Self-Organized Morphology in Nanostructured Materials

Springer Series in materials science 99 Springer Series in materials science Editors: R. Hull R. M. Osgood, Jr. ...

Author: Katharina Al-Shamery | Jürgen Parisi

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Springer Series in

materials science

99

Springer Series in

materials science Editors: R. Hull

R. M. Osgood, Jr.

J. Parisi

H. Warlimont

The Springer Series in Materials Science covers the complete spectrum of materials physics, including fundamental principles, physical properties, materials theory and design. Recognizing the increasing importance of materials science in future device technologies, the book titles in this series ref lect the state-of-the-art in understanding and controlling the structure and properties of all important classes of materials. 88 Introduction to Wave Scattering, Localization and Mesoscopic Phenomena By P. Sheng 89 Magneto-Science Magnetic Field Effects on Materials: Fundamentals and Applications Editors: M. Yamaguchi and Y. Tanimoto

96 GaN Electronics By R. Quay 97 Multifunctional Barriers for Flexible Structure Textile, Leather and Paper Editors: S. Duquesne, C. Magniez, and G. Camino

90 Internal Friction in Metallic Materials A Reference Book By M.S. Blanter, I.S. Golovin, H. Neuh¨auser, and H.-R. Sinning

98 Physics of Negative Refraction and Negative Index Materials Optical and Electronic Aspects and Diversiﬁed Approaches Editors: C.M. Krowne and Y. Zhang

91 Time-dependent Mechanical Properties of Solid Bodies By W. Gr¨afe

99 Self-Organized Morphology in Nanostructured Materials Editors: K. Al-Shamery and J. Parisi

92 Solder Joint Technology Materials, Properties, and Reliability By K.-N. Tu 93 Materials for Tomorrow Theory, Experiments and Modelling Editors: S. Gemming, M. Schreiber and J.-B. Suck 94 Magnetic Nanostructures Editors: B. Aktas, L. Tagirov, and F. Mikailov 95 Nanocrystals and Their Mesoscopic Organization By C.N.R. Rao, P.J. Thomas and G.U. Kulkarni

100 Self Healing Materials An Alternative Approach to 20 Centuries of Materials Science Editor: S. van der Zwaag 101 New Organic Nanostructures for Next Generation Devices Editors: K. Al-Shamery, H.-G. Rubahn, and H. Sitter 102 Photonic Crystal Fibers Properties and Applications By F. Poli, A. Cucinotta, and S. Selleri 103 Polarons in Advanced Materials Editor: A.S. Alexandrov

Volumes 40–87 are listed at the end of the book.

K. Al-Shamery J. Parisi (Eds.)

Self-Organized Morphology in Nanostructured Materials With 8 4 Figures

123

Professor Dr. Katharina Al-Shamery Universit¨at Oldenburg, Fakutlt¨at V and Center of Interface Science Carl-von-Ossietzky-Str. 9–11, 26129 Oldenburg, Germany E-mail: [email protected]

Professor J¨urgen Parisi Universit¨at Oldenburg, Fachbereich Physik, Abteilung Energie- und Halbleiterforschung Carl-von-Ossietzky-Str. 9–11, 26129 Oldenburg, Germany E-mail: [email protected]

Series Editors:

Professor Robert Hull

Professor J¨urgen Parisi

University of Virginia Dept. of Materials Science and Engineering Thornton Hall Charlottesville, VA 22903-2442, USA

Universit¨at Oldenburg, Fachbereich Physik Abt. Energie- und Halbleiterforschung Carl-von-Ossietzky-Strasse 9–11 26129 Oldenburg, Germany

Professor R. M. Osgood, Jr.

Professor Hans Warlimont

Microelectronics Science Laboratory Department of Electrical Engineering Columbia University Seeley W. Mudd Building New York, NY 10027, USA

Institut f¨ur Festk¨orperund Werkstofforschung, Helmholtzstrasse 20 01069 Dresden, Germany

ISSN 0933-033X ISBN 978-3-540-72674-6 Springer Berlin Heidelberg New York Library of Congress Control Number: 2007929993 All rights reserved. No part of this book may be reproduced in any form, by photostat, microﬁlm, retrieval system, or any other means, without the written permission of Kodansha Ltd. (except in the case of brief quotation for criticism or review.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media. springer.com © Springer-Verlag Berlin Heidelberg 2008 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data prepared by SPI Kolam using a Springer TEX macro package Cover concept: eStudio Calamar Steinen Cover production: WMX Design GmbH, Heidelberg Printed on acid-free paper

SPIN: 12067794

57/3180/SPI

543210

To Uschi

Preface

While scientists still marvel about the economical potential of the omnipresent nanomaterials promising a billion dollar market, nature is still miles ahead of us in nanotechnology. Already 500 million years ago, co-development of predator and prey coloration as well as the development of visual systems were a consequence of diversiﬁcation of life forms and were produced from mesoscopically ordered nanostructures. For example, arm ossicles from light-sensitive species of brittlestar consist of arrays of calcite microlenses with each lens minimizing spherical aberration and birefringence when focussing light towards nerve bundles. Spectacular are iridescent colors of certain insects, birds, and ﬂowers to make them visible on ultra-long ranges. The metallic blue of the wings of the tropical butterﬂy called “morpho rhetenor” is produced when light is diﬀracted at regularly ramiﬁed nanorods ordered with deﬁned distances. However, there are many other complex examples basing on the same principles: nature uses the properties of so-called photonic bandgap materials, consisting of dielectric media with periodical index modulation that inhibit propagation of light with certain colors over a range of scattering angles. People, therefore, have started to produce materials consisting of mesoscopically ordered molecules or nanoparticles, exhibiting intriguing new properties as compared with the single building blocks. The latter is also known as a novel “bottom-up” approach for nanolithography. First examples on how knowledge of the fabrication of such new materials is transferred to commercial products within a few years exist. But also, from the basic research aspect, these materials rise a lot of new questions to be dealt with in the future. The inﬂuence of morphological changes developing at a nanometer scale on the optical near ﬁeld with implications on the far ﬁeld is by far well understood. Artiﬁcial materials not known to nature, such as metamaterials with a negative refractive index, may be produced by bottom up methods soon. They will allow to build the perfect lens allowing to look at objects with light at atomic resolution or to make material invisible within a certain wavelength regime. In this volume, the question will be addressed, how to manufacture mesoscopically ordered materials. Special emphasis will be put on to compare ordering

VIII

Preface

phenomena under nonequilibrium situations, usually called self-organized structures, with those arising under situations close to equilibrium via selfassembly. Analogies are pointed out, diﬀerences are characterized, and eﬀorts will be made to ﬁnd common features in the mechanistic description of those phenomena. Of major importance is the question concerning the role of spatial and temporal order, in particular, the application of concepts developed on macroscopic and microscopic scales to structure formation occurring on nanoscales, which stands in the focus of interest on the frontiers of science. How optical properties of materials can be tuned is demonstrated in a ﬁrst example on the formation of one-dimensional waveguides from nanoaggregates of single organic building blocks. The formation of highly ordered twoand three-dimensional supramolecular structures is related to the chemical properties of the single building blocks in a second chapter. Furthermore, selfassembly of surfactants is used to produce nanomaterials of high monodispersity, enabling the self-organization of hexagonal networks of “supra” crystals, rings, tubes, dots, and labyrinths (Chaps. 2.5 and 4). Properties of the so formed mesoscopical materials can be tuned also by changing the size of the nanoscopic building blocks. The volume ﬁnally ends with treating how spatially periodic, temporally stationary turning patterns can be constructed out of nanodroplets, thus, combining elements of self-assembly with aspects of self-organization in the nonequilibrium pattern formation, arising out of the interplay between reaction and diﬀusion embedded in the self-assembled pattern. In a second example, it is shown how honeycomb carbon networks can be formed when applying the proper knowledge on transport and structuring. Finally, the book ends with a description how waves are transported in living systems. The editors would like to thank all authors for constructive eﬀorts to prepare their manuscripts and to contribute to the rich variety of topics included in this volume. Special thanks are due to Claus Ascheron and others from Springer Heidelberg for continuous commitment, eﬃcient support, and skillful technical assistance. The editors would like to thank our colleague Stefan C. M¨ uller (University of Magdeburg) for fruitful collaboration throughout drafting the concept of the book, for valuable discussions, input and support. Without him the realisation of the book would not have been possible. Furthermore the editors are grateful to all authors for constructive eﬀorts. Oldenburg July 2007

Katharina Al-Shamery J¨ urgen Parisi

Contents

1 Organic Crystalline Nanoﬁbers 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Growth of Ultrathin Films: Molecular Orientation Control . . . . . . . . . 2 1.3 Needle Films on Dedicated Templates: Mutual Orientation and Morphology Control of Nanoaggregates . . . . . . . . . . . . . . . . . . . . . . 6 1.3.1 Plain Mica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.2 Au-Modiﬁed Mica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.3 Water-Treated Mica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Selected Applications in Nano- and Microoptics . . . . . . . . . . . . . . . . . . 9 1.5 Summary and Outlook: Future Devices From Organic Nanoﬁbers . . . 14 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 Titanium-Based Molecular Architectures Formed by Self-Assembled Reactions 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Formation of Molecular Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Molecular Architectures Accompanied by Radical Induced C–C Coupling Reactions . . . . . . . . . . . . . . . . . . . . . 2.4 Molecular Architectures Based on C–C Coupling Reactions Initiated by C–H Bond Activation Reactions . . . . . . . . . . . . . . . . . . . . . 2.5 Conclusion and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Self-Assemblies of Organic and Inorganic Materials 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Structure of Colloidal Self-Assemblies Made of Surfactants and Used as Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Production of Nanocrystals by Using Colloidal Solutions as Templates and Their Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Self-Organization of Nanocrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 19 19 33 38 42 43 47 49 51 55

X

Contents

3.5 Colloidal Nanolithography by Using Nanocrystals Organized in a Given Structure as Masks [83] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4 Self-Assembled Nanoparticle Rings 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Experimental Formation of Nanoparticle Rings . . . . . . . . . . . . . . . . . . . 4.2.1 Spreading of Polymer Solution on Water Surface . . . . . . . . . . . . . 4.2.2 HDA Pancake Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 CoPt3 Nanoparticle Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Model for the Formation of HDA Pancakes . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Phase Separation of Binary Solution . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Rupture of Thin HDA Film into Micrometer-Size Pancakes . . . 4.4 Formation of a Nanoparticle Ring at the Edge of an HDA Pancake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Pinning of an HDA Micrometer-Size Pancake . . . . . . . . . . . . . . . . 4.4.2 Forces Acting on the Nanoparticle Located in the Interior of Pancake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Forces Acting on the Nanoparticle Located at the Edge of Pancake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 68 68 69 72 74 74 78 81 81 82 84 85 86

5 Patterns of Nanodroplets: The Belousov–ZhabotinskyAerosol OT-Microemulsion System 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2 The BZ-AOT System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.2.1 The BZ Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.2.2 AOT Microemulsions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.2.3 The BZ-AOT System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.3.1 Experimental Conﬁguration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.3.2 Turing Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.3.3 Patterns Associated with a Fast-Diﬀusing Activator . . . . . . . . . . 97 5.3.4 Complex Patterns – Dashes and Segments . . . . . . . . . . . . . . . . . . 100 5.3.5 Localized Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.4 Theoretical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.5 Constructing a Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.5.1 Linear Stability Analysis and Types of Bifurcations . . . . . . . . . . 106 5.5.2 Results of Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.6 Conclusion and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Contents

XI

6 Honeycomb Carbon Networks: Preparation, Structure, and Transport 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.2 Experimental Formation of Polymer Honeycomb Structures . . . . . . . . 118 6.2.1 Spreading of One Liquid on Another . . . . . . . . . . . . . . . . . . . . . . . 118 6.2.2 Production of Polymer Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.2.3 Structural Forms of Nitrocellulose Networks . . . . . . . . . . . . . . . . . 120 6.2.4 Structural Forms of Poly(p-phenylenevinylene) and Poly (3-octylthiophene) Networks . . . . . . . . . . . . . . . . . . . . . . 123 6.3 Model for the Formation of Honeycomb Structures in Polymer Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.3.1 Water Droplet on the Fluid Polymer Layer . . . . . . . . . . . . . . . . . . 125 6.4 Nitrocellulose Networks as Precursor for Carbon Networks . . . . . . . . . 132 6.4.1 Temperature Dependence of Hopping Transport in Carbon Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.4.2 Electrical Field Dependence of Hopping Transport in Carbon Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7 Chemical Waves in Living Cells 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.2 Waves of Metabolic Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7.3 Calcium Signaling Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Contributors

R. Beckhaus Institute of Pure and Applied Chemistry, University of Oldenburg 26111 Oldenburg Germany ruediger.beckhaus@ uni-oldenburg.de

H.R. Petty Departments of Ophthalmology and Visual Sciences and of Microbiology and Immunology, The University of Michigan Medical School Ann Arbor, MI 48105 USA [email protected]

I.R. Epstein Department of Chemistry and Volen Center for Comlex Systems, MS 015, Brandeis University Waltham, MA 02454 USA [email protected]

M.P. Pileni Laboratoire LM2N, URA CNRS 7070, Universit´e P. et M. Curie (Paris VI) BP 52, 4 place Jussieu 75252 Paris cedex 05 France

L.V. Govor Department of Physics, University of Oldenburg 26111 Oldenburg Germany [email protected]

H.-G. Rubahn Mads Clausen Institute, NanoSYD, University of Southern Denmark Alsion 2 6400 Soenderborg, Denmark

J. Parisi Department of Physics, University of Oldenburg 26111 Oldenburg Germany [email protected]

V.K. Vanag Department of Chemistry and Volen Center for Comlex Systems, MS 015, Brandeis University Waltham, MA 02454 USA [email protected]

1 Organic Crystalline Nanoﬁbers H.-G. Rubahn

Summary. Organic crystalline nanoﬁbers are a new class of nanoscaled organic materials that bear high potential as model systems for optics and photonics at the diﬀraction limit. In addition, due to the possibility to tailor to a large extent morphology as well as optoelectronic properties, organic nanoﬁbers are promising elements for future integrated devices. In this chapter the speciﬁc growth conditions are discussed that make the fabrication of this kind of matter possible as well as a range of applications in nano- and microoptics.

1.1 Introduction Nanooptics is about understanding and mastering the interface between the micro- and the macroworld using optical methods. In doing so new optical properties are found which are based on the dimensional conﬁnement that is a characteristic of nanoscaled materials. Metallic “quantum dots” such as Au nanoclusters are a good example of this domain, where changes in the size of the objects result in drastic changes of the optical properties [1]. These quantum dots have been well-studied in order to understand the fundamentals of the optoelectronic response in the nanodomain. In the meantime, they are also already used for, e.g., enhancing the brightness and stability of ﬂuorophores for biological imaging and are as such commercially available [2]. This illustrates the speed with which basic research results transfer into industrial products in this ﬁeld. Another example – now based on dielectric materials – is photonic band gap (PBG) materials [3]. Here, a periodic index modulation is manufactured in dielectric slabs (e.g., by laser- or electron-beam-drilling a matrix of submicronsized holes) which in the following inhibits the propagation of light of certain colors over a large range of scattering angles. In analogy to solid state physics this is called an “optical band structure.” Again, the commercialization occurred on the very short time scale of a few years, and optical ﬁbers implementing the PBG eﬀect are now available for a wide range of applications [4]. In contrast to the above-mentioned quantum dots the PBG eﬀect

2

H.-G. Rubahn

can be quantitatively understood using classical electrodynamics. And indeed the possibility to model the optical behavior of submicron-scaled materials using classical methods is often encountered in the context of nanooptics. The above examples from metallic and dielectric nanoscaled systems should not give the impression that optics in the subwavelength size regime is well understood. The inﬂuence of morphological changes on a nanometerscale on the optical near ﬁeld and from that on the resulting far ﬁeld needs to be as well investigated as the corresponding inﬂuence on the spectroscopic properties, the waveguiding or the intrinsic dynamics of optical excitations in nanoaggregates. On the application side, a thorough understanding of morphology dependencies is an important prerequisite for the controlled build-up of new nanoscaled optoelectronic elements such as light emitting devices or ﬁeld eﬀect transistors. In this chapter we describe generation and control of organic crystalline nanoﬁbers, which constitute a recently developed model system that bears a high application potential. Using organic molecules instead of inorganic compounds to build up nanostructures has the advantage of being able to work with higher luminescence eﬃciency per material density, higher ﬂexibility in terms of spectroscopic properties as well as easier and cheaper processing since controlled self assembled growth can be implemented. This chapter begins with a discussion of the growth of ultrathin organic ﬁlms on well deﬁned, single crystalline substrates. It will be shown that the growth in general depends on both molecular parameters such as chain length as well as substrate parameters such as surface free energy, polarity, roughness, etc. Depending on the exact growth conditions such as substrate temperature or growth rate, ﬁlms can be generated that consist of molecules with diﬀerent orientations with respect to the surface normal, namely nearly parallel (“upright”) or perpendicular (“laying”). Further optimization of the growth conditions results in the generation of needle-like but nonoriented structures on alkali halide crystals. Finally, by the use of the most appropriate substrate and fully optimized growth parameters, either dense arrays of nanoﬁbers or isolated nanoﬁbers are grown. Once this has been achieved, the substrate surface serves mainly as a template for producing tailored nanoaggregates, which in a next step are transferred onto other substrate surfaces.

1.2 Growth of Ultrathin Films: Molecular Orientation Control In the past, various light emitting organic molecules have been investigated in terms of their abilities to form well-organized, ultrathin ﬁlms for applications in optics or organic electronics [5]. Among those are thiophenes [6], PTCDA [7], pentacene [8], para-phenylenes [9], or anthraquinone [10] (Fig. 1.1). Up to now para-phenylenes have been found to provide us with the most

1 Organic Crystalline Nanoﬁbers

3

Fig. 1.1. Some organic molecules that have been used for building up ultrathin light-emitting ﬁlms or nanostructures are a, α-4T; b, PTCDA; c, pentacene; d, para-hexaphenylene; e, anthraquinone

promising nanoaggregates, and therefore we will concentrate here on this class of molecules. The short-chain para-phenylenes (p-nP, n = 4–6) form semiconducting ﬁlms or aggregates with rather delocalized π-electrons, and they emit blue light after excitation with either UV light (around 360 nm) or electrons. Films of this material have attracted a great deal of attention over the last years [11]. Because of their promising chemical and optical properties [12–14] they are well-suited candidates for building up active layers in organic light-emitting diodes (OLEDs) [15, 16], organic ﬁeld eﬀect transistors (OFETs) [17], and other electronic and optoelectronic devices [18].

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In principle, the molecules might be oriented either normal to the substrate or they might be aligned parallel to the substrate surface. The orientation has a strong inﬂuence on the device properties of the resulting ﬁlms or nanoaggregates. For example, the electrical conductivity for p-6P is highest perpendicularly to the molecules long axes since the HOMO has highest electron density near the middle of the molecule. As a result, ﬁlms of oriented molecules conduct also better in the direction perpendicular to the molecular axes [19]. Therefore one needs to characterize and optimize the molecular orientation. The characterization is rather simple in the case of a single crystalline ﬁlm with large domains since the individual molecules possess a well-deﬁned orientation with respect to the surface plane. In addition the main optical transition dipole moment is usually oriented along the long axes of the molecules and thus polarized absorption and emission studies can reveal the molecular orientations. For example, if all the molecules are oriented upright on the surface, then excitation under normal incidence will not result in light absorption and thus also not in luminescence. Figure 1.2 demonstrates that characteristic features in the absorption spectra allow one to distinguish between upright and laying molecules. A pronounced absorption maximum at 280 nm characterizes upright molecules. The position of this maximum is independent of the substrate material. A maximum at 340 nm characterizes laying molecules. Both maxima shift with chain length of the molecules to the red spectral regime, in agreement with theoretical predictions [20]. Another way to determine the orientation of the molecules is atomic force microscopy (AFM). Examples for a continuous ﬁlm of upright oriented p-4P molecules and a part of a needle-like structures (“nanoﬁber,” see later) made of laying p-6P molecules are shown in Figs. 1.3a, b. Height scans indicate in the case of the continuous ﬁlm terraces with height distances that correspond to tilted, normal oriented molecules (1.8 nm eﬀective length). For the nanoﬁber (Fig. 1.3b) no domains with characteristic corrugation of 2.5 nm (the length of a normal oriented and tilted p-6P molecule) are found. These conclusions agree with local optical measurements. In order to obtain well-deﬁned single crystalline organic ﬁlms on dielectric substrates subtle deposition control is needed. We have used a high vacuum system (base pressure 10−9 mbar) equipped with a fast entry lock and a multi-channelplate low energy electron diﬀraction spectrometer for the deposition of organic molecules and characterization of the resulting ﬁlms. Muscovite mica and alkali halide single crystals were cleaved in air, transferred into the apparatus, and were outgassed thoroughly at temperatures around 370 K. The substrates could be heated by a tungsten ﬁlament, and deposition took place at substrate temperatures between room temperature and 420 K via sublimation of the organic compounds from a Knudsen cell. Deposition rate and ﬁnal thickness of the organic ﬁlms were controlled via a goldplated and water-cooled quartz microbalance. Following deposition, LEED patterns were recorded in situ to verify the growth of single crystalline ﬁlms or nanoaggregates. Depending on deposition rate, substrate type, and substrate

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5

1.0

p-6P/KCI

Absorption [arb.units]

0.8

0.6

0.4

0.2 p-4P/NaCI 0.0 250

300

350

400

450

Wavelength [nm]

a)

Height [nm]

Fig. 1.2. Measured absorption spectra for ﬁlms of upright oriented p-4P molecules on NaCl and laying p-6P molecules on KCl. The absorption is given in arbitrary units and thus not to be mutually compared

b)

10 5 0 0

500 1000 Width [nm]

Fig. 1.3. (a) AFM image (2.17 × 2.17 µm2 ) of a ﬁlm of standing p-4P molecules on lithium ﬂuoride. Height information is given as a linescan in the inset. (b) Same as (a) but an AFM image (0.4 × 0.4 µm2 ) of a part of a nanoﬁber on mica (height scale 30 nm). We observe a modulation with 30 nm periodicity (black lines) which is due to the crystalline phase of the aggregates and not correlated to height variations due to individual molecules

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Fig. 1.4. Needle ﬁlm of p-5P on NaCl, generated at surface temperature of 330 K. The inset is an AFM image (6.5 × 6.5 µm2 )

temperature, ﬁlms with molecules oriented normal or parallel to the surface can be obtained. For example, if one adsorbs para-phenylenes at high deposition rates (0.5 nm s−1 ) ﬁlms of molecules oriented parallel to the surface are found in general [21]. At low deposition rates (0.02 nm s−1 ) the molecules are oriented normal to alkali halide surfaces if the deposition is performed at high temperatures. At room temperature ﬁlms of parallel oriented molecules are generated on alkali halides, whereas on mica molecules are oriented normal to the surface. If one modiﬁes the mica surface by, e.g., rinsing it in water or methanol, then a wetting layer of normal oriented molecules is generated even at high temperatures and at low deposition rates [22]. A closer look at the alkali halides (Fig. 1.4) shows that in addition to a continuous ﬁlm of p-5P molecules needle-like aggregates can be generated. The needles have widths of the order of a few hundred nanometers and heights of the order of a few ten nanometers. They are statistically distributed over the surface. Similar needle growth is observed on other surfaces such as TiO2 , too [23]. Since the needles show waveguiding properties we will use the term nanoﬁbers as an acronym.

1.3 Needle Films on Dedicated Templates: Mutual Orientation and Morphology Control of Nanoaggregates 1.3.1 Plain Mica Under certain growth conditions oriented needle growth is observed on muscovite mica surfaces (Fig. 1.5) [24, 25]. Whereas height and width of the needles are similar to those found on alkali halides, they are much longer

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Fig. 1.5. Hexaphenyl nanoﬁbers on mica, generated at a surface temperature of 400 K. Left-hand-side: epiﬂuorescence image. Right-hand-side: AFM image (6.4 × 6.4 µm2 , height scale 80 nm)

(up to several hundred micrometers) and they are very well mutually oriented. Detailed investigations via electron diﬀraction and using optical methods have shown that the growth mechanism of the needles is inﬂuenced by strong electric dipole ﬁelds that are induced on the mica surface upon cleavage [25]. These dipole ﬁelds possess two possible orientations on a cleavage plane of mica (three, if one takes into account lower lying planes), and they do not exist on alkali halide surfaces. The dipole ﬁelds induce a dipole moment in the polarizable organic molecules, leading to an attraction via dipole-induced dipole forces and thus to an alignment of the individual organic molecules along the surface dipole orientations. Subsequent molecules grow side-by-side on the adsorbed molecules on the mica surface if they possess enough surface mobility (i.e., if the surface is warm enough), leading to the generation of aligned, needle-like aggregates with very well-deﬁned molecular orientations. The needle growth process is thus dictated by the strength and orientation of the dipole ﬁelds on the surface, the polarisability of the molecules and their mobility on the surface. Domains with speciﬁc dipole directions can be huge on mica (of the order of square millimeters to centimeters) and consequently huge domains with parallel oriented nanoaggregates can be formed. The temperature window for the growth of long needles within which the surface has to be kept is only of the order of 20–30 K; at lower temperatures quasicontinuous ﬁlms consisting of very short, dense needles are formed. This strong temperature dependence allows one to grow the needles at predeﬁned spots on the surface via, e.g., local laser heating [26], and enables control over the environment of individual needles. In other words, large areas with equally distributed needles of the same morphology can be generated, but also

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Fig. 1.6. Inﬂuence of an ultrathin Au ﬁlm on the morphology (height (a), length (b)), and orientation (c) of para-hexaphenyl nanoﬁbers on mica. (d) Epiﬂuorescence image of the sample with 2 nm Au [27]

areas that contain just a few, uniquely identiﬁable needles. Both extremes are interesting in terms of fundamental nanooptical studies. 1.3.2 Au-Modiﬁed Mica Besides control over mutual orientations and density, control over the morphology of individual nanoaggregates is also important. In general it is found that height and width of the nanoaggregates can be set rather independent of each other, i.e., needles of constant height but with variable width can be generated by adjusting the growth conditions on plain mica surfaces. However, one usually pays for this variability with a very wide length distribution. More controllable results are obtained by modiﬁcation of the mica surface before starting the organic ﬁlm growth with, e.g., an ultrathin ﬁlm of gold nanoclusters. Figure 1.6 demonstrates the inﬂuence of such a layer on the morphology and orientation of p-6P nanoﬁbers. The plotted height has been determined by atomic force microscopy, whereas length and mutual orientation have been determined with the help of epiﬂuorescence microscopy. As seen, increasing the Au thickness results in a drastic decrease of lengths of the nanoﬁbers as well as a signiﬁcant increase in heights. The degree of mutual orientation of the nanoﬁbers, however, described via the standard deviation from a global orientation angle with respect to the substrate orientation does degrade only weakly. If one further increases the Au thickness to 5 nm, this behavior changes drastically, the spread in orientation angle increases to 40◦ and the length decreases to a few micrometers. Thus especially the slight modiﬁcation with a very thin ﬁlm leads to the most useful results. It is also noted that the overall luminescence eﬃciency of the needle ﬁlm depends on the Au cluster decoration. With increasing Au ﬁlm thickness the luminescence ﬁrst decreases, but then for ﬁlm thicknesses larger than 5 nm it increases again and becomes even stronger than the luminescence without Au decoration [27]. Apparently

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9

Fig. 1.7. Bent organic nanoﬁbers (p-6P) on hydrophilized mica: (a) AFM image (60 × 60 µm2 , height scale 150 nm) and (b) epiﬂuorescence image (17.5 × 25 µm2 ). Each ring in the optical image has a diameter between 4 and 5 µm and wall widths of the order of 100 nm [22]

the Au ﬁlm acts as a rough mirror and channels the emitted light along the surface normal. 1.3.3 Water-Treated Mica As demonstrated before, a strong modiﬁcation of the mica surface with, e.g., a Au cluster ﬁlm results in less straight nanoﬁbers. However, we do not observe signiﬁcantly bent nanoaggregates such as rings. Ring-formation occurs if one rinses the mica surface before organic ﬁlm growth with water and thus changes the surface hydrophobicity. In that case curved needles and rings or bent organic nanoﬁbers of various sizes are observed (Fig. 1.7). AFM images reveal that these structures grow on a wetting layer of upright oriented molecules and that the height to width ratio is diﬀerent from that found for straight needles [22]. Typical rings have widths of around 100 nm, i.e., smaller than that of the straight needles at similar growth conditions (around 300 nm), but they are signiﬁcantly higher (a few hundred nanometers). Especially the circular rings show rather narrow size distributions. Optical measurements reveal that the molecules making up the rings are oriented radially, i.e., the rings are truly bent nanoﬁbers. Following achievement of a high degree of growth control, the nanoﬁbers are used for investigating optical pecularities in the nanodomain. In the following, various applications are only brieﬂy discussed. For a more complete description the reader is referred to the original literature.

1.4 Selected Applications in Nano- and Microoptics Light emitting nanoﬁbers are an interesting model system for demonstrating the resolution limit of optical microscopy at the interface between micro- and macrocosmos. In Fig. 1.8 dark ﬁeld and epiﬂuorescence images of the same hexaphenyl nanoﬁber are shown. Structures with characteristic dimensions of

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Fig. 1.8. Comparison of dark ﬁeld (a) and epiﬂuorescence (b) images of the same nanoﬁber. The inset is an AFM image (1 × 1.5 µm2 , height scale 60 nm) of a break in the nanoﬁber that gives rise to a bright luminescence spot

a few ten nanometers such as breaks in the nanoﬁbers (exempliﬁed by an AFM image in Fig. 1.8b) are barely visible even in dark ﬁeld images since the diﬀerence in indices of refraction of nanoﬁbers and underlying substrate is small. In dark ﬁeld microscopy (Fig. 1.8a) one illuminates the sample under nearly grazing incidence, thus enhancing the visibility for structures on the surface that scatter light. Consequently such structures appear bright on a dark background. Note that the structures seen in Fig. 1.8 have heights of less than 100 nm, i.e., much smaller than the wavelength of the light used for scattering. Much better contrast and visibility of subwavelength structures is obtained in epi-ﬂuorescence microscopy (Fig. 1.8b). In such a set up UV light irradiates the nanoﬁbers under normal incidence and the resulting luminescence is observed under normal incidence, too. Excitation and luminescence light are separated with the help of a wavelength selective beam splitter and color ﬁlters. At the breaks in the needles the internally generated luminescence is scattered into the far-ﬁeld and thus submicron structures become easily visible. The true dimensions of the breaks, of course, cannot be determined via optical far ﬁeld microscopy. The possibility to separate the nanoﬁbers widely from each other (i.e., with distances that are larger than the wavelength of the emitted light) as well as their macroscopic long axes allow one to investigate in detail the inﬂuence of morphological changes in the nanometer-range on the optical properties. As an example we show in Fig. 1.9 spectra obtained from a single nanoﬁber (circles) and from an ensemble of nanoﬁbers (solid line). The spectrum from the single nanoﬁber has been obtained by illuminating the nanoﬁber inside a microscope with UV light and sampling the emitted light also inside the microscope with an optical ﬁber, connected to a miniature spectrometer. The relatively sharp spectral lines (given that the light is emitted from organic aggregates and that the samples are hold at room temperature) are due to a vibronic progression

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Luminescence Intensity [arb.units]

2.0 isolated needle

1.5

1.0

needle film 0.5

0.0 2.0

2.2

2.4

2.6

2.8

3.0

3.2

Energy [eV]

Fig. 1.9. Room temperature luminescence spectra obtained from an isolated nanoﬁber (open circles) and an ensemble of nanoﬁbers (solid line). The equidistant lines on top of the graph represent the expected vibronic progression due to the C-C stretching vibrations of all carbon atoms of the individual molecules in the nanoﬁber. Due to reabsorption the highest energy (0-0) mode is relatively weak. It becomes stronger if one cools the sample [28]

of the exciton emission (perpendicular lines on top of the graph). In the case of the single nanoﬁber spectrum the highest energy (0,0) band is not visible due to a cut-oﬀ-ﬁlter in the microscope. Nevertheless, comparison with the spectra from the needle ensembles reveals that the light emission becomes more focussed to a narrow color range (namely 420 ± 5 nm) if an individual nanoaggregate is considered. More extended spectroscopic measurements along a nanoﬁber show that the spectral width of this residual line depends on the morphology of the aggregate and that it becomes narrower if the nanoﬁber width decreases, e.g., at the tip of the nanoﬁber [28]. If one increases the intensity of the excitation light, nonlinear optical eﬀects can be observed in the nanoﬁbers. The collective nonlinear optical response of oriented arrays of para-hexaphenylene nanoﬁbers has been studied using femtosecond laser pulses [29]. At excitation wavelengths between 770 and 786 nm contributions to the two-photon signal intensity from both two-photon luminescence (TPL) and second harmonic generation (SHG) have been observed, where ISHG /ITPL ≈ 0.015. More recent studies of SHG from nanoﬁbers transferred onto glass substrates have shown that the SHG signal observed in [29] must have resulted from the wetting layer on the mica substrate. If one modiﬁes para-quaterphenylene by adding electron donor and acceptor groups (e.g., methoxy- and amino-groups) and grows nanoﬁbers from these functionalized

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molecules, then the increased hyperpolarizability of the molecules results in strong SHG from the nanoﬁbers [30]. The next logical step is to use the nonlinear optical signal to obtain spatially resolved information on molecular properties of the nanoaggregates. By use of a two-photon microscope the local polarized two-photon intensity along individual p-6P nanoﬁbers could be determined [31]. Figure 1.10 shows polarized 10×10 µm2 two-photon images of nanoﬁbers on mica. The nanoﬁbers were excited again with a femtosecond laser at 780 nm. From a comparison of the intensity distributions at diﬀerent polarization directions and employing the tensorial nature of the respective optical response one can deduce local orientations of the hexaphenyl molecules along the nanoﬁbers. Essentially, just as in the linear case the absorption (and luminescence) is maximum if the electric ﬁeld vector is parallel to the long molecular axis (which in turn is parallel to the optical transition dipole moment) and minimum if it is oriented perpendicular to it. Using the two-photon luminescence instead of the one-photon luminescence increases the spatial resolution and the signal-to-noise-ratio of the method. That way for all of the nanoﬁbers shown in Fig. 1.10 molecular orientations could be determined with a spatial resolution of less than 1 µm [31]. The results agree with possible molecular orientations predicted from bulk growth of a para-hexaphenyl crystal. Finally, organic nanoﬁbers are also a nice testing ground for methods that aim to deduce directly properties of the near ﬁeld such as scanning near ﬁeld optical microscopy (SNOM), Fig. 1.11 [32]. To obtain the images in Fig. 1.11 an inverted epiﬂuorescence microscope was mounted on a (x,y,z)-movable table and was used for focussed UV (360 nm) illumination of the sample outside

Fig. 1.10. Polarized 10 × 10 µm2 two-photon (400 nm) images of nanoﬁbers on mica. The nanoﬁbers were excited with a femtosecond laser at 780 nm with a total power of 25 mW and with its electric ﬁeld vector directed as shown with respect to the nanoﬁber axes. The detection was always polarized parallel to the individual molecules [31]

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Fig. 1.11. Near ﬁeld images (45 × 45 µm2 ) of waveguiding nanoﬁbers on mica. (a) Topographical image, (b) optical image in the near ﬁeld at contact with the nanoﬁbers. The nanoﬁbers have been excited with UV light on the left-hand side outside the viewing area shown in the plots. They are not visible in the optical far ﬁeld [32]

the direct viewing area of the SNOM. That way waveguiding through the nanoﬁbers could be measured. It is to be noted that the low transfer rate of 425 nm photons from the luminescing nanoﬁbers via the 160 nm diameter SNOM tip to the detector of less than 10−6 made rather dense needle arrays and strong focussing of the exciting UV light necessary. This, in turn, resulted in photobleaching of the samples, which limited the possible data integration time and thus the signal-to-noise ratio. A 45 × 45 µm2 scan along the sample at a constant distance of a few nanometers maintained by shear force feedback is shown in Fig. 1.11, both as topographic image (Fig. 1.11a, from the shear force feedback) and as optical image (Fig. 1.11b, from the measured counts in the photomultiplier). First of all, it is interesting to note that the optical image shows some structures at all since the observation point of the SNOM is outside the illumination area by the excitation light. This can only be explained by waveguiding of light through the nanoﬁbers, which then is transferred within the near ﬁeld into the SNOM. Second, individual nanoﬁbers show quite diﬀerent brightnesses, although they look topographically very similar and the far ﬁeld images reveal indeed almost the same brightness (not shown here). Again, since the UV excitation of the nanoﬁbers occurs outside the viewing area of the SNOM one could argue that some nanoﬁbers are not visible in the SNOM since they do not guide blue light of 425 nm. Measurements and calculations for waveguiding in individual nanoﬁbers indeed have shown that the critical minimum width for waveguiding is about 220 nm [33, 34]. The waveguiding is damped mainly by reabsorption in the nanoﬁbers. If one measures the scattered intensity as a function of distance from the excitation point, then one obtains the imaginary part of the dielectric function of an individual nanoaggregate, which is an important quantity since it determines the light-matter interaction [33].

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One should also recall that the SNOM images result from a coupling between waveguiding modes in the nanoﬁber and waveguiding modes in the SNOM ﬁber. The SNOM tip acts as a scatterer which transforms the wave vector of the nanoﬁber mode into diﬀerent wave vectors of scattered waves. Some of those scattered waves can be coupled to the propagating ﬁber modes. This process is most eﬀective if there is a phase matching between the waves. Therefore, it depends on the mutual position between nanoﬁber and the SNOM tip. The nanoﬁbers for which this condition accidentally is fulﬁlled are seen as more bright. It is tempting to assume that both waveguiding efﬁciency and phase matching are responsible for the strong selectivity of the SNOM.

1.5 Summary and Outlook: Future Devices From Organic Nanoﬁbers In this chapter growth and growth control of quasi single crystalline, ﬁber-like organic nanoaggregates on speciﬁc template surfaces have been discussed. By now it has thoroughly been demonstrated that organic molecular beam epitaxy of polarizable, rod-like molecules with large delocalized π-electrons (viz., para-phenylenes) on single crystalline, ﬂat substrate surfaces with large electric dipole domains (viz., muscovite mica) leads to the well organized growth of organic nanoﬁbers with remarkable optical properties. These nanoﬁbers have been used within the last ﬁve years for a series of benchmark experiments on static and dynamic, linear and nonlinear optics as well as morphology in the mesoscopic size regime. A few applications are detailed in this chapter. The fact that para-phenylenes plus muscovite mica constitute an unique combination from a crystallographic and growth dynamic point of view has resulted in unique nanoaggregates but obviously also limits the potential range of applications of these nanoﬁbers. However, two recent developments have opened the door to a much wider application potential of organic nanoﬁbers: (1) the possibility to transfer the nanoﬁbers from the original growth substrate to any other substrate or into liquids [35]; and (2) the possibility to functionalize a para-quaterphenylene block with speciﬁc groups and the generation of aligned nanoﬁbers from these functionalized molecules [36]. In terms of implementation of nanoaggregates into working devices the former development (1) has enabled electrical conductivity [37] as well as mechanical deformation measurements [38] on single nanoﬁbers, whereas development (2) resulted in the growth of tailored nanoscaled frequency doubling elements [30]. Further device development thus seems to be well in reach in the nearest future [39].

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Acknowledgments The author is indebted to the Danish Research Agencies FNU and FTP and to the TMR program FASTNet of the European Community as well as the Danish National Advanced Technology Foundation for ﬁnancial support. He would like to acknowledge Frank Balzer, Humboldt-University, Berlin as the coinventor of the organic p-6P nanoﬁbers. Although many more people are involved in various aspects of organic nanoﬁber research this article is based primarily on work performed together with Jonas Beerman, Jonathan Brewer, Vladimir G. Bordo, Sergey I. Bozhevolnyi, Manuela Schiek, and Valentyn Volkov.

