity diﬀusion away from the particle, which will then follow the diﬀusion equation ∂v (17) = ν∇2 v ∂t The moving colloid can be regarded as an initial, point-like perturbation on this diﬀusion equation of magnitude equal to the initial colloid velocity, v0 , thus leading to a ﬂow ﬁeld which will decay as a Maxwellian, v(r, t) =

v0 e−(r−ro )/(2νt) (2πνt)d/2

(18)

showing that its amplitude decays as 1/td/2 , d being the dimensionality of the system. Colloid diﬀusion will modify the amplitude, but not the functional form of the decay [56]. This algebraic decay of the vacf leads to memory eﬀects that have been measured experimentally, since this behavior persists at ﬁnite concentrations [52]. In the presence of conﬁning solid walls, the dynamic response of a suspension is modiﬁed. In this case the solid interfaces will absorb part of the momentum imparted by the colloids on the ﬂuid. As a result, the velocity diﬀusion will decay exponentially in a characteristic time L2 /ν, with L being the characteristic opening of the conﬁning geometry. Nevertheless, part of the sound that in unbound suspensions would become irrelevant on time scales τs , is now reﬂected by the solid walls and generates a counterﬂow. As a result,

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at intermediate times, the colloid moves in an environment with opposed velocity. In Fig. 1 I show the velocity, vorticity and density proﬁles for a disk moving between two slits at diﬀerent times after the colloid started moving to the right. One can see how the ﬂow ﬁeld develops collective ﬂuid motion opposed to the colloid, which eventually reverses its direction of motion. Such a reverse is clearly seen in Fig. 2.a, where the vacf becomes negative. Moreover, the decay is slow, and the vacf develops an algebraically long time tail ∗ of compressible origin which decays as 1/td /2+1 where d∗ is now the number of unbounded degrees of freedom (e.g. one for a cylindrical tube, two for two parallel plane walls). Such a decay may be understood with a reasoning analogous to that leading to (18). In this case it is sound reﬂection which leads to density diﬀusion [57,58].

Fig. 1. Flow ﬁeld, vorticity and density ﬁelds for a ﬂow surrounding a disk conﬁned between two parallel slits from top to bottom, respectively. The particle has a radius R = 5/2, is placed in a tube of diameter 17 and has an initial velocity to the right in a ﬂuid at rest. In contour plots the darkest color corresponds to the highest value of the corresponding quantity. a) Fields at a time t/τν = 0.47 after the colloid starts moving. b) Fields at a later time, t/τν = 0.78. Reprinted with permission from [58]. Copyright (1998) by the American Physical Society.

The integral of the vacf provides the suspension diﬀusion coeﬃcient. The algebraic decay under conﬁned geometries does not contribute to the diﬀusion coeﬃcient because it has a compressible origin; it only implies that the diﬀusion coeﬃcient is recovered after the colloid has properly coupled to the compressible induced ﬂow. Figure 2.b displays the diﬀusion coeﬃcient of a suspension of disks between two parallel slits for diﬀerent widths, L, in units of the colloidal diameters, λ ≡ L/(2R). The diﬀusion decreases as the slit becomes narrower. However, if one scales this obvious dependence by normalizing the diﬀusion coeﬃcient by its single particle value at the corresponding slit width, then two diﬀerent scenarios can be distinguished. When the slit can accommodate no more than one particle layer, λ < 2, then diﬀusion is

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controlled by particle-wall interactions; in this regime diﬀusivity shows a very weak dependence on suspension concentration. On the contrary, if more than two layers are allowed (λ > 2), the eﬀect of the conﬁning walls aﬀects only the scale of the diﬀusivity, the volume fraction dependence of the colloid mobility being mostly determined by particle-particle interactions.

Fig. 2. a) Velocity autocorrelation function for a sphere of radius R = 5/2 in a tube of diameter L = 9/2. Time is expressed in units of the time it takes momentum to diﬀuse the colloidal radius, τν = R2 /ν. Inset Log-log plot of the long time decay of the vacf to depict the asymptotic algebraic decay. The dashed line corresponds to the theoretical prediction τ −3/2 . (Reprinted with permission from [57]. Copyright (1997) by the American Physical Society). b) Diﬀusion coeﬃcients for colloidal suspensions of disks of radii R = 5/2 on slits of variable widths L, (λ ≡ 2R/L) as a function of the colloidal volume fraction φ. The diﬀusion coeﬃcients are expressed in units of their low concentration limit to scale out the obvious hindering of walls on particle mobility. (Reprinted with permission from [58]. Copyright (1998) by the American Physical Society).

It is possible to study the dependence of diﬀusivities on the distance to the solid walls, rather than its average behavior, as described in the previous paragraph. The conﬁning geometry gives rise to anisotropic diﬀusion and for planar surfaces the diﬀusion along the walls is non-monotonous; it exhibits a minimum in the center of the slit. These diﬀusion proﬁles give rise to significant non-Gaussian eﬀects in the dynamics of colloidal particles in conﬁned geometries [58]. 4.3 Colloids Under External Fields LB has been used to study the motion of colloids at long times and on long scales under the inﬂuence of external ﬁelds. Already the study of suspensions under the action of uniform external ﬁelds gives rise to unexpected behavior. Experiments have shown that a sedimenting suspension is characterized by the appearance of vortices and well deﬁned structures on distances that are

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long compared with the colloidal sizes [59], as shown in Figs. 3 and 4. Such structures and correlations have a purely hydrodynamic origin, because in its absence, e.g. if we assume that the ﬂuid medium simply applies an eﬀective friction on the moving colloids, the particles would settle with a mean velocity without signiﬁcant distortions in the initial uniform colloid distribution. Hence, hydrodynamic interactions induce deep modiﬁcations in the structure and behavior of suspensions when reacting to external ﬂows. The correlation length that characterizes such structures grows with volume fraction. The appearance of this characteristic length implies the existence of hydrodynamically induced particle correlations. In the absence of any correlations, the amplitude of velocity correlations for sedimenting suspensions are predicted to diverge with system size [60]. A big eﬀort has been devoted to assess both theoretically and numerically how sedimenting colloids correlate due to their long-range dissipative interactions, and in general terms on the physical mechanisms behind the observed structures and their characteristic sizes. LB simulations have been used and a careful analysis has been performed to elucidate the role played by boundary conditions. Simulations of systems containing up to 70000 colloids over long timescales (up to 2000 times the characteristic time it takes a colloid to displace its own size) and representative geometries and dynamic conditions (Reynolds numbers of order 5/100) have been performed [61]. The simulations have shown the sensitivity of the suspension structure to boundary conditions, and their relevance to velocity ﬂuctuations. Moreover, they have shown that cell geometry, volume fraction and polydispersity affect the behavior of sedimenting suspensions, and hence, a careful assessment in experiments is required to disentangle their diﬀerent eﬀects. LB studies have also been used for comparison with theoretical proposals which predict the possibility of screened and unscreened sedimentation depending on the relation between the eﬀective diﬀusivity and particle ﬂux ﬂuctuations [62]. Numerical data is also consistent with the existence of a correlation length observed experimentally. However, larger system sizes are still required to get unambiguous data and to resolve the remaining open issues in this area. Other techniques have been used to characterize the eﬀect of colloidal diffusion on sedimentation, such as stochastic rotation dynamics (see Sect. 5.1). These complementary studies have shown that diﬀusion plays a role on short time correlations, but that the structures discussed above are not modiﬁed by thermal ﬂuctuations [63]. 4.4 Colloids and Binary Mixtures In the previous examples the solvent was relevant due to the hydrodynamic coupling to the colloids and the eﬀective interactions such coupling gives rise to. However, there are other situations that lead to relevant coupling between suspended colloids and non-ideal solvents. An example of this happens if one