References 1. U. Kreibig, in Optics of Nanosized Metals, ed. by R.E. Hummel, P. Wißmann, Handbook of Optical Properties, Vol. II, Optics of Small Particles, Interfaces and Surfaces, (CRC, Boca Raton, 1997), p. 145 2. B.J. Butkus, Biophoton. Int. 5, 34 (2004) 3. J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton Press, Princeton NJ, 1995) 4. See, e.g. the website of the company ‘crystal ﬁbre’: www.crystal-ﬁbre.com/ 5. G. Witte, Ch. Woell, J. Mater. Res. 19, 1889 (2004) 6. G. Ziegler, in Thin Film Properties of Oligothiophenes, ed. by H.S. Nalwa, Handbook of Organic Conductive Molecules and Polymers: Vol. 3. Conductive Polymers: Spectroscopy and Physical Properties, (Wiley, New York 1997) 7. B. Krause, A.C. D¨ urr, K.A. Ritley, H. Dosch, D. Smilgies, Phys. Rev. B 66, 235404 (2002) 8. M. Brinkmann, S. Graﬀ, C. Straupe, J.C. Wittmann, C. Chaumont, F. Nuesch, A. Aziz, M. Schaer, L. Zuppiroli, J. Phys. Chem. B 107, 10531 (2003) 9. R. Resel, Thin Solid Films 433, 1 (2003) 10. G.I. Distler, Kristall und Technik 5, 73 (1970) 11. E. Zoyer et al., Phys. Rev. B 61, 16538 (2000) 12. L. Athouel, G. Froyer, M.T. Riou, Synth. Met. 55–57, 4734 (1993) 13. L. Athouel, G. Froyer, M.T. Riou, M. Schott, Thin Solid Films 274, 35 (1996) 14. M. Era, T. Tsutsui, S. Saito, Appl. Phys. Lett. 67, 2436 (1995) 15. F. Meghdadi, S. Tasch, B. Winkler, W. Fischer, F. Stelzer, G. Leising, Synth. Met. 85, 1441 (1997) 16. K. Erlacher, R. Resel, J. Keckes, G. Leising, Mat. Sci. For., 321–324, 1086 (2000) 17. M. Ichikawa, H. Yanagi, Y. Shimizu, S. Hotta, N. Suganuma, T. Koyama, Y. Taniguchi, Adv. Mat. 14, 1272 (2002) 18. F. Quochi et al., Appl. Phys. Lett. 84, 4454 (2004) 19. H. Yanagi, S. Okamoto, T. Mikami, Synth. Met. 91, 91 (1997) 20. A. Niko, F. Meghdadi, C. Ambrosch-Draxl, P. Vogl, G. Leising, Synth. Met. 76, 177 (1996) 21. F. Balzer, H.-G. Rubahn, Surf. Sci. 548, 170 (2004)

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22. F. Balzer, J. Beermann, S. Bozhevolnyi, A.C. Simonsen, H.-G. Rubahn, Nano Lett. 3, 1311 (2003) 23. G. Koller et al., Adv. Mat. 16, 2159 (2004) 24. A. Andreev et al., Adv. Mat. 12, 629 (2000); Thin Solid Films 403–404, 444 (2002) 25. F. Balzer, H.-G. Rubahn, Appl. Phys. Lett. 79, 3860 (2001); Surf. Sci. 507–510, 588 (2002); Adv. Funct. Mat. 15, 17 (2005) 26. F. Balzer, H.-G. Rubahn, Nano Lett. 2, 747 (2002) 27. F. Balzer, L. Kankate, H. Niehus, R. Frese, C. Maibohm, H.-G. Rubahn, Nanotechnology 17, 984 (2006) 28. A.C. Simonsen, H.-G. Rubahn, Nano Lett. 2, 1379 (2002) 29. F. Balzer, K. Al-Shamery, R. Neuendorf, H.-G. Rubahn, Chem. Phys. Lett. 368, 307 (2003) 30. J. Brewer, M. Schiek, A. Luetzen, K. Al-Shamery, H.-G. Rubahn, Nano Lett. 6, 2656 (2006) 31. J. Beermann, S.I. Bozhevolnyi, V.G. Bordo, H.-G. Rubahn, Opt. Comm. 237, 423 (2004) 32. V.S. Volkov, S.I. Bozhevolnyi, V.G. Bordo, H.-G. Rubahn, J. Microsc. 215, 241 (2004) 33. F. Balzer, V.G. Bordo, A.C. Simonsen, H.-G. Rubahn, Appl. Phys. Lett. 82, 10 (2003) 34. F. Balzer, V.G. Bordo, A.C. Simonsen, H.-G. Rubahn, Phys. Rev. B 67, 115408 (2003) 35. J. Brewer, C. Maibohm, L. Jozefowski, L. Bagatolli, H.-G. Rubahn, Nanotechnology 16, 2396 (2005) 36. M. Schiek, A. Luetzen, R. Koch, K. Al-Shamery, F. Balzer, R. Frese, H.-G. Rubahn, Appl. Phys. Lett. 86, 153107 (2005) 37. J. Kjelstrup-Hansen, H.H. Henrichsen, P. Boggild, H.-G. Rubahn, Thin Solid Films 515, 827 (2006) 38. J. Kjelstrup-Hansen, P. Bogild, H.-G. Rubahn, Small 2, 660 (2006) 39. K. Al-Shamery, H.-G. Rubahn, H. Sitter (ed.), New organic nanostructures for next generation devices. Springer Ser. Mater. Sci., Berlin (2007)

2 Titanium-Based Molecular Architectures Formed by Self-Assembled Reactions R. Beckhaus

Summary. The design of highly ordered supramolecular structures has gained more and more interest within the last few decades. The concept of self-assembly chemistry takes a key position in this ﬁeld and a multitude of supramolecular compounds have been synthesized by combining simple building blocks to two- and three-dimensional structures [1–3]. Due to their electronic and steric versatility aromatic N -heterocycles play a prominent role as classical ligands in coordination compounds, [4,5] as bridging ligands in binuclear derivatives [6–8] and as building blocks for supramolecular compounds [9–17]. Beyond their capability to connect metal centers by forming ligand to metal bonds they provide the opportunity of π-backbonding and thereby may aﬀect delocalization and transport of electrons [18]. Compared with the highly developed late transition metal supramolecular chemistry, only a few attempts have been made to use the well deﬁned coordination modes and the reducing properties of early transition metals [19, 20]. In recent years the research in the area of monometallic compounds has been extended to polymetallic supramolecular systems, which may have a considerable potential to design new materials for use in photochemical molecular devices. Further more, several polynuclaer compounds have been created, which possess functionality such as nonlinear optics, molecular magnetism, anion trapping, that means e.g., to act as molecular receptors, DNA photoprobe and other photophysical properties, successfully reﬂection potential advantages of multinuclear derivatives. In the course of our studies on the reactions of low-valent titanium compounds and aromatic N -heterocycles we succeeded in the syntheses of various tetranuclear complexes for which single crystal X-ray structure analyses conﬁrmed structures of molecular squares and rectangles. Here we wish to report on the syntheses of these novel self-assembled polynuclear titanium complexes and their properties. [21, 24, 25, 31, 71, 83]

2.1 Introduction The design of highly ordered supramolecular structures has gained more and more interest within the last few decades. The concept of self-assembly chemistry takes a key position in this ﬁeld and a multitude of supramolecular

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R. Beckhaus

Scheme 2.1. Binuclear low valent titanium complexes

compounds have been synthesized by combining simple building blocks to two- and three-dimensional structures [1–3]. Due to their electronic and steric versatility aromatic N-heterocycles play a prominent role as classical ligands in coordination compounds [4, 5], as bridging ligands in binuclear derivatives [6–8], and as building blocks for supramolecular compounds [9–17]. Beyond their capability to connect metal centers by forming ligand to metal bonds they provide the opportunity of π-backbonding and thereby may aﬀect delocalization and transport of electrons [18]. Compared with the highly developed late transition metal supramolecular chemistry, only a few attempts have been made to use the well-deﬁned coordination modes and the reducing properties of early transition metals [19, 20]. The formation of molecular squares and rectangles requires 90◦ angles at the vertices, as typical for square planar or octahedral late transition metal species [1]. Hence, only a few examples are known using distorted tetrahedral geometries at the corners [21–23]. In the course of our studies on the reactions of low-valent titanium nitrogen complexes (1) [24], which are characterized by strong magnetic coupling of both titanium centers leading to a diamagnetic properties of 1 [25], we are interested in the behavior of complexes of type 2 exhibiting bisazines as bridging ligands between low valent titanium centers. Here we wish to report on the syntheses of these novel self-assembled polynuclear titanium complexes and their properties, employing diﬀerent RR H Ti

Ti

Ti

Ti H RR

3

4

Titan(II) d2

5

6

R R Ti

Ti R R

R

R 7

Titan(I) d3

8

Titan(0) d4

Scheme 2.2. Low valent titanium “corners” in diﬀerent oxidation states

2 Titanium-Based Molecular Architectures

19

N

N N

N

N

N

N

N

N

N

N

N

N

N

pyridine

pyridazine

pyrimidine

pyrazine

triazine

tetrazine

9

10

11

12

13

14

N N

N

N N

4,4'-bipyridine 2-methylpyrazine 15

16

N quinoxaline

17

Scheme 2.3. Selected N-Heterocycles useful for formation of multinuclear titanium complexes

types of low valent titanium “corners” and typical bisazine ligands. The diﬀerent titanocene (d2 ) fragments are in situ generated by liberation of bistrimethylacetylene from the corresponding acetylene complexes, [26] whereas the titanium (d3) fragments become available from the nitrogen complex 1. The titanium (d4) species can be prepared in form of the bisfulvene complexes [27] and can be used in a direct manner [28]. 2.1.1 Results and Discussion It was found in our investigations that reactions of low valent early transition metal fragments with potentially bridging bisazines leads to formation of welldeﬁned molecular architectures (A), due to the strong reducing properties to accompanied radical induced C–C coupling reactions (B) and by primary C–H bond activation reactions to multifold dehydrogenative C–C couplings forming large surface aromatic systems (C). Details are given in the next chapters.

2.2 Formation of Molecular Architectures We recently reported on the reaction of the excellent titanocene precursor [Cp2 Ti{η2 -C2 (SiMe3 )2 }] (for Cp2 Ti (3) [26]) with pyrazine (12) that leads to the formation of the ﬁrst structurally characterized molecular square with titanocene(II) corner units [Cp2 Ti(µ-C4 H4 N2 )]4 (20) [21]. Using diﬀerent starting materials (5) for the metal compound as well as for the ligand (14, 15) further neutral molecular squares with titanocene corner units can be synthesized. Scheme 2.5 shows the formation of the molecular squares 18, 19, 20, and 21, that could be characterized by single crystal X-ray analysis, elemental

20

R. Beckhaus

Ti N

N

Ti

N

3x Ti

N

N H H

Ti

N N

Ti N

H

H

N H H

H H

H

N

H

H

N H

H

N

N

N

N

H Ti

N H

N

Ti

H

N

A

B

L H

Ti L

H

3x

N N

-3H2

H

H

C

Ti

N

N

N

N

N

Ti

Ti

N

Scheme 2.4. Reaction pathways of low valent titanium complexes with N-Heterocyles (A formation of molecular architectures, B radical induced C–C coupling reactions, C dehydrogenative coupling)

analysis, and IR. All compounds are intensely colored and highly sensitive to air and moisture. The reaction of [Cp2 Ti{η2 -C2 (SiMe3 )2 }] with 4,4-bipyridine (15) in toluene leads after a few minutes to a color change from yellow to dark blue and after 48 h at 60◦ C dark blue crystals of 19 can be isolated in yields of about 50%. The tetrazine bridged complex 18 can be isolated from a dilute reaction solution of 3 and tetrazine (14) in toluene after 48 h as a crystalline solid. The dark blue crystals of both complexes show an intense metallic lustre. They are only sparingly soluble in aliphatic and aromatic solvents and ethers and do not melt below 250◦ C. In the mass spectra (EI 70 eV) no molecular peaks could be observed. Due to their low solubilities no recrystallizations are possible. Therefore, suitable single crystals for the X-ray analysis were grown from the reaction solutions. Single crystals of 19 can be obtained at 60◦ C from toluene and in better quality from tetraline at room temperature. The molecular structure of 19, crystallized from tetraline (19a), is shown in Fig. 2.1. 19a crystallizes in the space group P42 /n with four solvent molecules per tetramer. The metal atoms are coordinated tetrahedrally by two Cp ligands and two heterocycles. As the titanium atoms are located in one plane the

2 Titanium-Based Molecular Architectures N N

Ti N N

N

N N Ti N N N

N Ti

N N

N N

N N

N

N

N

14

12

R R

R

N

Ti N

N

N

Ti

N

N

R R

Ti

R R

R = t-Bu

Ti

R

N

SiMe3

R=H

18

N

Ti

- C2(SiMe3)2

- C2(SiMe3)2

Ti

R

N

N

21

20 SiMe3 R

Ti N

N

N

Ti N

N

15

R=H R = t-Bu

N

N

Ti N

N

Ti

R

R N

- C2(SiMe3)2

3 5

N

Ti N

N

15

R

N

Ti

N

N

N

N

R

- C2(SiMe3)2 R

Ti

N

R

N Ti

19 R

R

21

Scheme 2.5. Reactions of the titanocene complexes 3 and 5 with pyrazine (12), bipyridine (15), and tetrazine (14)

complex forms a nearly perfect square with the bent metallocene moieties as corner units. Single crystals of the tetrazine bridged complex 18 can be grown from dilute reaction mixtures in toluene. Figure 2.2 shows the molecular structure of 18. 18 crystallizes in the space group P421 c and the crystal contains no solvent molecules. In contrast to the analogue tetrameric pyrazine bridged complex [Cp2 Ti(µ-C4 H4 N2 )]4 (20) [21] the tetrazine complex 18 does not really form a molecular square since the four titanium atoms do not lie in one plane but rather form a tetrahedron. As it is mostly observed with tetrazine the heterocycle coordinates as a bismonodentate ligand similar to pyrazine and not as a bisbidentate ligand [7]. In order to obtain analogue complexes with higher solubilities [(t-BuCp)2 Ti{η2 -C2 (SiMe3 )2 }] was used as a source for a titanocene fragment (4) with bulky substituted Cp ligands. The reactions of 4 with pyrazine and bipyridine proceed more slowly but show the same color changes to violet and blue that are observed when using [Cp2 Ti{η2 -C2 (SiMe3 )2 }]. If the reactions with pyrazine and 4, 4 -bipyridine are carried out in n-hexane, crystals of 20 and 21 can be isolated from the reaction mixture in yields of 65% and 79%, respectively. Compared to the analogue complexes with unsubsituted Cp ligands they show a considerably increased solubility in aromatic solvents and THF. Furthermore they have lower melting points (20 197–200◦ C; 21 203–206◦ C), but again no molecular peaks could be observed in the mass spectra (EI, 70 eV).

22

R. Beckhaus Ct2 Ct1

Ti1 N1 C5

C1

C4

C2

C3 C8 C9

C7 C6

C10 N2 Ti1a

Fig. 2.1. Structure of 19a (50% probability, without H-atoms). Selected bond lengths in ˚ A and angles in ◦ : Ti1–N1 2.2132(17), Ti1–N2a 2.1976(17), Ti1–Ct1 2.086, Ti1–Ct2 2.092, N1–C1 1.358(3), N1–C1 1.374(3), N2–C6 1.363(3), N2–C10 1.370(3), C1–C2 1.366(3), C2–C3 1.418(3), C3–C4 1.418(3), C3–C8 1.424(3), C4– C5 1.366(3), C6–C7 1.368(3), C7–C8 1.424(3), C8–C9 1.423(3), C9–C10 1.364(3), N1–Ti–N2a 84.83(6), Ct1–Ti–Ct2 132.39, Ct1 = ring centroid of C11–C15, Ct2 = ring centroid of C16–C20, symmetry transformation for the generation of equivalent atoms: a = −y + 1/2, x, −z + 1/2

Single crystals of 20 could be obtained from n-hexane, single crystals of 21 were grown by slow diﬀusion of n-hexane into a THF solution. Figures 2.3 and 2.4 show the molecular structures of 20 and 21. 20 crystallizes in the space group P21 /n and the crystal contains two nhexane molecules per molecular square. 21 crystallizes in the space group P-1 and contains 11 molecules THF per tetranuclear unit. Both complexes show a more or less square conﬁguration. The sterically demanding t-butyl groups take nearly the same position in both complexes. The Ti–N distances in 19, 20, and 21 lie in the upper limit for Ti–N bonds and correspond to values expected for titanium coordinated N-heterocycles [21]. Bond lengths and angles of the titanocene units correspond to known values for tetrahedral coordination geometry.

2 Titanium-Based Molecular Architectures

23

C12 C11 C8 C1

C10 N1

Ti1

C9

N4

N3 C6

C3

C7 N3a

C2 N2

C5 C4

Fig. 2.2. Structure of 18 in the crystal (50% probability, without H-atoms). Selected bond lengths in ˚ A and angles in ◦ : Ti1–N1 2.028(5), Ti1–N3a 2.132(5), Ti1–Ct1 2.086, Ti1–Ct2 2.075, N1–C1 1.377(7), N1–N2 1.420(5), N2–C2 1.305(7), N3–C2 1.337(7), N3–N4 1.412(7), N4–C1 1.298, N1–Ti1–N3a 88.84(19), Ct1–Ti–Ct2 130.24. Ct1 = ring centroid of C3–C7, Ct2 = ring centroid of C8–C12, symmetry transformation for the generation of equivalent atoms: a = −y + 1, x + 1, −z + 2

The successful syntheses of molecular squares with the diﬀerent bridging ligands lead to the attempt to synthesize a mixed-bridged complex that contains bridging ligands of diﬀerent lengths and exhibits the structure of a molecular rectangle. Generally, only a few molecular rectangles are hitherto known because most attempts to synthesize them in one step reactions resulted in the preferred formation of the two homobridged molecular squares [3, 14]. Therefore, a reaction with two subsequent steps was used to coordinate the two diﬀerent ligands to the titanocene moiety. Scheme 2.6 shows possible synthetic routes starting from a titanocene chlorine complex in the oxidation state +III (11). In the ﬁrst reaction step the ﬁrst bridging ligand is coordinated between two [Cp2 TiCl] units whose last coordination site is blocked by the chlorine

24

R. Beckhaus

Ct07

Ct05

Ct08 C64 C63

Ct06

N6 N5 Ti3

Ti4 C65 N7 C85 C86 Ct02

C66

C88

C43

N4 C42

C87

C44

C41

N8

N3

Ti1

C22 C21

Ct04

N2 Ti2

N1 C19 C20 Ct01

Ct03

Fig. 2.3. Structure of 20 in the crystal (50% probability, without H-atoms). Selected bond lengths in ˚ A and angles in ◦ : Ti1–N8 2.132(3), Ti1–N1 2.186(3), Ti1–Ct1 2.115, Ti1–Ct2 2.118, N1–C19 1.378(4), N1–C22 1.381(4), N2–C21 1.388(4), N2–C20 1.391(4), C19–C20 1.352(4), C21–C22 1.359(4), N8–Ti1–N1 84.30(10), Ct1–Ti1–Ct2 133,49, N2–Ti2–N3 85.08(10)

atom. This reaction can be carried out successfully with pyrazine as well as bipyridine and complexes 24 and 23 can be isolated as green crystals in yields of 55% and 43%, respectively, and characterized by X-ray analysis, IR, and elemental analysis [28]. In the mass spectra of 24 and 23 only peaks of the free ligands and [Cp2 TiCl] are observed showing the low stability of the dimeric compounds. For similar monomeric compounds [Cp2 TiClL] with L = pyridine, PPhMe2 a complete dissociation into the ligand and 22 has been observed at higher temperature (130◦ C in vacuo for [Cp2 TiClPPhMe2 ]) [29]. In the second step an abstraction of the chloride ligand by reduction of the titanocene(III)complexes 24 and 23 and a coordination of the second bridging ligand has to take place. Lithium naphthalenide is used as a soluble reducing agent and the sparingly soluble rectangle 25 precipitates from the reaction mixture. To inhibit dissociation of complexes 24 and 23 and avoid an exchange of the ligands during the reduction the reaction was carried out at −78◦ C.

2 Titanium-Based Molecular Architectures

25

Ct6 Ct8 Ct5

C75

C76

Ti3

C77 C83 C84

C79 C78

N4

Ti4

N6

C82

N5 C52

C81

Ct7 C80

N7 C107

C103

C56 C106

C53

C55

C104

C105

C54 C49 C50

C47

C51

Ct3

C110 C111

C48

C109

C112 N3

C24

C20

C25

C19

C26 C28 C27

C22

Ct1

N1

C21

Ti2 N2

C108 N8

C23

Ti1

Ct2 Ct4

Fig. 2.4. Structure of 21 in the crystal (50% probability, without H-atoms). Selected bond lengths in ˚ A and angles in ◦ : Ti1–N8 2.19(3), Ti1–N1 2.22(4), Ti1–Ct1 2.039, Ti1–Ct2 2.104 Ti2–N2 2.19(4), N1–C23 1.37(5), N1–C19 1.37(5), N2–C24 1.38(6), N2–C28 1.38(5), C19–C20 1.36(6), C20–C21 1.42(6), C21–C26 1.42(6), C21–C22 1.43(6), C22–C23 1.37(6), C24–C25 1.36(7), C25–C26 1.43(6), C26–C27 1.43(6), C27–C28 1.35(7), N8–Ti1–N1 83.7(13), Ct1–Ti1–Ct2 134.18

If pathway (b) is used and 24 is reduced in the presence of pyrazine the reaction does not lead to the rectangular complex 25, instead the formation of the molecular square 20 (R:H) is observed accompanied by a further product (probably 19). However, if 23 is reduced in presence of 4, 4 -bipyridine (pathway a) 25 can be isolated as needle-shaped blue violet crystals with an intense metallic lustre. The complex could be characterized by X-ray analysis, elemental analysis, and IR spectroscopy. The synthesis of 25 can be further simpliﬁed so that starting from [Cp2 TiCl2 ] neither 22 nor 24 have to be isolated and the molecular rectangle is easily accessible from simple, commercially available starting materials. If titanocene dichloride is reduced

26

R. Beckhaus Cl

N

Ti

Ti

Cl

N Ti

15

Cl

N

N

24

22

Ti

Cl

-78˚C

pathway b)

+2 LiC10H8 N

N

12

N

pathway a)

12

Ti

-78˚C N

Cl Ti

N

N

Cl

23

Ti

N

N

15 +2 LiC10H8 -2 LiCl +2 C10H8

-2 LiCl +2 C10H8

N

N

Ti

N

N

N

N

1/2

Ti

N

N

Ti

25

Scheme 2.6. Possible synthetic routes to the molecular rectangle 25

in presence of pyrazine with one equivalent of lithium naphthalenide and then a 4, 4 -bipyridine solution and the second equivalent of lithium naphthalenide are added after cooling to −78◦ C, 25 can be isolated in 45% yield. Crystallization from THF yields crystals of 25 that are suitable for X-ray diﬀraction. The molecular structure of 25 is shown in Fig. 2.5. 25 crystallizes from THF in the trigonal space group P31 21 containing two solvent molecules per tetrameric molecule in the crystal. Each titanocene unit is coordinated by a pyrazine molecule and a 4, 4 -bipyridine molecule and with the planar conﬁguration of the four titanium atoms a rectangular geometry results for the complex. The eﬃcient synthesis of 25 requires the absence of free pyrazine (12). If 25 is reacted with 12, 20 is formed by ligand exchange. Therefore only pathway (a) is successful. On the other hand no ligand exchange reactions occur between 25 and 15. Disproportion reactions of 25 itself to 19 and 20 do apparently not take place. A similar but more soluble rectangular complex 26 can be obtained by the same procedure using [(t-BuCp)2 TiCl2 ] instead of [Cp2 TiCl2 ] as starting material. After evaporating the reaction mixture and dissolving the residue in toluene, 26 can be obtained by ﬁltration from LiCl and subsequent addition of n-hexane. Again the blue-violet crystals show an intense metallic lustre. Single crystals of 26 can be grown by recrystallisation from benzene from which 26 crystallizes in the space group P-1. The coordination geometry of 26

2 Titanium-Based Molecular Architectures C19

C21

C22

C18 C15

C20 C29

C17 Ti2

C14

C30 Ti1a

C32 N4

C27

N3 C25

N2

C31

C28

C24

C23

C16

27

C26

C33

C34

C12

C13

C11

N1 C5

C6

C1

C10 N4a Ti1

C7

C4 C9

C2

C3

C8

Fig. 2.5. Structure of 25 in the crystal (50% probability, without H-atoms). Selected bond lengths in ˚ A and angles in ◦ : Ti1–N1 2.166(4), Ti1–N4a 2.215(4), Ti1–Ct1 2.111, Ti1–Ct2 2.068, Ti2–Ct3 2.095, Ti2–Ct4 2.094 Ti2–N2 2.128(4), Ti2–N3 2.220(4), N1–C13 1.375(6), N1–C11 1.385(6), N2–C14 1.379(6), N2–C12 1.396(6), N3–C29 1.353(6), N3–C25 1.359(6), N4–C30 1.352(7), N4–C34 1.356(6), C11–C12 1.359(7), C13–C14 1.351(7), C25–C26 1.371(7), C26–C27 1.426(7), C27– C28 1.425(7), C27–C32 1.432(7), C28–C29 1.359(7), C30–C31 1.347(7), C31–C32 1.427(7), C32–C33 1.411(7), C33–C34 1.363(7), N1–Ti1–N1a 83.68(15), N2–Ti2–N3 84.10(15), Ct1–Ti–Ct2 131.48, Ct3–Ti2–Ct4 131.00. Ct1 = ring centroid of C1–C5, Ct2 = ring centroid of C6–C10, Ct3 = ring centroid of C15–C19, Ct4 = ring centroid of C20–C24, symmetry transformation for the generation of equivalent atoms: a = x − y + 1, −y + 2, −z + 2/3

shows no signiﬁcant diﬀerences to the already discussed structures of the other tetranuclear complexes. The molecular structure of 26 is shown in Fig. 2.6. The Ti–N distances to the pyrazine bridge in 25 and 26 are found shorter by 0.05–0.09 ˚ A compared to the Ti–N distances of the titanium-bipyridine bond. In contrast to most of the known molecular rectangles with basically diﬀerent sides of the rectangle [3] the titanium-based compounds 25 and 26 contain two bridging ligands of similar type. Molecular rectangles with pyrazine and 4, 4 -bipyridine bridges has become available in the case of octahedrally coordinated rhenium corners [30], exhibiting comparable, L-M-L angles (83.5◦ , 25: A, 25: 7.20×11.52 ˚ A, 26: 83.9◦ , 26: 83.8) and sizes of the cavities (7.21×11.44 ˚ 7.22 × 11.38 ˚ A). However, in 25 and 26 the rectangular geometry is realized by tetrahedrally coordinated corner atoms.

28

R. Beckhaus

Ct7 Ct5 C78

C70

C77

C69

C71

Ti4

Ti3

N5

C76

N6 Ct8 N7 C99 C100 Ct2

C74

C75

C72

C73

N4

Ct6

C48

C97

C50 C47

C98

C49 Ct4

N8

C23

C27

C22

N3

C28

C26 Ti1

N1 C19

C21 C20

C24

N2

Ti2

Ct1 Ct3

Fig. 2.6. Structure of 26 in the crystal (50% probability, without H-atoms). Selected bond lengths in ˚ A and angles in ◦ : Ti1–N8 2.121(5), Ti1–N1 2.204(5), Ti1–Ct1 2.104, Ti1–Ct2 2.093, Ti2–N2 2.153(5), Ti2–N3 2.155(6), Ti2–Ct3 2.100, Ti2–Ct4 2.088, N1–C19 1.364(8), N1–C23 1.379(7), N2–C28 1.391(8), N2–C24 1.411(7), N3–C47 1.367(8), N3–C49 1.385(7), N4–C50 1.396(8), N4–C48 1.402(7), C19–C20 1.397(8), C20–C21 1.410(8), C21–C22 1.422(8), C21–C26 1.431(7), C22–C23 1.390(8), C24– C25 1.377(8), C25–C26 1.416(8), C26–C27 1.422(8), C27–C28 1.380(8), C47–C48 1.346(9), C49–C50 1.346(9), N8–Ti1–N1 84.82(18), Ct1–Ti1–Ct2 134.66, N2–Ti2– N3 82.51(19), Ct3–Ti2–Ct4 134.36

Except for 18 all complexes contain solvent molecules that can be removed by drying the crystalline solid in vacuum. In 19a the tetraline molecules are located in canals that are formed by the molecular squares. Figure 2.7 shows a greater section of the structure of 19a including the solvent molecules. The importance of the solvent molecules for the solid state structure and the relatively great conformational freedom of the tetranuclear compounds is shown by the structures of 19 obtained by crystallization from tetraline (19a) and toluene (19b). Figure 2.8 shows the conﬁguration of the four titanium atoms in 19a and 19b in the side view onto the tetramers. Whereas the conﬁguration of the bicyclic bridged complexes 19a and 19b is inﬂuenced by the solvents used for crystallization, the monocyclic bridged complexes 18 and 20 exhibit diﬀerent conﬁgurations as well (Fig. 2.9). The diﬀerence becomes visible in the arrangement of the titanium centers. Whereas

2 Titanium-Based Molecular Architectures

29

Fig. 2.7. Solid state structure of 19a with tetraline molecules Ti Ti Ti Ti

Ti

Ti

Ti Ti

Fig. 2.8. Side view on a structure of 19a (tetraline) and 19b (toluene), Cp rings are omitted for clarity

30

R. Beckhaus Ti

Ti

Ti

Ti

Ti

Ti Ti

Ti

Fig. 2.9. Side view on 20 (left) and 18 (right)

Table 2.1. Selected geometric data of complexes 19a, 19b, 18, 20, 21, 25, and 26 Ti–Ti (˚ A) 19a 19b 18 20 21 25 26

11.586 11.500 6.691 7.203 11.521 7.203 11.516 7.219 11.380

N–Ti–N (◦ )

Ti–Ti–Ti (◦ )

Σ Ti–Ti–Ti (◦ )

84.83(6) 83.4(4) 84.3(3) 84.23(10) 83.45(13) 83.89(15)

89.75 68.05 71.24 89.82 88.15 89.37

359.0 272.2 285.0 359.3 352.6 357.5

83.79(18)

89.95

359.8

in 20 all titanium atoms lie in one plane, a “butterﬂy-like” arrangement is found for 18. Although all N–Ti–N angles lie in the relatively small range between 83.2◦ and 86.6◦ the complexes exhibit the quite diﬀerent conformations shown in Figs. 2.8 and 2.9. This is indicated by the diﬀerent Ti–Ti–Ti angles and their sum. If all corner atoms lie in the same plane a sum of 360◦ results for the quadrangle, whereas every distortion of the planar conﬁguration leads to a decrease of the sum. Table 2.1 gives the average values for the N–Ti–N angles, the Ti–Ti–Ti angles and their sum and the Ti–Ti distances for all complexes. The values of the sum of the Ti–Ti–Ti angles show that 19a, 20, 21, 25, and 26 take a nearly planar conﬁguration whereas 19b and 18 are distorted from this planarity. As the titanocene fragment has proved to reduce some aromatic Nheterocycles by C–C-coupling reactions [21] and C–H and C–F bond cleavage [31] as well as to form complexes with stable heterocyclic radical anions by

2 Titanium-Based Molecular Architectures

31

2N N

N

+e-

N N

N

N

N

+e-

N N

N N 2-

N

N

+e-

N

N

+e-

N

N

Scheme 2.7. Two electron reductions of tetrazine and bipyridine

electron transfer [32, 33] it seems reasonable to presume an electron transfer to the ligand also for this reactions and some prove for this is given by the molecular structures in the solid state. In numerous complexes of low valent metals and heterocyclic ligands the change of bond lengths and angles that are sensible to the transfer of electrons has been used to evidence the electronic structure. With tetrazine derivatives as bridging ligands for example a lengthening of the N–N bond of 0.068 ˚ A and 0.073 ˚ A has been observed by Kaim et al. indicating the formation of a radical anion in two copper complexes [34, 35]. In a Fe(0)-complex of 2, 2 bipyridine the electron transfer to the heterocycle leads to a decrease of the bond length between the two pyridyl rings for 0.083 ˚ A, in a dimeric Yb(II)A complex of 2, 2 -bipyrimidine with a dianionic ligand a shortening for 0.142 ˚ is observed [6, 36, 37]. Scheme 2.7 shows the two reduction steps for tetrazine and bipyridine [7, 8]. A two electron reduction of tetrazine leads to a single bond between the two nitrogen atoms and the reduction of 4,4-bipyridine to a double bond between the two aromatic rings and an angle of 0◦ between the two pyridyl rings. These bonds and angles, respectively, show the greatest change upon reduction. To compare our data to the values of the free ligand and a fully reduced species the crystal structures of free 4,4-bipyridine (15) and the two electron reduced bis(trimethylsilyl)dihydro-4, 4 -bipyridine (27) [28] were determined. The asymmetric units of both structures contain two independent molecules and for 4,4-bipyridine the important values are given for both since the angle between the pyridyl rings diﬀer signiﬁcantly in the structures of the two molecules. In 27 no signiﬁcant diﬀerences exist so that here the average is given. Table 2.2 shows the angles and bond lengths between the two pyridyl rings in 4, 4 -bipyridine for the free ligand, for the two electron reduced species 27, and for the titanium coordinated ligands. These values show a signiﬁcant shift toward the two electron reduced species for the coordinated ligands. Nevertheless, the decrease is less than that observed for the two electron reduced bipyrimidine in the ytterbium complex of Andersen that contains bipyrimidine as a dianionic ligand [6]. So the changes in bond lengths and angles may indicate for the 4, 4 -bipyridine complexes an electronic structure with a radical ligand and titanium centers

32

R. Beckhaus

Table 2.2. Bond lengths and angles for free, reduced, and coordinated 4, 4 -biypridine C − C (˚ A)a

Angle of twist (◦ )

1.4842(19) 1.4895(18) 1.381(3) 1.424(3) 1.425(30) 1.432(7) 1.438(7)

15 27a 19a 21 25 26

34.39(6) 18.14(8) 0 4.23 7.60 4.89 9.80

a)

Me3Si N

N SiMe3 27

in the oxidation state +III. Scheme 2.8 shows the eﬀect of shortening and lowering of the twist angles in 4,4-bipyridyne complexes depending on the oxidation states of titanium centers [28]. A change of bond lengths toward the reduced species is also observed in the lengthening of the N–N bond in tetrazine from 1.321 ˚ A in free tetrazine [38] to 1.416(6) ˚ A in 18. Another sign for the reduction of the ligand is given by the change in the conformation of the ligand in 18 that does no longer exhibit the planarity of the free ligand (Fig. 2.2). The electronic structure of the radical complexes formed by electron transfer from titanium to N-heterocyclic ligands has been thoroughly investigated with the 2,2-bipyridine complexes that show a ground singlet and a thermally accessible triplet state [32, 33, 39]. Alike to the monomeric 2, 2 -bipyridine complexes for the 4, 4 -bipyridine complex 6 an antiferromagnetic behavior is observed for the temperature dependence of the magnetic susceptibility. Owing to the special geometry of the frontier orbitals in metallocene units, [40,41] the 2, 2 -bipyridine cannot act as a classical π-acceptor ligand but forms complexes with two remote electrons in interaction that are not diamagnetic as the π-acceptor complexes of titanocene with carbonyl or phosphane ligands. If an overlap of metal and ligand orbitals is possible a ground singlet state results for the two remote electrons [39]. This situation occurs if the ligand orbitals lie in the L-Ti-L plane. For the overlap with the π∗ -orbitals of the heterocycle in the tetrameric complexes it is thus important how far the ligand is rotated around the Ti–N bond. For this the tetrazine complex 18 shows two diﬀerent conformations of the bridging ligands toward the titanium atoms. The two respective bond lengths diﬀer in a characteristic manner as also known for dπ–pπ interactions in titanium amides [42]. With the ligand in the straight position (N/N/C plane of 14 to N/Ti/N 55.32◦ ) an overlap of metal acceptor and ligand donor

2 Titanium-Based Molecular Architectures

33

1.392(7) Å

1.487(3) Å Ti(III) - d1

Ti(II) - d2 Ti(II) - d2

Ti(III) - d1

29.4 ˚

0˚ 28

24

1.424(3) Å 4.2 ˚ 21

Scheme 2.8. Comparison of internal C–C distances and twist angles of 4,4-bipyridine ligands in 21, 24, and 28

˚ is orbitals becomes possible and in this case a short Ti-N-bond of 2.028(5) A found, the other bond length is with 2.132(5) ˚ A considerably longer (N/N/C plane of 5 to N/Ti/N 12.30◦ ) (Fig. 2.10). Due to the low solubility of ’19, 18 and 25, NMR spectra are not available. The more soluble complexes 20 (R: t-Bu), 21 (R: t-Bu), and 26 exhibit sharp 1 H NMR signals of the t-Bu groups (δ: 21 1.16; 20 1.12; 26 1.19, 1.44). Further signals show more or less strong broadenings and chemical shifts over a wide range as characteristic for nondiamagnetic derivatives.

2.3 Molecular Architectures Accompanied by Radical Induced C–C Coupling Reactions Low valent titanium complexes are characterized by strong reducing properties. Here we wish to report on the reactions of six-membered N-heterocycles which leads, by the way of selective C–C coupling, to the formation of polynuclear titanium compounds. Reactions of the titanocene complexes Cp2 Ti(η 2 -C2 (SiMe3 )2 (3) and Cp∗2 Ti(η 2 -C2 (SiMe3 )2 (4) with triazine (13) at 25 (3) or 60◦ C (4), respectively, led, after 48 h, to the dinuclear chelate complexes 29 or 30, respectively, which can be isolated in 45% or 32% yield (Scheme 2.9) [21]. Complexes 29 and 30 are sparingly soluble in aliphatic and aromatic hydrocarbons as well as in THF and decompose above 300◦ C, without melting. They display the extended molecular peak at m/z 518 (90) (29) and 799(40) (30) in the mass spectrum (EI, 70 eV). Owing to the low

34

R. Beckhaus Ct1 2.028(5)

Ti1

N3a

N1 2.132(5)

Ct2

Fig. 2.10. Positions of the two ligands at the titanium atoms and respective bond lengths in ˚ A in 18

solubility, single crystals of 29 and 30 were grown directly from the reaction solutions. Figure 2.11 shows the molecular structure of 29 determined by X-ray crystallography. The reactions of the alkyne complexes 3 and 4 with pyrazine (12) displayed varied behavior. As expected, in both cases the alkyne ligand is displaced: the tetranuclear pyrazine-bridged titanium complexes 20 is formed starting from 3, whereas 4 reacts under spontaneous threefold C–C coupling to give the trinuclear chelate complex 31 (Scheme 2.9). The reaction of 3 with 12 occurs in THF at room temperature, as is evident from the color change from yellow-brown to deep purple; after 48 h crystalline 20 can be obtained in 43% yield from dilute reaction solutions. Complex 31 is formed at 60◦ C in THF or toluene in 60% yield. The molecular structure of 31 (Fig. 2.12) like 20, 31 is also C2 -symmetrical: Ti2 lies on the twofold axis. The three new C–C bonds generated lead to the formation of an ideal cyclohexane ring in the chair conformation. The reaction of 3 with pyrimidine (11) gives after 2 h at 64◦ C the octanuclear titanium complex 32 in the form of yellow, needle-like crystals in 80% yield (Scheme 2.9). Crystals suitable for the X-ray structure analysis were obtained by crystallization from toluene. The resulting molecular structure is shown in Fig. 2.13. Similar to 29, 30, 20, and 31 no NMR spectra could be recorded for 32 because of its low solubility. In the crystal ﬁve equivalents of toluene were found per molecule of 32. The octanuclear compound crystallized in the polar space group P c. The C(sp3 ) centers formed by the C–C coupling show typical disorders. The formation of 29, 30, and 32 is characterized in each case through the linkage of one C–C bond of the N-heterocycle (11, 13) used. Whereas in the reaction with 13, only 29 or 30, respectively, is formed even when 3 and 4 are used in excess, that is only the chelate positions are occupied, the reaction

2 Titanium-Based Molecular Architectures

35

N N

N

N

N

1/2 L2Ti

TiL2

N N L: Cp 29 L: Cp* 30

N -C2(SiMe3)2 13

N N

TiCp*2

N

N

SiMe3

N 12

1/3 Cp*2Ti N N

31

-C2(SiMe3)2

N

N

L2Ti L: Cp 3 SiMe3 L: Cp* 4

TiCp*2

N +

N

-C2(SiMe3)2 11

Cp2Ti

N

1/8

TiCp2

N N

N

N

N

N

N

N

N

TiCp2

Cp2Ti TiCp2

N Cp2Ti

N N

N

N

TiCp2

N 32

Scheme 2.9. Molecular architectures accompanied by radical induced C–C coupling reactions

Ct1c

N1a C2d C5b

C4b

Ti1 C2 C3

N1

Ct1 C5

C1 C4

N2

Fig. 2.11. Structure of 29 in the crystal (50% probability). Selected bond lengths (˚ A) and angles [◦ ]: Ti1–N1 2.2108(16), Ti1–Ct1 2.072, N1–C1 1.299(3), N1–C2 1.468(2), N2–C1 1.352(2), C2–C2c 1.501(6), N1–Ti–N1a 76.13(9) Ct1–Ti1–Ct 133.09

36

R. Beckhaus

C29 C30 C28 C31 C27 Ti2

C32 N3

C5

N3a C4

C6 C4a C13

C3 N2

C8

C19 C10 C20

C18

C2a C2

C9 Ti1

N1 C17 C11 C7 C21

C1

C1a

Fig. 2.12. Structure of 31 in the crystal (50% probability, without H-atoms). Selected bond lengths (˚ A) and angles (◦ ): Ti1–N1 2.212 (3), Ti1–N2 2.193 (2), Ti2–N3 2.184 (2), Ti1–Ct1 2.149, Ti1–Ct2 2.146, Ti2–Ct3 2.158, N1–C1 1.343(4), N1–C2 1.458 (4), N2–C3 1.446 (4), N2–C6 1.349 (4), N3–C4 1.460 (4), N3–C5 1.342 (4), C1–C1a 1.413 (6), C2 −C2a 1.508 (7), C2–C3 1.496 (4), C3–C4 1.510 (4), C4–C4a 1.484 (7), C5–C6 1.413 (4), N1–Ti–N2 76.97 (9), N3–Ti–N3a 77.89 (12), Ct1–Ti1–Ct2 137.53, Ct3–Ti2–Ct3a 140,18

of 3 with 11 leads to the saturation of the terminal N atoms and thus to the octanuclear molecular aggregate 32. In 32 the eight linked titanium centers form a puckered ring (see Fig. 2.13), and the chelate centers are located above the plane of the terminal Cp2 Ti units. Two of the chelate positions (Ti2, Ti6) point toward the center of the ring. The formation of 29, 30, 31, and 32 is accompanied by a loss of the aromaticity of the N-heterocycle employed. Along the new C–C bonds, the H atoms always adopt a trans position. Particularly notable is the formation of 31 from 4 in comparison to that of 20 from 3. The Ti–N distances in 29, 30, 31, and 32 lie at the upper limit for pure Ti–N σ bonds without pπ − dπ interactions, [43] and correspond to the values expected for titanium-coordinated N-heterocycles [39, 44, 45]. The

2 Titanium-Based Molecular Architectures C34 C31

C38

C27a C26a

C13

C32 C37

C30 Ti7 Ti3 C29 C35 N4 C33 C36 N5 C39

C25a

C19 C20

C45 C41 N6 C42a C46 Ti4 C52 C50 C47 C53a N7 C71a C54 C51 C72a C56 C55 N8 C76a C58 N10 C66 C57 C67 Ti5 C59 C65 N9

C48

C64

C61

C62 C60

C7 Ti1 N1

Ti6N3C16N2

C40

C44 C43 C49

C9

C14a

C28 C15

C68

Ti2 C21

C11

C1

C24

C3 C10 C2

C6

N16

C17 C111 C22

C18

C8

C4

C12 C5

37

C100 b C101 C23 C109a C100 C98a C75a C97 C74a

C112 C102 C108 N15 C103 C107 Ti8 C104 C106 C99 N14 C105 C95

C73a C96 C77a N13 C78a C91 N11 C84 C86 C80a C90 N12 C92 C79a C81a C70a C94 C93 C85 C83 C88 C82 C69 C89

Fig. 2.13. Structure of 32 in the crystal (50% probability, without Hatoms). Selected bond lengths (˚ A) and angles at the terminal Ti5 and Ti7 centers as well as at the chelate positions Ti6 and Ti8: Ti5–N9 2.193(6), Ti5–N8 2.205(6), Ti6–N10 2.155(6), Ti6–N11 2.157(5), Ti7–N12, 2.182(6), Ti7–N13 2.190(6), Ti8–N15 2.153(6), Ti(8)–N(14) 2.158(6), N9–C67 1.334(9), N9–C68 1.430(9), N10–C67 1.272(9), N10–C70B 1.526(16), N10–C70A 1.569(15), N11–C84 1.301(9), N11–C81A 1.468(9), N11–C81B 1.565(10), N12–C(84) 1.367(8), N12–C83 1.378(9), N13–C95 1.361(9), N13–C96 1.415(9), N14–C95 1.313(9), N14–C98B 1.54(3), N14–C98A 1.552(10), N15–C112 1.317(9), N15–C109B 1.48(2), N15–C109A 1.496(11), N16–C112 1.338(9), N16–C111 1.422(8), C68–C69 1.302(11), C69–C70A 1.529(18), C70A–C81A 1.602(19), C81A–C82 1.420(9), C82–C83 1.306(10), C70B–C81B 1.462(19), C96–C97 1.287(11), C97–C98A 1.502(12), C98A–C109A 1.500(14), C98B–C109B 1.63(3), C109A–C110 1.481(11), C(110)–C(111) 1.314(10), C113–C119 1.362(16), N9–Ti5–N8 82.0(2), Ct–Ti5–Ct 131.72, N10–Ti6–N11 75.4(2), Ct–Ti6–Ct 133.48, N12–Ti7–N13 83.4(2), Ct–Ti7–Ct 131.88, N15–Ti8–N14 75.8(2), Ct–Ti8–Ct 135,16

Ti–N distances within 29, 30, and 31 do not diﬀer signiﬁcantly, such that mesomeric forms can be assumed. Bond lengths and angles of the titanocene units correspond to those for tetrahedral coordination geometry. The formation of the C–C coupled polynuclear complexes 29, 30, and 32 can be explained by the initial reduction of the heterocycles to radical anions, which has been described for several N-heterocycles in reactions with lowvalent metal complexes [6, 46–49] or by electrochemical reactions [50, 51]. The C–C coupling in the reaction to give 32 occurs regioselectively at the C4 atom

38

R. Beckhaus

of the heterocycle, the same is true for the electrochemical coupling [50]. The formation of 31 is particularly surprising, since pyrazine (12) is considered to be a typical bridging ligand and no deﬁned C–C coupling at metal centers has been described to date for this heterocycle [8, 52].