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Fig. 3. Particle image velocimetry data for a sedimenting suspension of polystyrene spheres of size R = 7.8μm in water. (a)-(d) velocity ﬁelds at two volume fractions ; (b)-(e) velocity ﬂuctuations (velocity relative to the mean sedimentation velocity), obtained from (a) and (d); (c)-(f) Velocity ﬂuctuation ﬂow lines. The distance bars are the characteristic correlation length obtained from correlation functions as depicted in Fig. 4. Reprinted with permission from [59]. Copyright (1997) by the American Physical Society.

Fig. 4. Velocity correlation functions of the component of the velocity of the particles parallel to gravity (z) a) in the direction of the ﬁeld and b) perpendicular to the direction of the ﬁeld. Diﬀerent symbols correspond to diﬀerent volume fractions. The distance is scaled by the factor φ−1/3 . In b) an anticorrelation is clearly visible, signaling the appearance of vortices of a well-deﬁned size which is controlled by the suspension volume fraction. Reprinted with permission from [59]. Copyright (1997) by the American Physical Society.

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considers a colloidal suspension in a symmetric binary mixture. Above its critical temperature, the two ﬂuids are miscible and the colloids are uniformly dispersed. If the system is suddenly quenched below its critical temperature in the region below the spinodal, the two ﬂuids become immiscible and start to phase separate through spinodal decomposition, as described in Sect. 3.1. A colloidal suspension will interfere with the kinetics of phase separation. Asymmetric mixtures with a minority component have been used to produce colloidosomes [64], spherical drops covered with colloids. The properties of such armoured droplets have been explored [65], and their sintering and further processing opens the possibility to obtain compact, rigid structures with sizes on the micron scale and beyond. For concreteness, we will discuss now the case of a ﬂuid mixture with similar amounts of the two ﬂuids when neutrally wetting colloids are suspended. The initial stages of spinodal decomposition are controlled by concentration diﬀusion. In this period domains develop and well deﬁned interfaces appear. Once domains are characterized by their bulk concentrations, domains will start to grow. In the case of equimolar mixtures domain coarsening is controlled hydrodynamically. During domain growth colloids will be captured at domain interfaces and they will remain kinetically trapped. In fact, neutrally wetting colloids do not have an energetic tendency to be at the ﬂuid interface, but its presence releases surface free energy. Indeed, if we call σ the ﬂuid/ﬂuid surface tension, a spherical colloid of radius R decreases the energy of the system by a factor πR2 σ if it stays at the ﬂuid/ﬂuid interface, rather than in the bulk of either phase. In units of

the thermal energy, kB T , one can deﬁne an eﬀective capillary length, R∗ = kB T /(πσ). This length is of a few ˚ A for a normal alcohol mixture, which means that colloids of a few nanometers have an energy decrease of tens of kB T . For practical purposes nanocolloids are already essentially irreversibly adsorbed if neutrally wetting. This energy gain is much larger than the entropy colloids lose by restricting their motion to the ﬂuid interface, and as a result ﬂuid interfaces will increase monotonously their colloid density during the spinodal decomposition process. As a result, at some point interfaces will be completely packed with colloids, in a kinetically arrested state (remember that the lack of amphiphilic character of neutrally wetting colloids implies that the ﬁnal equilibrium state of two ﬂuids with a planar interface is not modiﬁed by the colloidal suspension) [67]. This means that, contrary to molecular surfactants, colloidal suspensions may produce textures which are controlled by the kinetics of the process, rather than by the thermodynamic properties of the mixture. As a result, these materials oﬀer the possibility to control their properties, such as the mean pore size, by modifying the colloidal volume fraction. In fact, if we assume that all colloids, of size R, initially suspended in the mixture end up being captured at the ﬂuid -ﬂuid interface, the characteristic domain size is given by L∗ = R/φ, which can span several orders of magnitude by modifying colloidal size and suspension concentration. One can then have wide control of these systems’ properties based on their kinetics.

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Fig. 5. Time evolution of a suspension of neutrally wetting colloids in a symmetric binary ﬂuid after a quench at volume fraction φ = 0.2. Only the ﬂuid-ﬂuid interface and the colloids are shown. The former is viewed as dark or light depending on the ﬂuid it is in contact with. Only a small section of the whole system size is shown. One can see how, as time evolves, colloids are captured at the interfaces. At some point the ﬂuid interface is completely covered by colloids, and its further coarsening is arrested. The images also show the bicontinuous character of the structures formed. Reprinted with permission from [67]. Copyright (2005) by the AAAS.

In numerical simulations, one can observe how the presence of colloids slows down domain growth. It is hard to establish if the remaining slow growth will eventually stop or whether it will evolve on much longer time scales. Once the interfaces are covered by colloids, slow coarsening can only take place by the appearance of surface stresses (induced by colloid-colloid interactions; e.g. excluded volume for hard-sphere colloids) which are larger than the gain in surface free energy. Hence, colloids may be expelled from the interface, but only at high enough colloidal surface packings. Careful studies on well deﬁned geometries suggest that colloids do indeed arrest the relaxation of ﬂuid interfaces on simulation time scales [66]. For symmetric mixtures the colloids will arrest phase separation leading to the formation of bicontinuous interfacially jammed emulsions (bijels). Since the arrested structure percolates, bijels behave as solids, and their elastic properties will be characterized by the gel characteristic pore size, L∗ . Hence, by tuning the suspension volume fraction one can modify the elastic modulus of these materials. Since the elastic energy should scale with the interfacial energy density, σ/L∗ , (σ being the interfacial tension) it is possible to get static elastic modulus ranging between a few tens to 105 Pascal. Bijels will have peculiar solid properties; for example, they can sustain strains up to a certain threshold beyond which they will melt and recover their ﬂuidity. Once the force is removed, bijels are formed again (possibly with some hysteresis due to their kinetic origin). The bicontinuous character of bijels makes them appropriate materials to be use as nanoreactors. If one injects chemicals soluble in opposite ﬂuids, then they will only meet at the spaces left available at the interfaces by the colloidal particles as they are convected through the ﬂuids.

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LB has been used to predict the feasibility of bijels and to explore their main properties. This shows that one can use numerical simulation to predict new non- equilibrium structures which are controlled by the kinetics of nanoscopic-size particles. These structures start from nanoscopic objects and can generate highly heterogeneous mesoscopic structures with characteristic sizes between nano and micrometers.