2.4 Molecular Architectures Based on C–C Coupling Reactions Initiated by C–H Bond Activation Reactions C–C formation reactions are of fundamental interest in various applications of organometallic substrates. A great deal of work has been devoted to the combination transition metal initiated C–H bond activations (a, b) and subsequent C–C bond formation (c) [53, 54].

R H

R

LnM

H a)

R

R

+ R H LnM

LnM

- H2 b)

L nM R

R c)

The primary C–H bond activation step is well established [55–58]. From the practical point of view, a dehydrogenative coupling reaction from two C–H bonds make synthetic procedures shorter and more eﬃcient [59]. While reductive coupling of aryl ligands is well documented for group 10 biarlys [60], the analogous process on biaryl zirconocene derivatives can only be induced under photochemical conditions [61], and a radical decomposition seems to be preferred when diorganyltitanocenes are irradiated [62]. The most commonly observed carbon–carbon reductive elimination process involving complexes of group 4 metals is the coupling of 1-alkenyl groups [63–68] or iminoacyl ligands [69]. We have found that by reacting pyrazine, triazine, or pyrimidine with the titanocene acetylene complexes Cp2 Ti{η 2 -C2 (SiMe3 )2 } (3) or its permethylated analogues (4), as excellent titanocene sources [26, 70], multinuclear titanium complexes are formed [71], often accompanied by simultaneously occurring C–C couplings of the primary formed radical anions [21]. In this chapter we report the spontaneous coupling of N-heterocycles, initiated by C–H bond activation reactions. The reaction of quinoxalines 17 and its dimethyl analogues 17a with 4 results in the formation of 33 and 33a, respectively. The compounds can be isolated in yields of 17% (33) and 62% (33a) as crystalline products in one-pot syntheses at 60◦ C (24 h). These hexaazatrinaphthylene (HATN) titanium complexes are thermally stable (mp > 350◦ C 33, 353◦ C 33a), but very sensitive to air and moisture. The molecular peaks can be observed (33 m/z 919 (3%) [M+ ], 33a m/z 1002 (4%) [M+ ]). 33 is nearly insoluble in common solvents, whereas for 33a a paramagnetic behavior is proved by Evans method [72]. Products formed by

2 Titanium-Based Molecular Architectures

3 Cp2Ti{h 2-C2(SiMe3)2}

+

N

R

N

R

3

4 - 3 C2(SiMe3)2

Cp2 Ti N N

R R N Cp2Ti

60 ˚C

- 3 H2

R

R

N

A

N

TiCp2

R

33a

Cp2Ti

R: H R: CH3

R R

N

N

33 R

R :H 17 17a R: CH3

Cp2 Ti

RR N

N

39

N N

N

TiCp2 B

R

R

Scheme 2.10. Formation of the trinuclear HATN Titanium complexes 33 and 33a

reactions of the free HATN ligands [73] with 4 are in every respect identical to 33 and 33a, respectively. However, due to the general poor solubility of HATN ligands, there complexation often ends up in poor yields and reveals signiﬁcant disadvantages compared to the presented route. Suitable crystals for X-ray diﬀraction are obtained directly from the reaction solutions (33, Fig. 2.1). Disorder problems in 33a prohibit further discussion of structural parameters. 33 is Cm-symmetrical with the Ti1 centre on the mirror plane. The HATN ligand of 33 is nearly planar with a slight deviation of the outer fused benzene rings. Bond distances and angles in 33 suggest a reduced Nheterocyclic system with characteristic patterns of low valent N, N -chelated titanium complexes [21, 33, 39]. Whereas uncomplexed hexaazatrinaphthylene (39) shows for the central six membered ring three long (average 1.479 ˚ A) and three short C–C bonds (average 1.425 ˚ A) [73] for 33 shorter and more balanced distances (1.411(9)–1.438(9) ˚ A) are found. The central C–C bonds of the chelate positions in 33 (1.411(9)–1.426(7) ˚ A) appear shorter compared to the free ligand 39 (1.472(6) − 1.491(6) ˚ A) [73]. The C–N distances in 33 are elongated (1.396(6)–1.352(6) ˚ A) compared to 39 (1.318(5)–1.382(5) ˚ A [73], ˚ 1.323(3)–1.363(3) A [74]) indicating contributions from the amid mesomeric form A. The Ti–N distances (2.170(4)–2.195(3) ˚ A) lie in the upper limit for Ti–N σ bonds without pπ –dπ interactions and correspond to the values expected for titanium-coordinated N-heterocycles in agreement with the mesomeric form B [21].

40

R. Beckhaus

Ti23 C113 C18

C123 C17

Ti1 N1

C15

C133 C14

C13

C6

C12 C11

C26

N2 C9

C7

C10 N3

C2 C4

C5

C3

C1

C24

Ti2

C27 C19

C8

C23 C22

Fig. 2.14. Molecular structure of 33 (hydrogen atoms omitted). Ellipsoids are drawn at 50% probability. Selected bond length () and angles (deg): Ti1–N1 2.187(4), Ti2–N2 2.170(4), Ti2–N3 2.195(3), C1–C2 1.420(7), C2–C3 1.426(7), C1–N1 1.353(6), C2–N2 1.370(6), C3–N3 1.352(6), C4–N1 1.396(6), C9–N2 1.389(6), C10–N3, 1.384(6), C1–C1 1.411(9), C3–C3 1.438(9), C10–C10 1.435(9), C12–C12 1.359(11), C4–C5 1.379(7), C5–C6 1.386(6), C6–C7 1.390(7), C7–C8 1.369(7), C8–C9 1.406(6), C4–C9 1.434(7), Ti1–Ct1 2.080, Ti1–Ct2 2.096, Ct1–Ti–Ct2 135.57, N1–Ti–N1 75.7(2), N2–Ti–N3 75.97(14)

Reactions of pyridine (9) and 3 lead to stable binuclear pyridyl titanium hydrides through C–H bond activation and ortho titanation. Subsequent C–C bond formations are not observed [75]. Dehydrogenative coupling proceeds if benzannelated N-heterocycles with at least one ortho C–H bond are reacted with 3. The simplest representative of this type of heterocycles is quinoline (34). With the formation of biquinoline 37 another example for the dehydrogenative coupling is given what allows us to present the potential mechanism of the reactions to 33 and 33a in a concise manner. The assembly of 37 can be explained by a twofold primary C–H bond activation, leading to 35 and 36 followed by C–C coupling through reductive elimination (Scheme 2.11). Corresponding intermediates in the reactions of 4 and 17 enables further C–H activation and subsequent C–C bond formation steps to give 33. However, attempts to isolate intermediates like 35 or 36 even at lower temperatures have not been successful yet. The 2, 2 -biquinoline complex 37 can be isolated as crystalline product (61%), exhibiting comparable structural

2 Titanium-Based Molecular Architectures

N C6H12 60 ˚C

41

34 N H

+ 3 Cp2Ti

-C2(SiMe3)2

N

+ 34 -H2

Cp2Ti

Cp2Ti

N N

N

37

36

35

Scheme 2.11. C–C coupling reactions induced by C–H bond activation

characteristics as 2,2-bipyridine titanium complexes, proved by X-ray diﬀraction (Fig. 2.15) [39]. The shortening of the C9–C10 bond (1.432(2) ˚ A) e.g., A [76], 1.490(3) ˚ A [36]) indicates the compared to free 2, 2 -bypyridine (1.50 ˚ reduced nature of the chelating ligand. C8

C7

C12

C11

C13

C6

C5

C9

C10

C4

C1

C18 N2

N1

C14

C15

C24 C3

C2

Ti1 C26

C17

C16

C25 C21

C19 C23

C22

Fig. 2.15. Molecule of 37 in the crystal (hydrogen atoms omitted). Ellipsoids are drawn at 50% probability. Selected bond length (A) and angles (deg): Ti1–N1 2.1920(14), Ti1–N2 2.1960(12), C1–N1 1.390(2), C1–C2 1.417(2), C2–C3 1.380 (2), C3–C4 1.395(3), C4–C5 1.375(2), C5–C6 1.402(2), C1–C6 1.432(2), C6–C7 1.438(2), C7–C8 1.348(3), C8–C9 1.427(2), C9–N1 1.3748(19), C9–C10 1.432(2), N2–C10 1.372(2), C10–C11 1.435(2), C11–C12 1.347(3), C12–C13 1.429(3), C13–C14 1.403(2), C14–C15 1.371(3), C15–C16 1.402(2), C16–C17 1.381(2), C17–C18 1.413(2), C13–C18 1.433(2), C18–N2 1.384(2), Ti–Ct1 2.094, Ti1–Ct2 2.093, Ct1–Ti1–Ct2 133.76, N1–Ti1–N2 76.30(5)

42

R. Beckhaus

N

N N

N

+3x1

- 3 C2(SiMe3)2 N - H2

N

I2

N

N

N

N

33 - Cp2TiI2

N

38

39

Scheme 2.12. Formation of 33 by dehydrogenative C–C Coupling employing 38

The HATN complex 33 is also formed by dehydrogenative coupling of 2, 3 − (2 , 2 )-diquinoxalylquinoxaline (38) with 4, which is in agreement with the proposed mechanism. Reacting 33 with I2 (3eq) in n-hexane as solvent gives 39. With the selective formation of carbon–carbon bonds, in combination with C–H bond activation processes, particularly using commercial starting materials, an eﬃcient route for the coupling of N-heterocycles has been established. Molecular self-assembly with cyclic topology is one of the most useful methods to design and to build up nanometer-sized multinuclear transition metal complexes [1, 77].

2.5 Conclusion and Future Directions For the investigations described herein, we employed a novel synthetic method, in which metal-linked complexes of polyvalent N-donor ligands are formed from readily accessible building blocks. The good solubility of the starting materials allows a smooth separation of the (in most cases) sparingly soluble products. Further work on these systems should be able to show whether the record of the linkage of eight [Cp2 Ti] units can be beaten, and to what extent the observed self-assembly principles can be developed by using lowvalent early transition metals and redox-active acceptor ligands [3, 78]. The synthesis of 31 described herein could be the beginning of a new area of chemistry comparable to the chemistry of the HAT ligands which are diﬃcult to access [79–82]. In recent years the research in the area of monometallic compounds has been extended to polymetallic supramolecular systems, which may have a considerable potential to design new materials for use in photochemical molecular devices. Further more, several polynuclaer compounds have been created, which possess functionality such as nonlinear optics, molecular magnetism, anion trapping, that means e.g., to act as molecular receptors, DNA photoprobe, and other photophysical properties, successfully reﬂecting potential advantages of multinuclear derivatives.

2 Titanium-Based Molecular Architectures

43

Abbreviations Cp C5 H5 anion C5 Me5 anion Cp∗ ct in single X-ray structures ct = centroid of the carbon atoms of the respective Cp ring HAT 1,4,5,8,9,12-Hexaazatriphenylene HATN 1,6,7,12,13,18-Hexaazatrinaphthylene Me CH3

Acknowledgments I have to express my thanks to my coworkers A Dr. Axel Scherer, Dr. Susanne Kraft, Dr. Ingmar Piglosiewicz for excellent preparative work and fruitful discussions. I have also to thank Marion Friedemann, Edith Hanuschek, Wolfgang Saak, and Detlev Haase for technical and analytical assistance. This work was supported by the Fonds der Chemischen Industrie, by the Karl Ziegler Stiftung der Gesellschaft Deutscher Chemiker and by the Deutsche Forschungsgemeinschaft (BE 1400/4-1, /5-1).

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20. L.L. Schafer, T.D. Tilley, J. Am. Chem. Soc. 123, 2683 (2001) 21. S. Kraft, R. Beckhaus, D. Haase, W. Saak, Angew. Chem. 116, 1609 (2004); Angew. Chem. Int. Ed. 43, 1583 (2004) 22. P.J. Stang, J.A. Whiteford, Res. Chem. Intermed. 22, 659 (1996) 23. P. Schinnerling, U. Thewalt, J. Organomet. Chem. 431, 41 (1992) 24. A. Scherer, K. Kollak, A. L¨ utzen, M. Friedeman, D. Haase, W. Saak, R. Beckhaus, Eur. J. Inorg. Chem. 1003 (2005); Eur. J. Inorg. Chem. 1991 (2005) 25. F. Studt, N. Lehnert, B.E. Wiesler, A. Scherer, R. Beckhaus, F. Tuczek, Eur. J. Inorg. Chem. 291 (2006) 26. U. Rosenthal, V.V. Burlakov, P. Arndt, W. Baumann, A. Spannenberg, Organometallics 22, 884 (2003) 27. M. Diekmann, G. Bockstiegel, A. L¨ utzen, M. Friedemann, D. Haase, W. Saak, R. Beckhaus, Organometallics 25, 339 (2006) 28. S. Kraft, Dissertation, Universit¨ at Oldenburg, 2004 29. M.L.H. Green, C.R. Lucas, J. Chem. Soc. Dalton Trans. 1000 (1972) 30. T. Rajendran, B. Manimaran, F.-Y. Lee, G.-H. Lee, S.-M. Peng, C.M. Wang, K.-L. Lu, Inorg. Chem. 39, 2016 (2000) 31. I.M. Piglosiewicz, S. Kraft, R. Beckhaus, W. Haase, W. Saak, Eur. J. Inorg. Chem. 938 (2005) 32. A.M. McPherson, B.F. Fieselmann, D.L. Lichtenberger, G.L. McPherson, G.D. Stucky, J. Am. Chem. Soc. 101, 3425 (1979) 33. P.T. Witte, R. Klein, H. Kooijman, A.L. Spek, M. Polasek, V. Varga, K. Mach, J. Organomet. Chem. 519, 195 (1996) 34. M. Schwach, H.-D. Hausen, W. Kaim, Inorg. Chem. 38, 2242 (1999) 35. M. Glockle, K. Hubler, H.J. Kummerer, G. Denninger, W. Kaim, Inorg. Chem. 40, 2263 (2001) 36. M.H. Chisholm, J.C. Huﬀman, I.P. Rothwell, P.G. Bradley, N. Kress, W.H. Woodruﬀ, J. Am. Chem. Soc. 103(16), 4945 (1981) 37. L.J. Radonovich, M.W. Eyring, T.J. Groshens, K.J. Klabunde, J. Am. Chem. Soc. 104(10), 2816 (1982) 38. T. Eicher, S. Hauptmann, The Chemistry of Heterocycles (Georg Thieme Verlag, Stuttgart, 1995), pp. 1–504 39. R. Gyepes, P.T. Witte, M. Horacek, I. Cisarova, K. Mach, J. Organomet. Chem. 551, 207 (1998) 40. J.W. Lauher, R. Hoﬀmann, J. Am. Chem. Soc. 98, 1729 (1976) 41. J.C. Green, Chem. Soc. Rev. 27(4), 263 (1998) 42. W.W. Lukens Jr., M.R. Smith III, R.A. Andersen, J. Am. Chem. Soc. 118, 1719 (1996) 43. J. Feldman, J.C. Calabrese, J. Chem. Soc. Chem. Commun. 1042 (1991) 44. A. Ohﬀ, R. Kempe, W. Baumann, U. Rosenthal, J. Organomet. Chem. 520, 241 (1996) 45. U. Thewalt, K. Berhalter, J. Organomet. Chem. 302, 193 (1986) 46. D.R. Corbin, G.D. Stucky, W.S. Willis, E.G. Sherry, J. Am. Chem. Soc. 104, 4298 (1982) 47. D.R. Corbin, W.S. Willis, E.N. Duesler, G.D. Stucky, J. Am. Chem. Soc. 102, 5969 (1980) 48. W.J. Evans, D.K. Drummond, J. Am. Chem. Soc. 111, 3329 (1989) 49. L.D. Durfee, P.E. Fanwick, I.P. Rothwell, K. Folting, J.C. Huﬀman, J. Am. Chem. Soc. 109, 4720 (1987)

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50. G. Tapolsky, F. Robert, J.P. Launay, New. J. Chem. 12, 761 (1988) 51. H.-S. Chien, M.M. Labes, J. Electrochem. Soc. Electrochem. Sci. Techn. 133(12), 2509 (1986) 52. W. Kaim, Acc. Chem. Res. 18, 160 (1985) 53. W.V. Konze, B.L. Scott, G.J. Kubas, J. Am. Chem. Soc. 124(42), 12550 (2002) 54. G. Dyker, Angew. Chem. 111(12), 1808 (1999); Angew. Chem. Int. Ed. 38, 1698 (1999) 55. R.H. Crabtree, J. Organomet. Chem. 689, 4083 (2004) 56. A.S. Goldman, K.I. Goldberg, ACS Symp. Ser. 885, 1 (2004) 57. J.A. Labinger, J.E. Bercaw, Nature 417, 507 (2002) 58. R.H. Crabtree, J. Chem. Soc. Dalton Trans. 2437 (2001) 59. Z. Li, C.-J. Li, J. Am. Chem. Soc. 127, 3672 (2005) 60. R.K. Merwin, R.C. Schnabel, J.D. Koola, D.M. Roddick, Organometallics 11, 2972 (1992) 61. G. Erker, J. Organomet. Chem. 134, 189 (1977) 62. H.G. Alt, Angew. Chem. 96, 752 (1984); Angew. Chem. Int. Ed. 23, 766 (1984) 63. R. Beckhaus, J. Oster, J. Sang, I. Strauß, M. Wagner, Synlett 241 (1997) 64. R. Beckhaus, K.-H. Thiele, J. Organomet. Chem. 317, 23 (1986) 65. R. Beckhaus, K.-H. Thiele, J. Organomet. Chem. 268, C7 (1984) 66. P. Stepnicka, R. Gyepes, I. Cisarova, M. Horacek, J. Kubista, K. Mach, Organometallics 18, 4869 (1999) 67. U. Rosenthal, Angew. Chem. 116, 3972 (2004); Angew. Chem. Int. Ed. 43, 3882 (2004) 68. G. Erker, Acc. Chem. Res. 17, 103 (1984) 69. J. Campora, S.L. Buchwald, E. Gutierrez-Puebla, A. Monge, Organometallics 14, 2039 (1995) 70. U. Rosenthal, V.V. Burlakov, P. Arndt, W. Baumann, A. Spannenberg, V.B. Shur, Eur. J. Inorg. Chem. 4739 (2004) 71. S. Kraft, E. Hanuschek, R. Beckhaus, D. Haase, W. Saak, Chem. Eur. J. 11, 969 (2005) 72. D.F. Evans, J. Chem. Soc. 2003 (1959) 73. M. Alfonso, H. Stoeckli-Evans, Acta Crystallogr. Sect. E: Struct. Reports Online E57, o242 (2001) 74. M. Du, X.H. Bu, K. Biradha, Acta Crystallogr. Sect. C: Cryst. Struct. Commun. C57(2), 199 (2001) 75. H.S. Soo, P.L. Diaconescu, C.C. Cummins, Organometallics 23(3), 498 (2004) 76. L.L. Merritt Jr., E.D. Schroeder, Acta Cryst. 9, 801 (1956) 77. M. Fujita, J. Yazaki, K. Ogura, J. Am. Chem. Soc. 112, 5645 (1990) 78. P.J. Stang, Chem. Eur. J. 4(1), 19 (1998) 79. D.Z. Rogers, J. Org. Chem. 51, 3904 (1986) 80. C.G. de Azevedo, K.P. Vollhardt, Synlett 1019 (1019) 81. J.T. Rademacher, K. Kanakarajan, A.W. Czarnik, Synthesis 378 (1994) 82. B.F. Abrahams, P.A. Jackson, R. Robson, Angew. Chem. 110, 2801 (1998); Angew. Chem. Int. Ed. 37, 2656 (1998) 83. I.Piglosiewicz, R. Beckhaus, W. Saak, D. Haase, J. Am. Chem. Soc. 127, 14190–14191 (2005)

3 Self-Assemblies of Organic and Inorganic Materials M.P. Pileni

Summary. The use of self-assemblies of surfactants to produce colloidal solutions for nanomaterial preparation is reviewed. This involves the properties and roles of normal and reverse micelles, size, and shape control of nanocrystals, their selforganization. Nanocrystals are self-organized either in hexagonal network or in FCC “supra” crystals, rings, tubes, dots, labyrinth. These organizations is used for nanolithography.

3.1 Introduction During the past 20 years there has been much work devoted to the study of oil, water, and surfactant mesophases [1,2]. Microstructured, three-component ﬂuids provide well-deﬁned reproducible model systems. Elementary concepts, that are derived from formal statistical mechanics, yet capture the essence of the self-assembly process, have been developed [3–5]. Here the microstructure, which varies dramatically within a single phase, and phase boundaries are predictable. This system thus forms an ideal porous medium. The approach requires only the local curvature of the surfactant at the oil–water interface and global packing constraints set by component volume fractions. The microstructure of mixtures containing surfactant, oil, and water has been the subject of many debates [1, 2]. As recently as a decade ago, some of the most active workers in the ﬁeld of complex ﬂuids held that microstructures did not exist. Emulsions, by deﬁnition, have always been regarded as thermodynamically unstable systems. But this must now be qualiﬁed. Following our paper [6] in 1991, a great deal of work [7–12] has been done with divalent bis (2-ethylhexyl) sulfosuccinate, X(AOT)2 . There is also an interest in such systems as vehicles for templating and catalysis and for the characterization of random, porous microstructured matter, which are central issues in chemical engineering and cell biological applications.

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During the last decade, due to the emergence of a new generation of high technology materials, the number of groups involved in nanomaterials has increased exponentially [13,14]. Nanomaterials are implicated in several domains such as chemistry, electronics, high density magnetic recording media, sensors, and biotechnology. This is, in part, due to their novel material properties, that diﬀer from both the isolated atoms and the bulk phase. An ultimate challenge in materials research is now the creation of perfect nanometer-scale crystallites, identically replicated in unlimited quantities, in a state that can be manipulated and that behave as pure macromolecular substances. Thus the ability to systematically manipulate these is an important goal in modern materials chemistry. The electrical, optical, and magnetic properties of inorganic nanomaterials vary widely with their sizes and shapes. The major contribution to date has been to produce spherical nanocrystals with a very low size distribution. Deposition processes include using microwave plasma [15], lowenergy cluster beam deposition [16], inorganic chemistry [17], ball milling [18], sonochemical reactions [19], gel-sol [20], a ﬂame by vapor phase reaction, and condensation [21]. In 1986, we developed a method based on reverse micelles (water-in-oil droplets) to prepare nanocrystals [12, 13]. Normal micelles make it possible to produce ferrite magnetic ﬂuids [22]. In 1993 and again in 1995, we were able to regulate, in some cases, the shape of nanocrystals by using colloidal solutions as templates [23]. We demonstrated that the template is not the major parameter in controlling nanocrystal shapes. Adsorptions of ions and molecules on various faces enable producing anisotropic nanomaterials [24, 25] but all the various parameters that are able to control the shape of nanocrystals have yet to be identiﬁed [26]. Self-assembled nanocrystals have attracted increasing interest over the last ﬁve years [27–35]. As with other nanomaterials, the level of research activity is growing exponentially, fueled in part by the observation of physical properties that are unique to the nanoscale domain. Interestingly, it has been recently demonstrated that the physical (optical, magnetic, transport) properties [36–51] of nanocrystals organized in 2D and/or 3D superlattices diﬀer from those of isolated nanoparticles. These changes in the physical properties are due to the close vicinity and to the ordering of nanocrystals at a given distance from each other. Such collective properties are attributed to dipole– dipole interactions. Furthermore, the electron transport properties drastically change with the nanocrystal organization. In this chapter we describe various microstructures employed as templates. The present concept of “phase” in mesostructured ﬂuids is too restrictive and the existence of thermodynamically stable states of self-assembled supraaggregates is demonstrated. Spontaneous emulsions form in which lamellar phases contain and are surrounded by a bicontinuous microemulsion. These are self-assembly equilibrium states that require no energy input for their formation. These microstructures make possible a partial control of the size and shape of nanocrystals. However, the shape of the template is not the only

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parameter needed to produce well-deﬁned nanocrystals. Nanocrystals formed with small size distributions self-assemble in 2D and 3D superlattices with a long-range ordering or other mesoscopic structures.

3.2 Structure of Colloidal Self-Assemblies Made of Surfactants and Used as Templates Surfactants are molecules with a polar hydrophilic head (attracted by water) and a hydrophobic hydrocarbon chain (attracted by oil). If we solubilize a surfactant in water, the chains tend to self-associate to form various aggregates [3]. Of course, if the solvent is able to solubilize simultaneously the polar head and the alkyl chains, no aggregates are formed. The shape of the surfactant plays an important role in forming the assembly. If the surfactant molecules have a very large polar head and a small chain (Fig. 3.1a), the surfactant is then cone-shaped (like an ice cream cone), the chains tend to self-associate and form a spherical aggregate which is called a direct micelle (Fig. 3.1b). When the direct (or normal) micelle is formed at low concentration, it is spherical and the length of the hydrocarbon chain and the size of the polar head ﬁx its diameter. The system is dynamic. However, the micellar structure is always retained and the surfactants involved in formation of micelles leave the aggregate and are replaced by others that freely move in the aqueous bulk phase. After a few microsecond others replace all the surfactant molecules constituting a normal micelle. If the surfactant has the shape of a champagne cork (small polar head and branched hydrocarbon chains), spherical water-in-oil droplets are formed (Fig. 3.1c). These are usually called reverse micelles [1] (Fig. 3.1d). These droplets are displaced randomly and are subjected to Brownian motion, they exchange their water contents and reform two distinct micelles. In contrast to direct micelles, their size increases linearly with the amount of water added to the system and goes from 4 to 18 nm [52]. In the case of divalent surfactants such as Cu(AOT)2 , with AOT called diethylhexyl sulfosuccinate, in oil-rich regions and with a rather large amount of water, there are changes in the shape and dimension of the aggregates with formation of interconnected water channels (Fig. 3.1e). The space is divided into two volumes and regularly overlapped so that at any point the surface has the shape of a saddle. These are designated interconnected cylinders and are doubly continuous structures both in water and oil [2]. Adding more water induces a new phase transition. The system becomes opalescent and birefringent (the light is not extinguished between two crossed polarizers). It is a mixture of planar (Fig. 3.1f) and spherulite (Fig. 3.1g) lamellar-like phases. A further increase in the water content induces formation of a spontaneous, thermodynamically stable emulsion [53–55]. Previously the concept of emulsion demanded metastability. The phase, which appears to be a single one, is heterogeneous. The interior collapses to an interconnected-cylinder microemulsion and the exterior of the

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b

c

d

-

a

water

eA

f

g

Fig. 3.1. Surfactant shapes and various self-assemblies in colloidal solution: a and c, Various shapes of surfactants; b, normal (direct) micelles; d, reverse micelles with control of their size by the water content; e, interconnected cylinders; f, planar lamellar phase; and g, onion-like lamellar phase

spherulites is composed of the same isotropic microemulsion phase. The whole system is in equilibrium with this isotropic microemulsion. This is a “supraaggregate” containing interconnected cylinders in its external and internal phases and is represented in Fig. 3.2. The volume fraction of components and the detailed microstructured parameters of the entire system can be predicted from elementary packing considerations. The lamellae, which separate an interior and exterior phase made of interconnected cylinders, then adjust their spacing and component fractions to allow equilibration. The system collapses to a supra-aggregate with isotropic phases outside and inside the spherulites. A further increase in the water content induces an antipercolation process with a transition from interconnected cylinders to reverse micelles [55, 56]. Such unusual behavior can be predicted by taking into account the surfactant– water interface of interconnected cylinders and spheres [55]. In the region where spherical water-in-oil droplets are formed, we can ask what happens when the excess of oil is not suﬃcient to accommodate the curvature required by the surfactant chains. This occurs on increasing the surfactant concentration. Oil cannot ﬁll up the gaps between entities and the spheres must interdigitate. The geometrical conditions to reach interdigitated reverse micelles are given by the limiting value of the oil to surfactant volume fraction ratio. It is demonstrated that interdigitated reverse micelles can be formed by increasing the Cu(AOT)2 concentration [55]. More details can be found in [57].

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Fig. 3.2. FFEM pattern of supra-aggregate and sketch in the insert

3.3 Production of Nanocrystals by Using Colloidal Solutions as Templates and Their Limitations Fifteen years ago [58], we discovered that the reverse micelles [59] described earlier are good candidates for templates. Their two properties, i.e., exchange of water contents in Brownian motion collisions and their increase in size with the amount of water added to the system enable their use as variable size nanoreactors. Let us consider A and B solubilized in two micellar solutions. On mixing them, and because of the exchange process, A and B are in contact and react. It is thus possible to fabricate a very wide range

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of spherical nanomaterials [13] such as semiconductors, metals, oxides, and alloys (in this case some of them cannot be produced at the nanoscale whereas they are formed in the bulk phase [15]). The control of the template size, by changing the water content, enables regulating the spherical nanocrystal size (Fig. 3.3a) [60, 61]. It is of interest to note that this nanoreactor makes it possible to produce metal nanoparticles without any detectable oxide. There is a rather large consensus that reverse micelles are good nanoreactors for obtaining spherical nanomaterials. In most cases, a spherical template produces nanospheres. The particle size is controlled by hydration of the water pool. However, production of various species during the chemical reaction and/or presence of impurities play an important role in the nanocrystal growth, inducing formation of particles having various shapes [26]. The control of the nanocrystal shape is a real challenge and more data are now needed to ascertain the general principles that determine this shape. This is probably due to the fact that anisotropic materials are not in their thermodynamically stable state. Because the results obtained with reverse micelles are quite convincing and because the same three-component system produces self-assemblies of surfactants having various shapes, a number of groups tried to demonstrate that they could be used as templates. The ﬁrst results obtained are rather persuasive. Our work demonstrates that the shapes of colloidal solutions made of functionalized surfactants partially control those of the nanocrystals [23,57,60]. Figure 3.3b shows that an interconnected-cylinder template makes it possible to form cylindrical and spherical copper nanocrystals. The crystallinity of these nanomaterials is very high and the cylinder structure is characterized by a ﬁvefold symmetry [62]. In the phase diagram region made up of an onion phase containing both internal and external interconnected cylinders, a large variety of shapes is observed (Fig. 3.3c). This control in the nanocrystal shape by that of the template has been recently conﬁrmed by Simmons et al. [63]. However, the role of the template is not as obvious as described earlier. Adsorptions of ions and molecules have to be taken into account [26]. In the region of interconnected cylinders, small cylinders of copper metal nanocrystals are obtained (Fig. 3.3b1). Addition of less than 2.10−3 M chloride ions induces formation of copper nanorods [23] (Fig. 3.3b2) with an aspect ratio controlled by the amount of chloride ions in the microphase. The crystalline structures of these nanorods are similar to those of small cylinders (ﬁvefold symmetry). On replacing NaCl by NaBr, nanorods smaller than those obtained with NaCl and a rather large amount of cubes are produced [24, 25]. However, the aspect ratio does not change with bromide concentration. Furthermore, the shape of copper metal nanocrystals drastically changes with the type of salt added to the template [25] (Fig. 3.4). This cannot be due to changes in the colloidal structure by salt addition. In fact, the structure of the template remains the same in the presence of various salts (at ﬁxed concentration) [23]. Hence, the template is not the key parameter in

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Fig. 3.3. Copper nanocrystals produced in colloidal self-assemblies diﬀering by their shapes and obtained from Cu(AOT)2 –H2 O-isoctane solution. (a) Reverse micelles: Control of nanocrystal size with the water content, w, i.e., the size of water-inoil droplets. (b) Sketch of interconnected cylinders: (1) formation of spherical and cylindrical nanocrystals, (2) cylindrical particle composed of a set of deformed fcc tetrahedra bounded by (111) faces parallel to the ﬁvefold axis with an additional plane. (c) “supra-aggregates”: (1) Various nanocrystals, (2) particle composed of ﬁve deformed fcc tetrahedrals bounded by (111) planes, (3) Large ﬂat nanocrystals [111]-oriented and limited by (111) faces at the top, bottom and edges

Fig. 3.4. Various shapes of copper nanocrystals produced in interconnected cylinders in presence of various anions and same cation (Na+ ) having the same concentration. [Na2 SO4 ]==[Na2 CO3 ]==[NaHSO3 ]==[Na2 HPO4 ]==[NaCl]==[NaBr] ==[NaNO3 ]==]== [NaClO4 ]10−3 M

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controlling the nanocrystal morphology. This is explained in terms of selective ion adsorption on facets during the crystal growth. Other examples clearly indicate that selective adsorption of molecules during the crystal growth is also a key parameter in controlling the nanocrystal shape. Nanodisks [64–66] are produced in the presence of surfactants which no longer form well-deﬁned templates whereas nanospheres are produced in reverse micelles. The nanodisk size depends on the amount of hydrazine present in solution [65]. Note that this is the ﬁrst example where it is possible to control the silver nanodisks size, and then their optical properties, with changes in their color from red to gray (Fig. 3.5). Structural investigations show that nanodisks have a face centered cubic (f.c.c.) crystal structure with (111) ﬂat surfaces. A unique (111) stacking fault model parallel to the ﬂat (111) disk explains the observed 1/3{422} forbidden reﬂections in [111]. This suggests that the presence of the stacking faults may be the key in the formation and growth of the disk morphology [66].

3.4 Self-Organization of Nanocrystals At the end of the synthesis, the powder made of coated nanocrystals is dispersed in a nonpolar solvent and the nanocrystal concentration is controlled. In 1995 we ﬁrst demonstrated self-organization of nanocrystals with formation, on a mesoscopic scale, of a monolayer in a compact hexagonal network (Fig. 3.6A) [27, 28, 32]. When a drop of the solution containing nanocrystals is deposited on a TEM grid with a ﬁlter paper underneath, the solution migrates from the substrate to the ﬁlter paper and after a few seconds the solvent is totally evaporated. To obtain a well-deﬁned self-organization on a mesoscopic scale, attractive interactions between particles and repulsive ones between nanocrystals and the substrate are needed. This has been well demonstrated with silver sulﬁde nanocrystals deposited on HOPG and MoS2 [67]. On increasing the nanocrystal concentration, aggregates of nanocrystals are formed instead of a monolayer (Fig. 3.7A). At higher resolution (Fig. 3.7B), nanocrystals organized in fourfold symmetry are observed, indicating an ordering of nanocrystals in an FCC structure [32]. By controlling the evaporation rate and substrate temperature, large “supra” crystals of silver nanoparticles, on a mesoscopic scale with a long-distance order, are produced [33, 34] (Fig. 3.7C) with a very sharp edge (Fig. 3.7D). Similarly it is possible to make “supra” crystals with an FCC structure of cobalt nanoparticles [35] (Figs. 3.7E and F). In the latter case, all reﬂections characteristic of an FCC structure are observed. This result was not expected because of the dipolar interactions between nanocrystals. In fact they are weak enough to permit formation of “supra” crystals in a FCC structure. Instead of using the procedure described earlier, let us deposit a drop of solution with an anticapillary tweezer on a TEM grid. It remains on its support until the solvent is totally evaporated (solvent cannot escape from the

Fig. 3.5. Nanodisks: Sketch (G) and TEM pictures (A–F) of faceted particle with a ﬂat single crystal, two (111) faces at the top and the bottom, limited at the edges by three other (111) faces and at the corners by more or less extended (100) faces. Insert G HRTEM pattern in the middle and at the edge of the nanodisks

56 M.P. Pileni

Fig. 3.6. A: TEM patterns obtained by deposition of a drop of solution containing cobalt nanocrystals on HOPG deposited deposited on ﬁlter paper. B and C: Deposition of a drop of solution of silver nanocrystals (B) and cigar-like ferrite nanocrystals (C) by using an anticapillary tweezer

3 Self-Assemblies of Organic and Inorganic Materials 57

Fig. 3.7. “Supra” crystals of nanoparticles at various enhancements. A and B: “supra” crystals of silver sulﬁde nanocrystals obtained by deposition of a concentrated solution of particles on amorphous graphite. C and D: “Supra” crystals of silver nanocrystals obtained by controlling evaporation rate and substrate temperature. E and F: “Supra” crystals of silver nanocrystals obtained by controlling evaporation rate and substrate temperature

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grid). Figure 3.6B shows the formation of rings of nanocrystals [68,69]. This is obtained with a large variety of nanomaterials and depends neither on the substrate used nor on the shape of the nanomaterial (Fig. 3.6C). It is attributed to Marangoni instabilities [70] induced by a fast evaporation process. With increasing particle concentration, more complex organizations made of closepacked structures are produced. These are similar in shape to those observed in B´enard’s experiment with liquid ﬁlms and are attributed to a Marangoni eﬀect. To self-organize nanocrystals in lines an applied external force is needed. This was achieved by applying a magnetic ﬁeld, parallel to the substrate, during the evaporation process. Maghemite nanocrystals with 10 nm average diameter and coated with octanoic acid (C8 ) are deposited in a magnetic ﬁeld (0.59T) on HOPG substrate. Superimposed tubes with 3-µm average diameter are produced (Fig. 3.8A). At low nanocrystal concentrations the nanocrystals are self organized in lines (insert Fig. 3.8A). Conversely, with no applied magnetic ﬁeld, a rough ﬁlm made of spherical highly compact agglomerates with an average diameter of 2 µm is formed (Fig. 3.8B). By decreasing the nanocrystal concentration, clusters of nanoparticles are produced (inset Fig. 3.8B). The alignment of nanocrystals under an applied magnetic ﬁeld is rather surprising. In fact, the calculation of the dipolar parameter λ, deﬁned as the ratio of the magnetic dipolar to thermal energies, gives less than 1 for 10-nm γ-Fe2 O3 nanocrystals whereas chains are formed for λ larger than 3 [71–74]. Due to the low size distribution, the chain formation also cannot be explained by the presence of large particles. To explain the formation of the chain-like structures in spite of the weak dipolar interaction, both van-derWaals attraction and magnetic dipoles have to be taken into account: when the contact distance between particles is close enough “clusters” are formed due to the van-der-Waals forces as observed in the Fig. 3.8B insert. Due to their long-range order, the cluster formation considerably enhances the dipolar forces compared to isolated particles with creation of a “macro dipolar moment” leading to an anisotropic organization of the nanoparticles. To support this claim, the distance between nanocrystals is increased by changing the coating to dodecanoic acid (C12 ) instead of octanoic acid (C8 ). The SEM patterns observed with (Fig. 3.8C) and without (Fig. 3.8D) an applied magnetic ﬁeld show ﬂat and compact surfaces. At low particle concentration, no nanocrystal alignment is observed under a magnetic ﬁeld (inset Fig. 3.8C). From these data it is concluded [75] that elongated assemblies are obtained even for weakly dipolar particles by a careful choice of the coating agent. Since the origin of these structures is quite diﬀerent from that in highly dipolar ﬂuids, the structural and dynamical study can give fundamental insights into dipolar ﬂuids in general. The maghemite nanocrystals provide model systems to explore the inﬂuence of other interaction terms, such as hydrophobic attraction or the electrostatic repulsion between charged nanoparticles, on the nanoparticle organizations.

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Fig. 3.8. SEM patterns obtained by evaporation of a solution containing maghemite nanocrystals diﬀering by their coating with dodecanoic acid (A and B) and dodecanoic ou octanoic? acid (C and D) and subjected (A and C) or not (B and D) to an applied magnetic ﬁeld (H = 0.59 T) parallel to the substrate

With cobalt nanocrystals solubilized in hexane and subjected to an external magnetic ﬁeld perpendicular to the substrate, a large variety of mesoscopic structures characterized by various shapes and depending on the strength of the ﬁeld is observed [76] (Fig. 3.9). At very low applied ﬁelds, there are large dots with a rather wide size distribution (Fig. 3.9B). On increasing the applied magnetic ﬁeld, well-dispersed dots are observed with formation of well-deﬁned columns with a very sharp interface (Fig. 3.9C). A further increase in the applied ﬁeld induces a decrease in the dot height and the dot interdistance. At higher applied ﬁelds, the patterns show worm-like and labyrinth structures. Hence, well-deﬁned 3D superlattices of cobalt spherical nanocrystals on a very large scale are produced. Such structures were previously observed for a ferroﬂuid conﬁned with an immiscible nonmagnetic ﬂuid between closely-spaced parallel plates and subjected to a normal magnetic ﬁeld [77, 78]. The major diﬀerence is due to the fact that in our case the structures are maintained after

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Fig. 3.9. SEM patterns obtained by evaporation of a solution containing cobalt nanocrystals and subjected to various applied magnetic ﬁelds, H, perpendicular to the substrate (A) H = 0, (B) H = 0.10 T, (C), H = 0.27 T, (D), H = 0.45 T, (E) H = 0.57 T and (F) H = 0.85 T

total evaporation of the solvent. A simple model [79], assuming that the demagnetizing ﬁeld within the ferroﬂuid is uniform, is proposed and compared to the experiments. Rosenweig [79] claimed good agreement between the model and experiments. However, very recently Richardi et al. [80] demonstrated that the model needs to be revised as this agreement is no longer obtained. A new model [81,82] has been developed from which the free energy functionals of hexagonal and labyrinthine structures are derived.

3.5 Colloidal Nanolithography by Using Nanocrystals Organized in a Given Structure as Masks [83] The use of the self-assemblies described earlier as masks in order to make small devices in microelectronics, communication, and date storage is described.

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Fig. 3.10. Various organizations of 10-nm ferrite nanocrystals: (A) SEM patterns obtained by deposition of ferrite coated with dodecanoic acid with an anticapillary tweezer; (B) TEM pattern obtained by deposition of ferrite coated with citrate ion in a 0.59 T magnetic ﬁeld applied during the deposition process

It is shown in Fig. 3.6 that demonstrated that nanocrystals coated with alkyl chains can self-organize in ring-like structures. With γ Fe2 O3 nanocrystals, coated with dodecanthiol and dispersed in hexane, rings are formed (Fig. 3.10A). When the same nanocrystals are coated with citrate ions and dispersed in aqueous solution, the nanocrystals are aligned along the direction of the applied ﬁeld to which the solution is subjected during the deposition process. The average distance between these lines is 3 µm (Fig. 3.10B). In both cases, γ-Fe2 O3 nanocrystals are used as the mask in nanolithography. The γ Fe2 O3 nanocrystals are deposited on a polymethylmethacrylate thick ﬁlm spine coated on an SiO2 substrate. When γ-Fe2 O3 nanocrystals coated with citrate ions are subjected to 0.59 T during the deposition process, the AFM image conﬁrms the formation of long tubes (Fig. 3.11A) and, from the cross section, the height of the lines is around 250 nm (Fig. 3.11B). This pattern is in good agreement with that shown in Fig. 3.10B. The EDS pattern is that of γ Fe2 O3 in powder form. The sample is then subjected to a reactive ion etching (RIE) process [84] using O2 and SF6 . After etching and washing the sample with ethanol, the AFM image clearly shows the same lines (Fig. 3.11C) with 40 nm average depth, (Fig. 3.11D). As can be seen in the image, the precision of the lines remains almost equivalent before and after etching. This clearly shows that the tubes are reproduced into a silicon wafer. When γ Fe2 O3 nanocrystals, coated with lauric acid, are dispersed in hexane and deposited on the substrate by using an anticapillary tweezer, the AFM pattern shows that nanocrystals self-organize in micrometer rings (Fig. 3.11E) as shown in Fig. 3.10A. Because of the size distribution of the rings and the

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Fig. 3.11. AFM images of the substrate before (A, E) and after (C, G) the RIE process. Cross section of the substrate before (B, F) and after (D, G) the RIE process

fact that AFM is a local measurement, we focus on a given ring before and after etching. From the cross section of the ring, its diameter is 15 µm and its height is 200 nm (Fig. 3.11F). After etching and washing the sample with ethanol, the SEM pattern (Fig. 3.11G) is similar to that observed by AFM.