5 Mesoscopic Particle-Based Methods: Eﬀective Interactions The use of kinetic modelling, such as that described in detail in previous sections, allows the development of simpliﬁed models which capture correctly the collective hydrodynamic modes of a solvent. In this way it is possible to focus the attention on the detailed interactions between dissolved nanoscopic particles and their interactions mediated by the solvent, as well as their interactions mediated by the embedding ﬂuid. In the previous sections we have shown how such a description helps us to understand the dynamics of nanocolloids in a variety of situations, covering the diﬀerent relevant time scales ranging from the inertial to structural time scales. Kinetic models neglect the structural details of the solvent as well as the details of solvent-colloid interactions. The development of particle-based mesoscopic models has opened the possibility to address the collective modes of a ﬂuid from a complementary perspective. These models capture the proper hydrodynamic modes of a solvent. It is unclear when a simple hydrodynamic picture will fail; hence these particle-based models should be used with care. Recently, a detailed comparison between LB and MD for steady states showed that agreement of collective hydrodynamic modes persists down to nanometer scales if appropriate boundary conditions are considered [68]. Although the scales down to which agreement is achieved with an eﬀective mesoscopic model will also depend on the quantity and situation of interest, it is surprising that they may be of use at such small scales. There exist diﬀerent mesoscopic methods that have been proposed which account for momentum conservation. Among these, dissipative particle dynamics (DPD) [69] and stochastic rotation dynamics (SRD) [70] have become quite popular. In both cases the solvent is regarded as a set of point particles that evolve according to eﬀective dynamics. In general, the dynamics in these systems can be written quite generally as dri = vi ; dt

mi

dvi Fij = dt

(19)

j=i

where mi , ri and vi stand for the mass, the position and velocity of the ith particle, respectively, while Fij stands for the pairwise force between particles

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i and j. Although in atomistic simulations the forces between particles derive from a potential energy, in coarse-grained methods non-conservative force contributions are generally present to account for the disregarded microscopic degrees of freedom. Accordingly, the interparticle forces can be decomposed NC as the sum of conservative, FC ij , and a non-conservative, Fij , contributions : NC Fij = FC ij + Fij . Conservative Forces The eﬀective conservative forces should account for the neglected microscopic degrees of freedom, which give rise to eﬀective interaction between the remaining particles (the particles in this case do not represent microscopic entities and their physical interpretation is still under discussion). Traditionally, such interactions are modelled as eﬀective soft pair-wise forces; for example in standard DPD : FC ij = Aw(rij )eij , with eij = (ri − rj )/|ri − rj | where w(r) is a smooth monotonous, function of r. Repulsive forces of this kind lead to a quadratic dependence of the pressure on density [71], and hence do not allow for a liquid/gas phase separation. The absence of a short range divergence in the interaction potential makes it possible to use larger time steps than if atomistic potentials were used; when combined with multicomponent ﬂuids such soft potentials lead to a ﬂexible technique to explore soft materials. This feature has been exploited in the study of membranes, as described in detail in the contribution of Shillcock in this volume. Density dependent potentials constitute an alternative in mesoscopic modelling to soft pair-wise forces [72], although they are restricted to homogeneous systems due to their dependence on the global density. As a way to overcome such limitations and allow the modeling of more general phase diagrams, a recent proposal [73] suggests the use of potentials which vary according to the local density surrounding each particle. In this case the excess energy of the system is expressed as a sum of one-body energies, uex (ni ), which depend on particles’ positions only through local density averages, ni , e.g ni = j=i wρ (rij ) where wρ (r) is a weight function of range rc normalized such that in the absence of correlations ni reduces to the ﬂuid density. In this case, the free energy of the ﬂuid is given by βf (ρ) = ln ρ + uex (ρ),

(20)

with β ≡ 1/kB T the inverse thermal energy, showing that general equations of state can be achieved with soft-potentials. The use of simple potentials, as for example the double well excess one-body energy uex (n) = K(n − ρ1 )2 (n − ρ2 )2

(21)

(with ρ1 > ρ2 without lost of generality) where K sets the energy scale and controls the ﬂuid compressibility is enough to produce liquid/gas phase separation. This fact can be understood because the forces that derive from the

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excess energy (21) may be either repulsive or attractive depending on the local density; as a result, uex (n) itself induces condensation. A critical point is found for a particular Kc and liquid/gas coexistence develops afterwards [74]. Other one-body energies have been explored to obtain liquid/gas and binary mixture models; the ﬂexibility of these models allow for a coupling of both phase transitions. Non-Conservative Interactions Diﬀerent proposals exist for FNC ij in various coarse-grained approaches which lead to the appropriate hydrodynamic collective behavior at intermediate and long scales, e.g. DPD [69], Lowe–Andersen thermostat [11] or smoothed particle hydrodynamics [75]. In DPD, non conservative forces are also taken as pairwise additive forces which depend on the relative velocities of the two particle. To ensure appropriate thermal equilibrium, additional random forces are needed to compensate C D R for the energy lost by the dissipative forces, FN ij ≡ Fij + Fij . Explicitly, D FD ij = −γω (rij )vij · eij eij ,

FR ij =

2γkB T ω D (rij )ξij

(22)

where γ is directly related to the ﬂuid viscosity, ξij is a Gaussian variable of unit variance, and usually wD (r) = w(r) for simplicity’s sake. The same function wD (r) must appear in the dissipative and random contribution to C FN to ensure that in equilibrium the ﬂuctuation-dissipation relation is satij isﬁed. It is possible to show that hydrodynamics is recovered with these two types of non-conservative forces. Hence, in the absence of conservative forces, FC ij = 0, one models a dissipative ideal gas which, as was the case for LB, is enough to capture appropriate hydrodynamic behavior with tunable transport properties decoupled from the interactions between particles. An alternative to DPD has been introduced by Lowe [11], who proposes to regard the pairwise non-conservative force as a local Anderson thermostat which modiﬁes the relative velocity of a pair of selected particles. Since this is a momentum conserving thermostat, it recovers the proper hydrodynamic modes of the solvent. This method has the advantage of avoiding some of the technical subtleties of DPD and allows a wider control on the ﬂuid Schmidt number. Examples The use of these soft potentials oﬀers the possibility to simulate complex ﬂuids with reduced local structure. This is an important feature if one is interested in studying systems at the nanoscale ensuring a proper length scale separation. This is particularly relevant when considering conﬁned ﬂuids, or thin ﬁlm dynamics. Depending on the length scales of interest the local structure

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of a ﬂuid may be an inconvenience in particle-based simulations. Hence, for sizes above the nanometer, the use of these methods allows one to couple a hydrodynamically resolved solvent with tunable interactions that do not develop signiﬁcant local structure. This is important to study wetting and other interfacial phenomena on the nanoscale [74], although physical insight is required to propose appropriate eﬀective potentials. Similar advantages and contingencies are present in the use of DPD to model membranes, as discussed in the contribution of Schillock. Colloidal suspensions have been modeled traditionally by gluing a set of particles together in the shape required. Such an approach reproduces appropriate colloidal hydrodynamics at low and intermediate volume fractions [76], although it should be treated with care at high volume fractions. 5.1 Stochastic Rotation Dynamics In the previous models, the non-conservative forces can be regarded as local, momentum-preserving thermostats [77]; hence, they can be used to address isothermal situations. Malevanets and Kapral [70] put forward an alternative method, stochastic rotation dynamics (SRD), based on (19). Point particles of mass m are advanced during a ﬁxed time interval Δt, and at the end all particles are subject to eﬀective collisions. Rather than using pairwise forces, the system is partitioned in cells of a given size a30 . The velocity of all particles in a cell, relative to the center of mass velocity of the cell vcm , are rotated with respect to a randomly chosen axis (diﬀerent for each cell) vi → vcm + R(vi − vcm )

(23)

where R is a rotation matrix which rotates velocities by a ﬁxed angle around a randomly oriented axis. This collision procedure ensures momentum conservation and scales linearly with the number of particles. Hence, this method makes it possible to simulate large system sizes. Afterwards, particles are displaced during Δt integrating Newton’s equations of motion, (19), where the total force acting on particle i, j=i Fij , reduces to the external force, fi . This method is inspired by Direct Simulation MonteCarlo (DSMC), an original approach to simulate the Boltzmann equation, introduced by Bird [78]. It conserves energy and reproduces also proper hydrodynamics. Moreover, it is relatively easy to obtain explicit expressions for the transport coeﬃcients as a function of the basic SRD parameters [79]. The use of a ﬁxed grid to execute the rotations violates Galilean invariance. Hence, it is necessary to move the grid in order to restore translational invariance. Under the action of external forces, either walls or a thermostat are required to reach a steady state. Suspensions Colloids or polymers are modeled within this scheme in the same way as in MD. In order to preserve the computational advatange of SRD one disregards

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conservative forces between the solvent particles, which are point particles, but includes solute-solute and solute-solvent interactions from a prescribed potential energy. Up to now, the potentials used have been truncated Lennard– Jones, in which the corresponding amplitudes and ranges diﬀer for the solute and solvent particles [80]. Colloids evolve according to a standard MD algorithm, such as Verlet. For the ﬂuid particles, the net colloid- ﬂuid interaction is treated as an external force in the propagation step of the SRD dynamics. The larger mass of the colloids, together with the need to resolve the steepness of the potentials require the introduction of smaller time steps for the colloid dynamics, ΔtC . Hence, two diﬀerent time steps are considered. The ratio between this colloid and the original ﬂuid time steps depends sensitively on the colloid size as well as on the potentials considered. For nanocolloids and Lennard–Jones potentials it is argued that ΔtC /Δt ∼ 4 is enough to get reasonable results. However, the introduction of eﬀective attractions, such as happens with the standard DLVO, gives rise to new length and energy scales, and ratios up to a few hundreds have been reported as required to get reasonable results [81]. Nonetheless, a judicious choice of eﬀective attractions may allow one to reach smaller ratios, as argued in [82]. In fact, since one is not resolving the molecular composition of the nanocolloids, one can introduce eﬀective potentials, that can be borrowed from molecular type simulations. The use of such potentials may alleviate some of these limitations. Colloid-ﬂuid interactions give rise to ﬂuid structuring around the colloids; a feature that must be analyzed with care. Moreover, the large ﬂuid densities at which SRD operates leads to signiﬁcant depletion interactions between nanocolloids, and induces a spurious tendency to colloid aggregation. A careful tuning of the colloid-colloid and colloid-ﬂuid interaction range is needed to ensure that the magnitude of colloid attraction is not larger than a few kB T . In any case, it should be kept in mind when interpreting colloidal equilibrium structures and non-equilibrium response. Reference [82] has a detailed and complete description of the regimes that can be achieved with SRD and how to choose the range of parameters to reproduce realistic regimes for ﬂuid dynamics. Such an analysis has made it possible to study the sedimentation of nanocolloids with realistic parameters to assess the role of colloid diﬀusion (characterized through the suspension’s P´eclet number) on sedimentation, as mentioned in Sect. 4.3. The same strategy can be used to simulate polymers suspended in a ﬂuid in and out of equilibrium. In this case the polymer is regarded as a chain of springs in which beads interact through Lennard–Jones type potentials. This method has been used to analyze the eﬀects of hydrodynamics on the kinetics of polymer collapse, and has served to show that hydrodynamics speeds up polymer collapse, although it becomes less eﬃcient for proteins, where the existence of many competing free energy minima limits the development of cooperative ﬂow [83]. SRD has also been applied to analyze the dynamics of star polymers under shear ﬂow [84]. The ﬂexibility of SRD has allowed the sampling of a signiﬁcant range in the number of arms per star, and the

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rheological behavior as a function of the arm number has shown a change in behavior ranging from a response similar to that of a linear polymer to a behavior reminiscent of the tank-treading motion observed for elastic capsules. In fact, a coupling of SRD to a model of a membrane, regarded as a triangulated surface characterized by an elastic hamiltonian has also been succesfully applied to study the behavior of closed capsules in capillaries and the diﬀerent shapes cells can attain [85]. The coupling of the membrane and the embedding solvent makes it possible to analyze the stresses the membrane is submitted to in the diﬀerent regimes.

6 Conclusions In this chapter I have discussed several approaches to model the dynamics of colloid and polymer suspensions with proper hydrodynamic coupling between the dispersed particles and the embedding solvent. In all these mesoscopic approaches some degree of physical insight is needed because one has to decide the relevant degrees of freedom. Even if a connection to a basic theoretical description exists (as is the case of Lattice Boltzmann), it is not always clear how far one can push the model to small scales. The advantage of these models is that they are naturally suited to following the evolution of colloids and other suspended objects of nanoscopic size, while capturing a proper coupling with the embedding solvent. This ensures a realistic description of the dynamics of these materials. The combined use of kinetic and particle based models gives us the possibility to explore the relevance of the microscopic details associated with the eﬀective particle-solvent interactions on the collective dynamic properties of colloidal suspensions of various sizes. I have discussed a few signiﬁcative examples to show the predictive nature of mesoscopic simulations, the system sizes and time windows that can be achieved. The results obtained clearly show that these techniques are competitive and can reach regimes no other numerical methods can achieve.