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The EDS pattern shows presence of Si without any iron. This clearly indicates that the rings are made of silicon. This is a direct proof that ferrite nanocrystals are highly eﬃcient in being used as a mask. Hence mesostructures can be reproduced in a given substrate through a lithographic mask.

3.6 Conclusion In this review we demonstrate that reverse micelles constitute an eﬃcient nanoreactor for controlling the size of spherical nanocrystals whereas only a partial control in the shape is obtained. The major parameters in the shape control of nanocrystals seem to be the selective adsorptions on various faces during the particle growth. This question is still open. Normal micelles are used to produce well-deﬁned ferrite nanocrystals. Nanocrystals are self-organized in 2D either in a compact hexagonal network or in rings. By controlling temperature and evaporation time, “supra” crystals in FCC structures are produced even for magnetic nanocrystals. By applying a magnetic ﬁeld, various mesoscopic structures of nanocrystals are obtained. Collective properties due to these organizations are pointed out.

References 1. Reactivity in Reverse Micelles, Pileni, M.P.; Ed.; Pub.Elsevier: Amsterdam New-York, Oxford, Shannon, Tokyo 1989 2. The language of shape Ed.Hyde, S.; Andersoon, S.; Larsson, K.; Blum, Z.; Landh, T.; Lidin, S.; Ninham. B.W. Pub. Elsevier: Amsterdam, New-York, Oxford, Shannon, Tokyo 1997 3. D.F. Evans, D.J. Mitchell, B.W. Ninham, J. Phys. Chem. 90, 2817 (1986) 4. S.J.Chen, D.F. Evans, B.W. Ninham, D.J. Mitchell, F.D. Blum, S. Pickup, J. Phys. Chem. 90, 842 (1986) 5. I.S. Barnes, S.T. Hyde, B.W. Ninham, P.J. Derian, M. Driﬀord, T.N. Zemb, J. Phys. Chem. 92, 2286 (1988) 6. C. Petit, P. Lixon, M.P. Pileni, Langmuir 7, 2620 (1991) 7. J. Eastoe, G. Fragneto, B.H. Robinson, T.F. Towey, R.K. Heenan, F.J. Leng, J. Chem. Soc. Faraday Trans. 88, 461 (1992) 8. J. Eastoe, D.C. Steytler, B.H. Robinson, R.K. Heenan, A.N. North, J.C. Dore, J. Chem. Soc. Faraday Trans. 90, 2479 (1994) 9. J. Eastoe, B.H. Robinson, R.K. Heenan, Langmuir 9, 2820 (1993) 10. J. Eastoe, T.F. Towey, B.H. Robinson, J. Williams R.K. Heenan, J. Phys. Chem. 97, 1459 (1993) 11. D. Fioretto, M. Freda, S. Mannaioli, G. Oniri, A.J. Santucci, Phys. Chem. B 103, 2631 (1999) 12. D. Fioretto, M. Freda, G. Oniri, A.J. Santucci, J. Phys. Chem. 103, 8216 (1999) 13. M.P. Pileni, J. Phys. Chem. 97, 9661 (1993) 14. M.P. Pileni, Langmuir 13, 3266 (1997) 15. D. Vollath, D.V. Szabo, R.D. Taylor, J.O. Willis, J. Mat. Res. 12, 2175 (1997)

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16. A. Perez, P. Melinon, V. Dupuis, B. Prevel, L. Bardotti, J. Tuallon-Combes, B. Maselli, M. Treilleux, M. Pellarin, J. Lerme, E. Cottancin, M. Broyer, M. Janet, M. Negrier, F. Tournus, M. Gaudry, Mater. Trans. Special Issue on Nano-Metals 42, 1460 (2001) 17. C.B. Murray, D.J. Norris, M.G. Bawendi, J. Am. Chem. Soc. 115, 8706 (1993) 18. G.F. Goya, H.R. Rechenberg, J. Mag. Mag. Mat. 196–197, 191 (1999) 19. K.V.P.M. Shaﬁ, Y. Koltypin, A. Gedanken, R. Prozorov, J. Balogh, J. Lendvai, I. Felner, Phys. Chem. B 101, 6409 (1997) 20. T. Sugimoto, Y. Shimotsuma, H. Itoh, Powder Tech. 96, 85 (1998) 21. W.C. Elmore, Phys. Rev. 54, 309 (1938) 22. N. Moumen, M.P. Pileni, J. Phys. Chem. 100, 1867 (1996) 23. J. Tanori, M.P. Pileni, Adv. Mater. 7, 862 (1995) 24. A. Filamkenbo, M.P. Pileni, J. Phys. Chem. 104, 5867 (2000) 25. A. Filamkenbo, S. Giorgio, I. Lisiecki, M.P. Pileni, J. Phys. Chem. 107, 7492 (2003) 26. M.P. Pileni, Nat. Mater. 2, 145 (2003) 27. M.P. Pileni, J. Phys. Chem. 105, 3358 (2001) 28. L. Motte, F. Billoudet, M.P. Pileni, J. Phys. Chem. 99, 16425 (1995). 29. C.B. Murray, C.R. Kagan, M.G. Bawendi, Science 270, 1335 (1995) 30. M. Brust, D. Bethell, D.J. Schiﬀrin, C. Kiely, Adv. Mater. 9, 797 (1995) 31. S.A. Harfenist, Z.L. Wang, M.M. Alvarez, I. Vezmar, R.L. Whetten, J. Phy. Chem. 100, 13904 (1996) 32. L. Motte, F. Billoudet, E. Lacaze, J. Douin, M.P. Pileni, J. Phys. Chem. B 101, 138 (1997) 33. A. Courty, C. Fermon, M.P. Pileni, Adv. Mat. 13, 58 (2001) 34. A. Courty, O. Araspin, C. Fermon, M.P. Pileni, Langmuir 17, 1372 (2001) 35. I. Lisiecki, P.A. Albouy, M.P. Pileni, Adv. Mat. 15, 712 (2003) 36. A. Taleb, C. Petit, M.P. Pilen, J. Phys. Chem. B 102, 2214 (1998) 37. C. Petit, A. Taleb, M.P. Pileni, Adv. Mater. 10, 259 (1998) 38. C. Petit, A. Taleb, M.P. Pileni, J. Phys. Chem. B 103, 1805 (1999) 39. A. Taleb, V. Russier, A. Courty, M.P. Pileni, Phys. Rev. B 59, 13350 (1999) 40. V. Russier, M.P. Pileni, Surf. Sci. 425, 313 (1999) 41. C. Petit, T.D. Cren, Adv. Mater. 11, 1358 (1999) 42. V. Russier, C. Petit, J. Legrand, M.P. Pileni, Phys. Rev. B 62, 3910 (2000) 43. T. Ngo, M.P. Pileni, Adv. Mat. 12, 276 (2000) 44. S. I-Sun, C.B. Murray, D. Weller, L. Folks, A. Moser, Science 287, 1989 (2000) 45. C.T. A-Black, C.B. Murray, R.L. Sandstrom, S. Sun, Science 290, 1131 (2000) 46. A. Taleb, F. Silly, O. Gusev, F. Charra, M.P. Pileni, Adv. Mat. 12, 119 (2000) 47. T. Ngo, M.P. Pileni, J. Phys. Chem. 105, 53 (2001) 48. C.B. E-Murray, S. Sun, W. Gaschler, T.A. Betley, C.R. Kagan, IBM J. Res. Dev. 45, 47 (2001) 49. J. Legrand, C. Petit, M.P. Pileni, J. Phys. Chem. B 105, 5643 (2001) 50. T. Ngo, M.P. Pileni, J. Appl. Phys. 92, 4649 (2002) 51. C. Petit, V. Russier, M.P. Pileni, J. Phys. Chem. 107, 10333 (2003) 52. M.P. Pileni, T. Zemb, C. Petit, Chem. Phys. Lett. 118, 414 (1985) 53. P. Andr´e, A. Filankembo, I. Lisiecki, C. Petit, J. Tanori, T. Gulik-Krzywicki, B.W. Ninham, M.P. Pileni, Adv. Mater. 12, 119 (2000) 54. I. Lisiecki, P. Andr´e, A. Filankembo, C. Petit, J. Tanori, T. Gulik-Krzywicki, B.W. Ninham, M.P. Pileni, J. Phys. Chem. 103, 9176 (1999)

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55. A. Andr´e, B.W. Ninham, M.P. Pileni, New J. Phys. 25, 563 (2001) 56. I. Lisiecki, P. Andr´e, A. Filankembo, C. Petit, J. Tanori, T. Gulik-Krzywicki, B.W. Ninham, M.P. Pileni, J. Phys. Chem. 103, 9168 (1999) 57. M.P. Pileni, Langmuir 17, 7476 (2001) 58. C. Petit, M.P. Pileni, J. Phys. Chem. 92, 2282 (1988) 59. M.P. Pileni, Reverse Micelles (Elsevier, Amsterdam, 1989) 60. J. Tanori, M.P. Pileni, Langmuir 13, 639 (1997) 61. I. Lisiecki, M.P. Pileni, J. Am. Chem. Soc. 115, 3887 (1993) 62. I. Lisiecki, A. Filankembo, H. Xasck-Kongehl, K. Weiss, M.P. Pileni, Phys. Rev. B. 61, 4968 (2000) 63. B.A. Simmons et al., Nanoletters 2, 263 (2002) 64. M. Maillard, S. Giorgio, M.P. Pileni, Adv. Mat. 14, 1084 (2002) 65. M. Maillard, S. Giorgio, M.P. Pileni, J. Phys. Chem. B. 107, 2466 (2003) 66. V. Germain, J. li, D. Ingert, Z.L. Wang, M.P. Pileni, J. Phys. Chem. B 107, 8717 (2003) 67. L. Motte, E. Lacaze, M. Maillard, M.P. Pileni, Langmuir 16, 3803 (2000) 68. M. Maillard, L. Motte, M.P. Pileni, Adv. Mat. 13, 200 (2001) 69. M. Maillard, L. Motte, M.P. Pileni, J. Phys. Chem. 104, 11871 (2000) 70. P. Manneville, Dissipative Structures and Weak Turbulence (Academic press, Boston, 1990) Perspectives in Physics 71. R.W. Chantrall, A. Bradbury, J. Popplewell, S.W. Charles, J. Appl. Phys. 53, 2742 (1982) 72. T. Tlusty, S.A. Safran, Science 290, 1328 (2000) 73. J.M. Traves, J.J. Weis, M.M. Telo da Gamma, Phys. Rev. E 59, 4388 (1999) 74. H. Morimoto, T. Maekawa, Int. J. Mod. Phys. B 13, 2085 (1999) 75. Y. Lalatonne, J. Richardi, M.P. Pileni, Submitted for publication 76. J. Legrand, T. Ngo, C. Petit, M.P. Pileni, Adv. Mat. 13, 254 (2001) 77. R.E. Rosensweig, Ferrohydrodynamics (Cambridge University Pres, Cambridge, UK, 1985) 78. L.T. Romanikew, M.M.G. Slusarczyk, D.A. Thompson, IEEE Trans. Magn. Mag. 11, 25 (1975) 79. R.E. Rosenweig, M. Zhan, R. Shumovich, J. Magn. Magn. Mater. 39, 127 (1983) 80. J. Richardi, I. Ingert, M.P. Pileni, J. Phys. Chem. B 106, 1521 (2002) 81. J. Richardi, I. Ingert, M.P. Pileni, Phys. Rev. E. 66, 46306 (2002) 82. J. Richardi, M.P. Pileni, Phys. Rev. E, in press 83. D. Ingert, M.P. Pileni, J. Phys. Chem. B 107, 9617 (2003) 84. The reactive ion etching is performed as follows for all the samples : 20 s with O2 10cc, 5µbar, 30 W and then 30 s with SF6 10cc, 5µbar, 30 W.

4 Self-Assembled Nanoparticle Rings L.V. Govor

Summary. Formation of self-assembled rings of CoPt3 nanoparticles was achieved on the surface of water by spreading a binary mixture composed of two solutions: nitrocellulose dissolved in amyl acetate and CoPt3 particles stabilized by hexadecylamine dissolved in hexane. The self-assembly process of the nanometer-sized particles into micrometer-sized rings results from phase separation in a thin ﬁlm of the mixed solutions leading to a bilayer, and the subsequent decomposition during solvent evaporation of the top hexadecylamine-rich layer into pancakes. The subsequent evaporation of the remaining solvent from these pancakes gives rise to a retraction of their contact line. The CoPt3 particles located at the contact line follow its motion and self-assemble along this line.

4.1 Introduction In the last few years, more and more research groups concentrate on nanotechnology as the technology for future markets. For nanofabrication with the sizes below 30 nm, conventional electron-beam lithography becomes exceedingly diﬃcult. Therefore, the nanoparticles and the mesoscopic structures formed with the nanoparticles have gained increasing interest. Nanoparticles often possess novel properties that are diﬀerent from the properties of the bulk materials and that dependent on size, shape, and surface composition [1]. The ordering of nanoparticles into micrometer-size structures that retain the unique nanoparticle properties can be used for the development of new functional materials and for the fabrication of a variety of nanodevices [2]. A fundamentally diﬀerent alternative (compared to electron-beam lithography) that is not limited by the spatial resolution of the pattern of the resist, can be the natural self-assembled processes into certain media that lead to form ordered structures on micrometer length. One from that is self-assembly of nanoparticles in a ringlike structure which occurred during the formation of a thin ﬁlm of the polymer solution containing nanoparticles. The resulting

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structures can be formed by using diﬀerent forces that lead to some ordering of the nanoparticles. Recently, several papers have reported the formation of nanoparticles to ringlike structures. Ohara and Gelbart [3] describe such patterning which self-assembles from a solution of nanometer-sized metal particles on a solid substrate. The rings having a diameter of 0.1–1 µm were formed during the drying process and resulted from holes nucleating in wetting thin liquid ﬁlms that contain the particles. Kurikka et al. [4] have observed that barium ferrite nanoparticles can combine to the well-known structure of the olympic rings (intersection of rings, ring diameter 0.6–5 µm). The mechanism of ring building was explained in terms of the formation of holes in an evaporating thin ﬁlm and interparticle dipolar forces. Maillard et al. [5, 6] found out that the formation of rings of nanoparticles (silver, copper, cobalt, silver sulﬁde) in the micrometer range is related to Benard–Marangoni instabilities in deposited liquid ﬁlms. Tripp et al. [7] have shown that Co nanoparticles can self-assemble in similar rings as a consequence of the following processes: dipole-directed self-assembly (typically 5–12 particles, ring diameter between 50 and 100 nm) and evaporation-driven hole formation in viscous wetting layers (ring diameter ranging from 0.5 to 10 µm). Wyrwa et al. [8] have described a one-dimensional arrangement of metal nanoparticles formed by a selfassembly process at the phase boundary between water and dichloromethane. In the present work, experimental evidence of how phase separation in a thin ﬁlm of a binary mixture of solutions that includes a polymer and CoPt3 nanoparticles with a stabilizer is given. Phase separation leads to a bilayer structure and a subsequent decomposition of the top layer into micrometersized pancakes, what correspondingly leads to the self-organized formation of rings of nanoparticles (ring diameter ranging from 0.6 to 1.5 µm, particle diameter 6 nm). As an underlying physical mechanism, a self-assembly process of the nanoparticles at the pancake contact line is proposed as a consequence of the shrinking of the pancake during evaporation, i.e., the particles located at the contact line follow its motion. This study may extend the class of self-assembling processes which involve a variety of forces governing the formation of ring structures with micrometer size [3–8].

4.2 Experimental Formation of Nanoparticle Rings 4.2.1 Spreading of Polymer Solution on Water Surface A blend (B) containing 50% of a 1% nitrocellulose solution (NC) in amyl acetate and 50% of a solution of CoPt3 particles stabilized with hexadecylamine (HDA) and dissolved in hexane was used in the present investigation. The CoPt3 particles with a diameter of 6 nm were prepared via the simultaneous reduction of Pt(acac)2 and thermal decomposition of Co2 (CO)8 in the presence of 1-adamantanecarboxylic acid and HDA both functioning as stabilizing and size-regulating agents [9]. As prepared CoPt3 particles were

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thoroughly washed in ethanol. Repeated cleaning processes make the CoPt3 particles insoluble in either nonpolar or polar solvents due to the removal of the surfactant from the surface of the nanoparticles. Addition of HDA (7 mg ml−1 ) to the suspension of the CoPt3 particles in hexane (17 mg ml−1 ) leads to the formation of a clear stable colloidal solution of particles. Spreading of a binary mixture of the solutions onto a water surface in a Petri dish of 90 mm diameter was used for the preparation of the thin ﬁlm, which provided the basis for the pancake formation [10, 11]. Total spreading of a drop of the blend solution B on the water surface is obtained due to the positive spreading coeﬃcient [12], SB/W = γW/A − γB/A − γB/W = 25.3 mN m−1 . Here, γW/A = 72.5 mN m−1 , γB/A = 22.8 mN m−1 , and γB/W = 24.4 mN m−1 give the surface tension at the interfaces between water and air (W/A), between blend and air (B/A), and between blend solution and water (B/W), respectively. The interfacial tension γB/W was derived from the diﬀerence between the surface tension of water saturated with the polymer solution and that of the polymer solution saturated with water. The values of the surface tensions were determined using a stalagmometer [13]. Since the volume of the spread drop was 3 µl, the total thickness of the resulting spread ﬂuid layer (with a diameter of 90 mm) can be estimated to be about 600 nm. After evaporation of amyl acetate and hexane, the dry thin ﬁlm was transferred onto a glass substrate. The topography of this solid thin ﬁlms was analyzed by atomic force microscopy (AFM, model Burleigh Vista 100) and the arrangement of the CoPt3 particles by transmission electron microscopy (TEM, model Zeiss EM 902). 4.2.2 HDA Pancake Structures According to the AFM investigation (ac mode, topography), a sea-island-like phase separated structure of HDA islands containing CoPt3 particles (hereafter called HDA pancakes) on the cellulose thin ﬁlm could be found [14, 15]. The size of the HDA pancakes depends on the location along the radius of the spreading area, but the thickness of the cellulose ﬁlm was more or less constant on the full spreading area. The areas with equal HDA pancake size represented circular regions with a width of 2–4 mm, located symmetrically in the spreading center. In the center of the spreading area, we observed the smallest HDA pancakes with the height hd = 5 nm and the diameter Dd = 650 nm. In the circular region that was at a distance of 20 mm from the spreading center, we found the largest HDA pancakes with hd = 23.5 nm and Dd = 1, 500 nm, respectively. All other circular regions with hd in the range from 5 to 23.5 nm (corresponding to Dd in the range from 0.65 to 1.5 µm) were located between these circular regions. Figures 4.1a, 4.2a, and 4.3a show typical AFM images of the structure of the HDA pancakes located on the surface of the cellulose ﬁlm, diﬀering in the average pancake height hd of 5.0 nm, 14.9 nm, and 23.5 nm, respectively. The corresponding values of the average diameter Dd are 650 nm, 950 nm, and

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Fig. 4.1. AFM image of smallest-size HDA pancakes on a cellulose ﬁlm. (a) Threedimensional representation; (b) and (c) are histograms of the diameter and height distribution of the pancakes, respectively. The mean size of the pancakes is given by the diameter Dd = 650 ± 7 nm with a standard deviation of ±62 nm and the height hd = 5.0 ± 0.2 nm with a standard deviation of ±1.7 nm

1,500 nm, respectively. It should be noted that the pancakes in all samples investigated are aligned with some curvature, extending over a length of a few tens of µm. The pancakes of small size can be connected with or separated from each other (see Fig. 4.1a). The distance between the pancakes increases with the size of the pancakes, so that the large pancakes became more isolated (see Fig. 4.3a). The uniformity of the pancake distribution increases with the pancake size, as clearly illustrated in Figs. 4.1b, 4.2b, and 4.3b. The small pancakes are well characterized by the two maxima in the histogram of the diameter distribution. Apparently, the pronounced ﬁrst maximum in Fig. 4.1b diminishes in Fig. 4.2b and, ﬁnally, almost disappears in Fig. 4.3b. A similar behavior can be observed in the histogram of the height distribution displayed in Figs. 4.1c, 4.2c, and 4.3c. The latter indicates that the lateral growth of the pancake size is approximately proportional to its vertical growth. It turns out that the distribution of the pancake diameter shown in Fig. 4.3b is comparatively narrow, centered around the mean value Dd = 1,500 nm with a standard

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Fig. 4.2. AFM image of the middle-size HDA pancakes on a cellulose ﬁlm. (a) Three-dimensional representation; (b) and (c) are histograms of the diameter and height distribution of the pancakes, respectively. The mean size of the pancakes is given by the diameter Dd = 950 ± 13 nm with a standard deviation of ±119 nm and the height hd = 14.9 ± 0.5 nm with a standard deviation of ±4.4 nm

deviation of ±191 nm. This means that about 70% of all values of the pancake diameter are conﬁned to the interval 1,500 ±191 nm. The error range of the mean value of Dd is estimated to ±26 nm. The corresponding distribution of the pancake height in Fig. 4.3c unveils a mean value hd = 23.5 ± 0.6 nm with a standard deviation of ±4.3 nm. The distribution of the pancake diameter and height for the both samples shown in Figs. 4.1 and 4.2 can be found in the caption of corresponding ﬁgures. Figure 4.4a demonstrates a two-dimensional AFM picture of the same sample shown in Fig. 4.1. It can be clearly seen that some of the pancakes have an M -shaped height proﬁle what illustrates Fig. 4.4b, but the large pancakes were always round and convex to some extent (Fig. 4.3a). For a detailed understanding and a modeling of the self-assembly mechanism, it is important to know how the HDA pancakes penetrated into the cellulose ﬁlm. For determination of this penetration depth, the HDA pancakes were removed by immersing the specimen in hexane acting as selective solvent for HDA for a period of 5 min. Figure 4.4c shows a typical AFM picture of the remaining

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Fig. 4.3. AFM image of the largest-size HDA pancakes on a cellulose ﬁlm. (a) Three-dimensional representation; (b) and (c) are histograms of the diameter and height distribution of the pancakes, respectively. The mean size of the pancakes is given by the diameter Dd = 1,500 ± 26 nm with a standard deviation of ±191 nm and the height hd = 23.5 ± 0.6 nm with a standard deviation of ±4.3 nm

cellulose layer. The measured thickness of the underlying cellulose ﬁlm was more or less constant at about 3–4 nm. Figure 4.4d clearly illustrates that the depth of the voids that remain in the cellulose ﬁlm after removal of the HDA pancakes was about 1.0 nm only, indicating that the thickness of the cellulose ﬁlm under the voids amounts to at least 2–3 nm. Consequently, the HDA pancakes have no contact with the substrate surface, i.e., the substrate is completely covered by the cellulose ﬁlm. Figure 4.4d quantiﬁes the height of a rim that surrounds the HDA pancake to about 5 nm. 4.2.3 CoPt3 Nanoparticle Rings Figure 4.5 shows TEM images of the HDA pancakes which clearly demonstrate that the CoPt3 particles with a radius of 3 nm self-assemble into a ring pattern

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Fig. 4.4. (a) AFM image of HDA pancakes on a cellulose ﬁlm with average diameter Dd = 650 nm and average height hd = 5.0 nm. (b) AFM proﬁle of the scan line indicated in a. (c) AFM image of the cellulose layer after removal of the HDA pancakes. (d) Proﬁle of the scan line indicated in c

located at the perimeter of the pancakes. Figures 4.5a, b display a prototypical particle ring with a diameter Dd ≈ 930 nm that can be recognized as a twodimensional (2D) ring structure. Figure 4.5c shows a particle ring with a diameter of 860 nm and a conﬁguration that can be called one-dimensional (1D). In both cases, the distance between the individual particles within the ring is roughly constant. Moreover, three-dimensional (3D) particle clusters occur. In the spreading area, it can be found only a few regions where the pancakes are situated near the segments of the solid HDA ﬁlm, greatly extending the pancake size. Figure 4.5d exhibits a typical example of such a segment edge where the CoPt3 particles assemble at the contact line. It is interesting to note that Fig. 4.5d also displays an unﬁnished process of pancake formation. The HDA ﬁlm segments were usually essentially thicker than the height of the pancakes. It must be noted that the samples shown in Figs. 4.5a–d have been imaged before and after removal of HDA from the samples. It turned out that both number and position of the individual CoPt3 particles did not change, which indicates that all particles are situated at the interface between HDA and the cellulose layers and are somewhat embedded in the cellulose layer. In other words, the CoPt3 particles in the interior of the dry HDA pancake are

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Fig. 4.5. TEM images of a CoPt3 particle ring formed at the edges of HDA pancakes. (a) 2D assemblage; scale bar 116 nm. (b) Magniﬁcation of the area indicated by the bright box in a; scale bar 15 nm. (c) 1D assemblage; scale bar 98 nm. The detailed structure is magniﬁed in the inset; scale bar 20 nm. (d) Segment of the HDA ﬁlm with CoPt3 particles assembling at its edge; scale bar 270 nm (70 nm in the inset)

only located on the interface with the cellulose layer, but not in the rest of the droplet volume. If the particles were not partially embedded in the cellulose layer, they would have been washed oﬀ during HDA removal.

4.3 Model for the Formation of HDA Pancakes 4.3.1 Phase Separation of Binary Solution According to the experimental results shown in Fig. 4.4, it can be assumed that the initial thin layer of mixed solutions on the water surface transforms into a bilayer structure which consists of a hexane–hexadecylamine-rich (HDA-rich) phase at the solution–air interface with thickness ≤300 nm and an amyl acetate–cellulose-rich (NC-rich) phase at the solution–water interface

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with thickness ≤300 nm. Figure 4.6 sketches a pictorial scenario of the whole mechanism of the ring formation proposed in this work, and Fig. 4.6a shows the formation of the bilayer structure. Assuming complete phase separation, the thickness of both layers may be determined from the spread drop volume, the covered area, and the volume ratio of the two phases in the initial blend solution. Note that at the beginning of evaporation, the HDA-rich phase contains 99.4% hexane and only 0.6% HDA, the corresponding NC-rich phase 99% amyl acetate and only 1% cellulose. The number of CoPt3 particles in each layer was about 6×1012 . One reason for the formation of the bilayer is that the HDA-rich phase with its lower surface free energy (18.4 mN m−1

Fig. 4.6. Schematic illustration of the development of the phase-separated structure and the corresponding CoPt3 particle rings. (a) Formation of phase-separated layers (bilayer structure). (b) Rupture of the HDA-rich layer into pancakes. (c) Formation of the particle ring in a separated HDA-rich pancake. The contact line moves from point s to point p before it is pinned. Jf indicates the radial outward solvent ﬂow. (d) Forces which act on the CoPt3 particle in the interior of the HDA pancake. The media (1), (2), (3), and (4) are NC-rich layer, HDA-rich layer, CoPt3 particle, and air, respectively. (e) Forces which act on the CoPt3 particle located at the contact line of the HDA pancake. fth is the thickening force per particle. (f) Assembling of the particles at the contact line during its motion (top view of the sketch shown in c)

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for the HDA-rich component and 24.6 mN m−1 for the NC-rich component) wets the surface region, in order to minimize the free energy at the interface between air and solution [16–19]. The role of CoPt3 particles in the formation of HDA pancake patterns was analyzed with a blend that contains 50% of a 1% NC solution in amyl acetate and 50% of a 1% HDA solution in hexane without nanoparticles. Not an exactly identical, but a similar HDA pancake pattern was found. This means that in the HDA pancake formation the CoPt3 particles do not play a crucial role and, in the following discussion, both the attraction between nanoparticles in the solution and that between nanoparticles and HDA solution will be not considered. The condition for the wetting of the HDA-rich layer on the NC-rich one is given by [12] SHDA/NC = γNC/A − γHDA/A − γHDA/NC > 0, where SHDA/NC designates the spreading coeﬃcient. Here, γHDA/A and γNC/A mean the surface tension of the HDA-rich phase and that of the NC-rich phase at the boundary between the corresponding phase and air, respectively; γHDA/NC is the surface tension at the interface between the HDA-rich and the NC-rich phases. For a careful analysis of the evaporation process of the bilayer structure, we recorded the values of the surface tension of both HDA and NC layers in the solid state (in the following, referred to as HDAs and NCs, respectively). They were obtained from the so-called Zisman plot [20], where the contact angle θc of the droplets (volume 1–3 µl) of water, glycerol, formamide, piridine, cyclohexanone, decalin, and n-decane on the surface of thin HDAs and NCs ﬁlms located on a glass substrate was measured (Fig. 4.7). In choosing the ﬂuids, two demands were pursued: ﬁrst, a possibly large interval of surface tension for the Zisman plot and, second, the liquids must not dissolve HDAs or NCs layers. The extrapolation of the approximation lines (line 1 for the NCs ﬁlm and line 2 for the HDAs ﬁlm) shown in Fig. 4.7 to cos θc = 1 gives

Fig. 4.7. Zisman plot of the NCs (ﬁt line 1) and the HDAs ﬁlms (ﬁt line 2) measured with the liquids described in the text. The intersection of the extrapolated best-ﬁt line with cos θc = 1 gives a critical surface tension of the corresponding ﬁlm

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Fig. 4.8. Time dependence of the mass evaporation of three solvents: two-phase solvent contained amyl acetate and hexane as well as pure amyl acetate and pure hexane. The corresponding slope of the approximation lines (evaporation rate) is indicated in the ﬁgure

the critical surface tension of the ﬁlm to be investigated. The term “critical” is used because any liquids taken on the Zisman plot in Fig. 4.7 whose surface tension enlarges the “critical surface tension” deﬁne a ﬁnite contact angle with the ﬁlm investigated. The resulting values of the surface tension of both HDAs and NCs layers in the solid state amount to γHDAs/A = 25.8 ± 0.9 mN m−1 and γNCs/A = 28.6 ± 1.2 mN m−1 , respectively. According to the result received by the Zisman plot, the relation γHDAs/A < γNCs/A is valid in the solid state, but it is not known when in the course of evaporation this relation is valid and this inequality is applicable during the entire process. Thus, in the following, the alteration of both parameters γHDA/A and γNC/A during evaporation will be discussed. First, the evaporation rate of 24 ml hexane was determined experimentally by monitoring the mass losses vs. time under similar geometric and temperature conditions as they were in the self-assembly experiment. Figure 4.8 indicates that this rate amounts to 3.30 ± 0.06 mg s−1 . The analogous experimentally estimated evaporation rate of 24 ml amyl acetate was 0.16 ± 0.01 mg s−1 . Second, the evaporation rate of the two-phase solvent that contained 24 ml hexane and 24 ml amyl acetate was determined experimentally. In the latter case, evaporation process divides into two distinct regimes, ﬁrst, the evaporation of hexane with 2.45±0.11 mg s−1 and, second, that of amyl acetate with 0.13±0.01 mg s−1 (see Fig. 4.8). On the one hand, the evaporation rate of hexane in the two-phase solvent is 26% lower than that of pure hexane. The corresponding decrease of the evaporation rate of amyl acetate amounts to 19%. On the other hand, the evaporation rate of hexane (pure or in the two-phase solvent) is about

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20 times larger than that of amyl acetate. The existence of such a relation between the evaporation rates of hexane and amyl acetate can be assumed in our self-assembly experiment, despite the fact that here the top HDA layer acts as diﬀusion barrier for amyl acetate evaporation. From the latter data of the evaporation rates in the two-phase solvent, the determined time t0 necessary to evaporate 1.0 mg hexane amounts 0.4 s and that to evaporate 1.3 mg amyl acetate amounts 10 s in our self-assembly experiment. With the mass loss of each solvent during the evaporation time t for 0 < t < t0 , described as ms = mso (1 − t/t0 ), where mso denotes the mass at the starting point of the evaporation process, the time dependence of the surface tension γ of the binary HDA solution that contains hexane (γs/A ) and HDA (γHDAs/A ) can be calculated as [12] γ = γs/A Ns + γHDAs/A NHDAs − βNs NHDAs ,

(4.1)

where β is a semiempirical constant. Here, Ns = (1 − t/t0 )/(1 − t/t0 + α) and NHDAs = α/(1 − t/t0 + α) are fractions of the corresponding component in the binary solution, where α = mHDAs /mso , and mHDAs is the mass of HDA in the top layer. Accordingly, (4.1) can be used for the NC-rich phase that contains amyl acetate and cellulose. Figure 4.9a displays the results calculated for both values of the surface tension vs. the evaporation time. Line 1 characterizes the evaporation process of the HDA-rich layer with γs/A = 18.4 mN m−1 (hexane), γHDAs/A = 25.8 mN m−1 , mso = 1.0 mg, mHDAs = 4.2×10−3 mg, and β = 1. Line 2 describes the evaporation of the NC-rich phase with γs/A = 24.6 mN m−1 (amyl acetate), γNCs/A = 28.6 mN m−1 , mso = 1.3 mg, mNCs = 7.5×10−3 mg, and β = 1. The corresponding spreading coeﬃcient SHDA/NC and the change of γHDA/NC vs. evaporation time are depicted by the lines 3 and 4 in Fig. 4.9b. In Fig. 4.9c, they are presented vs. the surface tension of the HDA-rich layer. Obviously, SHDA/NC is positive until γHDA/A = γNC/A and before dewetting of the HDA-rich layer on the surface of the NC-rich layer starts. Line 4 shows the interfacial tension at the interface between the two phases which was calculated via γHDA/NC = γHDA/A + γNC/A − 2(γHDA/A γNC/A )0.5 [21]. 4.3.2 Rupture of Thin HDA Film into Micrometer-Size Pancakes The characteristic equilibrium thickness de of the wetting HDA-rich layer on the NC-rich layer results from a competition between attractive long-range forces (as measured by γHDA/A with the tendency to make the ﬁlm thicker) and short-range forces (as characterized by SHDA/NC with the tendency to thin the ﬁlm) and is determined according to [22–24] de = a(3γHDA/A /2SHDA/NC )1/2 ,

(4.2)

4 Self-Assembled Nanoparticle Rings

79

where a2 = |A|/6πγHDA/A is a molecular length, derived from the ratio of the eﬀective Hamaker constant A = A12 −A22 and the surface tension of the HDArich layer. Here, the NC-rich layer was denoted as body 1 and the HDA-rich layer as body 2. The Hamaker constant A12 = (A11 A22 )1/2 [25, 26] describes the dispersive interaction between NC-rich and HDA-rich layers, where A11 and A22 are the individual Hamaker constants [27] of the NC-rich and HDArich layers, respectively. Based on these considerations, the HDA-rich layer cannot become thinner than de . If there is not enough material to cover the whole substrate with a ﬁlm of thickness de , islands or pancakes [22–24] will be formed. Figure 4.9a, b indicates that, during the solvent evaporation from the ﬁlm, the surface tension γHDA/A of the upper layer rises (line 1), while the corresponding spreading coeﬃcient SHDA/NC becomes smaller (line 3). According to Fig. 4.9b, c, the equilibrium thickness de (4.2) strongly varies with the evaporation time following the transient of the spreading coeﬃcient SHDA/NC , caused by the variation of γHDA/A from 18.4 to 24.6 mN m−1 . This change leads to a decrease of SHDA/NC by more than two orders of magnitude. For example, with γHDA/A ≈ 24 mN m−1 and, correspondingly, SHDA/NC ≈ 0.25 mN m−1 , and further using a = 0.3 nm [23], the equilibrium thickness of the wetting HDA-rich layer on the NC-rich layer amounts to about de ≈ 4 nm. Here, the length a was considered as a constant. Indeed, the molecular length a2 = [|(A11 A22 )1/2 − A22 |]/6πγHDA/A will change on the altering of the Hamaker constant A22 and γHDA/A during the evaporation time 0 < t ≤ 0.4 s (the change of A11 in this time interval is very small, see line 2 in Fig. 4.9a). The change in A22 is proportional to that of γHDA/A , which varies by a factor of about 1.4. In this case, the change of a will be very small and cannot inﬂuence essentially the evaluation of de . Based on these considerations, it can be expected that the equilibrium thickness de of the HDA-rich layer, when it decomposes into pancakes, was in the range of the diameter of a CoPt3 particle. In this case, the particles were located near the interface between the HDA-rich and the NC-rich layers. This assumption is in good agreement with our experimental result, which indicates that all nanoparticles shown in Fig. 4.5 were located on the cellulose layers. Such argument follows from the ﬁnding that both number and position of the individual CoPt3 particles were unchanged by the removal of HDA pancakes from the samples (see Sect. 4.2.3). The HDA-rich layer decomposes into pancakes at a layer thickness d < de , more precisely, at a layer thickness being about 10–15% [3] thinner than de . Below this value, the HDA-rich layer becomes unstable and the formation and growth of the dry patches occur, in order to achieve the equilibrium thickness de . Consequently, under these conditions, the layer decomposes into pancakes [23, 28] (see Fig. 4.6b). In support of this model for the underlying mechanism sketched in Fig. 4.6a, b, the HDA ﬁlms were spin-coated onto a glass substrate covered with a cellulose ﬁlm (thickness 2–4 nm). The concentration of the HDA solution (in hexane) was varied in the range from 0.015 to 0.5 wt%. It was found that, for all concentrations, HDA forms pancake-like

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Fig. 4.9. (a) Calculated surface tension vs. evaporation time for the HDA-rich layer (line 1) and the NC-rich layer (line 2). (b) and (c) calculated spreading coeﬃcient SHDA/NC (line 3) and that of the interfacial tension between two phases, γHDA/NC , (line 4) as a function of time and as a function of the surface tension of the HDA-rich layer, respectively

4 Self-Assembled Nanoparticle Rings

µm

nm

8

4

4

2

(a) 0

81

4

8 µm

0

(b) 0.2

1.0

µm

Fig. 4.10. (a) AFM image of HDA clusters spin-coated from an 0.062% HDA solution on a cellulose ﬁlm. The diameter of the clusters is about 700 nm, their height about 3 nm, respectively. (b) Proﬁle of the scan line indicated in a

clusters on the NC-coated substrate. The diameter and height of these clusters depend on the HDA concentration in the solution. For example, Fig. 4.10 visualizes the clusters that were spin-coated from the 0.062% HDA solution onto a NC ﬁlm. The diameter of the clusters is about 700 nm; their height amounts to 3 nm. These results are in good agreement with those described above for the pancake formation illustrated in Fig. 4.6. It must be emphasized that, at the end of the dewetting process, the resulting HDA pancakes still contain a solvent, since without solvent, HDA would crystallize and the dewetting stop before the quasiequilibrated pancakes have formed a circular shape. The HDA pancakes also contain CoPt3 particles.

4.4 Formation of a Nanoparticle Ring at the Edge of an HDA Pancake 4.4.1 Pinning of an HDA Micrometer-Size Pancake The drying process of a pancake of the HDA-rich phase that contains CoPt3 particles is presented in Fig. 4.6c. As long as the pancake contact line (interface between air, liquid, and substrate) is not pinned, the reduction of the pancake volume by evaporation leads to a motion of its interface from the dashed to the solid line, i.e., the contact line moves from point s to point p. Those particles that are located at the contact line follow its shrinking and assemble into a ringlike pattern. The nature of the force governing such ordering can be some capillary attraction, arising when the particle size is comparable to the thickness of the HDA-rich pancake during dewetting [29–31]. The cited papers clearly demonstrate that, in all experiments, the 2D ordering of the particles always started when the tops of the micrometer and submicrometer particles protrude from the liquid layer. The capillary attraction energy is proportional to r2 (r is the particle radius) and can be much larger than the

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thermal energy (kT ), even with particles of a diameter of about 10 nm [30]. When the contact line gets pinned (point p), the pancake shape changes from the solid to the dotted line in Fig. 4.6c, and an outward ﬂow Jf of the solvent develops, since the solvent removed via evaporation from the edge of the pancake must be replenished by a ﬂow from the interior [32–34]. The ﬂow Jf can transfer up to 100% of the solute to the contact line [34]. 4.4.2 Forces Acting on the Nanoparticle Located in the Interior of Pancake For the understanding of the mechanism responsible for the ring formation, it is interesting to know the correlation between the number of CoPt3 particles which are assembled at the contact line during its motion and the number of those particles which move with the ﬂow Jf . The gravity force mg of a single particle with a diameter of 6 nm and covered with a 1.4-nm thick HDA shell amounts to 2.2×10−20 N (Fig. 4.6d), and the corresponding buoyancy force in the HDA-rich layer to fb = ρVp g = 2.6 × 10−21 N. The parameters m and Vp are mass and volume of a particle, g denotes the gravitational acceleration, ρ = 0.79 g ml−1 and ρ = 0.80 g ml−1 are the densities of the HDA- and the NC-rich layer, respectively. The resulting relation mg > fb also allows that during the formation of the HDA- and NC-rich layers, the CoPt3 particles can already move downward into the water substrate. During evaporation of the solvent, the viscosity of both layers increases. If the values of the viscosity become large enough, in order to prevent the motion of the particles into the water substrate, the particles will be dispersed in both layers. For modeling the nanoparticle ring arrangement, our starting point consists of a homogeneous lateral distribution of the CoPt3 particles in the solution before the pancake formation occurs. It was found that the number of nanoparticles that were ﬁnally detected on the cellulose layer is lower than that expected from the concentration of the initial solution. For example, the expected numbers of CoPt3 particles contained on the surface inside the rings (excluding the particles at the contact line forming the ring) for the samples shown in Fig. 4.5a, c must be about 1,400 and 1,200, respectively, (determined from the concentration of particles in the initial blend solution). Experimentally, about 540 particles for the sample shown in Fig. 4.5a and about 220 particles in the sample shown in Fig. 4.5c were counted, i.e., about 39% and 18% of the expected number of particles were found, respectively. It can be assumed that the missing particles have dropped into the water substrate or the particles had a dissimilar distribution in the spread ﬁlm. According to the experimental results demonstrated in Fig. 4.5 which indicate that the CoPt3 particles in the interior of the dry HDA pancake are only located on the interface with the cellulose layer, but not in the rest of the droplet volume, it can be assumed that, before the contact line gets pinned and the ﬂow Jf sets in, the majority of the CoPt3 particles were dispersed in a laterally homogeneous way near or onto the cellulose layer. The possibility of

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83

their motion from the center to the contact line in a ﬂuid HDA pancake via the solvent ﬂow Jf can be qualitatively estimated by the equation ff = ffr , where other contributions are negligible compared to the ﬂow force ff and the friction force ffr (see Fig. 4.6d)]. The Reynolds coeﬃcient for the motion of a CoPt3 particle with the radius rs in the ﬂow Jf is Re = ρvrs /η = 5.9 × 10−3 v s m−1 . Here ρ = 0.79 g ml−1 , η = 5.9 × 10−4 Ns m−2 , and v are density, viscosity, and velocity of the ﬂow Jf , respectively; rs = r + δ = 4.4 nm is the radius of the CoPt3 particle overcoated with a HDA monolayer. For example, for the velocity v = 1 m s−1 , the Reynolds coeﬃcient amounts to Re = 5.9 × 10−3 , and the motion of the nanoparticle in such ﬂow is laminar. The force ff = 6πηrs v

(4.3)

describes the interaction of the solvent ﬂow Jf (including the HDA molecules) with a nanoparticle. The counterforce ffr = Kfz derives from the lateral friction force acting on the particle, where K is a dimensionless coeﬃcient of order unity [3] and fz is the attractive dispersion force along the vertical z-axis between each particle and the NC-rich layer. The interaction energy between a small CoPt3 particle and the NC-rich layer can be described as W (D) = −Ar/6D [21], the corresponding interaction force becomes fz = ∂W (D)/∂D = Ar/6D2 , where r = 3 nm is the radius of the CoPt3 particle, D is the distance between the particle and the NC-rich layer, and A is the Hamaker constant appropriate to the CoPt3 particle, interacting through HDA with the NC-rich layer. In our case, the thickness of the HDA layer (body 2 in Fig. 4.6d) between the CoPt3 particle (body 3) and the NC-rich layer (body 1), as well as between the CoPt3 particle and air (body 4), is very small. That means, all interaction components between the surrounding materials across the CoPt3 particle will contribute to the total interaction energy between each particle and the NC-rich layer [35, 36]. In this case, from standard arguments, the interaction force between the CoPt3 particle 3 and the NC-rich layer 1 across the HDA layer 2 can be expressed by [21] √ √ √ A121 A323 A424 A323 A424 A121 r A232 − − + , (4.4) fz = 6 (2r)2 (2r + δ)2 (2r + δ)2 (2r + 2δ)2 where δ = 1.4 nm is the thickness of the HDA monolayer adsorbed on the CoPt3 particle. The eﬀective Hamaker constants A232 , A121 , A323 , and A424 can be composed of the respective Hamaker constants of each medium [25, 26]. 2 2 2 A232 = A323 =

A22 −

A33

, A121 =

A11 −

A22

, A424 =

A44 −

A22

.