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Index

α helices, 573 β strands, 573 θ solvent, 304, 314 3D simulation methods, 269 ab-initio, 437, 439 acrylonitrile, 35 AFM, 57, 70 height image, 73, 76 tapping-mode, 72 agglomeration, 29 aggregation, 67 number, 312 air matrix, 64 alignment, 171, 252 large scale, 201 parallel, 180, 209 perpendicular, 180, 209 alkoxide, 39 alkyl chain, 119 alkyltrimethylammonium halide, 39 AMBER, 561, 568 amphiphiles, 3, 8, 397, 529 amplitude equation, 374 Anderson thermostat, 589 annealing, 90 aqueous medium, 67, 99 Asakura–Oosawa–Vrij model, 424, 425 Babinet’s principle, 104 bacteria, 5 Bancroft rule, 12 bath variables, 440 BCC-spheres, 46, 363

beads, 531, 539, 541 bending elastic modulus, 15 benzyl alcohol, 61 Berendsen’s barostat, 565 thermostat, 565 Bessel function, 105 bicontinuous, 46, 236, 272 double gyroid (GYR) structure, 479 interfacially jammed emulsions (bijels), 606 microemulsions, 3, 36 bilayers, 530, 543 bimodal distribution, 60 binary mixtures, 399, 593, 603 hard-sphere, 425 binary systems, 6 binary water-surfactant system, 8 bio-inspired systems, 529 biomimetic, 5 biomineralisation, 5 biopolymers, 563 birefringence, 217 Bjerrum length, 304 blends, 159, 356 blob size, 134, 136 block copolymer, 199, 273, 283, 287, 301, 327, 371, 478, 515 ABA, 234, 247, 482, 516 ABC, 79, 344, 356, 360, 371, 519 larger-molecular-weight, 59 multi-component systems, 349 smaller-molecular-weight, 59

618

Index

solutions, 61 star, 483 thin ﬁlms, 68 boundary conditions, 478 stick, 597 boundary velocity, 377, 385 Bragg condition, 108 branching chain, 471 Brazovskii (Hartree) ﬂuctuation corrections, 331, 335 Brazovskii energy, 373 Brazovskii equation, 373 Brownian dynamics (BD), 258, 536, 589 butadiene, 35 butterﬂy patterns, 160 canonical ensemble, 468, 472, 564 capacitor, 209 capillary length, 605 cells, 529 central miscibility gap, 19 chain stiﬀness, 542 Chapman–Enskog expansion, 592, 595 charge, 304, 397 charge annealing eﬀect, 137 CHARMM, 561, 568 CHCl3 , 210 chemical potential, 280, 471, 486 chloroform, 61, 71, 208, 245 cluster analysis, 574 cluster expansion, 426 coagulation, 25, 28 coalescence, 25, 28, 65 coarse-graining, 396, 439, 446, 535, 609 coarsening, 375 kinetics, 238 coil regions, 573 colloid, 26, 396, 440, 587, 598 charged, 413 nanoparticle mixtures, 421 polymer mixture, 422, 424 soft, 429 colloidal hydrodynamics, 599 colloidal suspensions, 35, 397, 596 complex ﬂuids, 587 conductivity, 24 conﬁnement, eﬀects, 242 high, 599

nano, 496, 498, 519 conformational asymmetry parameter, 166 conformational space, 580 connectivity, 273, 278, 285 conservative force, 541, 608 convection-diﬀusion equation, 595 convex bodies, 275 core, 133, 140, 302, 312, 317 corona, 133, 135, 140, 144, 153, 302, 310, 312 correlation function, 329 direct, 351, 410 correlation volume, 315 cosurfactant, 12 counterion, 131, 135, 142, 302, 413, 417 density proﬁle, 418 free, 311 crew-cut structures, 129 critical charge fraction, 135, 147 critical micelle concentration, 9, 310, 312 crosslinking, 38, 54 crystalline solid, 108 curvature, 546 spontaneous, 6, 13, 18 energy, 6 Gaussian, 7, 13, 100 mean, 100, 283 natural, 7 radii of, 100 cutoﬀ criterion, 561 cylinders, 33, 174, 231, 235, 272, 283, 504, 522 core-shell, 80 parallel, 505 perpendicular, 505 cylindrical, geometry, 15 mesh, 478 mesophase, 220 micelles, 320 microdomain, 201 pores, 520 D22 diﬀractometer, 140 Daoud–Cotton blob, 134, 146 Debye screening length, 15 defect, 195, 219, 223, 249, 254, 272

Index annihilation, 254 coalescence, 182 dynamics, 254, 257 horse-shoe, 255 motion, 225, 256 open ends, 256 removal, 293 topological, 254, 273, 375 degree of microphase separation, 286 degree of neutralization, 138, 147 degree of polymerization, 328 density correlations, 131 density functional theory, 407, 474 dynamic, 212, 218, 242, 246, 258, 461, 505, 520, depletion forces, 27 deuterium, 118, 244 nucleus, 117 dewetting, 238 diblock copolymers, 45, 159, 171, 287, 332, 371, 480, 531, 544 amphiphilic, 129 asymmetric, 172 cylinder-forming, 511 lamellae-forming, 70, 507 symmetric, 176 thin ﬁlms, 499 dielectric constant, 191, 201, 209 dielectric contrast, 204 dielectric permittivity, 303 diﬀerential scanning calorimetry (DSC), 168 diﬀusion, 25 coeﬃcient, 601 constant, 578 model, 65 dilute salt free solution, 304 dimers, 25 dimyristoylphosphatidylcholine, 531 dipalmitoylphosphatidylcholine (DPPC), 546 direct simulation Monte Carlo (DSMC), 610 Dirichlet boundary condition, 478 disclination, 255, 376 displacement ﬁeld, 209 dissipative force, 541 dissipative particle dynamics (DPD), 258, 537, 589, 607

distribution function theory, 409 DLVO theory, 418, 420, 587, 611 domain evolution, 375, 606 double gyroid, 49, 54, 62, 334, 348 channel, 67 double wave pattern, 53 drug delivery vehicles, 544 DSSP, 573 dynamic behavior, 231, 236, 253 dynamic exchange, 119 dynamic GRPA, 487 dynamic moduli, 155 dynamic shear moduli, 164 Edwards equation, 470 eﬀective Hamiltonian, 400, 405 eﬀective interaction, 397, 404, 607 eﬀective potential, 557 eigenvector, 580 El Nino current, 272 elastic energy, 306 electric charge screening, 10 electric ﬁeld, 171, 199, 209 DC, 201 induced alignment, 226 oblique, 221 threshold strength, 179, 220 electro-optical birefringence, 16 electrodes, 203 electron densities, 110 electrostatic blobs, 304 electrostatic interactions, 414 ellipsoidal domains, 186 emulsiﬁcation, 25 emulsion polymerisation, 34 emulsion preparation, 27 entanglement, 437, 449 entropy, 9 conﬁnement, 308 conformational, 46, 306 driven surface segregation, 236 eﬀects, 22 translational, 22, 46 epitaxial transition, 480 equilibria three-phase, 7 two-phase, 7 ergodic hypothesis, 569