(4.5) The individual Hamaker constants Aii can be extracted from experimentally determined data of the surface tension γii as [21] Aii = 24πγii (D0 )2 ,

(4.6)

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with a cutoﬀ intermolecular separation D0 = 0.165 nm [21]. Equation (4.6) yields A22 = 5.3 × 10−20 J for the experimentally obtained value of γHDAs/A = 25.8 mN m−1 taken from the Zisman plot. For the NC-rich layer with γNC/A = 24.6 mN m−1 at the moment of dewetting of the HDA-rich layer, the corresponding Hamaker constant is A11 = 5.0 × 10−20 J. Upon further considering the characteristic value of the Hamaker constant for most metals, A33 ≈ 4 × 10−19 J [21] and A44 = 0, the eﬀective Hamaker constants A232 ≈ 1.6 × 10−19 J, A121 ≈ 0.4 × 10−22 J, and A424 ≈ 5.3 × 10−20 J could be determined. For the above values attributed to the parameters in (4.4), the calculated interaction force becomes fz ≈ 1.3 × 10−12 N. Assuming furthermore K ≈ 0.5 [3], the friction force is quantiﬁed as ffr ≈ 6.6 × 10−13 N. From the equation ff = ffr , the velocity v of the ﬂow Jf that is necessary for the motion of the CoPt3 particle from the interior to the contact line would be v ≈ 6.7 × 103 µm s−1 , which means that only at such a high velocity the ﬂow force would be suﬃciently strong to overcome the frictional forces and drag the nanoparticles. At lower velocities, the nanoparticles do not move. Experimental studies or results on the ﬂow velocity in micrometer-sized droplets with a pinned contact line on the ﬂuid substrate are not available. For macrosized droplets on a solid substrate, only a few studies have been published including the approach for the extraction of the lateral ﬂow velocity [34, 37]. Deegan et al. [34] have measured the velocity in the ﬂuid by tracking the motion of polystyrene microspheres with a diameter of 1 µm inside a drying water droplet with a radius of 2 mm, when the contact line was pinned. They found that the velocity of the water ﬂow Jf near the contact line was about 6 µm s−1 . The evaporation rate in the diﬀusion-limited regime was proportional to the diameter of the evaporating droplet. This happens because the evaporation rate is lowered due to the ﬁnite probability that an evaporated molecule will return to the droplet [37]. In our case where the evaporating pancake has a diameter of about 1 µm, the velocity of the ﬂow Jf must be essentially smaller than 6 µm s−1 . In conclusion, the value v ≈ 6.7 × 103 µm s−1 , claimed above to be necessary for the motion of a CoPt3 particle from the interior to the contact line, cannot be reached in the HDA-rich pancakes investigated here. 4.4.3 Forces Acting on the Nanoparticle Located at the Edge of Pancake The above results clearly indicate that it is not very plausible that the particles found at the perimeter of the pancake have been transferred from its interior. That means, practically all CoPt3 particles in a ring are assembled due to the retraction of the contact line caused by evaporation. This is graphically illustrated in Fig. 4.6e, f, where the dashed line represents the initial ﬂuid pancake, immediately after the HDA-rich layer was ruptured into pancakes (radius Rs ), and the solid line represents the dry pancake (radius Rp ). The force acting radially on the contact line, associated with a force to thicken the

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85

HDA-rich pancake, is given by [23] Fth = 2πRSHDA/NC (de /d)2 − 1 ,

(4.7)

where R denotes the radius of the ﬂuid HDA-rich pancake. The thickening force Fth drags the CoPt3 particles with the retraction of the pancake perimeter (the three-phase contact line), until it is balanced by the total friction force Ffr . The latter results from the superposition of the friction forces of individual nanoparticles, located at the contact line of a pancake with the radius R = Rp , and can be formulated via the friction force per particle, ffr = Kfz (see (4.4)), (4.8) Ffr = Kfz φ(Rs2 − Rp2 )/rs2 . Here, φ denotes the area fraction covered by particles. The expected value φ = 126 × 10−3 was derived from the concentration of particles in the initial blend solution. Experimentally, from the concentration of particles in the interior of the rings illustrated in Fig. 4.5a, c, it was found that not all particles remained at the interface between the HDA and the cellulose layer, i.e., the experimental values of φ for the rings shown in Fig. 4.5a, c amount to 48×10−3 and 23×10−3 , respectively. The term φ(Rs2 − Rp2 )/rs2 in (4.8) accounts for the number of particles located between the dashed and the solid lines in Fig. 4.6f. The radius Rp can be extracted from (4.7) and (4.8) as 1 Rp = 2

1/2 2 2πrs2 SHDA/NC [(de /d)2 −1] πr2 SHDA/NC [(de /d)2 −1]) 2 . +4Rs − s Kfz φ Kfz φ (4.9)

The validity of (4.9) can be checked with the rings shown in Fig. 4.5a, c, the number Nd = 131 of which was observed on the sample with an area As = 18.2 × 21.4 µm2 . From these data, the value Rs = (As /4Nd )0.5 = 860 nm is estimated. The predominant parameters in (4.9) are SHDA/NC , d, and K. With SHDA/NC = 0.25 mN m−1 , d = 0.85de [3], and K = 0.5 [3], the corresponding radius Rp calculated from (4.9) for the ring shown in Fig. 4.5c becomes 560 nm (experimentally, Rp = 430 nm). The corresponding value of Rp of the ring shown in Fig. 4.5a, derived from (4.9) with SHDA/NC = 0.25 mN m−1 , d = 0.85de , and K = 0.5, is 690 nm (experimentally, Rp = 465 nm). Obviously, the above agreement between the experimentally measured and theoretically calculated values of Rp supports the proposed model that CoPt3 particle rings were formed by the retraction of the contact line of the HDA pancake during evaporation.

4.5 Summary and Conclusions Self-assembly of rings of CoPt3 nanoparticles in ultrathin polymer ﬁlms derives from phase separation of a binary solution on the water surface.

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This process favors to the formation of a bilayer structure which consists of an HDA-rich layer at the solution–air interface and an NC-rich layer at the solution–water interface. The subsequent dewetting of the HDA layer on the surface of the NC layer leads to its decomposition into micrometer-sized pancakes. The simultaneous evaporation of the HDA pancakes gives rise to a shrinking of their perimeter, and the CoPt3 particles located at the contact line follow this retraction. The self-assembly of CoPt3 particles along the contact line in ordered one- or two-dimensional rings strongly beneﬁt from the attraction between the particles. Acknowledgments The author would like to thank G.H. Bauer, G. Reiter, and J. Parisi for valuable discussions of the experimental results and the proposed model of nanoparticle ring formation, E. Shevchenko and H. Weller for the preparation of CoPt3 nanoparticles and discussion of the experimental results.

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20. W.A. Zisman, in Contact Angles, Wettability, and Adhesion, Advanced Chemistry Series, vol 43 (ACS, Washington DC, 1991) 21. J.N. Israelachvili, Intermolecular and Surface Forces (Academic, London, 1991) 22. P.G. de Gennes, Rev. Mod. Phys. 57, 827 (1985) 23. F. Brochard-Wyart, J. Daillant, Can. J. Phys. 68, 1084 (1990) 24. F. Brochard-Wyart, J.M. di Meglio, P.G. de Gennes, Langmuir 7, 335 (1991) 25. J. Visser, Adv. Colloid Interface Sci. 3, 331 (1972) 26. D. Bargeman, F. van Voorst Vader, J. Electroanal. Chem. Interfacial Electrochem. 37, 45 (1972) 27. H.C. Hamaker, Physica 9, 1058 (1937) 28. F. Brochard-Wyart, P. Martin, C. Redon, Langmuir 9, 3682 (1993) 29. N.D. Denkov, O.D. Velev, P.A. Kralchevsky, I.B. Ivanov, H. Yoshimura, K. Nagayama, Nature 361, 26 (1993) 30. P.A. Kralchevsky, N.D. Denkov, V.N. Paunov, O.D. Velev, I.B. Ivanov, H. Yoshimura, K. Nagayama, J. Phys.: Condens. Matter 6, A395 (1994) 31. A.S. Dimitrov, C.D. Dushkin, H. Yoshimura, K. Nagayama, Langmuir 10, 432 (1994) 32. E. Adachi, A.S. Dimitrov, K. Nagayama, Langmuir 11, 1057 (1995) 33. R.D. Deegan, O. Bakajin, T.F. Dupont, G. Huber, S.R. Nagel, T.A. Witten, Nature 389, 827 (1997) 34. R.D. Deegan, O. Bakajin, T.F. Dupont, G. Huber, S.R. Nagel, T.A. Witten, Phys. Rev. E 62, 756 (2000) 35. S. Nir, C.S. Vassilieﬀ, in Thin Liquid Films: Fundamentals and Applications, ed. by I.B. Ivanov (Marcel Dekker, New York, 1988), p. 207 36. G. Reiter, A. Sharma, A. Casoli, M.-O. David, R. Khanna, P. Auroy, Langmuir 15, 2551 (1999) 37. J.T. Davies, E.K. Rideal, Interfacial Phenomenon (Academic, New York, 1963)

5 Patterns of Nanodroplets: The Belousov–Zhabotinsky-Aerosol OT-Microemulsion System V.K. Vanag and I.R. Epstein

Summary. A reverse microemulsion composed of octane, water and the surfactant aerosol OT (AOT) consists of nanometer-sized droplets of water surrounded by a monolayer of AOT molecules ﬂoating in a sea of oil. If one adds to this system the components of the Belousov–Zhabotinsky (BZ) oscillating chemical reaction, one can observe a remarkable variety of complex patterns. These include spirals and traveling concentric circular waves that may move either toward or away from their centers, segmented traveling waves, chaotic waves, stationary Turing patterns and standing waves, and localized patterns. The wavelengths of the observed structures are typically of the order of 200 µm, i.e., about 20,000 droplet diameters. The type of pattern obtained can be controlled by varying the microemulsion composition, which determines the size and spacing of the droplets, and the concentrations of the BZ reactants, which determines the chemical kinetics. The behavior can be simulated numerically using relatively simple reaction-diﬀusion models.

5.1 Introduction In a recent letter to Chemical and Engineering News, a correspondent [1] wrote, with tongue in cheek, “We have recently prepared a nanomaterial in our laboratory that is less than 1 nm across but is capable of sensing the pH of an aqueous solution. In the presence of suﬃcient base, it absorbs light at 550 nm, thus emitting a visible light signal. We plan to call this nanomaterial “phenolphthalein,” subject to a patent search to see if the name is not already in use.” The writer was responding to what he perceived as an unhealthy tendency on the part of the magazine, and scientists in general, to trumpet the virtues of nanoscale phenomena and technology when, of course, all of chemistry involves the behavior of nanoscale objects – atoms and molecules. The question of interest, addressed in this volume, is whether there are materials and phenomena whose properties depend critically on structures and events at the tiny length scales that one thinks of as the “nanoworld.” In this chapter, we examine a system in which a complex chemical reaction takes place in nanodroplets of water situated in a sea of oil, a reverse

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microemulsion. We have entitled the chapter “Patterns of Nanodroplets” rather than “Patterns in Nanodroplets,” since, as we shall point out, no pattern formation occurs within an individual droplet, and it takes many, many droplets to construct a pattern. How many remains a matter for continuing research. Our system combines elements of self-assembly, in the structure of the microemulsion, with aspects of self-organization, in the nonequilibrium pattern formation arising out of the interplay between reaction and diﬀusion embedded in the self-assembled structure. In Sect. 5.2, we describe the Belousov–Zhabotinsky reaction-aerosol OT system (BZ-AOT), focusing on the properties that give rise to pattern formation. Section 5.3 contains a catalog of experimental results, illustrating the vast array of patterns possible when complex reaction dynamics are combined with a nanostructured environment. We then consider some theoretical aspects and summarize the results of our eﬀorts to model the BZ-AOT system. This chapter concludes with a discussion of some thoughts about possible future directions for research on this system.

5.2 The BZ-AOT System In this section, we outline the key features of the chemistry of the Belousov– Zhabotinsky (BZ) reaction, the structural properties of AOT microemulsions, and the preparation and properties of the combined BZ-AOT system. 5.2.1 The BZ Reaction The BZ reaction [2, 3], the metal ion-catalyzed oxidation of an organic substrate, e.g., malonic acid (MA), by bromate in concentrated aqueous sulfuric acid, is the most versatile and thoroughly studied oscillating chemical reaction [4]. In a stirred closed (batch) system, like a simple beaker, it gives rise to temporal oscillations with a period of about 1 min. At a well-chosen set of reactant concentrations, these oscillations can persist with only small changes in amplitude and period for an hour or more before the system reaches equilibrium. It is this ability to sustain batch oscillations over many cycles that gives the BZ reaction its unique role in nonlinear chemical dynamics. If the reaction is run in an unstirred system, particularly in a thin layer of solution, one observes striking spatial patterns of traveling concentration waves [5], typically concentric circles (“target patterns”) or spirals, which arise spontaneously after a brief induction period. Since the catalysts 2+ employed – cerium, ferroin (tris(1,10-phenanthroline)iron(II)), or Ru(bipy)3 (tris(2,2 -bipyridyl)ruthenium(II)) – change color as they oscillate between their reduced and oxidized states, the temporal and/or spatial patterns in the BZ system are easily monitored by eye, spectrophotometrically, or with a video camera.

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Most studies of BZ waves have been carried out in aqueous solution in a closed system, e.g., a Petri dish. The waves travel at a constant speed, determined by the reactant concentrations, on the order of several mm per min. When two waves collide, they annihilate and disappear. More sophisticated studies, in which conditions can be held constant over long times, are carried out in open reactors (CFURs) [6], which allow for a continuous input of fresh reactants, thereby maintaining the necessary far-from-equilibrium conditions indeﬁnitely. Several studies of pattern formation in the BZ reaction in structured media have been carried out using gels [7], membranes [8], beads of ion exchange resin [9] and mesoporous glasses [10]. The reactants in the BZ system – bromate, malonic acid, catalyst – are all quite polar, as are the majority of the intermediates and products generated during the course of the reaction. Some nonpolar intermediates are produced, however, notably elemental bromine, Br2 , which acts as an inhibitor, and bromine dioxide BrO2 , which is related to the autocatalytic species HBrO2 and behaves as an activator. 5.2.2 AOT Microemulsions The structure of the surfactant sodium bis(2-ethylhexyl) sulfosuccinate, more commonly know as aerosol OT (AOT) is shown in Fig. 5.1. Combined with water and oil, it forms a ternary mixture that can exhibit a variety of

Fig. 5.1. (upper left) Structure of AOT. (upper right) Phase diagram of the AOT– water–oil system. L2 – reverse microemulsion phase (or water-in-oil microemulsion), LH , hexagonal phase; LC, Liquid crystal (or lamellar) phase. The structure of each phase is shown below the diagram

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microstructures, including discrete spherical water droplets, interconnecting bicontinuous water channels, interacting rods, etc. [11]. In Fig. 5.1, we show a simpliﬁed version of the phase diagram of the AOT–water–oil system. The region of interest for this chapter is labeled L2 , the reverse microemulsion phase, characterized by droplets of water surrounded by a monolayer of AOT in a sea of oil. The polar head groups of the AOT molecules project in toward the water core, while the nonpolar tail groups are directed out into the oil. By varying the composition of the mixture within region L2 , one can control or tune the structure of the microemulsion. The average radius of the water core is determined by the ratio of water to surfactant molecules ω = [H2 O]/[AOT] and is approximately given in nm by Rw = 0.17ω [12]. Since we typically work with mixtures in which ω lies in the range 9–25, we are dealing with droplets that are several nanometers in diameter. The spacing between droplets is controlled by the volume fraction ϕd of the dispersed phase (water plus surfactant). This quantity is related to the volume fraction of water ϕw as ϕd ≈ ϕw (1 + 21.6/ω), and ϕw is related to the concentration of droplets cd as ϕw = V1 cd NA , where V1 = 4πRw 3 /3 is the volume of the water core of a droplet, and NA is Avogadro’s number. The average distance between nanodroplets can be estimated as (cd NA )−1/3 . As ϕd increases, clusters of droplets begin to grow [13], and at a critical value ϕp (around 0.5), percolation takes place, resulting in the formation of long dynamic channels of water, an increase of about three orders of magnitude in the conductivity of the microemulsion [14], and an increase of one to two orders of magnitude in the diﬀusion coeﬃcient of water molecules [15]. Because of the small size of the droplets and the small number of solute molecules in each droplet (see below), diﬀusion within a single droplet plays essentially no role in pattern formation in the BZ-AOT system. Communication between droplets can take place via two distinct diﬀusive processes. Highly polar species, which are largely conﬁned to the water droplets, are exchanged when droplets collide and undergo ﬁssion and fusion. The characteristic time for this process, (cd kex )−1 , is several milliseconds for typical values of ϕw , where kex is the mass exchange rate between droplets. Less polar or nonpolar species, which dissolve in the oil phase diﬀuse through the oil as single molecules and display diﬀusion coeﬃcients typical of single molecules, on the order of 2 × 10−5 cm2 s−1 . The diﬀusion of polar molecules at ϕd < ϕp is equivalent to the diﬀusion of water droplets, a much slower process, since the diﬀusing species is essentially a macromolecule containing perhaps 104 monomers. According to the Stokes–Einstein equation, the diﬀusion coeﬃcient Dd of a droplet of radius Rd is given by Dd = kT /6πηRd , where η is the viscosity of the medium. Measurements of diﬀusion rates in the oil and water phases of an AOT microemulsion [15] conﬁrm that the former process occurs one to two orders of magnitude faster than the latter as long as ϕd is well below the percolation threshold, while at ϕd ≥ ϕp both diﬀusion coeﬃcients of oil and water molecules are approximately equal to 4 × 10−6 cm2 s−1 .

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5.2.3 The BZ-AOT System When a chemical reaction takes place in a microemulsion, the behavior of the system may be modiﬁed considerably from that observed in aqueous solution as a result of the diﬀerent microenvironment encountered by the reactant molecules or due to mass exchange processes between water droplets. For example, for the simple prototype reaction in aqueous solution X + Y → products

(5.1)

with [X] [Y] and rate constant kr , the decay of [X] is described by the simple exponential function [X] = [X]0 exp(−kr [Y]t),

(5.2)

where the subscript zero signiﬁes the initial concentration, and the change in [Y] over time is taken to be negligible. The same reaction carried out in a microemulsion with nX < 1 nY , where nX and nY are the average numbers of X and Y particles, respectively, per water droplet, is characterized by the “stretched exponential” kinetics of the Infelta–Tachiya equation [16,17]:

nY kex kr nY kr2 t− [1 − exp(−(k + k )t)] . nX (t) = nX 0 exp − ex r kex + kr (kex + kr )2 (5.3) We shall see below that the condition nX < 1 actually holds for several species in the BZ-AOT system. If kex >> kr , (5.3) reduces to (5.2) if we replace average numbers of molecules per droplet by concentrations. For small droplets (1–2 nm), kex ∼ = 107 M−1 s−1 , which implies that rate constants 6 −1 −1 should remain unchanged but that fast reactions smaller than 10 M s (with kr > 107 M−1 s−1 ) should proceed more slowly in the microemulsion. Several such fast elementary processes are thought to be important in the BZ reaction. Early experiments on the BZ reaction in a stirred AOT microemulsion [18] showed that the oscillatory behavior of the BZ system was preserved, but the period and the oscillatory concentration range were modiﬁed by the presence of the surfactant. Vanag and Boulanov [19] subsequently demonstrated that the behavior of the system depended upon both the concentration and the size of the droplets. They suggested that the changes result from the eﬀective modiﬁcation of key rate constants in the microemulsion system. In Fig. 5.2 we show the results of light scattering experiments on a freshly prepared AOT microemulsion loaded with some of the BZ reactants. We observe two peaks in the radius distribution, corresponding to two sorts of objects: small water droplets and larger droplets or possibly clusters of small droplets. The mass exchange rate between large clusters can be as small as (103 –105 )M−1 s−1 [20, 21]. In this case, (5.3) implies that even relatively slow elementary reactions of the BZ system will be slowed in the microemulsion. As the microemulsion “ripens” over a period of 1–2 days, the two peaks coalesce, and the system becomes increasingly monodisperse.

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4

Scattered Light, %

3

1

2

1

2

0 1

10 Rh, nm

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Fig. 5.2. Distribution of radii of water nanodroplets. Curves 1 and 2 were obtained in light-scattering experiments for fresh and 1-day old microemulsions (ω = 15, ϕd = 0.55), respectively, loaded with H2 SO4 (0.4 M) and MA (0.6 M)

5.3 Experimental Results The BZ-AOT system displays a remarkable array of pattern formation phenomena, many of them not previously observed in other reaction–diﬀusion systems. In this section, we ﬁrst describe the preparation of the system and our experimental conﬁguration. We then present the results obtained as we vary both the structure of the microemulsion, i.e., the droplet fraction ϕd , and the chemistry of the BZ system, which we characterize by the ratio of initial concentrations [H2 SO4 ][NaBrO3 ]/[MA], which constitutes a rough measure of the relative strengths of activation and inhibition in the reaction mixture. An overview of the kinds of patterns we have found is given in Fig. 5.3. 5.3.1 Experimental Conﬁguration Our experiments are generally performed at room temperature in an apparatus consisting of two ﬂat optical windows 50 mm in diameter separated by an annular Teﬂon gasket with inner and outer diameters 20 and 47 mm, respectively, and a thickness of 0.1 mm. The reaction volume is thus a closed cylinder of radius 10 mm and height 0.1 mm. This reaction layer is illuminated by a 40 W tungsten source passing through an interference ﬁlter [450 nm for 2+ 2+ Ru(bipy)3 , 510 nm for Fe(phen)3 ]. The patterns are observed through a microscope equipped with a digital CCD camera connected to a personal computer. Two stock microemulsions, ME I and ME II, are used in the experiments. They contain the same size and concentration of water droplets and are

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Fig. 5.3. Overview of pattern formation in the BZ-AOT system

prepared by mixing 1.5 M solutions of AOT in octane with aqueous solutions of the appropriate concentrations of H2 SO4 and MA for ME I and of the catalyst and sodium bromate for ME II. We have carried out experiments with ferroin, 2+ Ru(bipy)3 and bathoferroin (tris(4,7-diphenyl-1,10-phenanthroline)iron(II)) as the catalyst. The reactive microemulsion is generated by mixing equal volumes of ME I and ME II. Microemulsions with the same droplet size but different droplet concentration can be prepared by simple dilution with octane. High grade commercially available reagents give satisfactory results, with the exception that the octane must be puriﬁed by stirring with concentrated sulfuric acid for 2–3 h in order to remove unsaturated organic impurities that would otherwise react with the bromine produced in the BZ reaction. 5.3.2 Turing Patterns The initial motivation for studying the BZ-AOT system was to create spatially periodic, temporally stationary Turing patterns in the BZ reaction. Turing patterns, initially proposed as a mechanism for morphogenesis in biological systems [22], were ﬁrst observed experimentally in the cholorite-iodide-malonic acid (CIMA) oscillating reaction [23] in a gel reactor. Attempts to generate these patterns in the BZ reaction were unsuccessful because it proved impossible, at least in aqueous media, to meet the criterion delineated by Turing that the activator species must diﬀuse signiﬁcantly more slowly than the inhibitor species. In the CIMA reaction one can retard the eﬀective diﬀusion of the

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Fig. 5.4. Turing patterns in the BZ-AOT system. Initial concentrations (in M) in the aqueous phase and parameters ω and ϕd of the AOT ME: (a–f) [MA] = 0.25; [H2 SO4 ] = (a, b) 0.2, (c,d) 0.175, (e) 0.14, (f) 0.142; [NaBrO3 ] = (a, b) 0.18, (c, d) 0.15, (e) 0.14, (f) 0.142; [ferroin] = (a–f) 4.2 × 10−3 , [Ru(bpy)3 2+ ] = (e)0.25 × 10−3 and 0 in all other cases, Ru(bpy)3 2+ was added to make the BZ-AOT system photosensitive; ω = (a, b) 15, (c,d,f) 18.3, (e) 18; ϕd = (a, b) 0.355, (c–f) 0.48; frame size, mm × mm, (a,b,f) 5 × 3.75; (c) 7.6 × 5.7; (d,e) 2.61 × 1.96. (a) 10 min after sandwiching, (b) 50 min after sandwiching; frame (d) is an enlargement of the white rectangle in frame (c). White corresponds to maximum value of catalyst Z (ferriin) concentration, black to minimum

activator iodine-containing species, because I− and I2 form a complex with the starch indicator that is immobilized in the gel. No analogous complexing agent is available to interact with the bromous acid, which serves as the activator in the BZ reaction. By running the BZ reaction in AOT microemulsion, we are able to slow the diﬀusion of the polar species by a factor of 10–100 as described above. Under appropriate conditions, the dominant species present in the oil phase is Br2 . Since this inhibitor species is able to diﬀuse signiﬁcantly faster than the activator, which is conﬁned to the slow-diﬀusing water droplets, the Turing criterion is satisﬁed, and we can obtain Turing patterns. Several examples of such patterns appear in Fig. 5.4. Frame (a) shows a transient as the system evolves to the ﬁnal stationary pattern of frame (b). At the concentrations used here, nearly stationary Turing patterns emerge throughout the reactive layer almost immediately after introducing the ME. Because the process occurs so rapidly, the Turing patterns are quite irregular. After about 1 hour these chaotic pseudostationary patterns transform to relatively well-organized spots. In other cases (usually at smaller ϕd ), hexagonal spots emerge gradually, spot by spot, occupying the homogeneous, unstructured area. Frames (b) and (c) illustrate the two most common types of Turing patterns: spots and stripes (labyrinth). An unusual pattern, circular stripes, is shown in (f), at a composition where the BZ-AOT system in a stirred reactor exhibits bulk oscillations. In the spatially extended conﬁguration, bulk

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oscillations and stationary Turing patterns compete. The homogeneous area in the lower right corner oscillates and produces phase waves that transform to stationary stripes when they reach the patterned area. If the initial stationary pattern is a spot or a small circle, then a Turing target pattern emerges. Frames (d) and (e) demonstrate another phenomenon not seen in Turing patterns in other media, stripes and spots of diﬀerent amplitudes in a single pattern. 5.3.3 Patterns Associated with a Fast-Diﬀusing Activator Turing patterns require that the inhibitor species diﬀuse more rapidly than the activator, a situation that prevails in the BZ-AOT system at low droplet fractions, where Br2 is the predominant species in the oil phase. As we discuss in Sect. 5.4, at higher ϕd bromine is replaced as the most prevalent species in the oil phase by BrO2 , so that instead of a fast-diﬀusing inhibitor, we now have a fast-diﬀusing activator. Under these conditions, the dynamics are quite diﬀerent. The instability that gives rise to pattern formation is now a wave bifurcation rather than a Turing bifurcation (see Sect. 5.4), and we obtain a new set of patterns that are periodic in both time and space. Standing waves, like Turing patterns, have a ﬁxed position and display an intrinsic wavelength determined by the rate and diﬀusion constants and the reactant concentrations. Standing waves, however, oscillate in time – at each point in the medium the concentration oscillates with the same period. The nodal points, at which the amplitude of oscillation is zero, remain ﬁxed and deﬁne the position of the standing waves. Figure 5.5 shows a standing wave pattern in a bathoferroin-catalyzed BZ-AOT system. Frame (e) demonstrates that the pattern recurs at periodic time intervals T , while frame (f) shows that the deviations from the average in patterns separated in time by half an oscillation period, T /2, cancel almost exactly. Another type of pattern, not seen in other reaction–diﬀusion systems, consists of groups, or packets, of traveling waves. While wave packets are common in quantum mechanics and in electromagnetic systems, their occurrence in dissipative systems is unexpected. In Fig. 5.6 we show several examples of packet waves in a ferroin-catalyzed BZ-AOT system [24]. The most frequently observed patterns in the aqueous BZ reaction consist of target patterns or spirals. These traveling waves move outward from the center of the pattern and disappear (annihilate) when two waves collide. Wave segments travel at a constant velocity given by the eikonal equation [25, 26]: v = c0 − D/r,

(5.4)

where v is the normal wave speed, c0 is the speed of a plane wave, D is the diﬀusion coeﬃcient of the activator species, and 1/r is the curvature of the wave segment. Such behavior is seen in the BZ-AOT system as well, but under certain conditions in that system, new phenomena arise. One of the most

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Fig. 5.5. Standing waves in the BZ-AOT system (a,b,d–f) and in model (19–22) (c,g). Frame size: (a) 5 × 3.7 mm2 , (b,d,e,f) 1.9 × 1.4 mm2 . Snapshots (b,d) show the same portion of (a) at times t = (b) 0 and (d) T /2, where the period of oscillation T = 90 s. (e) is a snapshot at t = T, (f ) = (b) + (d). ω = 15, ϕd = 0.473, [MA] = 0.4 M, [H2 SO4 ] = 0.2 M; [NaBrO3 ] = 0.18 M; [bathoferroin] = 5 mM. (g) is a full cycle of standing waves (with time interval T /8, from the top to bottom of the right column and then from the top to bottom of the left column) for a small area 20 × 20. For a large area, 150 × 100 (c), standing waves are irregular as in experiment. Parameters Dx = Dz = 0.01, Ds = Du = 1, f = 1.5; for (g): q = 4 × 10−4 , ε = 0.3, ε2 = 1.5, ε3 = 0.003, α = 0.3, β = 0.26, γ = 0.4, χ = 0; for (c) : q = 3 × 10−3 , ε = 0.385, ε2 = 3.2, ε3 = 0.0024, α = 6.3, β = 0.275, γ = 0.1, χ = 0.004

Fig. 5.6. Packet waves in the BZ-AOT system. ω = (a)15, (b,d) 16.4, (c)15.2. ϕd = (a) 0.57, (b,d) 0.64, (c) 0.45. [MA]/M = (a) 0.25, (b–d) 0.3. [H2 SO4 ]/M = (a, c) 0.2, (b,d) 0.3. [NaBrO3 ] = (a) 0.15 M, (b,d) 0.2 M, (c) 0.23 M. [ferroin] = 4 mM. Size (mm × mm) = (a)2.5 × 2.2, (b,d) 1.88 × 1.4, (c) 3.76 × 2.81. Figure (d) evolved from (b). Arrows mark direction of wave movement. Teﬂon border is seen in (c) at right bottom corner

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Fig. 5.7. Simulation of packet waves and antipacemakers from (5.19–5.22) for positive (a,b,d) and negative (c) d(Im(λ))/dk (cf. Fig. 5.6). Size = (a,d) 150 × 100, (b) 200 × 100, (c) R = 50. Narrow (L/100, L = 150) left vertical stripe was perturbed in (a); two vertical narrow lines from both sides were perturbed in (b); two points symmetric about the center (at x = ±R/2) were perturbed in (c). Parameters: For (a,b,d): q = 0.0015, f = 1.4, (a) ε = 0.34, (b,d) ε = 0.36, ε2 = 1.4, ε3 = 0.006, α = (a)6, α = (b,d) 7, β = 0.32, γ = 0.2, χ = 0, Dx = Dz = 0.01, Ds = Du = 1. For (c) q = 0.0033, f = 1.5, ε = 0.4, ε2 = 3.5, ε3 = 0.0016, α = 6.2, β = 0.28, γ = 0.1, χ = 0.00595, Dx = Dz = 0.01, Ds = Du = 0.9

Fig. 5.8. Fully developed inwardly moving spiral (antispiral) and target (antitarget or antipacemaker) patterns in a BZ-AOT microemulsion. (a to c) Antispirals, ϕd = 0.55. (d) Concentric waves (antipacemaker), ϕd = 0.59. For all microemulsions ω = 15, [MA] = 0.3 M, [H2 SO4 ] = 0.2 M, [ferroin] = 4 mM. [NaBrO3 ](M) = (a) 0.23, (b,c) = 0.2, (d) 0.21. Frame size (in mm × mm): (a) 5.1 × 3.75, (b) 3 × 2.25, (c) 1.8 × 1.5, (d) 2.7 × 2.5. Arrows mark directions of wave movement and antispiral rotation

startling is the occurrence of inwardly moving spirals and concentric circles, referred to as antispirals and antitargets [27], respectively. This behavior is shown in Fig. 5.8 (Fig. 5.7 is discussed in Sect. 5.5.2). Multiarmed versions of antispirals have also been seen. The patterns are sustained by the emergence of new waves at the periphery of the pattern. In contrast to spirals, which obey (5.4) and have a constant pitch, antispirals have a pitch that increases with the distance from the center and do not follow the eikonal equation. They are related to another phenomenon seen ﬁrst in the BZ-AOT system, accelerating waves. These traveling waves, illustrated in Fig. 5.9, have the property that they speed up as they approach one

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Fig. 5.9. Accelerating waves in the BZ-AOT system. Collision occurs in frame (c) within the circled region, after which the pair of waves that entered horizontally moves oﬀ vertically. ω = 18.8, ϕd = 0.74. [H2 SO4 ] = 0.2 M, [NaBrO3 ] = 0.15 M, [MA] = 0.3 M, [ferroin] = 4 mM; t = (a) 90 s, (b) 110 s, (c) 122 s, (d) 134 s. Frame size = 5.2 × 4.0 mm

another and, instead of being annihilated on collision, are deﬂected, apparently unchanged, at right angles to the original direction of approach [28]. The discovery of antispirals initiated a number of theoretical discussions [29–33]. We explain the antispiral behavior on the basis of a wave instability with negative dispersion dIm(λ)/dk . But others [29, 33] were able to obtain antispirals with the aid of the complex Ginzburg–Landau equation and simple two-variable models, which cannot produce wave instability. So, at present there are at least two diﬀerent interpretations of inwardly rotating waves. 5.3.4 Complex Patterns – Dashes and Segments At relatively high ratios of activator to inhibitor concentration, but at lower droplet fraction and hence lower interdroplet fractions of HBrO2 , we observe another set of phenomena yet to be seen in other reaction–diﬀusion systems. The ﬁrst of these [34] are dash waves, patterns that consist of lines of wave segments separated by gaps that move coherently normal to the wave. They display two wavelengths, the distance between segments along a wave, and the normal distance between waves. They exhibit properties characteristic of both Turing patterns and excitable systems. Dash waves arise only in freshly prepared microemulsions, which have a bimodal distribution of droplet sizes like the system shown in Fig. 5.2. An example of such a pattern is shown in Fig. 5.11b (the simulations in Fig. 5.10 are discussed below in Sect. 5.5.2). A related phenomenon is the occurrence of segmented spirals [35], illustrated in Fig. 5.12. These striking patterns evolve from ordinary spirals (Fig. 5.11a). Dashed waves emerge near the reactor edge and move toward the spirals, causing breaks to develop in the spirals and ultimately leading to the beautiful patterns shown in Fig. 5.12.

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Fig. 5.10. Accelerating waves in the Brusselator model with fast-diﬀusing activator, (5.28)–(5.30). Random initial conditions, size = 30 × 30, t = (a) 1.126, (b) 1.146, (c) 1.156, (d) 1.166. Circles show locations where acceleration of two oncoming waves before collision occurs (cf. Fig. 5.9). Zero-ﬂux boundary conditions. Parameters: a = 2.9, b = 3.2, c = 2, d = 1.5, ε1 = 0.02, ε2 = 0.2, Du = Dv = 1, Dw = 20

Fig. 5.11. Dash waves in the BZ-AOT system. (a) Dash waves approach an ordinary spiral, causing beginning of segmentation near the center of the spiral. (b) A welldeveloped dash wave pattern. Conditions: ϕd = 0.36, ω = 15. [MA] = 0.3 M, [H2 SO4 ] = 0.2 M, [NaBrO3 ] = 0.18 M, [bathoferroin] = 0.0049 M. Size (mm2 ) = 3.8 × 5.0

5.3.5 Localized Patterns The patterns we have described so far all have the property that once they are fully developed they occupy essentially the entire reactor. One can imagine another kind of pattern in which most of the medium is quiescent and only a

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Fig. 5.12. Segmented spirals in a BZ-AOT microemulsion. Stroboscopic composites (c–e) show trajectories built from spiral segments in (a,b). (c): superposition of snapshots taken every 50 s of segments of spiral turns initially closest to the cores; (d): trajectories of segments of the second turn of the spiral at the upper right; the two “eyes” are derived from the tip segments only. (e): summation of (c) and (d). White corresponds to higher concentration of bathoferriin in (a) and (b) and marks overlapping trajectories in (c) and (d). Frame size (mm2 ), (a–e) 3.72 × 4.82, (f–h) 2.11 × 2.63. Time between (a) and (b) is 66 s; between (f) and (g) 219 s, and between (g) and (h) 244 s. ϕd = (a, b)0.36, (f–g) 0.47, ω = 15. [MA] = 0.3 M, [H2 SO4 ] = 0.2 M, [NaBrO3 ] = 0.18 M, [bathoferroin] = 0.0049 M

portion exhibits spatial and/or temporal concentration changes, or in which a spatially patterned region resides on a background of homogeneously oscillating medium. Such localized patterns are found in a number of physical systems [36], theoretical models [37], and in the aqueous BZ reaction subjected to either global feedback [38] or periodic forcing [39], but until recently they had not been observed in autonomous reaction–diﬀusion systems. Figure 5.13 shows examples of two kinds of localized structures seen in the Ru(bipy)catalyzed BZ-AOT system [40]. Snapshots (a, b) illustrate localized Turing patterns, in which the rings are stationary and are surrounded by an area of spatially uniform medium. In the ferroin-catalyzed BZ-AOT system, we observe similar localized Turing structures. In Fig. 5.13c, d, we see another type of localized pattern, an oscillon. Here, the background is stationary and the pattern oscillates in time. Oscillons have been observed in vibrating granular materials [41]. Because they are spatially aperiodic, localized structures have the capacity to store considerably more information than Turing patterns and may provide an alternative model for biological development.

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Fig. 5.13. Localized Turing patterns (a,b) and oscillons (c,d) in the Ru(bipy)3 2+ catalyzed BZ-AOT system. (a) and (b) are snapshots taken in diﬀerent regions of the reactor. Period of the oscillons is 47 s, the diameter of the ring-oscillon is about 0.6 mm. ϕd = 0.41, ω = 15, [H2 SO4 ] = 0.25 M, [NaBrO3 ] = 0.2 M, [MA] = 0.25 M, [Ru(bpy)3 2+ ] = 4.2 mM. Frame size, mm2 = (a, b)5.06 × 3.73, (c,d) 2.13 × 1.87 2+

In a set of experiments on the photosensitive Ru(bpy)3 -catalyzed BZ-AOT system [42], we showed that if there is bistability between a localized Turing pattern and a homogenous steady state, the medium can be used to store an image like that shown at the lower left in Fig. 5.3. The image was imprinted by shining light of an appropriate intensity through a mask containing the image. If the light intensity is maintained in the range where the system is bistable, the image persists for an extended period of time, an hour or more, which is comparable to the lifetime of the ME in our experiments. One can erase the image by increasing the light intensity above the range of bistability. The image returns if the higher intensity is maintained only brieﬂy, but it is permanently eradicated by longer exposures to bright light. The system thus shows promise as an addressable, rewritable information storage device.

5.4 Theoretical Considerations In an eﬀort to understand how the patterns we have described above might arise in the BZ-AOT system, we have attempted to model the key elements of this system. Our modeling eﬀorts have for the most part employed the standard techniques used to model patterns in other, unstructured reaction– diﬀusion systems, i.e., partial diﬀerential equations (PDE), numerical simulation, and linear stability analysis. This approach neglects many details of the nanostructure of the medium, including the fact that individual droplets may have very few (or even no) molecules of a given species, rendering a deterministic treatment suspect. We note that a typical nanodroplet of radius 5 nm has a volume of about 5 × 10−22 L, which means that on average each droplet will contain only a single molecule of any species, e.g., the catalyst, which is present at 3–4 mM concentration. Intermediates, whose concentrations are often in the micromolar range, may be found in only 1 in 1,000 droplets. Fluctuations are clearly signiﬁcant in such a system, and one could attempt

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to treat them, perhaps with a stochastic cellular automaton model [43]. On the other hand, the structures observed to date have wavelengths of the order of a few hundred micrometers, i.e., perhaps 105 droplets, and, as we show below, our mean ﬁeld continuum PDE models are quite successful at reproducing at least the qualitative aspects of the observed patterns. PDE models are also considerably more eﬃcient computationally. For the moment, therefore, we restrict ourselves to trying to derive insights from these relatively simple models, though further experimental results or the desire for a more detailed understanding may require more elaborate modeling.

5.5 Constructing a Model Our eﬀorts to model the BZ-AOT system build upon the considerable success that has been achieved in modeling the aqueous BZ reaction. The most frequently employed BZ model is the Oregonator [44]. This three-variable model is a simpliﬁcation of the much more detailed Field–Kur¨ os–Noyes (FKN) mechanism for the full BZ reaction [45]. It does a remarkable job of reproducing many of the phenomena seen in the reaction with only ﬁve elementary steps: A+Y → X+P X + Y → 2P

(5.5) (5.6)

A + X → 2X + 2Z 2X → A + P

(5.7) (5.8)

Z → hY,

(5.9)

where A = BrO3 − , P = HOBr, X = HBrO2 , Y = Br− , Z = the oxidized form of the catalyst, e.g., ferriin, and h is a stoichiometric coeﬃcient. The concentrations of X, Y, and Z are allowed to vary, while that of A is assumed to be ﬁxed. To convert a BZ model like (5.5)–(5.9) into a model for the BZ-AOT system, we introduce reactions that describe the transfer of activator and/or inhibitor species between the aqueous and oil phases, variables that specify the concentrations of the relevant species in the oil phase, and terms for the diﬀusion of all variable species. We assume that the chemistry within the water droplets is governed by the “aqueous” steps (5.5)–(5.9) and that the only “reactions” in the oil phase are the transfer reactions, since the species in the oil will have no partners to react with, the key reactants being conﬁned to the aqueous phase. In order to decide whether we need to introduce an activator, an inhibitor or both in the oil phase, we carry out a simple analysis based on the partition coeﬃcients of the two candidate species, BrO2 and Br2 , between water and AOT and between octane and AOT. For each species, we have three linear

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equations to solve for its concentrations in the three phases (1 = water, 2 = surfactant, 3 = oil): ϕw x1 + ϕS x2 + ϕoil x3 = ϕw x0 x1 /x2 = KWS

(5.10) (5.11)

x3 /x2 = KOS

(5.12)

subject to the constraints ϕw + ϕS + ϕoil = 1 ϕS = kd ϕw

(5.13) (5.14)

Here ϕw , ϕS , and ϕoil represent the volume fractions of water, surfactant, and oil phases, respectively, the xi are the concentrations of the species of interest in the three phases after the transfer reactions reach equilibrium, x0 is the total concentration of the species if it were all conﬁned to the water droplets, and KWS and KOS are, respectively, the water-AOT and octaneAOT partition coeﬃcients. Equation (5.10) simply expresses the conservation of the solute; (5.11) and (5.12) deﬁne the partition coeﬃcients; (5.13) indicates that the volume fractions add up to unity; equation (5.14) gives the relationship between the volume fractions of AOT and water. At a typical value of ω = 15, kd (= 21.6/ω) is about 1.5. For Br2 , KWS = 0.002, KOS = 0.2; for BrO2 , KWS = 0.8, KOS = 1.25. Solving 5.10–5.14) with these values leads to the dependence of concentrations in the oil phase on the droplet fraction shown in Fig. 5.14. We thus expect the activator to be the dominant species in the oil phase at high values of ϕd , while the inhibitor should predominate at low droplet fractions. At intermediate ϕd , about 0.45, it is appropriate to include both activator and inhibitor in the oil phase. Since the Oregonator model does not explicitly contain Br2 or BrO2 , there is no rigorous way to take into account these species in the oil phase. Examination of the full FKN model shows that making some reasonable approximations we can represent the oil-soluble key species by introducing two simple

Fig. 5.14. Relative concentrations (x3 ) of Br2 (curve 1) and BrO2 (curve 2) in the oil phase as a function of droplet fraction ϕd = ϕw + ϕS . (solution of (5.10)–(5.14))

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pairs of reversible reactions X S Z U

→ S, →X → U, → Z,

(5.15) (5.16) (5.17) (5.18)

where S and U represent the oil phase species BrO2 and Br2 , respectively. Steps (5.15)–(5.18) describe the production of these species and their transfer between the aqueous and oil phases. “Elementary” reactions (5.5)–(5.9) and (5.15)–(5.18) with the introduction of diﬀusion terms and appropriate rescalings lead to the following four PDEs for the BZ-AOT system: ∂x/∂τ = [f z(q − x)/(q + x) + x − x2 − βx + s]/ε + dx ∆x,

(5.19)

∂z/∂τ = x − z − αz + γu + dz ∆z, ∂s/∂τ = (βx − s + χu)/ε2 + ds ∆s,

(5.20) (5.21)

∂u/∂τ = (αz − γu)/ε3 + ∆u,

(5.22)

where ε, ε2 , ε3 , α, β, γ, χ, and q are constants involving the rate constants, the concentration of A, the droplet fraction ϕd , f = 2h, and ε, ε3 1. The diﬀusion constants are scaled so that the diﬀusion constant of U (or u) is unity. The fact that species diﬀuse faster in the oil phase is expressed by the conditions dx , dz ds ≈ 1. At high droplet fractions, (5.22) may be dropped from the model, while deleting equation (5.21) is appropriate at low ϕd . Some experimental patterns, like oscillons, are diﬃcult to explain with a model based on simple “elementary” reactions, like model (5.19)–(5.22). For this case we have developed another model, which employs the same four variables, by reducing the full FKN mechanism: ∂x/∂τ = [y(q − x) + x − x2 − βx + s]/ε + dx ∆x

(5.23)

∂z/∂τ = x − z + dz ∆z ∂s/∂τ = (βx − s)/ε2 + ds ∆s

(5.24) (5.25)

∂u/∂τ = [y(q + 2x) − αu/y]/ε3 + ∆u,

(5.26)

where y = f z/(2q + 3x) + [αu/(2q + 3x)]1/2 represents [Br− ], and f = f0 + u/(K1 z + u). In Sect. 5.5.2, we present some results of numerical simulations with these and other models. 5.5.1 Linear Stability Analysis and Types of Bifurcations Before looking at the results of numerical simulations, we ﬁrst discuss some general considerations involving models of reaction–diﬀusion systems. It is useful to think of patterns in space and time as arising out of instabilities of the

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homogeneous steady state of a system. We imagine starting from conditions under which the homogeneous steady state is stable to temporal and spatial perturbations and changing a parameter (e.g., a reactant concentration or the droplet fraction) until the system undergoes a bifurcation, where the steady state loses its stability. Beyond this bifurcation point, an inﬁnitesimal perturbation will cause the steady state system to undergo a transition to another state that is temporally and/or spatially periodic. In mathematical terms, we start from a set of PDE like equations (5.19)–(5.22) and obtain the homogeneous steady state concentrations by solving the algebraic equations that result when we set all time and spatial derivatives to zero. We analyze the stability of this state by examining the behavior of inﬁnitesimal perturbations of the form exp(λt) exp(ikr). On discarding terms of second and higher order in the amplitude of the perturbation, we obtain a set of equations of the form Det(J − λI − k2 D) = 0,

(5.27)

where the elements of the Jacobian matrix J are the partial derivatives of the rate expressions, e.g., [f z(q − x)/(q + x) + x − x2 − βx + s]/ε in (5.19), with respect to the concentration variables, evaluated at the steady state; I is the identity matrix; and D is a diagonal matrix whose elements are the diﬀusion coeﬃcients of the variable species. The eigenvalue λ of (J − k2 D) with the most positive real part determines the rate of growth (if positive) or decay (if negative) of a perturbation to the steady state. Thus as we change the spatial character of a perturbation that causes Re(λ) to go from negative to positive, the system undergoes a bifurcation. As (5.27) implies, λ depends upon the wavenumber k, i.e., perturbations with some wavelengths will grow or decay more rapidly than those with other spatial scales. In reaction–diﬀusion systems, three types of bifurcation [46], distinguished by the values of Im(λ) and k at which Re(λ) ﬁrst becomes nonnegative, are of particular signiﬁcance: (a) Hopf bifurcation (Im(λ) = 0, k = 0): the steady state gives way to homogeneous (bulk) oscillation with a period 2π/Im(λ) (b) Turing bifurcation (Im(λ) = 0, k = 0): a temporally stationary, spatially periodic pattern emerges with wavelength 2π/k (c) Wave bifurcation (Im(λ) = 0, k = 0): the pattern is both temporally and spatially periodic The dependence of the eigenvalue λ on the wavenumber k is often summarized in a dispersion curve like the ones shown in Fig. 5.15. Dispersion curves, and the results of linear stability analysis in general, can tell us much about the behavior of a system near a bifurcation, but the pattern obtained well beyond the bifurcation point may be quite diﬀerent from the predictions of the linear analysis because of the dominant eﬀects of nonlinear terms.

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Fig. 5.15. Dispersion curves for model (5.19)–(5.22) demonstrating Hopf (curve 1), Turing (curve 2), and wave (curve 3) instabilities. Curves 1–3 are real parts, curves 4–6 are corresponding imaginary parts (divided by 90) of eigenvalues. Parameters: q = 0.0015, f = 1.4, α = 9, β = 0.34, γ = 0.2, χ = 0, ε2 = 1.3, ε3 = 0.006, dX = dZ = 0.01, ε = (1)0.36, (2,3) 0.37, dS = (1) 0.5, (2) 0.72, (3) 1, du = (1)0.72, (2,3) 0.82. Curves 1 and 3 have negative maxima at Im(λ) = 0 (around k = 5)

5.5.2 Results of Numerical Simulations In this section, we present a selection of results obtained by numerical simulations using the techniques and models discussed above. We shall see that, qualitatively at least, all of the experimentally observed phenomena are captured by the models, but that there is still much room for improvement. No single model to date reproduces all of the patterns found in the BZ-AOT system (compare models (5.19)–(5.22) and (5.23)–(5.26)), and whether it will prove possible to do so with a PDE model of manageable size remains an open question. At low droplet fraction, we obtain Turing patterns in our experiments. Under these conditions, the fast-diﬀusing inhibitor should be the major species present in the oil. The pattern shown in Fig. 5.16 was obtained with a 12-step FKN-type model augmented with reactions that transfer Br2 to and from the oil phase. It strongly resembles the Turing structure shown in Fig. 5.4b. Similar Turing patterns can be obtained with simpler models (5.19)–(5.22) or (5.23)–(5.26) that incorporate a rapidly diﬀusing inhibitor u. Linear stability analysis reveals the existence of a Turing bifurcation. At intermediate values of ϕd , taking into account both activator and inhibitor in the oil phase, we obtain a wave bifurcation. This instability can lead to the development of standing waves or various kinds of traveling (packet) waves. In Fig. 5.5c, g, we show an example of standing waves found in model (5.19)–(5.22). As in the experiments, temporal oscillations occur at each point, and the nodal lines separating regions of positive and negative deviation from the average concentration remain stationary. Figure 5.7 shows packets of plane traveling waves and a pair of antitarget patterns. We ﬁnd that the direction of motion of individual waves with respect to either the center, in the case of target or spiral patterns, or the center of mass of the packet, in the case of packet waves, is determined by the quantity d(Im(λ))/dk , analogous to the

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Fig. 5.16. Simulation of a Turing pattern in a BZ-AOT system on the basis of the FKN mechanism. White corresponds to higher ferriin concentration (z). All aqueous species have diﬀusion coeﬃcient (in dimensionless units) D = 0.05. Diﬀusion coeﬃcient of Br2 in oil is 1

group velocity in electromagnetic waves. If this quantity is positive, waves move in the same direction as the packet center, away from the point of perturbation (Fig. 5.7a, b, d). If it is negative, we obtain antispirals or antitargets (Fig. 5.7c) or packets in which individual waves (actually phase waves) travel against the direction of movement of the packet as a whole (to the center of perturbation) and have negative curvature in 2D. While some of the phenomena we have found in the BZ-AOT reaction have been seen experimentally only in this system to date, simulations suggest that they are of considerably more generality. Accelerating waves, for example, occur when the droplet fraction is high, i.e., when the predominant species in the oil phase is the activator. In Fig. 5.10, we show accelerating waves obtained in a simulation of the classic Brusselator model [47] with a fastdiﬀusing activator added, equations (5.28–5.30). The behavior is strikingly similar to that of the experiment seen in Fig. 5.9. du/dt = (a − (1 + b)u + u2 v − cu + dw)/ε1 + Du ∆u dv/dt = bu − u2 v + Dv ∆v

(5.28) (5.29)

dw/dt = (cu − dw)/ε2 + Dw ∆w

(5.30)

As a ﬁnal example, we show in Fig. 5.17 oscillons obtained in simulations in both one and two spatial dimensions of a reduced FKN model (5.23)–(5.26) with fast-diﬀusing activator and inhibitor species to take account of the fact that this behavior is observed at intermediate values of ϕd . The ability to simulate these structures enables us to examine how localized patterns interact with an eye toward developing possible uses for such patterns in applications like information processing [40].

5.6 Conclusion and Future Directions The BZ reaction has become the most powerful tool in nonlinear chemical dynamics. It exhibits a wide range of patterns in both time and space, from

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Fig. 5.17. Simulations of oscillons in one (a) and two (b) spatial dimensions with model (5.23)–(5.26). In (a), curve 1 is activator x, curves 2 and 3 are catalyst; curve 1 shows same time as curve 2, which is separated in time from curve 3 by half a period of oscillation. In (b), vertical direction shows concentration of activator as a function of position in the horizontal plane. Parameters: f0 = 0.52(0.5), q = 0.0004, K1 = 50, α = 0.1, β = 0.72, ε = 0.05, ε2 = 2, ε3 = 0.001, dx = 0.01, dz = 0.3, ds = 0.028

simple periodicity to complex periodic behavior to chaos. When the reactants of this remarkable reaction are dispersed in a water-in-oil microemulsion, the repertoire of behavior of this virtuoso system grows still wider, even under the simplest of laboratory conditions: ambient temperature and a closed batch reactor. After sandwiching the reactive microemulsion between a pair of glass windows, we see patterns that emerge after a few minutes and persist for hours. It is hard to imagine a more convenient or more productive experimental system for studying pattern formation in reaction–diﬀusion systems. In the several years that we have studied the BZ-AOT system, we have only scratched the surface of this rich vein of scientiﬁc information. Much remains to be done. We suggest here a few directions in which future research on the BZ-AOT and related systems might go. We have explored a single reaction with a single surfactant in a single region of that surfactant’s complex phase diagram. Other reactions, such as the CIMA system [48], which yields Turing patterns in aqueous gels [23] may be worth exploring in microemulsions. The use of surfactants whose properties diﬀer from those of AOT is another promising direction. Even within the BZ-AOT system, it may be worth investigating the behavior of mixtures that form other structures, e.g., liquid crystals. A surfactant molecule that contains a modiﬁcation of the Ru(bipy)2+ 3 catalyst has been reported [49] and might be an interesting candidate to replace some or all of the AOT in the standard BZ-AOT system. The structures we have observed in the BZ-AOT system have characteristic lengths of 100–500 micrometers and are pseudo-two-dimensional, since the layer thickness is less than one characteristic wavelength. Since individual droplets have sizes four to ﬁve orders of magnitude smaller, it is reasonable to ask whether structure exists at shorter scales as well. By using higher resolution techniques than the optical microscopy we have employed thus far, it may

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be possible to detect structures with smaller wavelengths, even perhaps at the true nanoscale, much as been done with sophisticated imaging techniques in viewing patterns on catalytic surfaces [50]. The use of ﬂuorescent catalysts may prove helpful here. By working with thicker layers and employing confocal or magnetic resonance imaging techniques, it should be possible to obtain and analyze three-dimensional structures and compare them with, for example, the three-dimensional Turing patterns predicted in simulations [51, 52]. An area for further theoretical study is the treatment of systems, like the BZ-AOT medium, consisting of large numbers of interacting subunits, each of which contains only a small number of particles and is therefore subject to large ﬂuctuations. Understanding the collective behavior of such ensembles would be important for treating not only the BZ-AOT system but biological systems as well. A potential practical application involves the use of stationary localized structures – Turing patterns or oscillons – in devices for information storage and/or processing. Coullet et al. [53] recently proposed, on the basis of a mathematical analysis that static localized structures can be utilized for storage and retrieval of information. Our preliminary simulations suggest that interaction between neighboring oscillons leads to collision and annihilation, adjustment of the interoscillon distance, birth of a new peak between them, or independent coexistence, depending on the initial distance between the 2+ oscillons. Since the Ru(bipy)3 -catalyzed system is photosensitive, we have the ability to imprint a BZ-AOT medium with oscillons at desired initial positions. These properties, combined with the stability of localized BZ-AOT structures make them excellent candidates for use in information applications. A ﬁnal, more speculative notion, is that an understanding of the BZ-AOT system might provide useful insights into certain social systems. Agent-based models, in which interacting entities, like droplets, have their own dynamic properties, have begun to play a signiﬁcant role in studies of social systems [54]. The wealth of patterns generated by the existence of two very diﬀerent diﬀusion rates in the BZ-AOT system may have some bearing on the spatiotemporal behavior that can arise in a mixed social community where diseases or ideas, for example, may propagate through diﬀerent subpopulations at diﬀerent rates because of diﬀerences in genetic or cultural factors. In summary, while we have learned much from this astoundingly rich system, there is still much more to be discovered. Acknowledgments We thank our colleagues, Milos Dolnik, Lingfa Yang, Akiko Kaminaga, and Anatol Zhabotinsky, for many helpful discussions and suggestions. We also thank Lingfa Yang for technical assistance in preparing this manuscript for publication. This work was supported by the Chemistry Division of the National Science Foundation and the Packard Foundation Interdisciplinary Science Program.

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40. V.K. Vanag, I.R. Epstein, Phys. Rev. Lett. 92, 128301 (2004) 41. P.B. Umbanhowar, F. Melo, H.L. Swinney, Nature 382, 793 (1996) 42. A. Kaminaga, V.K. Vanag, I.R. Epstein, Angewandte Chemie Int. Ed. 45, 3087 (2006) 43. V.K. Vanag, Physics-Uspekhi 42, 481 (1999) 44. R.J. Field, R.M. Noyes, J. Chem. Phys. 60, 1877 (1974) 45. R.J. Field, E.K¨ or¨ os, R.M. Noyes, J. Am. Chem. Soc. 94, 8649 (1972) 46. M.C. Cross, P.C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993) 47. I. Prigogine, R. Lefever, J. Chem. Phys. 48, 1695 (1967) 48. P. De Kepper, I.R. Epstein, K. Kustin, M. Orban, J. Phys. Chem. 86, 170 (1982) 49. G. Sprintschnik, H.W. Sprintschnik, P.P. Kirsch, D.G. Whitten, J. Am. Chem. Soc. 99, 4947 (1977) 50. J. Dicke, P. Erichsen, J. Wolﬀ, H.H. Rotermund, Surf. Sci. 462, 90 (2000) 51. A. De Wit, G. Dewel, P. Borckmans, D. Walgraef, Physica D 61, 289 (1992) 52. A. De Wit, P. Borckmans, G. Dewel, Proc. Nat. Acad. Sci. 94, 12765 (1997) 53. P. Coullet, C. Riera, C. Tresser, Chaos 14, 193 (2004) 54. J.M. Epstein, R. Axtell, Growing Artiﬁcial Societies (Brookings Institution, Washington, DC, 1996)

6 Honeycomb Carbon Networks: Preparation, Structure, and Transport L.V. Govor and J. Parisi

Summary. Honeycomb patterns with a single cell diameter of about 2 µm have been fabricated via spreading a drop of the initial polymer solution on the surface of cooled distilled water and following the subsequent inﬂuence of the water vapor on the resulting polymer thin ﬁlm. We introduce an advanced structuring model capable to describe the underlying physical mechanism. The electrical conductivity of nitrocellulose extending from insulator to metal behavior distinctly changes by orders of magnitude via vacuum heat treatment at temperatures ranging from 600 to 1,000◦ C. For the case of carbon nets, the conductivity of which is far beyond the metal–insulator

transition, the speciﬁc resistivity ρ depends on T as ρ(T ) ∝ T −b exp [T0 /T ]1/p in the range from 4.2 to 295 K. In the low-temperature regime, a Coulomb gap in the density of states located near the Fermi energy level occurs, i.e., p = 2. At high temperatures, the pre-exponential part ρ(T ) ∝ T −b dominates. In the intermediate temperature range, we disclose Mott’s hopping law with p = 3. The electrical ﬁeld dependence of variable range hopping is examined in its region of validity by ln ρ(T ) ∝ T −1/2 . We demonstrate the electrical conductivity σ caused by thermally nonactivated charge carriers at high ﬁelds to comply with ln σ(E) ∝ E −1/3 .

6.1 Introduction One of the many fundamental properties of nature’s beauty is the ubiquitous periodicity in self-assembled materials that can be observed everywhere in life science. Until recently, the term “self-assembly” was in most cases applied exclusively to biological structures. Experimental investigations today disclose various classes of self-assembled materials beyond those in biology. One example of self-assembled pattern formation on the macroscopic scale are ordered mesoporous solids with pore sizes ranging from 50 nm to 10 µm, which can have relevance for chemistry, photonics, optoelectronics, lightweight materials, and thermal insulation [1, 2]. Self-assembly involves forces such as

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dipolar, van der Waals, hydrophilic or hydrophobic interactions, chemisorption, surface tension, and gravity. For fabricating porous membranes, there exists a huge variety of preparation methods [1]. Self-assembly may oﬀer some advantages in the ﬁelds addressed above: the patterning process can function in diﬀerent media, and cost-intensive large-scale technology is not necessary. Nowadays, ordered mesoporous solids with nanoscale pore sizes are manufactured by self-organization of spherical micelles from diblock copolymers in a selective solvent [3]. The class of mesoporous materials can be formed by colloidal templating [4, 5]. Colloidal crystals of polystyrene or silica spheres are embedded with a ﬂuid that ﬁlls the space between the spheres. Next, the templating spheres are removed, and one awaits the creation of a porous solid where the dimension of the pores matches those of the templating spheres. Water-assisted formation of ordered mesoporous membranes has been described in various papers [6–11]. In that case, the membranes are built by condensation of water vapor on the ﬂuid polymer solution ﬁlm and subsequent evaporation of a solvent from the polymer solution. Recently, we have developed a preparation method that allows for membrane patterns with evenly shaped hexagonal cells having a diameter of about 1–2 µm [12–14]. There, the technology how to get mesoporous membranes out of diﬀerent polymers has been reviewed in detail. Simultaneously, some eﬀort was undertaken to interpret the self-organizing mechanism of patterning by thermodynamic processes that take place between water droplets on the ﬂuid polymer solution surface. We have suggested that the stabilization of water droplets on a ﬂuid surface is indispensable for ordered structure formation. In the following, the most important stabilization parameters are discussed that can inﬂuence the growth of condensing water droplets on the ﬂuid polymer solution layer and their interaction between each other. In general, the order grade of materials determines their chemical and physical properties. For example, the ordering on the atomic scale produces energy bands that dominate the electrical properties of crystalline materials. On the other hand, disordering can lead to distinctly diﬀerent properties that can be unique to noncrystalline materials. The above developments in fabricating ordered honeycomb carbon networks with noncrystalline pore walls, the electrical conductivity of which can be changed from insulator to metal behavior via heat treatment under vacuum conditions, allow us to study the role of periodicity of carbon networks in their conductivity. So far, it is well known that the low-temperature behavior of the speciﬁc resistivity ρ(T ) of amorphous semiconductors, granular thin ﬁlms, porous materials, and other disordered solid state media can usually be described by the elementary process of variable range hopping (VRH) of charge carriers. The degree of doping in those disordered systems is selected in such a way that the value of the electrical conductivity lies in the vicinity of the metal–insulator transition (MIT) on the insulating side. Such a dependence follows the equation

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ρ(T ) = ρ0 exp

T0 T

1/p

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,

(6.1)

where ρ0 and T0 denote material parameters which do not (strongly) depend on temperature; T0 is determined by the density of localized states, N (E), near the Fermi energy EF . The characteristic value p depends on both the dimensionality and the degree of disorder that governs the medium investigated [15]. The values p = 4, p = 3, and p = 2 correspond to VRH in three-, two-, and one-dimensional (3D, 2D, and 1D) disordered systems, respectively [16]. The characteristic value p = 2 can be observed also for 3D and 2D cases and is then explained with the model of Efros and Shklovskii [17–19]. The latter considers long-range Coulomb interactions between electrons that come along from diﬀerent centers of localization near the Fermi level. As a result, a Coulomb gap in the density of states, N (E), near the Fermi energy develops. For 3D, the gap obeys the dependence N (E) ∝ (E − EF )2 . At the Fermi energy EF , we have N (E) = 0. Theoretical investigations of the current–voltage characteristics of amorphous semiconductors, the ohmic conductivity of which in the low-temperature regime is described by (6.1), have been carried out in earlier work [20–24]. According to the calculations of Hill [20] as well as Pollak and Riess [21], the relation CeErm σ(E) = σ(0) exp (6.2) kB T should be fulﬁlled by the conductivity σ(E) = j(E)/E in case of the boundary condition eErm > kB T . Here, j is the current density, E the electrical ﬁeld, e the electron charge, l = Crm is the hopping length, rm = aξc /2 is the maximum hopping length on the percolation paths, a is the localization radius 1/p of the wave function, ξc = (T0 /T ) is the percolation threshold (at ξc >> 1), kB is Boltzmann’s constant, and C is a constant, the value of which was determined to be 0.8 (given in reference [20]) and 0.17 (given in reference [21]). On the other hand, the work of Apsley and Hughes [22] receives the dependence ln σ(E) ∝ E 2 for the same electrical ﬁeld range. At high electrical ﬁelds, the activationless hopping conductivity has been predicted by Mott [23]. Accordingly, the current density versus electrical ﬁeld characteristic was found to be [25] 1/m E0 j(E) ∝ exp − . (6.3) E Note that the value of m – according to reference [25] – corresponds with the value of p from (6.1) just at p = 4. The equality m = p = 4, therefore, provides evidence that, in the model examined, the high electrical ﬁeld plays the same role as the temperature does within the regime of ohmic conductivity. Experimental work [26, 27] has conﬁrmed the above conclusion, both for the case m = p = 4 (3D) and the case m = p = 3 (2D). Rentzsch et al. [28]

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observed experimentally the equality m ≈ p ≈ 1.3–1.6 for polycrystalline ZnSe ﬁlms having a thickness of 1 µm. VRH in a parabolic quasi-gap at high electrical ﬁelds was investigated in [29–31], where the equality m = p = 2 has been determined. On the other hand, the same measurements performed on Ge1−x Cux amorphous thin ﬁlms [32] led to the values p = 2 and m = 4. The prevailing ambiguity of the above statements concerning the character of the current–voltage (respectively current density versus electrical ﬁeld) characteristics of amorphous semiconductors, as described by (6.1), marked the starting point of the present work. Moreover, for our analysis of the charge transport process, we have chosen the micrometer-size porous carbon membranes with highly ordered honeycomb structures.

6.2 Experimental Formation of Polymer Honeycomb Structures 6.2.1 Spreading of One Liquid on Another For producing a self-organized honeycomb polymer network, we have developed a four-step method. Figure 6.1 illustrates these steps: (a) deposition of one drop of polymer solution (liquid F 1) on the cooled water surface (liquid F 2, 3–5◦ C); (b) spreading of one drop of polymer solution to an extremely thin layer; (c)–(d) interaction of water vapor (air with 100, 75%, 32%, or 19% relative humidity at 20◦ C) with the polymer thin ﬁlm surface for self-structuring the spatial distribution of the lateral water droplets and subsequently generating the polymer network; (e) mechanical removal of the structured network from the liquid and potential ﬁxation on a substrate by annealing. In general, the spreading of a drop of liquid F 1 on the surface of liquid F2 occurs when the criteria [33] γF2/G > γF1/G + γF1/F2 is valid. Here, γF1/F2 ,

Fig. 6.1. Formation of a self-assembled honeycomb polymer structure: (a) deposition of one drop of polymer solution F 1 on the cooled water surface F 2; (b) spreading of one drop of polymer solution to a thin layer; (c) water vapor condensing on the polymer ﬁlm surface; (d) growing of water droplets and building of the compact hexagonal structure, i.e., polymer network; (e) drying of the polymer network and transfer from the water surface to a ﬁxed substrate

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γF1/G , and γF2/G denote the surface tension between the liquids F1 and F2, between liquid F1 and gas G (in our case, G is air), and between liquid F2 and gas G, respectively. The degree of spreading of a ﬁlm of liquid F1 over liquid F2 characterizes the spreading coeﬃcient SF1/F2 = γF2/G − γF1/G − γF1/F2 . For SF1/F2 > 0, total spreading is achieved, i.e., the liquid drop F1 will cover the whole surface of the liquid F2 and, thereby, form a monomolecular layer at the edge. For SF1/F2 < 0, there is no spreading. 6.2.2 Production of Polymer Networks As liquid F 1 we have used three diﬀerent polymer solutions: (a) 1% nitrocellulose solution in amyl acetate; (b) 2% xylene solution of poly(p-phenylenevinylene) precursor (PPV); (c) 2% xylene solution of poly (3-octylthiophene) (P3OT). The value of the surface tension of each polymer solution was determined using a stalagmometer. This instrument consists of a capillary tube through which distilled water ﬂows with a volume of V0 . It enables the counting of the number of droplets, nF2 , passing through the capillary. Measuring the polymer solution F1 with a volume of V0 in an analogous way allows to determine the surface tension as γF1/G = γF2/G nF2 ρF1 /nF1 ρF2 , where ρF1 and ρF2 are the densities of the solution and water, respectively. nF1 is the number of droplets of the solution. The value of γF1/F2 was determined from the diﬀerence between the surface tensions of the water saturated with the polymer solution and that of the polymer solution saturated with water. Both surface tensions were measured in the manner described above. The demand for total spreading of one drop of the 1% nitrocellulose solution in amyl acetate on the cooled water surface is fulﬁlled, because we have SF1/F2 = 38.3 mN m−1 with γF1/G (20◦ C) = 24.6 mN m−1 , γF2/G = 74.9 mN m−1 , and γF1/F2 = 12 mN m−1 [13]. The size of the spread thin cellulose layer in the vessel (with a diameter of 93 mm) was 70 mm. Since the volume of the spread drop came to 15 µl, the thickness of the resulting spread liquid cellulose layer can be estimated to 3.9 µm. This thin ﬁlm was subject to the inﬂuence of water vapor which indicates the self-organized formation of a honeycomb network structure. The relative humidity of air was taken constant to about 75% at a temperature of 20◦ C. Depending on the time elapsed after the water vapor has started to aﬀect the polymer ﬁlm, one obtains a variety of network structures distinguished both in form and size. During our experiments, the above time span changed between 1 and 60 s. In a ﬁnal step, after having dried the network, it was transferred from the water surface to a sapphire substrate. The corresponding spreading coeﬃcients of both 2% xylene solutions of the PPV and P3OT are nearly equivalent and amount SF1/F2 = 8.8 mN m−1 with γF1/G = 30 mN m−1 and γF1/F2 = 36.1 mN m−1 [14]. Over 10 s after having spread the polymer solution ﬁlm on the surface of liquid water, it was then subject to water vapor (air with 19% relative humidity at a temperature of 20◦ C). It should be noted that the diameter of the initial spread liquid

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polymer layer in the vessel (with a diameter of 93 mm) at the beginning was about 50 mm. Then, the size of the liquid polymer layer decreases to about 20–30 mm. The thickness of the resulting dry polymer layer was not homogeneous. The edge of the polymer layer was thinner and in this area the formation of a hexagonal network takes place. The middle area of the layer was thicker and not structured. 6.2.3 Structural Forms of Nitrocellulose Networks After depositing the water vapor on the polymer layer and drying the latter, a fractal-like geometry appears, i.e., areas (in the form of stripes) with network structures and areas without them come to light. The majority of the structured stripes was distributed at the edge of the polymer layer and normally had a width of 0.5 mm and a length of 20 mm. The network stripes can be connected with or separated from each other. Figure 6.2a gives a typical scanning electron microscope (SEM) picture of a nitrocellulose network which contains segments with well-ordered hexagonal cells. One of these segments is illustrated in Fig. 6.2b. Note the strong spatial homogeneity and reproducibility of the individual cells inside the network structure. In this case, the water vapor was coated 10 s after having spread the polymer layer onto the cooled

Fig. 6.2. SEM images of a nitrocellulose network which was prepared via coating the water vapor 10 s after having spread the polymer layer onto the cooled water surface. Scale bars: (a) 30 µm; (b) 10 µm; (c) 2 µm; (d) 1 µm

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water surface. For a more detailed look at the geometry, blow-up pictures are displayed in Figs. 6.2c, d. The diameter of the cell (i.e., the distance between the pore wall centers) amounts to 2.6 µm and the width of the pore wall to about 0.4 µm. The cross-section of the pore wall perpendicular to the photograph plane oﬀers the view of a T-like shape. The latter means that each hexagonal cell lies on a hexagonal base also having extremely thin side walls. The height of the base is 0.5 µm. Each one of these six side walls represents some kind of a frame with various thickness along the circuit on which a thin polymer ﬁlm is stretched. In most cases, the polymer ﬁlm unveils an oval aperture in the middle of the frame, see Fig. 6.2c, d. It was found that the diameter of a single cell at the edges of the stripe is often smaller than that in the center. Figure 6.3a, b demonstrates SEM images obtained from the network located at the edge of the stripe after annealing for 1 h at a temperature of 950◦ C under vacuum conditions and which we have used afterwards for conductivity measurements. Obviously, the cross-section of the pore wall perpendicular to the photograph plane does not represent the T-like shape mentioned earlier (Fig. 6.2d), i.e., the pore wall merges after annealing. As a result, we observe an almost two-dimensional cell

Fig. 6.3. SEM images of diﬀerent areas of one carbon network with (a)–(b) a twodimensional elementary cell and (c)–(d) a three-dimensional elementary cell which were heated for 1 h at a temperature of 950◦ C under vacuum conditions. The network was prepared via coating the water vapor 10 s after having spread the polymer layer onto the cooled water surface. Scale bars: (a) 5 µm; (b) 2 µm; (c) 4 µm; (d) 2 µm

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conﬁguration. The corresponding Fig. 6.3c, d illustrates a network fragment and single cell in the center of the stripe after similar annealing. Here, we have three-dimensional elementary cells of a carbon network pattern obtained from a corresponding nitrocellulose structure after annealing. In case where the latter cells have a relatively large diameter, the total network extends over an area of about 30×30 µm2 . Apparently, the upper and lower hexagonal cells (one placed on the top of the other) – both with a diameter of 6 µm – are practically equal. The relating two cells are connected only at the corners (cf. Fig. 6.3d). The height of the junction between the upper and the lower cells is approximately 1.5 µm. The width of the upper part of the pore wall amounts to about 0.25 µm, but the lower part is smaller. A slightly other kind of nitrocellulose network (compared to the one of Fig. 6.2) is illustrated in Fig. 6.4a, b, the structure of which we have obtained when coating the water vapor 60 s after spreading the polymer layer onto the cooled water surface. Take note of a relatively strong spatial homogeneity and reproducibility of the individual cells inside the network pattern, similar to that illustrated in Fig. 6.2. For a closer look at the underlying geometry, a blow-up of Fig. 6.4a conﬁned to a single elementary cell is displayed in Fig. 6.4c. It turns out that the cross-section of the pore wall has a somewhat plate-shaped form. The width of the pore wall amounts to approximately

Fig. 6.4. SEM images of a nitrocellulose network with plate-shaped pore walls which were prepared via coating the water vapor 60 s after having spread the polymer layer onto the cooled water surface. Scale bars: (a) 5 µm; (b) 2 µm; (c) 1 µm; (d) 1 µm

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1.5 µm, and the height of the “plate walls” to about 0.25 µm. Inside the cell, one can perceive the breakthrough of the thin polymer ﬁlm. We point out that the cross-section of the pore wall of the latter structure does not have a T-like shape (compared to the one in Fig. 6.2). Figure 6.4d shows the corresponding single cell illustrated in Fig. 6.4c after annealing for 1 h at a temperature of 750◦ C under vacuum conditions. We have observed that the thin lower part of the pore wall was merged after annealing. 6.2.4 Structural Forms of Poly(p-phenylenevinylene) and Poly(3-octylthiophene) Networks A typical network of the PPV precursor is shown in Fig. 6.5a, c taken at diﬀerent magniﬁcation [14]. The strong periodicity and homogeneity of the individual cells can be clearly recognized. For a more closer look at the proﬁle of the network, atomic force microscopy (AFM) studies have been performed. The contact-mode topographic proﬁles of the PPV precursor network are displayed in Fig. 6.5b, d. The depth of the pores amounts 0.6 µm, the diameter about 1 µm, and the width of the pore walls 0.7 µm. To end up with a conjugated PPV network, the sample of the PPV precursor network was annealed for 2 h at a temperature of 160◦ C under nitrogen ﬂow atmosphere. The main ﬁnding is the following: the PPV precursor network converts to a conjugated

Fig. 6.5. (a) SEM image of a PPV precursor network. Scale bar: 25 µm; (b) AFM proﬁle analysis of the scan line (height vs. lateral coordinate) indicated in part a; (c) Magniﬁcation of part a. Scale bar: 2.5 µm; (d) AFM proﬁle along the line indicated in part c

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Fig. 6.6. SEM images of a P3OT network. Scale bars: (a) 15 µm; (b) 2 µm

PPV ﬁlm that consists of periodic hill-like structures with a translational periodicity of the precursor network, i.e., the network melts to the ﬁlm. The network of the conjugated P3OT is illustrated in Fig. 6.6a. A similar segment-like pattern (compared to the cellulose net in Fig. 6.2a) with wellordered single pores in separate segments was found. Note that the structural form of the P3OT pore wall shown in Fig. 6.6b is more similar to the cellulose wall shown in Fig. 6.4b. Our AFM studies quantify the depth of the pores to

Fig. 6.7. SEM images of a P3OT ﬁlm after subject to water vapor. (a) and (b) illustrate the cases where the water vapor acts on the P3OT layer being slightly dry. Scale bars: (a) 2 µm; (b) 2 µm. (c) shows the case where the water vapor acts on the already ﬂuid P3OT layer. Scale bar: 2 µm. (d) magniﬁes a single pore of the sample treated in Fig. 6.6. Scale bar: 0.6 µm

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0.5 µm, the diameter to about 1 µm, and the width of the pore walls to 0.8 µm. Figure 6.7 shows diﬀerent yet neighboring areas from one sample, which clearly illustrate the mechanism of network formation. Figure 6.7a demonstrates that the water droplets were initially condensed on the surface of the P3OT thin ﬁlm, when the latter was not ﬂuid enough to let the water droplets sink into the polymer layer. In Fig. 6.7b, the latter process has already started for the case of a few water droplet, indicating the formation of pores. The completed network shown in Fig. 6.7c could be obtained only by the sinking and ordering of water droplets in the ﬂuid enough P3OT layer. The sinking depth of the water droplets determines the pore diameter and, simultaneously, the width of the pore wall. For example, the width of the pore walls shown in Fig. 6.7c amounts to about the twofold pore diameter. In case of the network given in Fig. 6.7d (taken from the sample addressed in Fig. 6.6), however, the wall width turns out to approximately the pore diameter.