619

620

Index

erucyl bis-(hydroxyethyl)methylammonium chloride, 456 erythrocyte membrane, 533 Espresso code, 536 essential dynamics, 566, 580 etching, 76 ethylene oxide, 10, 15, 119 ethyleneoxide monoalkylether, 25 Euler characteristic, 275, 277, 283, 289 evaporation/condensation mechanism, 65 exchange kinetics, 22, 25 excluded-volume interactions, 310, 422 exponential relaxation time (ERT) model, 591 external ﬁelds, 602 FCC, 343 ferroelectric particle, 27 FF-TEM, 18 ﬁeld-emission gun (FE-SEM), 51 ﬁlaments, 33 ﬁrst-stage ordering process, 81 Flory–Huggins model, 279, 351, 470 ﬂow, elongational, 194 shear, 115 ﬂuctuation, 134, 146, 178, 181, 272, 454, 504, 566, 580, 603 composition, 183 conformal, 181 dissipation theorem, 446, 474, 541, 589, 609 induced attraction, 314, 320 non-conformal, 181 perpendicular cylinder, 515 shape, 22 force ﬁeld, 557, 559 forced Rayleigh scattering (FRS), 254 four-atom interaction potential, 558 four-point scattering pattern, 174 Fourier harmonics, 331 Fourier transform, 104, 132, 224, 329, 350, 403, 466 free energy, 329, 330, 473, 594 friction, 443, 447 functional derivative, 486 fusion, 457

G family, 347, 364 G2 lattice, 347 Galilean invariance, 610 Gauss–Bonnet theorem, 16 Gaussian chain, 64, 203, 279, 304, 463, 491 Gibbs adsorption isotherm, 8 Gibbs phase rule, 11, 17 Gibbs triangle, 4, 11, 19, 32 Gibbs–Thomson eﬀect, 66 Ginzburg parameter, 351 Ginzburg–Landau free energy, 372, 484 glycocalix, 129 grain, 255 boundaries, 49, 252, 376, 377, 383 rotation, 184, 218, 223 grand canonical ensemble, 469, 473 Green–Kubo formula, 578 GROMACS, 568 GROMOS, 560, 568 half-cylinders, 243 half-lamella, 75, 77 hard sphere, 133, 152 head group, 10, 110, 541 Helfrich free energy, 13, 24 Helfrich–Hurault undulations, 181 helices, 522 heterogeneous nucleation, 35 hexagonal arrays, 33 hexagonal doughnut pattern, 53 hexagonal symmetry, 333 hexagonal-lamellar transition, 168 hexagonally packed cylinder (HEX) structure, 46, 285, 479 hexagonally perforated lamellar structure (HPL), 479 hexagons, 73 hexanol, 12 hierarchical structuring, 5 Hildebrand approximation, 356, 359 holes, 236 hollow loops, 130 homo-atomic parameters, 559 homogeneous nucleation, 34 homopolymers, 159 honeycomb-like pattern, 73 Hookean springs, 542

Index hybrids, 62, 69, 522 hydrocarbon chain, 9 hydrodynamic ﬁelds, 591 hydrodynamic interactions, 65 hydrogen bond breaking, 10 hydrogen-terminated silicon, 71 hydrophilic, 7, 530 hydrophobic, 7, 9, 530 alkyl chain, 100 core, 130 hypernetted-chain approximation, 411 Ia¯ 3d space group symmetry, 112, 334 Im3m, 112 image functionals, 273 improper dihedral angle vibration, 558 in-situ SANS, 181, 216 in-situ synchrotron SAXS, 217 incompressibility condition, 471 incompressibility limit, 353 indium tin oxide, 69 initial order, 221, 223 inner-corona region, 148 inorganic particle-surfactant assemblies, 6 instability, 225, 379, 390 inter-micelle structure, 150 inter-particle interference, 107 interactions bonded, 558 electrostatic, 559 induced, 399 induced pair, 406 interfacial, 172, 176 long-range, 46 non-bonded, 558 non-conservative, 609 short-range, 46, 70 van der Waals, 46, 559 interdroplet exchange, 31 interface, 59, 293, 606 internal, 12 interfacial energy, 28, 177, 220, 233 interfacial tension, 7, 28, 34, 48, 159, 255, 389 ionic impurities, 189 ionization, 138, 139 ions, 130 IR spectra, 138, 192, 550

621

islands, 236, 238 Jarvis–Patrick algorithms, 574 Kapton ﬁlm, 176 kinetic behavior, 238 kinetic modelling, 607 kinetic pathways, 94 kinetically stabilised systems, 4, 7, 28 Kuhn length, 304 Lagrange multiplier, 475 lamellae, 47, 61, 234, 272, 283, 499 alternating, 56 double-mixed, 502 liquid crystals, 41 microdomain, 201 mixed, 502 P2VP, 57 parallel, 68, 70, 77, 81, 86, 502 perpendicular, 48, 68, 86, 88, 501 PI, 57 symmetry, 333 three phase coexisting, 79, 87 Landau Hamiltonian, 328 Landau instability, 332 Langevin equation, 445 Laplace pressure, 26 lattice Boltzmann (LB), 590 lattice disordered spheres, 189 lattice gas cellular automata (LGCA), 590 lattice models, 496 leap-frog method, 564 Legendre transform, 407 Lennard–Jones, 14, 559, 611 Li ions, 189 Liebler energy, 373 Lifshitz line, 358 Lifshitz number, 351 light scattering, 161 LINCS, 558 line tension, 547 linear response function, 404, 417 Linkam CSS450 shear cell, 163 lipids, 530 liposomes, 546 liquid crystalline arrays, 40 liquid-like reﬂection, 113

622

Index

liquid-like structure, 106 lithium chloride, 190 lithium-polymer complexes, 191 local electroneutrality, 308 London forces, 559 long range ordered features, 237 long-range eﬀect, 86 long-range order, 188 loss modulus, 155 Louiville operator, 442 Lowe–Anderson thermostat, 609 lyonematics, 114 lyophilic, 397 lyophobic, 397 macroemulsions, 7, 25, 28 macroions, 398, 415 macromolecules, 396 macrophase separation, 56 macroscopic director order, 121 magnetic ﬁeld, 115, 119 magnetic memory density, 70 Markov process, 547 Maxwell equation, 204 Mayer functions, 427 mean breadth, 275 mean-spherical approximation, 411 melt, 372 membrane, 62, 529, 543, 610 bending modulus, 536 fusion, 547 mesh-like structure, 81 MesoDyn, 220, 256, 489 mesoporous materials, 5 mesoscopic hydrodynamics, 386 mesoscopic modelling, 372, 530, 587 Mesoworm model, 457 metal ions, 29, 69 metastable states, 94, 280 methyl methacrylate (MMA), 36 Metropolis algorithm, 547 micellar aggregates, 4 micellar cubics, 112 micellar nucleation, 36 micelle center of mass structure factor, 132 micelle, 8, 302, 328 aggregation number, 132, 303 anisotropic, 105

charged, 144 crew-cut, 320 disk shaped, 114 double-tailed, 320 fully charged, 153 interpenetrating, 154 inverse, 10 polyampholyte, 314 polyelectrolyte, 302, 309 polyelectrolyte copolymer, 130 reverse, 10, 31, 323 rod-like, 40, 115 salt-free, 149 star-like, 316, 320 wormlike, 453 micellisation, 9 microcanonical ensemble, 564 microdomain spacings, 251 microemulsion, 3, 12, 28, 159 one-phase, 20 polymeric, 160 microion plasma, 420 microphase, 328 separation, 46, 56, 72, 327 Mie-scattering, 32 miktoarm ABC copolymers, 356 Millar indices, 108 Minkowski functionals, 273, 280 missing neighbor eﬀect, 501, 519 molecular dynamics (MD), 438, 534, 555, 588 coarse-grained, 535, 542 steered, 566, 581 trajectories, 569 molecular visualization, 570 momentum transfer, 132, 141, 153 monatomic gas, 105 monomolecular brush, 75 monomolecular ﬁlm, 73 Monte Carlo, 495 oﬀ-lattice, 536 morphological image analysis, 269 morphological tailoring, 5 morphology, 327 association, 139 checkerboard, 511 body-centered cubic sphere (BCC), 333, 479 lamellar, 71, 220, 479

Index metastable, 255 non-bulk, 240 non-conventional, 340 nonequilibrium, 194 multi-phase equilibria, 3 multiple scale approach, 374 NAMD, 568 nanochannels, 67 nanocolloids, 611 nanofabrication, 48 nanohybrids, 55 nanolithography, 372 nanoporous, 5 nanoreactor, 3, 323 nanorods, 33 nanostructure, 70, 258 nanostructured biopolymers, 555 nanotechnology, 199, 372 nanotomography, 244 Navier–Stokes equation, 589, 593 nearest neighbor, 496 nematic liquid crystals, 387 Neumann boundary condition, 478 neutron scattering, 23 Newtonian ﬂuids, 387 non-electrolyte plating, 67 non-equilibrium structure, 75, 84, 93 nonlinear response, 404, 417 Nose–Hoover thermostat, 564 nuclear magnetic resonance (NMR), 12, 116 nucleation and growth, 29, 217, 223 nucleation centers, 219 Ohta–Kawasaki energy, 373 oil-in-water, 3 one-component system, 400 Onsager kinetic operator, 373 Open Computational Tool for Advanced material technology (OCTA), 462 OPLS, 560 force ﬁeld, 570 order parameters, 125, 202 order-disorder concentration, 206 order-disorder transition, 169, 212, 233, 327, 371, 382 order-order transition, 186, 255 organosiloxane, 39

623

orientation, 199 lamellar, 248 long-range, 193 mixed, 176 parallel, 90, 172, 203, 217, 382 perpendicular, 203, 382 transverse, 382 Ornstein–Zernike relation, 411 orthogonal ﬁelds, 193 orthorhombic lattice, 366 osmotic pressure, 67, 135 osmotic regime, 306, 311, 319 Ostwald ripening, 26, 65 outer-corona region, 148 ozonolysis, 49 P2VP, 49, 62, 72 packing conﬁguration, 15 packing parameter, 9 Pake powder pattern, 118 palladium acetylacetonate, 61 palmitoyloleoyl phosphatidylethanolamine (POPE), 546 parallel computing, 562 partial melting, 293 partial molar volumes, 132 particles, amorphous, 29 ceramic precursor, 27 crystalline, 29 latex, 6, 34, 36 magnetic, 5, 27 metal, 31 metal nano, 55, 91 monodisperse nano, 27 nano, 5, 27, 396 non-interacting, 105 palladium nano, 56, 91 quasi-crystalline, 29 silver, 31 soft, 428 superconducting, 27 path integral, 469 pattern analysis, 270 pattern formation, 279 patterned-homogeneous surfaces, 507 PAXY diﬀractometer, 140 PDMS, 176

624

Index

pentadecylphenol, 337 Percus–Yevick approximation, 133 periodic boundary conditions, 500, 563 perturbation theory, 401 pH, 303, 323 phase F ddd, 336, 343, 366 behaviour, 12 bicontinuous, 11, 100 bicontinuous cubic, 100, 112 coexistence, 124 crystalline surfactant, 5, 39 cubic, 4 cylindrical, 232 dilute micellar, 105 disordered, 333 droplet, 11, 14, 21 gel, 124 gyroid, 232, 272, 283 hexagonal, 4, 40, 100, 105, 111, 124, 165, 376 hexagonally perforated, 113 intermediate, 100, 113 inversion temperature, 27 isotropic, 100 isotropic micellar, 105 lamellar, 4, 24, 100, 108, 165, 374, 375, 382 liquid crystalline, 6 lyotropic, 100 mesh, 100, 105, 113 micellar cubic, 100 nematic, 100, 105, 114 perforated lamellar, 243 PI, 62 random mesh, 113 ribbon, 100 second order transition, 328, 595 solid, 6 sponge, 79, 105 thermodynamically stable, 11 transition, 122, 162 transient, 255 PHEMA, 205 phospholipids, 541 photonic crystals, 372 PI, 72 PI globules, 74 PI homopolymer, 50

pinning, 375 pixel, 282 PMMA, 69, 235, 510 Pn3m, 112 Poisson statistics, 31 Poisson–Boltzmann equation, 308 poly(ethylene)-b-poly(ethylene oxide), 160 poly(ethylene-oxide) (PEO), 160, 545 poly(ethylene-oxide)-polyethylethylene (PEO-PEE), 531, 545 polyampholytes, 302 polybutadiene, 237 polydispersity, 22, 28, 33 polyelectrolyte, 302, 397, 411 brushes, 129 corona, 130 regime, 306, 311, 318 stars, 302, 305 tail, 309 polyethylene, 160, 445 polyethylene melts, 450 polymer, 397, 469 concentration, 215 networks, 6 polymersome, 531, 543 polymethacrylate, 35 polymorphism, 3 polystyrene (PS), 67, 172, 237, 510 homopolymer, 50 layers, 337 pore formation, 547 porous polymeric ﬁlms, 37 potential of mean force, 446 precipitation, 5 preferential attraction, 234 primitive model, 413 projection operator, 441 propagator, 442 protein, 572 secondary structure, 573 protocols of Luzzati, 105 PS-b-PMMA, 172, 180, 201 PS-PMMA, 509 pseudomacroions, 398 quadrupolar contributions, 117 quasi-conserved system, 67 quasimonomers, 351

Index quaternary ammonium salt, 37 R3m, 114, 124 radial density function, 106 radical polymerisation, 34 radio-frequency energy, 117 radius of gyration, 465, 573 random copolymers, 172 random force, 444, 541 random phase approximation, 330, 350, 412, 416, 484, 489 Rayleigh–Debye–Gans scattering pattern, 161 reaction kinetics, 6 reduced critical roughness parameter, 89 reduced Flory interaction parameter, 328 reorientation kinetics, 205, 214 reptation dynamics, 437, 475 repulsive screened Coulomb potential, 133, 152 response theory, 401 restricted model, 317 reverse micellar aggregates, 4 Reynolds number, 389 rheology, 159, 162 rhombohedral mesh, 113 rigidity, 6 bending constant, 15 rod-box models, 112 rodlike polymers, 130, 397 root mean square deviation (RMSD), 571 root mean square ﬂuctuation, 579 Rouse model, 454, 475 S47 H10 M43, 202 S49 M51 , 202 S50 I50 , 202 saddle splay modulus, 16 SALS, 164, 165 salt, 313, 413 concentration, 15, 136 induced contraction, 147 free, 414 SB1, 238 SB2, 238 SB3, 238