6.3 Model for the Formation of Honeycomb Structures in Polymer Films 6.3.1 Water Droplet on the Fluid Polymer Layer Water Droplet Before the Contact with the Polymer Layer Let us discuss in a more detailed way the condensation of water vapor on the surface of the ﬂuid polymer layer, representing one crucial factor for the patterning process. Since the cell diameter of our networks is about 2 µm, we investigate the shape of a water droplet with the same diameter, before it dissolves on the surface of the polymer thin ﬁlm. We have to compare two pressure quantities: the ﬁrst one is the capillary pressure PL which gives rise to the spherical shape of the droplet, the second one the so-called gravitation pressure PG causes the ﬂattening of the droplet. The capillary pressure can be calculated by the well-known formula PL = 2γF2/G /Rd , with Rd giving the droplet radius. In our case, we obtain PL = 1.4 × 109 Pa. If the contact plane between the droplet and the polymer thin ﬁlm is approximately πRd2 , we can determine PG = md g/πRd2 with the water droplet mass md = 4πRd3 ρF2 /3 (ρF2 is the water density). Finally, one ends up with PG = 1.3 × 10−2 Pa. If we compare both types of pressure, it turns out PL PG . That means that the water droplet has a spherical shape before laying on the surface of polymer solution. Water Droplet After the Contact with the Polymer Layer In Fig. 6.8a the ﬁrst moment the water droplet contacts the polymer layer is sketched: If the droplet comes into touch with the surface, the destruction of the droplet will start. The angles θ21 and θ23 , in this case, are nonequalized

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Fig. 6.8. Model for the lay on of the water vapor on the polymer layer: (a) process of envelopment of the water droplet by the polymer layer; the thin ﬁlm on the surface of the water drop indicates a monomolecular polymer layer; (b) moment of the ﬁrst contact of the water droplet with the cooled water surface; (c) model structure of the network according to (b) in the plane of the polymer layer

boundary angles, because they experience variations in the development of further physical and chemical processes that take place at the phase boundary. Water is a denser medium than amyl acetate (0.87 g cm−3 ), but the water droplets do not sink as a consequence of a subtle balance between buoyancy, droplet weight, and capillary forces [34]. The droplets are situated at the interface between the ﬂuid polymer layer and air. Only a small part of the droplet is located above the surface of the ﬂuid polymer layer. The water droplets on the ﬂuid polymer layer do not coalesce immediately after they touch, because they are separated by a thin ﬁlm of the polymer solution. We assume that a thin polymer ﬁlm (only a few monomolecular layers thick) develops on the top of the droplets, similar to the spreading of the polymer solution drop on the water surface. The latter thin ﬁlm on top of the water droplet might be a reason for the growth retarding of the water droplets and, therefore, a key parameter for the regulation of the droplet size. In addition, we suggest that water droplets in the polymer solution are covered with a solid polymer layer at the interface between the two liquids. Experimentally, these layers can be seen in Fig. 6.4a–c, which are characterized by bursting

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holes (black); the water ﬂows out of them after the solvent has evaporated. Such a layer prevented coalescence of water droplets, i.e., the precipitation of nitrocellulose at the interface between the polymer solution and the water droplets provides the basis for the formation of a compact hexagonal structure of water droplets. In our model scheme of Fig. 6.8a, the precipitation layer is illustrated as a solid line between the water droplets and the polymer solution. Similar results have been obtained for another experimental situation, where water droplets are deposited on the surface of a solution of poly(p-phenylene) block polysterene in carbon disulﬁde [9]. The corresponding forces which come into play at the above phase boundary are indicated in Fig. 6.8a. These forces converted to the length unit of the wetting line are equivalent to the corresponding quantities of the surface tension. From the condition that gives the balance of the surface tensions, the equilibrium state at the edge circumference of the contact between the water droplet and the polymer layer can be described by the two equations γF1/G cos θ13 = γF2/G cos θ23 + γF2/F1 cos θ21 ,

(6.4)

γF2/G sin θ23 + γF1/G sin θ13 = γF2/F1 sin θ21 ,

(6.5)

where the angles θij are deﬁned in Fig. 6.8a. Equation (6.4) treats the forces balance in the plane of the polymer layer, and (6.5) treats the forces balance directed perpendicular to it. The shape of the water droplet (the angles θij are deﬁned in Fig. 6.8a) can be qualitatively discussed only using the initial values of the surface tension γ. The water droplet shape on the surface of paraﬃn oil has been studied by Knobler and Beysens [35], and on the surface of carbon disulﬁde by Pitois and Francois [9]. They found that the water droplet has the form of a strongly asymmetric lens, the major part of which is nearly a complete sphere suspended from the surface. Based on these studies and our experimental results displayed in Figs. 6.2 and 6.4, the shape of the water droplet located on the surface of the polymer solution in amyl acetate has been accordingly sketched in Fig. 6.8a. In our case, the capillary pressure in the water droplet is drastically higher than the gravitation pressure (PL PG ), i.e., the inﬂuence of gravity forces on the droplet shape can be neglected. That means that the small water droplet on the ﬂuid polymer layer is composed of two spherical segments (see Fig. 6.8a), the angles θij and radii Rij of which can be determined by the surface tension forces [36] 2 2 γik γij 1 γjk − 1+ , (6.6) cos θij = 2 γij γjk γjk Rij /r = 1/ sin θij ,

(6.7)

hij /r = (1 − cos θij )/ sin θij ,

(6.8)

where θij , Rij , hij , and r are deﬁned as illustrated in Fig. 6.8a. It should be noted that, in the case of an extremely small lens, Princen [36] has assumed

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that angle θ13 is zero. In our case of a relatively thin polymer ﬂuid layer, the exact determination of the angles θij from (6.6) is not that easy, because initial values of the parameters γF2/G , γF1/G , and γF2/F1 alter with time via the evaporation of the solvent out of the polymer solution. On the other hand, for the case of a water droplet placed on the surface of paraﬃn oil, the angles θij were measured by Knobler and Beysens [35]. They observed values in the range θ21 = 135◦ –140◦ and θ23 = 20◦ –25◦ . The ﬂuid properties (i.e., surface tension, density) of amyl acetate and paraﬃn oil are comparable. The force γF1/G sin θ13 × 2πr in (6.5) does not permit the water droplet to sink into the polymer solution layer, i.e., angle θ13 cannot be zero. On the other hand, to provide this force for a water droplet with a diameter of 2 µm as in our case, the deviation from a horizontal line is necessarily not particularly strong. The force γF2/G sin θ23 for achieving the equilibrium state (described by (6.5)) tends to pull the ﬂuid polymer layer onto the water droplet. The water droplet then more and more penetrates the polymer layer. The envelopment of the water droplet by the polymer layer will take place as long as the angles θ21 , θ23 , and θ13 (or the radii of the surface curvatures, R21 and R23 , see Fig. 6.8a) experience variations such that the pressure inside the water droplet caused by the surface curvature of the boundary between water and air is equal to the pressure that results from the surface curvature at the boundary between water and polymer layer. Their balance can be expressed mathematically by γF2/G /R23 = γF2/F1 /R21 , where R23 and R21 are the radii of the water droplet at the boundary between water and air and between water and polymer layer, respectively. If we use the values γF2/G = 71 mN m−1 and γF2/F1 = 12 mN m−1 , we obtain the ratio R23 /R21 ≈ 6 for nitrocellulose solutions in amyl acetate. That means that pressure equality in the lower and upper part of the water droplet at thermodynamical equilibrium can only be guaranteed if radius R23 exceeds radius R21 by a factor of six (see Fig. 6.8a). The corresponding ratio R23 /R21 ≈ 2 can be determined for both 2% xylene solutions of PPV and P3OT with γF1/F2 = 36.1 mN m−1 . Formation of Breath Figures The self-assembled ordering of the water droplets on the surface of the ﬂuid polymer layer can be explained as follows. With the condensation of the water vapor on the cold surface of the liquid, the water droplets form into a pattern of breath ﬁgures [37–40], the geometry of which can be diﬀerent. The basic physical arguments supporting the development of breath ﬁgures on the ﬂuid surfaces were discussed by Knobler et al. [35] and Steyer et al. [6, 41]. They suggest that breath ﬁgures on ﬂuid surfaces evolve through three stages: (a) initial stage, when the water droplets are isolated such that they do not strongly interact and the average droplet radius < Rd > grows with time td 1/3 as < Rd > ∝ td ; (b) cross-over stage, when the surface coverage is high and the rate of the droplet growth increases; (c) coalescence dominated stage,

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when the surface coverage is high and constant, and the droplet radius grows as < Rd > ∝ td . Following the studies by Steyer et al. [6] and Chan et al. [34] the attractive force F between two water droplets on the surface of the polymer solution separated by a distance ld can be determined as 2 4π Rd6 ρa2 g 2 1 2 1.5 2 0.5 F = + 0.25(1 − k ) − 0.75(1 − k ) , 3 ld γF1/G ρs

(6.9)

where Rd denotes the radius of the droplets, ρa is the absolute density of the polymer solution, g is the earth’s gravitational acceleration, γF1/G is the surface tension between the polymer solution and air, ρs is the relative density of the polymer solution compared to air, k = r/R21 , where r and R21 are radii as illustrated in Fig. 6.8a. Equation (6.9) holds if the bond number B0 = Rd2 ρa g/γF1/G is small enough (i.e., <0.1, see reference [34]). In our case, B0 is of the order of 10−6 for all polymer solutions and water droplets with a diameter of 2 µm. Parameter k characterizes the contact angles of the water droplet with the surface of the polymer solution and is highly sensitive to the wetting properties between water and the polymer solution. Steyer et al. [6] have proposed that the formation of a hexagonal structure for the water droplets occurs when the value of parameter k ranges between 0.4 and 0.8. The hexagonal structure disappears when k varies from 0.2 to 0.4. The exact determination of parameter k and the corresponding force F is not that easy in our case, because initial values of the parameters γF2/G , γF1/G , γF2/F1 , and ρa alter with time. We are interested to consider only some tendency according to (6.9) with initial values of the surface tension γ. Equation (6.9) tells us that, upon altering k, the force can increase considerably, i.e., k can play the role of a key parameter for the formation of a hexagonal structure of water droplets. For the present case, the value of parameter k can be estimated from (6.7). Taking the angle θ21 = 135◦ , one calculates k ≈ 0.7. The corresponding experimental value of k which derives from the geometrical size of a single cell in Fig. 6.2 is equal to 0.8 for a nitrocellulose network. From (6.8), we determine for θ23 = 20◦ and r ≈ 0.7 µm (i.e., R21 = 1 µm) the parameter h23 ≈ 0.1 µm, and for θ21 = 135◦ the parameter h21 ≈ 1.7 µm. The increase in surface tension of the polymer solution, γF1/G , leads to a decrease of the attractive force F between the two water droplets. Accordingly, the corresponding distance ld between the droplets increases. Our ﬁnding can be well corroborated in Figs. 6.2 and 6.4. The width of the pore wall of the nitrocellulose network, obtained when the water vapor was coated 10 s after spreading the polymer ﬁlm, amounts to 0.25 µm (see Fig. 6.2). Turning to Fig. 6.4 where the time period between spreading and subsequent coating has been extended to 60 s, the relating width of the pore wall increases to about 1.0 µm, i.e., the surface tension of the nitrocellulose solution increases with the solvent evaporation time. Accordingly, the corresponding attractive force F between the two water droplets decreases (6.9).

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During the drying process of the polymer ﬁlm, the water droplets approach the cooled water surface. In Fig. 6.8b, the contact between the water droplet and the cooled water surface together with the corresponding variation of the radii R23 and R21 are displayed schematically. Between the water droplet and the cooled water surface, we ﬁnd a thin precipitation polymer ﬁlm. The radius R21 will be enlarged, since the lower part of the water droplet which is in contact with the cooled water surface (more precisely, with the thin polymer ﬁlm) will have a ﬂat boundary plane. Figure 6.8b features the crosssection of the pore wall of one cell of the polymer network which develops between two neighboring water droplets. Comparison of Fig. 6.2 and Fig. 6.8b yields a pronounced similarity between the cross-section of the experimentally prepared elementary cell of the polymer network and the one of the modeled cell. From these pictures, we conclude that the minimum thickness of the polymer layer in the middle of the frame (i.e., the base plate) corresponds to the minimum distance between the water droplets near the AA∗ line. For the case of nitrocellulose in amyl acetate, the above distance is probably a few monomolecular layers thick, but for the PPV network it is essentially larger. Formation of 2D and 3D Networks The appearance of small areas (30 × 30 µm2 ) in the central region of the nitrocellulose network stripes (see Fig. 6.2a) with large elementary cells of the form displayed in Fig. 6.3c derives from some size gradient of the water vapor droplets. The formation of a hexagonally arranged pattern of water droplets on the liquid polymer layer with the above size gradient can be explained as follows. Because of the water vapor condensation on the cold surface of the liquid in the ﬁrst moment, the water droplets conform to the islands which later on build the breath ﬁgures [37–40]. The islands attract each other in a much stronger way than it is the case for the single droplets. One can argue that the attractive force F between the islands is proportional to the sixth power of their radii, as demonstrated in (6.9). The intrinsic droplets in the stripe undergo a maximum shrinkage, and they can coalesce when the polymer ﬁlm between the water droplets is not large enough. In case we have an extension of the water vapor comparable to the thickness of the polymer layer (about 4 µm), the total polymer solution will even move to the space in between the water droplets. As a consequence, there only remains a quite thin polymer ﬁlm underneath the water droplets such that the polymer solution will be uniformly distributed with respect to the AA* line. The subsequent evaporation of amyl acetate out of the solution does not give rise to an essential redistribution of the polymer relating to the AA* line. We end up with the result that the depth of the cell structure (i.e., its dilation in the third spatial dimension) shown in Fig. 6.3c, d is larger (about 1.5 µm) than the one displayed in Fig. 6.3b (about 0.5 µm). Comparison of the experimental polymer networks shown in Figs. 6.2 and 6.4–6.7 and our model picture outlined in Fig. 6.8 allows us to assume that,

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at ﬁrst, solidiﬁcation of the polymer network (i.e., evaporation of the solvent) takes place and, afterwards, the water droplet will break through the thin polymer ﬁlm and ﬂow into the cooled water (see the dark bursting holes within the cells of Fig. 6.4b, c). As already stated above, the thickness of the liquid polymer layer is about 4 µm after dissolving on the water surface. During the drying process, the thickness of the polymer layer amounts to about 0.5 µm. As can be clearly recognized in Figs. 6.2 and 6.5, the whole polymer material dissolved is solidiﬁed in the region of the AA∗ line, more precisely, above that line. If the height h (see Fig. 6.8b) of the enveloping polymer layer is not too large, a hexagonal patterning can be achieved. The experimental realization is captured in Figs. 6.2 and 6.5, the underlying basic cell structure outlined in Fig. 6.8c. That means that, at a low height, also the width of the pore wall will be small. The junctions cross in the knots of the hexagonal network under an angle of 120◦ and usually form a continuous pattern. Apparently, hexagonal net structures develop at small values of h, as a result of the cross-over of three types of zig-zag lines which proceed to the directions 1, 2, and 3 (see Fig. 6.8c). It must be emphasized that the hexagonal shape of the elementary cells of the polymer network becomes more pronounced after total evaporation of the solvent out of the pore walls, what is clearly illustrated in Fig. 6.3b. The self-organized formation of the nitrocellulose network given in Fig. 6.4 sets in when the water vapor begins to aﬀect the polymer layer not until 60 s after spreading it onto the cooled water surface, and the thin ﬁlm gradually becomes dry in the meantime. In other words, evaporation of amyl acetate out of the polymer layer during the elapsed time of 60 s gives rise to an increasing concentration of the polymer inside the polymer ﬁlm. As a consequence, the distance between neighboring droplets of water vapor that precipitate onto the liquid surface of the polymer layer increases with the surface tension γF1/G (cf. (6.9), γF1/G = 24.6 mN m−1 at the beginning of evaporation). Accordingly, the same holds for the interspace between neighboring droplets of the water vapor, where the polymer material accumulates after the thin ﬁlm has completely become dry (in the vicinity of the AA* line). In this case, the cross-section of the pore wall will have the shape of a plate as shown in Fig. 6.9a, and the corresponding whole network resembles that one shown in Fig. 6.9b. The lifted edges in Fig. 6.9a are displayed together with the different pattern in Fig. 6.9. Upon comparing the experimental results in Fig. 6.4 with the modeled networks in Fig. 6.9, one recognizes a good correspondence. A similar structured form of the pore wall has been detected for the P3OT network shown in Fig. 6.6. However, it is diﬀerent compared to the case of the PPV network, although the values of the surface tension of both PPV and P3OT polymer solutions are the same (γF1/G = 30 mN m−1 ). The above difference can be understood owing to the peculiarity of the attraction between the polymer molecule in each individual solution.

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Fig. 6.9. Model structure of the network according to the experimental situation illustrated in Figs. 6.4 and 6.6; (a) shape of the cross-section of the pore wall; (b) form of the network. r1 is the radius of the deformed water droplets

6.4 Nitrocellulose Networks as Precursor for Carbon Networks For obtaining carbon networks with diﬀerent electrical conductivity, the nitrocellulose precursor samples were annealed for 2 h at process temperatures Tht ranging from 500 to 1,000◦ C under vacuum conditions (1 × 10−5 mbar). As a result, we end up with carbon net structures that predominantly have hexagonal elementary cells with a diameter ranging from 1 to 3 µm (see Fig. 6.3a). The width of the pore walls depends on the diameter of the hexagonal cells of the carbon network and, thus, varies between 100 and 350 nm for diﬀerent samples. The thickness of the pore walls in all samples is about 40 nm. For the sake of performing electrical measurements, four gold contacts were evaporated. The resulting active area of the carbon network fragment conﬁned between the contacts was about 40× 40 µm2 . To be able to reach the regime of linearity in the current–voltage characteristic by recording the dependence of the speciﬁc resistivity on temperature in the range from 4.2 to 295 K, the current varied from 0.1 nA (in the high ohmic regime) to 100 nA (in the low ohmic regime). The speciﬁc resistivity was calculated using the relation ρ = Rs/L; R is the resistance of the network (measured in the linear regime of the current– voltage characteristics); s is the overall net cross-section, equal to the sum over all cross-sections of parts of the carbon network that are connected with the Au contacts; and L is the length of the net, i.e., the distance between the Au contacts. Current–voltage characteristics have been recorded in the operation regime of the power source, each at constant temperature within the range from 4.2 to 290 K. To arrive at an as accurate as possible examination of the dependence of the conductivity on the electrical ﬁeld, current–voltage

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measurements were carried out separately in the following electrical ﬁeld regimes: 0–10, 0–100, 0–1000, and 0–9000 V cm−1 . 6.4.1 Temperature Dependence of Hopping Transport in Carbon Networks Experimental Temperature Dependence of the Speciﬁc Resistivity In Fig. 6.10, the dependence of the speciﬁc resistivity on inverse temperature, ρ(1/T ), is shown in a semi-logarithmic plot for the samples 1–12. The corresponding annealing temperature Tht and also the relating speciﬁc resistivity at room temperature, ρ(295 K), taken for each annealing temperature are given in Table 6.1. In contrast to our expectations, the increase of the annealing temperature of the starting cellulose net structures does not always lead to a higher electrical conductivity in the annealed carbon network. For example, the resistivity of a carbon net annealed at a temperature of 700◦ C (curve 5) is by orders of magnitude lower than that for a net annealed at a temperature of 725◦ C (curve 2). The latter phenomenon is more pronounced in the lower temperature range. The dependences of the speciﬁc resistivity on temperature displayed in Fig. 6.10 are characteristic for VRH, commonly described by (6.1). For obtaining an as precise as possible value of the characteristic parameter p, we turn to the local activation energy εa according to [19,42,43] εa =

d(lnρ) , d(1/kB T )

(6.10)

8 1

ln (ρ[Ωcm])

6

4

3

2

5

4 6

2 0

7

-2

8

-4

9-12

0.00

0.05

0.10

T

0.15 -1

0.20

0.25

-1

[K ]

Fig. 6.10. Dependence of the speciﬁc resistivity of carbon nets on inverse temperature, plotted on a semi-logarithmic scale. The numbers of the diﬀerent samples are indicated, their annealing temperature can be found in Table 6.1

134

L.V. Govor and J. Parisi -1 1 2

log10 (εa[eV])

-2

3 4 5 6 7 8

-3

B=1.0

-4 9 0.4

B=0.66

B=0. 5 0.8

1.2

1.6

2.0

2.4

log10 (T[K]) Fig. 6.11. Dependence of the local activation energy on temperature, plotted on a double-logarithmic scale. The number of the curves correspond to the numbers in Fig. 6.10. B gives the slope of the approximation line

where kB is the Boltzmann constant. Figure 6.11 shows a double-logarithmic plot of εa (T ), where four characteristic temperature ranges are indicated. A careful analysis of the dependence εa (T ) for each sample was carried out in Fig. 6.12, in order to determine the value of p and its corresponding temperature range. Upon decreasing the ambient temperature of the samples 8–12 from Tc = 295 K to Tb (Table 6.1), we recognize a rapid reduction of the local activation energy according to εa (T ) ∝ T α . The characteristic values α = c of these samples are also given in Table 6.1; they are about 1.3–1.5. Reducing the ambient temperature of the samples 2–12 from Tb to T13 leads to an overall dependence εa (T ) ∝ T α with α = αb ≈ 1.0 (see Table 6.1). Even lower temperatures from T13 to 4.2 K cause a signiﬁcant reduction of the dependence εa (T ). At temperatures varying from T13 to T12 , the exponent α is denoted by αM , while in the temperature range from T12 to 4.2 K, α is denoted by αCG . In Table 6.1, the corresponding numerical values of αM and αCG are listed. The values of αM give rise to the assumption that the temperature dependence of the speciﬁc resistivity from T13 to T12 can be described by Mott’s hopping law for the two-dimensional case. For diﬀerent samples, the mean value of αM is 0.63. Upon further decreasing the temperature from T12 to 4.2 K, VRH in the Coulomb gap is observed. The values of αCG diminish when approaching the MIT on the insulating side (see Table 6.1). In Table 6.2, the values of T0 introduced in (6.1) are given. They were extracted from the experimentally obtained dependences lnρ ∝ (T0 /T )1/p for p = 3 and p = 2 in the corresponding temperature ranges and denoted by TM and TCG , respectively. In the inset of Fig. 6.12, the dependence of TCG on TM is plotted on a double-logarithmic scale for the samples showing both,

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Table 6.1. Electrical parameters of carbon networks investigated (samples No. 1–12) No.

Tht ρ(295 K) c αb b αM pM αCG pCG Tc Tb T13 T12 ◦ C mΩ cm K K K K 1 600 8360 – – – 0.61 2.56 – – – – 295 – 2 725 200 – 1.10 3.13 – – 0.48 1.92 – 295 – 90 3 775 56 – 1.06 1.97 – – 0.44 1.78 – 295 – 85 4 775 35 – 1.00 1.20 0.66 2.94 0.44 1.78 – 295 140 75 5 700 28 – 0.91 0.80 0.57 2.32 0.38 1.61 – 295 135 55 6 750 20 – 0.94 0.62 0.63 2.70 0.40 1.66 – 295 120 40 7 800 15 – 0.98 0.48 0.62 2.63 0.33 1.49 – 295 85 15 8 950 12 1.43 0.91 0.33 0.64 2.78 – – 295 150 18 4.2 9 900 10 1.32 0.97 0.08 – – – – 295 17 4.2 – 10 1000 9 1.45 0.99 0.07 – – – – 295 17 4.2 – 11 900 9 1.49 0.91 0.06 – – – – 295 17 4.2 – 12 1000 8 1.41 1.06 0.04 – – – – 295 17 4.2 – Tht is the annealing temperature, ρ(295 K) the speciﬁc resistivity at 295 K, c the exponent in (6.13) which holds in the temperature range from Tc to Tb ; αb , αM , and αCG are the exponents of the local activation energy εa (T ) ∝ T α in the temperature ranges from Tb to T13 , from T13 to T12 , and from T12 to 4.2 K, respectively; b and pM are the exponents in (6.12) which hold in the temperature ranges from Tb to T13 and from T13 to T12 , respectively; pCG is the exponent in (6.11) which holds in the temperature range from T12 to 4.2 K Table 6.2. Parameters of VRH in the Coulomb gap of carbon networks investigated (samples no. 2–7) No. TM TCG εr a ∆ECG kB T12 εoptCG εoptM εa ρ0 ∗ ρ0 N0CG T = T12 T = T12 T = T12 K K nm meV meV meV meV meV mΩ cm mΩ cm eV−1 cm−2 2 3 4 5 6 7

– – 5352 2467 869 353

3637 1005 761 447 180 66

6 2 11 4 13 5 17 6 29 10 44 16

– – 25.6 19.2 8.8 2.9

7.7 7.3 6.4 4.7 3.4 1.2

24.6 12.6 10.3 6.7 3.6 1.3

– – 10.6 6.7 3.8 1.5

24.7 13.5 9.5 6.1 3.4 1.3

– – 2.4 3.3 6.2 8.6

15 15 64 4.7 11 18

5.4 × 1013 9.4 × 1013 1.1 × 1014 1.2 × 1014 1.9 × 1014 1.6 × 1014

TM and TCG are constants in (6.12) for Mott’s hopping law and in (6.11) for VRH in the Coulomb gap, respectively; εr is the dielectric constant, a is the radius of the localized states, ∆ECG is the Coulomb gap width at T = T12 , kB T12 is the thermal energy at T = T12 ; εoptCG is the optimum hopping energy in the Coulomb gap, εoptM is the optimum hopping energy in Mott’s hopping law, εa is the experimentally obtained local activation energy at T = T12 ; ρ0 and ρ0 ∗ are constants in (6.11) and (6.12), respectively; N0CG is the density of states at the edge of the Coulomb gap

Mott’s hopping law and VRH in the Coulomb gap, in diﬀerent temperature ranges. Obviously, the relationship between these two constants follows the phenomenological equation TCG = ATM t , with A ≈ 0.07 and t ≈ 1.

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3.0 log10 (TCG [K])

log10 (εa[eV])

-1.6

-2.0

2.5

αb=0.94

2.0 2.5

-2.4

6 t = 0.94

0.5

1.0

αM=0.63 T12 =40K

αCG=0.40

-2.8

T13 =120K

3.0 3.5 log10 (TM [K])

1.5

2.0

2.5

log10 (T[K]) Fig. 6.12. Dependence of the local activation energy on temperature of sample 6, plotted on a double-logarithmic scale. The dashed lines give the approximation of the dependence εa (T ) in the corresponding temperature range. The inset shows the dependence TCG = ATM t , plotted on a double-logarithmic scale

As aforementioned, nearly all samples investigated disclose a temperature range where the exponent of the dependence εa (T ) ∝ T α is α = αb ≈ 1.0 (see Table 6.1). This means that in the temperature range from Tb to T13 the dependence ρ(T ) ∝ T −b is observed. The values of b given in Table 6.1 were evaluated from the ﬁts of the curves lnρ(lnT ), experimentally obtained in the temperature range from Tb to T13 . At this point, some experimental results should be discussed in detail. For the samples 2 and 3, it is not that easy to specify exactly the range of the dependence εa ∝ T α , where Mott’s hopping law holds. The latter range becomes more evident for the case of sample 4. In contrast, Mott’s hopping law could even be observed for the whole temperature range from 80 to 295 K in sample 1. Analysis of the Speciﬁc Resistivity In connection with the above analysis of the temperature dependence of the local activation energy εa (T ), all dependences ρ(T ) of the carbon networks in the temperature range from 4.2 to 295 K could be represented analytically as the sum of four laws [44,45]. The ﬁrst one describes VRH in the Coulomb gap at temperatures varying from 4.2 K to T12 , according to (1/pCG ) TCG . (6.11) ρ(T ) = ρ0 exp T

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The second and the third law characterize VRH with a constant density of localized states in the temperature range from T12 to T13 and the power law of ρ(T ) in the temperature range from T13 to Tb for all samples, respectively. Analytically, they can be written as (1/pM ) TM ∗ −b ρ(T ) = ρ0 T exp . (6.12) T Take note of the fact that the dependence ρ(T ) ∝ T −b in (6.12) strongly dominates the transport behavior of all samples in the temperature range from T13 to Tb . In case of a low share of Mott’s hopping law at temperatures ranging from T12 to T13 (samples 2 and 3), the exponential part in (6.12) is negligible. The fourth law characterizes the temperature range from Tb to Tc according to c−1 T ρ(T ) = ρ1 exp − . (6.13) T1 The parameters b, c, pCG , pM , ρ0 , ρ∗0 , ρ1 , TCG , TM , and T1 in the above equations are constants. For all samples far from the MIT, (6.11) and (6.12) dominate the dependence ρ(T ) (see Table 6.1). Simultaneously, (6.13) and the power law part in (6.12) become more important for samples near the MIT. The temperature dependence of the speciﬁc resistivity of sample 8 represents some kind of threshold between the dependences ρ(T ) near and far from the MIT. For the samples 8–12, the values of T1 in (6.13) amount to 320, 1290, 1880, 1980, and 3190 K, respectively. The corresponding values ρ1 are calculated to 21, 17, 16, 15, and 13 mΩ cm, respectively. Following (6.10) and (6.11), the temperature dependence of the local activation energy in the range from 4.2 K to T12 , where VRH in the Coulomb gap is observed, can be described by εa (T ) = kB (1/pCG )TCG (1/pCG ) T (1−(1/pCG )) .

(6.14)

For temperatures ranging from T12 to Tb , the local activation energy is given by εa (T ) = kB bT + kB (1/pM )TM (1/pM ) T (1−(1/pM )) .

(6.15)

The second term characterizes the dependence εa (T ) for temperatures varying from T12 to T13 , the ﬁrst term from T13 to Tb . In the temperature range from Tb to Tc , the dependence εa (T ) reads c−1 T εa (T ) = (c − 1)kB T . (6.16) T1 A double-logarithmic plot of εa (T ), e.g., for sample 6, is shown in Fig. 6.12. Therefrom, the values of the parameters pCG and pM in the corresponding temperature ranges could be obtained. Following (6.14), the characteristic value pCG of (6.11) in the temperature range from 4.2 K to T12 is

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pCG = 1/(1 − αCG ). By approaching the MIT on the insulating side, the characteristic value pCG decreases (see Table 6.1). From (6.15) follows that the characteristic value pM in the temperature range from T12 to T13 is pM = 1/(1 − αM ). For diﬀerent samples, the mean value of pM in (6.12) is about 2.7. Coulomb Gap The appearance of the Coulomb gap in the density of states near the Fermi energy derives from the Coulomb interaction of charge carriers localized at diﬀerent impurity centers [17–19]. The transition from the Efros-Shklovskii (ES) law to Mott’s hopping law at the temperature T12 can be explained as follows. The charge carriers at T > T12 move via states that are located outside the Coulomb gap, where the density of localized states is only weakly dependent on energy. The presence of both, Mott’s hopping law and the VRH in the Coulomb gap for the samples 4–7, indicates that the impurity band width of these samples is much higher than the Coulomb gap width. For the samples 2 and 3, Mott’s hopping law is only weakly present. Hence, one can conclude that the density of localized states outside the Coulomb gap is not constant. From the values given in Tables 6.1 and 6.2, the change in the Coulomb gap width by approaching the MIT on the insulating side can be determined. Following reference [19], the expressions for calculating the constant T0 in (6.1) with p = 3 and p = 2 are TM =

βM kB a2 N0

(6.17)

βCG e2 , kB aεr

(6.18)

and TCG =

respectively, where a means the radius of the localized states, N0 is the constant density of states outside the Coulomb gap, and εr is the dielectric constant. According to reference [19], we have βM = 13.8 and βCG = 2.8. The dependence of the density of localized states on energy in the vicinity of the Coulomb gap for 2D is NCG (E) =

λε2r |E − EF | e4

(6.19)

with λ = 2/π. From (6.17), (6.18), and (6.19) follows that the Coulomb gap width at T = T12 can be expressed as ∆ECG =

βM kB TCG 2 . λβCG 2 TM

(6.20)

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For the samples 4–7, we can obtain the values of the Coulomb gap width at T = T12 by substituting the values TCG and TM (received from measurement, see Table 6.2) in (6.20). In addition, the Coulomb gap width can also be determined by the value of the optimum hopping energy in the Coulomb gap, εoptCG . Its temperature dependence is given by [19, 46] εoptCG = ACG kB (TCG T )1/2

(6.21)

with ACG = 0.5. For energy values lying outside the Coulomb gap, where Mott’s hopping law applies, the temperature dependence of the optimum hopping energy for 2D can be described by [19] εoptM = AM kB (TM T 2 )1/3

(6.22)

with AM = βM −1/3 = 0.4. In Table 6.2, the values of these optimum hopping energies at temperature T12 calculated from both sides, the Coulomb gap side (6.21) and the side of a constant density of states, N0 (6.22), are given. The values of the local activation energy εa at T = T12 , experimentally obtained from the dependences εa (T ) (see Figs. 11 and 12), are also shown in Table 6.2. A good correlation between these three parameters can be observed by comparing them. According to reference [19], the Coulomb gap width ∆ECG is twice as much as the values of the optimum hopping energy εoptCG , in agreement with our results in Table 6.2. The absolute values of the Coulomb gap width at temperature T12 obtained in a diﬀerent way are in good accordance with each other – certainly a proof for the reliability of these values. Comparing the values of εoptCG and kB T12 , one can see that, far from the MIT, the optimum hopping energy in the Coulomb gap is much higher than the energy kB T12 . The more one approaches from the insulating side of the MIT, the smaller becomes the diﬀerence between these two values. One can conclude that the optimum hopping energy at T ≤ T12 is smaller than the Coulomb gap width and, therefore, the ES law is observed in experiment at T ≤ T12 (6.11). On the other hand, the optimum hopping energy at T > T12 is larger than the Coulomb gap width. The latter means that Mott’s hopping law with a constant density of states, N0 , is dominant. Equating the value of 2εoptCG with the one of the Coulomb gap width, the value of N0 at the edge of the Coulomb gap can be calculated from (6.19). In Table 6.2, such values of N0 are listed and denoted as N0CG . It can be seen that the density of states near the Fermi energy enlarges by a factor of three during the transition from sample 2 to sample 7. Moreover, the values of the density of localized states, N0M , resulting from (6.17) in the validity range of Mott’s hopping law (Table 6.3), are nearly the same as those of N0CG , resulting from (6.19) at the edge of the Coulomb gap (Table 6.2). Such accordance of N0M and N0CG at the point of transition, T12 , again suggests a proof for the reliability of these values. Resulting from (6.19), the Coulomb gap width for T = T12 depends on the density of states at the edge of the Coulomb gap, N0CG , and the dielectric

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Table 6.3. Parameters of Mott’s hopping law of carbon networks investigated (samples no. 4–8) No.

kB T13 meV

εoptM T = T13 meV

εa T = T13 meV

Th

kB Th

K

meV

εoptM T = Th meV

εa T = Th meV

N0M eV−1 cm−2

4 12.1 16.1 14.2 107 9.2 13.4 11.7 1.2 × 1014 5 11.6 12.1 10.0 95 8.2 9.6 8.0 1.8 × 1014 6 10.3 7.9 6.7 90 7.7 6.6 5.2 1.8 × 1014 7 7.3 4.7 3.7 50 4.3 3.3 2.6 1.8 × 1014 8 1.5 0.68 0.5 10 0.86 0.46 0.4 – kB T13 and kB Th are the thermal energies at T = T13 and T = Th , respectively; Th is the intermediate point of the temperature range from T12 to T13 ; εoptM is the optimum hopping energy in Mott’s hopping law, εa is the experimentally obtained local activation energy, N0M is the density of states in Mott’s hopping law

constant εr . The decrease of the Coulomb gap width by approaching the MIT on the insulating side until its closure is mainly a consequence of the increase of the dielectric constant εr and its divergence at the MIT (εr → ∞) [47, 48]. So the decrease and closure of the Coulomb gap by approaching the MIT on the insulating side lead to a lowering of the critical temperature T12 (see Table 6.1) and the constant TCG (see Table 6.2). Furthermore, the decrease of T12 and TCG also depends on the localization length by approaching the MIT on the insulating side. As already mentioned earlier, a lowering of the value of pCG in (6.11) by approaching the MIT on the insulating side was observed. One reason for such a decrease might be related to the transformation of the form of the Coulomb gap (6.19), the latter becoming steeper at lower energies, e.g., N (E) ∝ E 3/2 (see reference [49]). Hopping Energy in the Validity Range of Mott’s Law In Table 6.3, the parameter values of the optimum hopping energy in Mott’s law, εoptM (6.22), the experimentally obtained local activation energy εa , and the thermal energy kB T13 at T = T13 , i.e., at the upper edge of the validity range of Mott’s hopping law, are listed. For the samples 4 and 5, the values of εoptM and εa are either higher or comparable to the values of kB T13 . The values of kB T13 become larger than that of εoptM and εa , starting at sample 6. Such tendency also holds at the intermediate point Th of the temperature range from T12 to T13 (Table 6.3). For all samples where Mott’s hopping law is observed, the overtop of the parameters εoptM and εa over the values of kB T is only valid near the temperature T12 . Too small values of the parameters εoptM and εa in relation to kB T in the temperature range from T12 to T13 could be explained by the inﬂuence of the pre-exponential part ρ(T ) ∝ T −b on the exponential part in (6.12). This inﬂuence is particularly important close to the temperature T = T13 . Indeed, a more strongly pronounced dependence

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εa ∼ T 1.0 at temperatures T ≈ T13 compared to the dependence εa ∼ T 2/3 (Fig. 6.12) diminishes the experimentally obtained values of εa . Radius of the Localized States For the determination of the radius of the localized states, the value of the dielectric constant εr of the carbon networks we have investigated has to be well known. Resulting from the inset of Fig. 6.12, the phenomenological equation is TCG = ATM t with A ≈ 0.07 and t ≈ 1. Furthermore, the correlations TM ∝ 1/a2 and TCG ∝ 1/aεr are a direct consequence of (6.17) and (6.18), respectively. Upon comparing these correlations with the experimental data, we ﬁnd TM /TCG ∝ εr /a → const or εr ∝ a. This means that the critical exponent for both εr and a by approaching the MIT on the insulating side is nearly the same. Our result is in good agreement with experimental data of amorphous Cr–SiOx thin ﬁlms [50] and indium doped CdSe crystals [51]. We have compared our values of TCG for sample 2, furthermost situated from the MIT, with the values of carbon ﬁbers [52] and come to the conclusion that the value of the dielectric constant for sample 2 is εr = 6. By using the correlations TCG ∝ 1/aεr and εr ∝ a, the change of εr diﬀering from sample to sample can be calculated. In Table 6.2, such values of εr for the samples 2–7 are listed. The radii of the localized states, a, can be determined from the resulting experimental data of TCG and (6.18), also given in Table 6.2. High-Temperature Conductivity Next, we analyze in detail the temperature ranges from T13 to Tb and from Tb to Tc , where the phenomenological correlation ρ(T ) ∝ T −b and (6.13) are valid. In the whole temperature range investigated, we did not ﬁnd hopping with a constant activation energy for any of the samples mentioned. That means, there is no hopping between nearest-neighbor localized states at temperatures varying from 4.2 to 295 K. Therefrom, one can suppose that in our porous carbon networks tails of localized states pulled out of the conduction and valence band as a result of disorder and some overlap between these tails occur [15, 53]. The nearer the states are situated at the mobility edge, the weaker they are localized. The mobility edge is deﬁned as an energy threshold which separates localized states from delocalized ones [53]. At high temperatures (from T13 to Tb ), tunneling of the charge carriers occurs via weakly localized states at the smallest barriers. The inﬂuence of the dependence ρ(T ) ∝ T −b (b → 0) decreases and shifts to the range of lower temperatures at the samples 8–12 that are near the MIT. The latter phenomenon is closely connected with the existence of the temperature range from Tb to Tc , where the dependence of ρ(T ) follows (6.13). Such stronger dependence of ρ(T ) possibly derives from the transfer of the charge carriers excited at the mobility edge.

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6.4.2 Electrical Field Dependence of Hopping Transport in Carbon Networks Experimental Field Dependence of the Conductivity In Fig. 6.13, typical conductivity versus electrical ﬁeld characteristics of sample 7 recorded at diﬀerent values of the ambient temperature (from 4.2 up to 290 K) are displayed in a semi-logarithmic representation [54]. The same functional dependences can be also obtained for the case of sample 6 (not shown here). In the inset of Fig. 6.13, we have plotted a blow-up of the conductivity vs. electrical ﬁeld characteristic of sample 7 in the low-ﬁeld regime. Obviously, a linear dependence between the conductivity and the electrical ﬁeld strength (i.e., the horizontal parts of the curves in the semi-logarithmic representation of Fig. 6.13) is present only in the limited ﬁeld interval indicated on the r.h.s. of the characteristics. Upon decreasing temperature, the extension of that ﬁeld range noticeably decreases. A still more accurate determination of characteristic changes in the dependence σ(E) at constant temperature can be achieved via analyzing the modiﬁcations in the slope of the curves σ(E). It is, therefore, helpful to use the parameter [44] wσ (E) = d ln σ(E) / dE,

(6.23)

which denotes nothing but a measure of the degree of nonlinearity in the conductivity vs. electrical ﬁeld characteristic. At constant temperature, (6.2) can be expressed as (6.24) σ(E) = σ(0) exp(AE n ).

3

290 200

250 150 80 60 40 30 20 15

2

1

sample# 7 90 70

ln (σ [ S / cm ])

ln ( σ [ S / cm ])

4

10 8 6 4.2

15 2

1

10 8 6 4 .2 0

0

2000 E [ V /cm ]

300 E [ V /cm]

600

4000

Fig. 6.13. Dependence of the conductivity on the electrical ﬁeld of sample 7, taken at diﬀerent temperatures (values indicated in K) and plotted in a semi-logarithmic scale. The inset magniﬁes the same dependence in the regime of lower electrical ﬁelds

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Here, the constants σ(0) and A are functions of temperature, but they do not depend on the electrical ﬁeld. The parameter wσ is then given by wσ (E) = AnE n−1 .

(6.25)

We emphasize that the case n = 0 means saturation of the conductivity, i.e., σ = const. Exponent n in (6.24) can be gained from plotting the experimental data of the dependence log wσ vs. log E and taking advantage of the slope of the approximation line. In Fig. 6.14, the electrical ﬁeld dependence of the parameter wσ for the σ(E) curves of sample 7 (Fig. 6.13) is displayed in a double-logarithmic scale. The diﬀerent ambient temperatures distinguishing the curves correspond to that in Fig. 6.13. For the sake of a vivid illustration of the experimental results, we have also marked in Fig. 6.14 the slope of the curve that corresponds to the value n = 1. It becomes obvious that, within the low electrical ﬁeld range, (6.24) with n = 1 is valid. For each curve in Fig. 6.14, the end of the latter regime is always indicated by an arrow. With increasing temperature, the validity range of n = 1 clearly shifts towards the regime of higher electrical ﬁelds. For temperatures T ≥ 70 K, the n = 1 interval was not at all attainable within the scope of ﬁeld strength applied by us. Upon closer looking at the inset of Fig. 6.14 (blow-up of the 4.2 K curve in the low electrical ﬁeld range), we uncover an interval where (6.24) with n = 1.4 is valid that

4. 2

-3

8

n= 1

n = - 0.33

10 15

-4

20 30 40

-2.4

log10 (wσ)

log10 (w σ)

sample#7

Eth

6

n = 1.4

60. . 90

-2.5

-5

150-290

T = 4.2 K -2.6

-6

1.2 1.6 2.0 log10 (E [ V/cm ])

1

2.4

2

3

4

log10 (E [ V/cm ]) Fig. 6.14. Dependence of the parameter wσ (6.23) on the electrical ﬁeld of sample 7, taken at diﬀerent temperatures (values indicated in K) and plotted in a doublelogarithmic scale. The data are extracted from the σ(E) curves of Fig. 6.13. Eth is the characteristic threshold electrical ﬁeld, deﬁning the transition from the low to the high ﬁeld regime. In practice, the intersection of the corresponding approximation lines (the slope of which is nothing but the exponent n of (6.24) is referred to as Eth . The upper limit of the low ﬁeld interval, where n = 1 ceases to be valid, is marked by an arrow. The inset focusses on the dependence wσ (E) at the temperature T = 4.2 K and lower electrical ﬁelds

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precedes the n = 1 part of the curve. The n = 1.4 interval also extends to the high ﬁeld range with increasing temperature. The apparent saturation of the conductivity vs. electrical ﬁeld characteristics (n = 0) and their subsequent complete merging into one curve (n < 0) are strongly pronounced in the functional dependence wσ (E), where the curves consolidate with the slope n = −0.35 (sample 6) and n = −0.33 (sample 7) – the latter case indicated in Fig. 6.14. There exists a characteristic threshold electrical ﬁeld (Eth ) during the transition from the low to the high ﬁeld range. For the case of sample 7 (see Fig. 6.14), the intersection point of the two approximation lines of the relationship wσ (E) between the interval of n = 1 and that of n = −0.33 was determined and marked by Eth . After a rather comprehensive determination of the electrical ﬁeld interval from the dependence wσ (E), where (6.24) with n = 1 applies and, thus, parameter A in (6.24) is given by A = Cerm /kB T = el/kB T (via (6.2) with l = Crm ), the parameter l can be extracted. As already stated in the introductory part of our contribution, the parameter l is proportional to the maximum hopping length rm . Hereto, the slope of the dependence lnσ(E) in the electrical ﬁeld range governed by n = 1 was taken, and the value of l could be calculated for diﬀerent temperatures. The resulting temperature dependence of the hopping length, l(T ), becomes evident in Fig. 6.17 for the case of the samples 6 and 7. Obviously, we end up with the law l ∝ T −ν , where ν = 0.9. In the inset of Fig. 6.17, the dependence of the threshold electrical ﬁeld Eth on temperature T is plotted in a double-logarithmic scale for the case of the samples 6 and 7. One plainly ﬁnds that the threshold increases with increase in temperature, following the law Eth ∝ T s , where s = 1.5. Next, for comparison, we will analyze the dependences σ(E) obtained experimentally at T = 4.2 K for the samples 4–7. The change of the character of σ(E) can still be read oﬀ more vividly from the functional dependence wσ (E), as strongly manifested in Fig. 6.15 for T = 4.2 K. It must be emphasized that the value of the threshold electrical ﬁeld Eth appreciably shifts to the higher ﬁeld range, when increasing the distance from the MIT on the insulating side. Since the interval of the dependence σ(E) with n = 1 is extremely small for the case of the samples 4 and 5, the relation l(T ) could not be gained with suﬃcient accuracy. Keep note of the fact that, for the samples 4 and 5 the ﬁeld range that precedes the interval with n = 1 [see (6.24)] excels by a larger slope, i.e., n = 2.6. The low electrical ﬁeld regime that precedes the interval with n = 2.6 could not be examined with suﬃcient accuracy for the case of the samples 4 and 5, as a consequence of inevitable large ﬂuctuations in the dependence wσ (E). In the low-temperature range at electrical ﬁeld values E ≥ 2 × 103 V cm−1 , the conductivity does not depend on temperature for the case of the samples 6 and 7. In the above ﬁeld regime described by (6.3), usually the dependence j(E) is analyzed. A more accurate value of the parameter m can be obtained in analogy to the determination of the characteristic value n in (6.24), as described earlier. Therefore, the parameter wj (E) that features the nonlinearity

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1.5 log10( E th [ V/cm])

4

#7

log10 (l [nm])

1.0 #6

#6

3

s = 1.5

#7

0.5

1.0

1.5 log10 (T [K])

0.5

ν = 0.9

0.0 0.5

1.0

1.5

2.0

log10 (T [K])

Fig. 6.15. Dependence of the parameter wσ (6.23) on the electrical ﬁeld of the samples 4, 5, 6, and 7, taken at the temperature T = 4.2 K and plotted in a doublelogarithmic scale. According to Fig. 6.14, the total electrical ﬁeld regime analyzed divides into intervals of diﬀerent slopes of the wσ (E) curves (i.e., diﬀerent values of the exponent n of (6.24)) n = −0.35

#7 #

n = 2.6

6

log10 (wσ)

-3

n=1 #

5

-4 T = 4.2 K 1

#

2

4

n=−0.33 3

4

log10 (E[ V/cm ])

Fig. 6.16. Dependence of the parameter wj (6.26) on the electrical ﬁeld of the samples 6 and 7, taken at the temperature T = 4.2 K and plotted in a doublelogarithmic scale. The inset shows the dependence of the current density on the electrical ﬁeld, taken at the temperature T = 4.2 K and plotted in a semi-logarithmic scale. The exponent 1/m of (6.3) can be immediately drawn from the experimental data (in analogy to the determination of the exponent n of (6.24) in Fig. 6.15)

of the dependence j(E) is introduced and accordingly deﬁned by wj (E) = d lnj(E)/ dE.