625

SBS, 246 scaling laws, 140 scaling theory, 301 scattering, angle, 132 centres, 102 factor, 329 atomic, 104 function, 219, 485, 491 intensity, 104, 132, 202 azimuthal, 206 length contrast, 132 pattern, 107 vector, 103, 183 Scherk’s ﬁrst surface, 50 scission energy, 455 screened Coulomb interactions, 398 second shell harmonics approximation, 344 second-stage ordering process, 81 secondary ion mass spectroscopy (SIMS), 244 segmental motion, 235 segregation parameter, 288 selectivity parameter, 357 self-assembly, 55, 86, 129, 302, 371, 542 monomers, 239 surfactant systems, 3 self-consistent ﬁeld (SCF) theory, 233, 278, 334, 462, 467, 476, 504, 542, 547 dynamic, 203, 223, 279, 476, 542 self-consistent potential, 471 self-diﬀusion coeﬃcient, 453 SETTLE, 558 SFM, 234, 243 height image, 84 SHAKE, 558 shape deformation, 533 shapes, 270 shear, 154, 159, 193, 223, 252, 280, 285, 293, 387, 481 alignment, 160, 381 relaxation modulus, 454 induced deformation, 251 induced demixing, 159 induced mixing, 159 induced phase separation, 164 silica, 5, 39

626

Index

siloxane, 39 Simulation Utilities for Soft and Hard Interfaces (SUSHI), 462 single gyroid, 364 small angle neutron scattering (SANS), 18, 23, 131, 143, 162, 174, 180, 244 smoothed particle dynamics, 609 SNARE-mediated fusion, 546 soft biomaterials, 542 soft materials, 159, 395 solubilisation limit, 22 solution, 205, 208, 211, 226, 302, 319, 563 structure factor, 133 solvent, 61, 396, 440, 468, 535, 563, 575, 589 apolar, 10 diluent, 34 accessible surface area (SASA), 572 casting, 208, 213 evaporation, 213, 245 non-ideal, 593 non-selective, 81, 209 vapor treatment, 81 SPC, 563 sphere-to-cylinder transition, 187 spheres, 130, 272, 283, 522, 602 spherical droplets, 23 spherical geometry, 15 spherical mesh, 478 spherical nanoparticles, 30 spin-casting, 69 spinodal condition, 357 spinodal decomposition, 596, 605 spontaneous emulsiﬁcation, 12 stabilisation, 27 stability, 320, 340, 391 star polymers, 134, 429 statistical mechanics, 7 statistical weight, 463 stochastic rotation dynamics (SRD), 607, 610 Stokes’ law, 12 storage modulus, 155 stress tensor, 387, 592 STRIDE, 573 strong segregation approximation, 177, 328 structural evolution, 12

structure factor, 104, 131, 143, 577 partial, 144, 145, 149 static, 403, 416 styrene, 35, 36 subchain, 471 substrate, 68, 71, 172, 235 carbon coated, 240 eﬀect, 87, 92 hexagonally-patterned, 512 ITO, 88 long-range eﬀect, 94 modiﬁed, 173 nano-patterned, 507 polyimide, 79, 86 silicon, 173 SiOx , 86, 239 square-patterned, 513 stripe-patterned, 513 supported thin ﬁlms, 235 supra-molecular assemblies, 156 surface antisymmetric, 502, 510 area, 275, 283 area per molecule, 110 ﬁelds, 235, 239 free, 239 homogeneous, 499 induced alignment, 70, 81, 178 induced orientation, 180 nanopatterned, 253 neutral, 86, 513, 514 non-neutral, 70 patterned, 86, 91 relief structures, 237 roughness, 88 stepped, 90 stripe patterned, 510 structures, 245 symmetric, 502, 510 tension, 8, 605 water/air, 8 surfactant, 3, 5 concentration, 8 ionic, 9 mediated synthesis, 39 non-ionic, 10, 15, 119 ternary water-oil system, 10 water systems, 99 suspension polymerisation, 34

Index suspensions, 610 swelling, 209 Swift–Hohenberg equation, 373 swollen ﬁlm, 245 swollen polymer network, 65 symmetric blend, 166 synchrotron SAXS, 202 T-junctions, 178 tadpole conﬁguration, 309 tadpole heads, 310 tails, 541 Tanford eﬀect, 9 template, 3, 48 tension induced fusion, 548 ternary phase diagram, 200 ternary system, 3, 6 terpolymer, 71 SVT triblock, 81 terrace, 90, 238, 255 thermal blobs, 309 THF, 82, 205 thickness quantization, 70 thin ﬁlms, 47, 74, 171, 183, 201, 233, 481, 498 triblock terpolymer, 78 three-body interaction, 314, 405, 542 function, 558 three-dimensional order, 193 three-phase equilibria, 4 three-phase region, 20 tilt boundaries, 378 time constants, 216 TIP4P, 563 toluene, 222 topography, 85, 241 topological equivalance, 277 topological invariants, 277 torque, 225 torus, 283, 522 transmission electron microscopy (TEM), 12, 32, 38, 50, 62, 68, 70, 82, 131, 139, 172, 180, 185, 234, 337 Twentanglement, 449 twist boundaries, 379 two-component mixture, 400 two-dimensional lattice, 111 two-phase region, 20

627

two-scale-length behavior, 354 two-spot pattern, 174 ultrathin ﬁlms, 83 undulation, 180, 219, 511 instabilities, 221, 235 unimers, 309, 314 unrestricted model, 317 UV-O3 treatment, 76 valence, 305, 312, 422 vapor pressure, 246 velocity autocorrelation function, 579, 600, 602 Verlet’s method, 564 vertices, 341 vesicle, 10, 130, 529, 543 fusion, 545 lipid, 544 multi-lamellar, 4 synaptic, 548 virtual free energy, 340 viscoelasticity, 155, 435, 455 viscosity, 154, 201, 205, 211, 387, 454, 591 zero-shear, 456 volume, 283 vorticity, 601 water, 303, 563 water cluster, 9 water-in-oil, 3 weak crystallization, 332 weak segregation approximation, 328, 374 weakly correlated systems, 408 wetting, 598, 606 asymmetric, 234, 241 symmetric, 233, 241 Wigner–Zeitz cell, 364 Winsor I, 16, 20 Winsor II, 18, 20 Winsor III, 17 X-ray, diﬀraction, 41 coherent scattering, 103 small angle scattering (SAXS), 101, 131, 142, 143, 162, 165, 179, 251 ex-situ, 217

628

Index grazing incidence (GISAXS), 189, 235, 245

Zeeman contributions, 117 zeolites, 5, 39 zwitterionic, 37