(6.26)

We point out that our method for determining 1/m was already applied in reference [28]. Figure 6.16 displays the resulting dependences wj (E) at

146

L.V. Govor and J. Parisi 12

-2.5

ln (j [ A/cm2 ] )

log10 ( wj)

-2.0

1/m=0.18

11

#7 #6

10 9 0.30

-3.0

E -0.16 [ V -0.16 cm0.16]

0.35

1/m=0.12 -3.5

#

T=4.2 K

3.00

#

3.25

3.50

6

7 3.75

log10 (E[ V/cm ])

Fig. 6.17. Dependence of the hopping length on temperature of the samples 6 and 7, plotted in a double-logarithmic scale. The inset displays the temperature dependence of the characteristic threshold electrical ﬁeld Eth . The exponents ν and s of the relations l ∝ T −ν and Eth ∝ T s , respectively, can be immediately drawn from the experimental data

T = 4.2 K in a double-logarithmic scale. The values 1/m for the case of the samples 6 and 7, both extracted from the slope of the above dependence, amounted to 0.18 and 0.12, respectively. On the basis of the obtained average value 1/m = 0.16, the dependence ln j vs. E −0.16 is plotted in the inset of Fig. 6.16. An ideal straightening of ln j(E) in those coordinates can be clearly seen. Analysis of the Conductivity The linear relation between the logarithm of the conductivity and the electrical ﬁeld at low temperatures that applies for samples near the MIT (i.e., samples 6 and 7) is present only in a relatively short ﬁeld range. With increasing temperature, the above range extends to the high ﬁeld regime. Such an enlargement of the region, where the linear σ(E) characteristic dominates, can be explained by (6.2). At low temperatures, the exponential part in (6.2) is large, and its inﬂuence on the dependence σ(E, T ) becomes already apparent in the low electrical ﬁeld range, compared to the pre-exponential part σ(0). Upon increasing the temperature, the exponential part in (6.2) diminishes and comes into play only in the high electrical ﬁeld range. In the following, let us more accurately analyze the low electrical ﬁeld range via (6.2). From the temperature dependence of the hopping length, l(T ), of the samples 6 and 7 plotted in Fig. 6.17, we obtain the law l = Crm ∝ T −ν , where ν = 0.9, i.e., l increases with decreasing T . Note that for both samples the dependence l(T ) unveils two characteristic temperature regimes. First, in the range from 4.2 to 15 K, we have hopping conductivity in the Coulomb gap with pCG = 1.49, see sample 7 in Table 6.1. In the temperature range from 15

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147

to 85 K, we observe Mott’s hopping law for the 2D case with pM = 2.63. In the range from 4.2 to 40 K, the characteristic value pCG = 1.66 was determined for sample 6. The interval with pM = 2.70 embraces temperatures between 40 and 120 K. For the ensuing discussion, we use as appropriate values pCG = 2 for VRH in the Coulomb gap and pM = 3 for Mott’s hopping law in case of all samples investigated. Following the experimentally gained dependence l(T ) shown in Fig. 6.17, it is diﬃcult to detect the diﬀerence between the dependence valid for hopping conductivity in the Coulomb gap and that for Mott’s hopping law. According to the references [19–21], in the range of validity of (6.1) with 1/p = 1/2 (or 1/p = 1/3), the parameter l should increase with decreasing temperature proportional to T −1/2 (or T −1/3 , respectively). However, the dependence obtained experimentally tells us that we have a much stronger rise of l during the reduction in temperature. A similar result has also been found in references [32, 55]. It can only be explained under the assumption that the constant C increases with decreasing temperature. The change of the value of the constant C upon varying parameter ξc (deﬁned above as percolation threshold ξc = (To /T )1/p ) was predicted theoretically in references [24, 56, 57]. Following reference [57], that modiﬁcation in the range 16 ≤ ξc ≤ 29 investigated is described by C(ξc ) = (14 ± 2) × 10−3 ξc .

(6.27)

For the temperature range of Fig. 6.17, where (6.1) with p = 2 and p = 3 applies, the values of the parameter ξc are limited within the interval 2 ≤ ξc ≤ 6. If one assumes that (6.27) also holds in the range of the parameter ξc investigated by us, then the dependence l(T ) can be expressed by l = Crm = Ac aξc 2 ,

(6.28)

where Ac is a constant. The maximum hopping length rm and the localization radius a were already introduced in the frame of the discussion of (6.2). Now the dependence l(T ) in the temperature range, where hopping conductivity in the Coulomb gap (further on marked by the index CG) with p = 2 in (6.1) is present, looks like (6.29) lCG (T ) = Ac aTCG /T. At temperatures, where Mott’s hopping law (further on marked by the index M ) with p = 3 is present, the dependence l(T ) can be described as 2/3

lM (T ) = Ac a(TM /T )

.

(6.30)

When comparing the analytically determined values of the dependence l(T ) in (6.29) and (6.30) with the experimentally gained ones in Fig. 6.17, good agreement can be observed. It should be emphasized that the intervals governed by the dependence lCG (T ) ∝ T −1 and the dependence lM (T ) ∝ T −2/3 of the experimental curves do not diﬀer practically. A possible reason can be derived from the shortness of the temperature ranges, where the above dependences have been examined experimentally.

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Threshold Electrical Field As already mentioned in the experimental part of this work, the conductivity of the samples 6 and 7 near the MIT on the insulating side does not depend on temperature at electrical ﬁelds E ≥ 2 × 103 V cm−1 in the lowtemperature range. The threshold electrical ﬁeld Eth serves as a parameter that characterizes the transition from thermally activated to activationless hopping conductivity. Shklovskii [25] has deduced theoretically the relation Eth ≈ kB T /ea, which points at a dependence on temperature like Eth ∝ T 1.0 . Except to an accuracy of the numerical coeﬃcient, the latter relationship applies both to the cases p = 2 and p = 3 in (6.1). The temperature dependence we have obtained for the samples 6 and 7 is of the form Eth ∝ T 1.5 (see inset of Fig. 6.17). Such deviation can be explained by the following argument. In a relatively high electrical ﬁeld, where the change of the potential energy of an electron, eEl(T ), along the hopping length l(T ) is equal to the extension of the energy range near the Fermi level, εopt (T ), that allows for hopping (i.e., the “optimum” hopping energy), the electron can move into the direction of the ﬁeld. The hopping energy in the Coulomb gap had been expressed by (6.21). Equating the energies εoptCG (T ) and eEthCG lCG (T ), where εoptCG (T ) and lCG (T ) are given by (6.21) and (6.29), respectively, results in a relation for the temperature dependence of the threshold electrical ﬁeld in the Coulomb gap as (6.31) EthCG = (ACG kB /Ac eaTCG 1/2 )T 3/2 . Outside the Coulomb gap, i.e., in the range of validity of Mott’s hopping law, the temperature dependence of the optimum hopping energy for the 2D case had been expressed by (6.22). Thus, the temperature dependence of the threshold electrical ﬁeld outside the Coulomb gap can be written as EthM = (AM kB /Ac eaTM 1/3 )T 4/3 .

(6.32)

When comparing the analytical relations (6.31) and (6.32) with the dependences obtained experimentally, a good correlation can be observed (see inset of Fig. 6.17). The radii of the localized states, a, for the case of the samples 6 and 7 have been determined from the dependence σ(1/T ) in Section 4.1.5. Their values amounted to 10 and 16 nm (see Table 6.2), respectively. The value of a can also be gained from the analytical relations (6.29) or (6.30) and the slope of the experimental dependences l(T ), plotted in a double-logarithmic scale, see Fig. 6.1715. A similar analysis is possible with the help of the analytical equations (6.31) or (6.32) and the slope of the experimental dependences Eth (T ), see inset of Fig. 6.17. Comparing the received values of a with the ones from Table 6.2 shows that the numerical characteristic value of C(ξc )/ξc in (6.27) should be equal to 0.07 in the examined parameter interval 2 ≤ ξc ≤ 6 (instead of the theoretical value [57] 0.014). From (6.31) and (6.32), it follows that EthCG and EthM with increasing temperature shift to the high electrical ﬁeld range. Such behavior can clearly

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be seen in the experimental dependences of Fig. 6.14. The increase of Eth with temperature simultaneously leads to an extension of the low electrical ﬁeld range, where (6.24) with n = 1 is valid, towards higher ﬁelds. The corresponding shift of the arrows in Fig. 6.14 provides direct experimental evidence. The shift of the threshold electrical ﬁeld EthCG into the high ﬁeld range with increasing distance from the MIT (i.e., turning from sample 7 via sample 6 and sample 5 to sample 4) at the temperature T = 4.2 K (see Fig. 6.15) mainly results from the rise of the parameter TCG . From the equality of the energies εoptCG (T ) and eEthCG lCG (T ), we end up with the relation EthCG ∝ TCG 1/2 /lCG for constant temperature. With increasing distance from the MIT on the insulating side, the parameter TCG ∝ 1/aεr increases, as a consequence of the reduction of both the radius of localized states, a, and the dielectric constant εr . Such a conclusion can also be conﬁrmed experimentally. For example, the values of EthCG at T = 4.2 K for the samples 7 and 6 (see Fig. 6.15) amount to 260 and 790 V cm−1 , respectively. That means that, when turning from sample 7 to sample 6, we have a threefold enlargement of the threshold electrical ﬁeld. The value of TCG becomes larger by approximately the same factor, i.e., TCG = 66 K for the case of sample 7 and TCG = 180 K for the case of sample 6 (see Table 6.2). High and Low Electrical Field Ranges In the experimental part of our work, it turned out that the electrical ﬁeld range with n > 1 (exponent n = 1.4 for the samples 7 and 6) precedes the one with n = 1 in (6.24). With increasing temperature, also the range with n > 1 shifts to higher ﬁelds (see Fig. 6.14). Upon growing distance from the MIT on the insulating side at T = 4.2 K, for the samples 5 and 4, we observe both the shift of the interval with n > 1 to the high ﬁeld range and the increase of the characteristic value of the exponent n up to n = 2.6 (see Fig. 6.15). We are familiar with only one work [22], where it is pointed out that the dependence lnσ(E) possesses a range of validity of the law lnσ(E) ∝ E 2 . The latter should precede the range with the dependence lnσ(E) ∝ E. Within the regime of high electrical ﬁelds and low temperatures, a saturation of the curves σ(E) ∝ E n takes place together with a fusion to one curve (see Fig. 13). It can be clearly seen by the functional dependences wσ (E) (Fig. 6.15), where the curves with the slope n = −0.35 (sample 6) and the curves with n = −0.33 (sample 7) merge. In case of VRH, following Mott’s hopping law for 3D in the high electrical ﬁeld range, the relation ln σ(E) ∝ E −1/4 was determined in the theoretical work of Apsley and Hughes [22]. It were straightforward to assume that the relation ln σ(E) ∝ E −1/2 applies for VRH in the Coulomb gap inside the high electrical ﬁeld range. However, our experimentally gained values of n are smaller, and the relation ln σ(E) ∝ E −1/3 is valid. In (6.24), we did not take into account the ﬁeld dependence of the pre-exponential part. But consideration of the latter relation

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will not lead to a substantial approximation between our values of n and the ones given in reference [22]. As already mentioned in the introduction, the current density vs. electrical ﬁeld relation j(E) is usually analyzed in the high electrical ﬁeld regime. The investigation of (6.3) by means of looking at the dependence wj (E) = d ln j(E)/dE shows that, for the case of the samples 6 and 7, (6.3) holds with the characteristic value 1/m (according to 0.18 and 0.12, respectively). Note that the above values of 1/m are substantially smaller than the value 0.5 taken in references [29–31]. Despite of the fact that the ohmic hopping conductivity (or resistivity, respectively) in the Coulomb gap is described by ln ρ(T ) ∝ T −1/2 , the activationless conductivity in the high electrical ﬁeld regime obeys the law ln σ(E) ∝ E −1/3 . Thereby, the current density changes like ln j(E) ∝ E −1/6 . It means that a strong electrical ﬁeld (in the activationless hopping conductivity regime) is not equivalent to a high temperature (in the ohmic hopping conductivity regime) such that it gives rise to a smaller extent to the increase in conductivity. A similar result was achieved in the experimental work of Aleshin and Shlimak [32].

6.5 Summary and Conclusions We have described an experimental preparation technique that is capable to produce macroscopic carbon network structures with diﬀerent shapes of the basic cell. There, a drop of the initial polymer solution spreads onto a cooled water surface, and the water vapor interacts with the resulting polymer thin ﬁlm. Following the self-assembly process of precipitating droplets of the water vapor on the polymer layer and subsequently evaporating the solvent, the originally homogeneous polymer ﬁlm proceeds to a hexagonal network pattern. By the help of an elementary model study on the self-organized structuring process in the liquid polymer ﬁlms, we succeeded in specifying and interpreting the morphology of the basic network cells observed experimentally. The dependence of the speciﬁc electrical conductivity of self-organized carbon nets on temperature was investigated. Four diﬀerent transport mechanisms were observed in the temperature range from 4.2 to 295 K. The temperature dependence of the speciﬁc resistivity of the carbon network structures, whose conductivity is situated far away transition, from the metal–insulator (1/p) . In the low-temperature can be described by ρ(T ) ∝ T −b exp [T0 /T ] range, a Coulomb gap in the density of localized states near the Fermi level occurs with the characteristic value p = 2. In the high-temperature range, the pre-exponential part ρ(T ) ∝ T −b dominates. At intermediate temperatures, Mott’s hopping law is observed with p = 3. For the samples, the speciﬁc resistivity of which is situated near the metal–insulator transition, their behavior in the low-temperature range obeys the power law ρ(T ) ∝ T −b . The value of b decreases from 3 to 0 when looking at samples with a conductivity

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far away from and near to the metal–insulator transition, respectively. In the high-temperature range,the speciﬁc resistivity of the carbon nets is characc−1 terized by ρ(T ) ∝ exp − [T /T1 ] , with c varying from 1.3 to 1.5. The existence of four charge transport mechanisms can be attributed to tails in the density of localized states, pulled out of the conduction and valence band, as a result of disorder and some overlap between these tails. In the regime of low temperatures, the charge carrier transport of the porous carbon networks investigated derives from hopping conductivity in the Coulomb gap, described by ln ρ(T ) ∝ T −1/2 . Investigation of the current density vs. electrical ﬁeld characteristic of samples near the metal–insulator transition on the insulating side unveils four distinct regions of ﬁeld dependence of the conductivity, σ(E). At low electrical ﬁelds, ohmic conductivity is observed, which does not depend on E. With increasing electrical ﬁeld, the conductivity rises, ﬁrst following the law ln σ(E) ∝ E n , where n changes from 1.4 to 2.6 with increasing distance from the metal–insulator transition on the insulating side. Upon further raising the electrical ﬁeld, the conductivity obeys the relation ln σ(E) ∝ E 1.0 . The temperature dependence of the hopping length in that ﬁeld range can be written as l(T ) ∝ T −0.9 . We have demonstrated that at temperatures, where ohmic conductivity in the Coulomb gap expressed by ln ρ(T ) ∝ T −1/2 is valid, activationless conductivity at high electrical ﬁelds follows the law ln σ(E) ∝ E −1/3 . Accordingly, the current density changes via ln j(E) ∝ E −1/6 . The disproportion between our experimental data and theory [25] shows that a strong electrical ﬁeld in the regime of activationless hopping conductivity is not equivalent to a high ambient temperature in the regime of ohmic hopping conductivity. The temperature dependence of the threshold electrical ﬁeld that characterizes the transition from the low to the high ﬁeld regime can be extracted by Eth ∝ T 1.5 . Acknowledgments We acknowledge our colleagues I.A. Bashmakov and G.H. Bauer for successful cooperation and fruitful discussions. We would like to thank V. Uchov and S. Martyna for taking the scanning electron microscopy pictures.

References 1. 2. 3. 4. 5. 6. 7. 8.

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7 Chemical Waves in Living Cells H.R. Petty

7.1 Introduction Propagating waves may be observed within the excitable matrices of living cells. The most familiar example of this is the movement of an action potential along a neuron [1]. In this case the excitable matrix is the diﬀerential distribution of ions across the neuron’s two-dimensional plasma membrane. Although small input signals (voltages) have no eﬀect on membrane channels, the non-linear current–voltage properties of ion channels cause large transmembrane currents or action potentials when the input is above a threshold voltage. This perturbation travels along a membrane as ion channels embedded in the membrane cycle between resting, activated (conducting), and inactivated (or refractory) states. Although action potentials travel over large distances, they serve as a useful model for other wave propagation phenomena a much smaller distance scales in cells. In addition to the plasma membrane, a cell’s cytoplasm is also an excitable matrix. In this case, however, the matrix is nominally a diﬀusion-coupled three-dimensional matrix, although in homogeneities may exist. For example, waves of intracellular calcium release have been previously observed [2, 3]), although this has been limited to very large cells, such as oocytes. Nonetheless, these waves appear to behave according to the same general principles as those found during action potentials as well as in the Belousov–Zhabotinsky reaction [1], an inorganic chemical process. Wave formation is, however, not limited to the macroscopic sizes associated with the Belousov–Zhabotinsky reaction and neuronal depolarization or to those of calcium waves in very large cells, but rather is a general property of all living cells. We have recently developed a high-speed microscopy system resembling high-speed spectroscopy methods, which allows images of cells or tissues to be captured at very brief shutter speeds (∼100 ns). At these shutter speeds, image blurring due to wave motion and molecular diﬀusion are insigniﬁcant. As modern electro-optic components can achieve duty cycles of ∼1 frame per ms, stop-action movies of chemical wave propagation are created. Using this approach, we have found

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Wave variables

NAD(P)H Flavoproteins pH Calcium Membrane potential Superoxide

Point of ignition Ignition time/threshold Propagation speed Propagation direction Ability to annihilate Shape Number Novel communication properties Surface-to-organelle transmission Organelle-to-organelle transmission Coherent transmission between cells

Found Cells Tissues

several types of waves in cells (Table 7.1). These waves may vary in their chemical composition, velocity, shape, intensity, and location. With respect to location, these waves may be associated with just one organelle such as the plasma membrane, or shared among intracellular organelles. In addition, certain chemical waves have been found to be released from cells in to the extracellular environment whereas others may be shared among cells by intracellular wave motion. These ﬁndings suggest that spatiotemporal chemical patterns are a key aspect of living matter.

7.2 Waves of Metabolic Activity Cell metabolism is an excitable matrix [4] generally assumed to be uniformly distributed throughout a cell. Using this homogeneous model of metabolism, early theoretical studies predicted the existence of spatial metabolic waves in cells [5–10]. Traveling waves of NADH autoﬂuorescence and pH were ﬁrst demonstrated experimentally in yeast extracts [4, 11–13]. We subsequently conﬁrmed the presence of these waves in individual living cells using highspeed microscopy [14–17]). Moreover, we found that these waves have several interesting physical and physiological properties. Neutrophils, initially spherical in solution, attach to surfaces by forming small adhesion sites at the substrate–cell interface that are rich in certain membrane receptors and signaling molecules. We found that metabolic waves form target patterns which begin as local increases in NAD(P)H ﬂuorescence intensity. After the metabolic perturbation lingers at the adhesion site for roughly 200 ms, a rapid expansion of the wave is observed (Fig. 7.1A). The critical radius (rcr ) estimated from the micrographs (∼1–1.5 µm) is not inconsistent with the value calculated from rcr = D/c, where rcr is the critical radius, D is the diﬀusion coeﬃcient, and c is the plane wave velocity [18].

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Fig. 7.1. High speed microscopy reveals intracellular metabolic waves of neutrophils. (a) Target patterns formed during cell adherence (100 ns exposure/50 ms duty cycle). (b) Longitudinal wave formed during cell polarization and motility (100 ns/25 ms). (c) By using an marker in the extracellular environment, the release of superoxide anions into the extracellular matrix can be detected as the NAD(P)H wave reaches the front of the cell (100 ns/100 ms). (d) Cell activation is accompanied by the formation of two traveling waves (100 ns/25 ms). (e) When cells are exposed to a chemotactic stimulus in a direction perpendicular to the direction of cell orientation, the metabolic waves reorient into the future direction of cell migration (100 ns/100 ms). For clarity, the locations of the cells are outlined in the ﬁrst frames of these series of images (×980)

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These metabolic waves may be initiated at the adhesion site due to the greater turnover of ATP (and hence heightened compensatory levels of NADH). After neutrophils adhere to surfaces (such as a tissue in vivo or glass slide in vitro), they become morphologically polarized (asymmetrical) and oriented to allow cell migration. The initial adherence site becomes the real (uropod) of the cell. When viewed by high-speed microscopy, metabolic waves are seen to travel from the uropod toward the front (lamellipodium) of the cell and are therefore traveling in the direction of cell migration (Fig. 7.1B). Although the waves nominally appear to be longitudinal, imaging using high shutter speeds suggests that these waves are spherical with a large radius of curvature [17]. These small, self-organized waves participate in cell functions, such as the synthesis of superoxide anions, which participate in host resistance to infectious disease and act as paracrine signaling molecules. NADPH provides electrons for the NADPH oxidase, which produces superoxide anions according to the relationship: 1/2 NADPH + O2 → 1/2 NADP+ + 1/2H+ + O2 − .

(7.1)

As the NADPH arrives at the front of the cell as a wave, there is periodic production of superoxide anions, which leads to large local concentrations of superoxide anions at the front of the cell for brief time periods [17]. This fact was demonstrated by experiments in which a reporter that becomes ﬂuorescent in the presence of superoxide was included in the extracellular environment. In this way, superoxide release from the cell can be detected just after the NAD(P)H arrives at the lamellipodium (Fig. 7.1C). The direction of the metabolic wave orientation may allow cells to direct oxidant release in a particular direction, for example, toward a target such as a tumor cell or bacterium. Superoxide delivery in concentrated and brief pulses may facilitate target destruction. When neutrophils become activated by an immunological stimulus, the single longitudinal metabolic wave splits into two waves (Fig. 7.1D). These two waves travel in opposite directions along the cell’s long axis and periodically coll de with one another near the center of the cell. Although some intracellular waves annihilate when they collide, these do not. This might be explained in several ways. For example, the waves could have a small refractory zone that would allow the waves to “skip” over one another without annihilation. It is also possible that the waves are of low density and pass by one another without interacting signiﬁcantly. Whatever the mechanism(s) of wave splitting and collision may be, these issues are of interest physically and correlate with crucial biological activities. Another curious aspect of cell stimulation was observed for these metabolic patterns. When cells were activated on a small region of their surface at one side of the cell, the propagating wave did not travel through the entire width of the cell, but traveled through only a corridor of cytoplasm after beginning at a domain along the side of the cell (Fig. 7.1E) [16]. Subsequent studies showed that certain metabolic enzymes, especially hexokinase, were concentrated at

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this domain where stimulation was occurring [19]. Consequently, local glucose ﬂux and metabolic activity would be higher in this region, which might explain why the autowave did not extend for the entire side of the cell. It therefore appears that certain features of immunologic activation of cells might be explained by physical theory. As biochemical rate constants and diﬀusion coeﬃcients are sensitive to temperature, wave velocity should also vary as a function of temperature. This has been demonstrated, for example, for calcium waves of cardiomyocytes [20]. Furthermore, the temporal dynamics of neutrophil metabolic oscillations have been shown to be a function of temperature (e.g., [21]). Importantly, the frequency of metabolic oscillations increases dramatically at temperatures above 37◦ C [21]. In this way, a small change in temperature can elicit a dramatic change in metabolic dynamics and the production of oxidants by neutrophils. We have proposed that the temperature-dependence of these oscillations and waves may underlie the beneﬁcial eﬀects of the higher body temperatures on host resistance to infection associated with the biology of fever [21]. This is particularly important because the mechanistic relevance of the thermal component of fever in host defense has been unknown. Although a homogeneous excitable metabolic matrix model may be applicable in many biological circumstances, the most interesting clinical conditions occur when certain constituent enzymes are heterogeneously distributed. A previous theoretical study suggested that metabolic enzyme location (cytosolic vs. bound to the cytoskeleton) aﬀects the spatiotemporal organization of metabolic activity [10]. Although the enzymes and enzyme distributions studied in our experiments diﬀer from those of Marmillot et al. [10], they are consistent with the broad implications concerning the eﬀect of enzyme location on wave formation. As mentioned above, NAD(P)H waves inﬂuence the production of superoxide anions by the NADPH oxidase. The primary source of NADPH formation is the hexose monophosphate shunt (HMS). Two enzymes of the HMS, glucose-6-phosphate dehydrogenase and 6-phosphogluconate dehydrogenase, produce NADPH. In neutrophils from normal healthy pregnant women, these two enzymes translocate to the cell center (or centrosome), whereas in cells from men and nonpregnant women these same two enzymes are found throughout the cytoplasm and at the cell periphery [22, 23]). These translocation events are responsible for reduced HMS activity and oxidant production. We have proposed that the translocation of HMS enzymes to the centrosome uncouples the HMS from its substrate, glucose-6-phosphate, which is formed at the cell surface [22]. Interestingly, cells from pregnant women do not display the wave splitting phenomenon described above and thereby only exhibit one traveling metabolic wave for both unstimulated and stimulated morphologically polarized neutrophils. This suggests that the activity of the HMS is tied to the underlying physical disposition of its constituent enzymes and the related metabolic wave pattern. As the metabolic waves are linked with superoxide production and superoxide production is linked to both target and host damage, these physical changes of neutrophils during pregnancy

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diminish inﬂammatory responses. The anti-inﬂammatory eﬀects of pregnancy may account for the ability of certain many women with autoimmune diseases such as multiple sclerosis, arthritis, and uveitis to go into remission during the second and third trimesters of pregnancy, but relapse after delivery. By dissecting these regulatory wave systems further, it might be possible to aﬀect the wave properties of cells in vivo to design better therapeutic methods; thus, one might use this dynamic physical information to design drugs to force patients into remission from multiple sclerosis.

7.3 Calcium Signaling Waves Calcium waves have been demonstrated in certain very large cells such as oocytes and myocytes [2,3]). Nonetheless, it has not been possible to visualize these waves in cells of typical size (<20 µm) because calcium signal motion and diﬀusive displacement occur at a timescale that precludes detecting their wave motion using conventional imaging techniques. In the absence of data to the contrary, calcium signaling in most cells is envisioned as a uniform rise and fall within cell cytoplasm. However, we have observed rich patterns of calcium signaling in typical eukaryotic cells. The calcium signaling matrix is very complex and likely contributes to the spatiotemporal calcium signaling patterns observed in living cells. In contrast to metabolic responses, calcium’s excitable matrix is not an isotropic three-dimensional structure. The multiple levels of heterogeneity that exist within this matrix are illustrated in Fig. 7.2. Calcium-conducting channels are found in the plasma membrane as well as many intracellular membranes. Although there are many types of plasma membrane calcium channels, they can be broadly classiﬁed as those that are activated by voltage changes (volt-

Fig. 7.2. Structural heterogeneity of calcium signaling apparatus. Calcium can be released into the cytosol of the cell from the plasma membrane for from intracellular calcium stored in intracellular compartments. The intracellular mediators shown are: inositol trisphophate (IP3), cyclic ADP ribose (cADPR), NAADP+)

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age gated channels) and those that are not (nonvoltage gated). Each of these very broad classes of membrane channels are made up of many individual channels which vary in amino acid sequences and biochemical properties. They can also vary in the spatial locations; some channels may cluster, for example at lipid rafts, whereas others are uniformly distributed at the cell surface [24]. A large number of membrane receptors generate second messengers, such as inositol trisphosphate (IP3) and sphingosine-1-phosphate (S1P), which interact with their intracellular receptors to provoke calcium release from intracellular stores. These intracellular stores may include a variety of structures including the endoplasmic reticulum, nuclear envelope, granules, and mitochondria. Not only do the organelles introduce spatial heterogeneity, but receptors for these second messengers may also vary spatially on the organelle. For example, the locations of the IP3 and ryanodine (cADPR) receptors vary in their spatial locations on the endoplasmic reticulum. Feedback loops among these signaling elements contribute to the regulation of calcium signaling; these include calcium (calcium-induced calcium release), physical (such as conformational coupling), voltage-mediated interactions, and other biochemical pathways such as metabolism. Furthermore, the endoplasmic reticulum’s membrane is known to interact with the plasma membrane, thereby providing an example of organelle-to-organelle route of communication. Like the formation of target patterns during leukocyte adherence to substrates discussed above, calcium target patterns also form during cell adherence. In these experiments cells were loaded with a calcium-sensitive dye, such as indo-1 or indo-5F, whose quantum yield improves dramatically in the presence of calcium. During cell adherence, which is generally mediated by members of the integrin supergene family, calcium signals are generated within cells, which rely upon the release of intracellular stores of calcium. Figure 7.3A shows an example of the spatiotemporal features of this calcium signal wave associated with cell adherence. A repetitive sequence of calcium signals is visualized in the form of a target pattern. These waves may play an important role in communicating the signaling events throughout the cytoplasm of the cell. Diﬀerent sorts of calcium waves are observed for morphologically polarized cells. In this case, plasma membrane calcium channels are believed to be principally involved. In contrast to the waves of metabolic activity seen for polarized cells, calcium waves were found to begin at the lamellipodium of the cell and travel unidirectionally around the perimeter of the cell in a counterclockwise direction [25], as viewed from the bottom of the cell (Fig. 7.3B). As the membrane channels open and close, the wave is observed to propagate around the cell. The origin of the unidirectional propagation is unknown, but apparently there is some inherent asymmetry built into the signaling apparatus. The waves appear to be restricted to the periphery of the cell. As the calcium enters the cell’s cytosol, it is rapidly pumped out of the cytosol by calcium-ATPases in the plasma and endoplasmic reticulum membranes. These waves appear rather two-dimensional in appearance. There are several reasons for this: the images were taken at just one focal plane, the images were deconvolved using

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Fig. 7.3. High speed microscopy shows complex calcium signal patterns in neutrophils labeled with a calcium-sensitive dye. (a) Target patterns formed during cell adherence (100 ns/50 ms, ×780). (b) On polarized cells, a single plasma membrane calcium waves travels about the periphery of the cell (250 ns/50 ms, ×800). (c) When polarized cells are exposed to an activating substance, two calcium waves are observed propagating about the membrane (250 ns/50 ms, ×800). (d) In this example, a neutrophil has internalized two red cells (arrows). One calcium wave travels about the perimeter of the cell. However, when the calcium wave passes by the region containing the internalized target cells, a calcium wave splits to travels around both internalized targets (150 ns/20 ms, ×820). (e) Multiple calcium waves, including both plasma membrane and intracellular waves about granules, are observed during the activation of rat basophilic leukemia cells (150 ns/60 ms, ×520). (f ) Expanding calcium waves are observed when neutrophils interact with a monolayer of endothelial cells (250 ns/200 ms, ×470)

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an approximate point-spread function (which performs in software many of the advantages that scanning confocal microscopy performs in hardware), and the membrane potential, which provides an electrostatic component potentiating the entry of calcium, is nonuniformly distributed about the periphery of adherent cells due to the spatial distribution of potassium channels which set the transmembrane potential of leukocytes. One of the earliest events in cell direction ﬁnding is the formation of two calcium waves in the cell. When migrating cells are stimulated, by a peptide or lipid derived from a bacterial cell for example, two calcium waves traveling in opposite directions are observed (Fig. 7.3C). These two waves do not annihilate when they collide on the cell [25]. This is likely due to the fact that the two waves are driven by diﬀerent underlying biochemical mechanisms, as the second calcium wave can be inhibited by pharmacological reagents that do not aﬀect the ﬁrst. When the plasma membrane is locally exposed to a cue that causes the cell to migrate in a new direction (a chemotactic factor), two calcium waves emerge from the site of chemotactic factor binding as the ﬁrst calcium wave reaches this site. This site will become the new lamellipodium of the cell, thus re-orienting the direction of cell migration. Calcium signaling is also observed within cells where it plays an important role in mediating the fusion of one cellular membrane with another. One example of this is the calcium-mediated fusion of phagosomes with lysosomes. Calcium waves have been found to travel from the plasma membrane to the phagosomes, which they encircle (Fig. 7.3D) [25, 26]. These events are crucial in the fusion of lysosomes with phagosomes to form phagolysosomes, which kill, internalize and digest targets. The process of phagocytosis is mediated by cell surface receptors for antibody molecules. Studies employing site-directed mutagenesis showed that an LTL sequence within this membrane receptor is required for the signal routing from the plasma membrane to the phagosome [25]. We believe that the LTL sequence is recognized by a protein at the surface of a small strand of the endoplasmic reticulum that connects the plasma membrane to the phagosome. The ﬁnding that peptides containing a sequence that directs the peptide to the cytoplasm of the cell (so-called Trojan peptides) coupled to the LTL sequence, bind speciﬁcally to the endoplasmic reticulum (Clark, Kindzelskii and Petty, unpublished) supports this concept. Furthermore, such Trojan-LTL peptides accumulate at sites of phagocytosis and inhibit both the routing of the calcium wave to the phagosome and phagolysosome fusion. This suggests that the LTL signal sequence is an important participant in regulating the routing of calcium waves within cells. An important event in allergic responses is the release of histamine, which is known to be regulated by calcium signaling. Although proteins participating in this process have been well-described in the literature, the signaling routes of the calcium waves have not been reported. In Fig. 7.3E we show one stop-action frames of calcium signaling in a rat basophilic leukemia (RBL) cell, which is similar to a mast cell, undergoing degranulation in response to membrane stimulation. As mentioned above for activated neutrophils, two

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peri-membrane calcium waves are observed. Moreover, in this case, additional calcium waves in the cytoplasm are observed. In contrast to the waves noted above, this wave takes the form of brief circular ﬂashes about the periphery of intracellular granules, which are driving their fusion with one another and the plasma membrane and, thereby, the release of histamine and other mediators from the cell. Calcium waves take many forms in living cells and participate in numerous critical functions in leukocytes, including intercellular communication. High speed microscopy has also revealed that chemical waves may be shared among cells. Leukocytes, which are generally found in the blood, must migrate across endothelial cells lining blood vessels to reach the extravascular tissues during injury or infection. To do this, leukocytes must ﬁrst bind to the endothelial cells and elicit calcium signals within the endothelial cells. Fig. 7.3F shows a leukocyte binding to endothelial cells. A calcium signal ﬁrst appears in the leukocyte, which is followed by the accumulation of calcium in the two neighboring endothelial cells. After some accumulation of calcium in the endothelial cells, a large expanding calcium wavefront is observed in both endothelial cells. Thus, the dynamics of calcium signaling in the leukocyte shares some temporal coherence with that of the endothelial cells. As a result of this signal, the endothelial cells relax their cytoskeletons thus forming an opening, which allows the leukocyte to pass this cell plane lining blood vessels.

7.4 Conclusions The self-organized microscopic chemical patterns described above lead to several important conclusions. Chemical waves are a rather general feature of cells; they are not restricted to certain large cell types. As a general feature of living cells, they demand more attention from experimentalists to decipher their underlying biochemical features and physical mechanisms. Theoretical studies have been useful in the analysis of these waves. For many years, physical theory was far more advanced than the biological observations; as experimental tools such as high speed gating, single photon-sensitive camera heads, computers, etc. improved in performance and declined in cost, high-speed microscopy became a possibility. It should now be possible to develop improved theoretical analyses and robust computational models of these chemical waves. Furthermore, these waves are not restricted to one chemical species, but rather have been found for numerous chemicals in cells. The types of waves listed in Table 7.1 are those associated with various research problems in my laboratory; they likely constitute only a fraction of the total number of waves found in cells. From the data summarized above as well as our published studies, it seems clear that intracellular waves link biological structures with their func-

7 Chemical Waves in Living Cells

165

tions. Several unambiguous correlations between wave properties and cellular responses have been shown. One example of this is the cellular production of superoxide anions. The NADPH oxidase uses electrons derived from NADPH to produce superoxide. High speed microscopy studies have shown that a plume of superoxide emanates from the cell’s lamellipodium within ms, of the arrival of the NAD(P)H wave at the lamellipodium [17]. Another example of the interaction of chemical dynamics and cell function is how cells chose a direction to migrate. When a region of the membrane is exposed to a chemotactic factor by the local discharge of a micropipet, two calcium waves form at this site, which will become the new lamellipodium of the cell and the direction of cell migration. In addition, the formation of two calcium waves, as well as the formation of two NAD(P)H waves, in response to receptor-mediated cell activation demonstrates, as expected, that these chemical waves are linked with the signal transduction apparatus of the cell. Furthermore, several types of ligands that similarly behave as activation stimuli, have similar eﬀects of the intracellular signaling patterns [16]. We propose that these dynamic chemical processes, which result from the perturbation of surface receptors by ligands, constitute an input of information to the cell which is distributed to some (calcium) or all (metabolism) parts of the cell. Our experiments suggest that chemical waves travel well-deﬁned pathways at speciﬁc times to mediate information transduction, processing and distribution – much like a computer chip. The heterogeneity of the calcium signaling apparatus (Fig. 7.2) allows the information to be shunted along speciﬁc routes at speciﬁc times. Sometimes chemical waves annihilate when coming into contact (null operation), other times there is no eﬀect (identity operation), and at other times the signal intensity is enhanced (additive operation). Such actions may represent a rudimentary form of chemical computation. As dozens or perhaps hundreds of chemical waves might be scurrying about a cell simultaneously, numerous channels exist for data input, processing, and distribution. As each channel has multiple physical variables (velocity, ignition point, amplitude, repetition frequency, etc.), the potential chemical “bandwidth” is far more impressive than current structural (on–oﬀ) structural models of cell signaling. The analogy that cells resemble analog computers merits further experimental and theoretical attention. Finally, we propose that nonlinear chemical dynamics will form a new genre of drug development [27, 28]. Despite the fact that new methodologies such as combinatorial libraries, robotics, genomics, and proteomics have been introduced in recent years, they have not increased the rate of drug discovery. Although enormous investments have been made, combinatorial libraries and genomics have yet to yield their ﬁrst drug. Traditional drug development techniques, such as rational and serendipitous drug discovery, generally rely upon the equilibrium binding of a candidate drug to a puriﬁed biopolymer. Such reductionist strategies miss the facts that: cells are never at equilibrium, cellular processes are dynamic, and intracellular biochemical pathways are connected via large numbers of feedback loops. By studying speciﬁc chemical

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subsystems (e.g., glycolysis and calcium signaling) in semiautonomous cells, such as leukocytes and tumor cells, we can simplify the system enough to collect physically and physiologically meaningful data. Several lines of evidence already in hand support this potentially revolutionary approach to drug development. First, several anti-inﬂammatories (indomethacin and dexamethasone) or compounds with some anti-inﬂammatory action (mibefradil) are known to aﬀect chemical waves. Moreover, the antimetastatic drug carboxyamido triazole (CAI) inhibits a calcium wave. As multiple gene products are required for calcium wave formation and propagation, we reasoned that combining multiple drugs acting on diﬀerent proteins participating in forming the same chemical wave would demonstrate enhanced ability to stop the migration and invasion of tumor cells [29]. We found that the combination of CAI with mibefradil demonstrated improved properties relative to either drug alone. Our dynamical assays have also identiﬁed the mechanism of inﬂammatory suppression associated with pregnancy [22, 23]. This may lead to the discovery of new drugs aimed at mimicking these eﬀects on cells. Drugs aimed at inhibiting wave communication between the plasma membrane and phagosome, such as the Trojan-LTL peptide mentioned above, may be particularly useful in managing inﬂammatory processes at the body surface, such as keratitis, a leading cause of blindness. These very small dynamic structures may have a profound inﬂuence on how we perceive biology and practice medicine in the twenty-ﬁrst century, as the gap between physics and physiology disappears. This work has been supported by grants from the National Cancer Institute, the National Institute of Allergy and Infectious Diseases and the National Multiple Sclerosis Society.

References 1. I.R. Epstein, J.A. Pojman, An Introduction to Nonlinear Dynamics: Oscillations, Waves, Patterns and Chaos (Oxford University Press, Oxford 1998) 2. A. Goldbeter, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behavior (Cambridge University Press, Cambridge 1990). 3. L.F. Jaﬀe, Cell Calcium 32: 217 (2002) 4. A. Boiteux, B. Hess: Ber. Bunsen-Ges. Phys. Chem. 84, 392 (1980) 5. A. Goldbeter, Proc. Natl. Acad. Sci. USA 70, 3255 (1973) 6. A. Goldbeter, R. Lefever, Biophys. J. 12, 1302 (1972) 7. M. Herschkowitz-Kaufman, G. Nicolis, J. Chem. Phys. 56, 1890 (1972) 8. B. Hasslacher, R. Kapral, A. Lawnziak, Chaos 3, 7 (1993) 9. B. Hess, A. Mikhailov, Ber. Bunsen-Ges. Phys. Chem. 98, 1198 (1994) 10. P. Marmillot, J.-F. Hervagault, G.R. Welch: Proc. Natl. Acad. Sci. USA 89, 12 103 (1992) 11. S.C. Muller, T. Mair, O. Steinbock, Biophys. Chem. 72, 37 (1998) 12. T. Shinjyo, Y. Nakagawa, T. Ueda, Physica D 84, 212 (1995)

7 Chemical Waves in Living Cells 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

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T. Mair, S.C. Muller, J. Biol. Chem. 271, 627 (1996) H.R. Petty, R.G. Worth, A.L. Kindzelskii, Phys. Rev. Lett. 84, 2754 (2000) H.R. Petty, A.L Kindzelskii, J. Phys. Chem. B. 104, 10952 (2000) H.R. Petty, A.L. Kindzelskii, Proc. Natl. Acad. Sci., USA 98, 3145 (2001) A.L. Kindzelskii, H.R. Petty, Proc. Natl. Acad. Sci., USA, 99, 9207 (2002) J.J. Tyson, J.P. Keener: Physica D 32, 327 (1988) J.-B. Huang, A.L. Kindzelskii, H.R. Petty, Cell. Immunol. 218, 95 (2002) J. Engel, A.J., Sowerby, S.A.E. Finch et al., Biophys. J. 68, 40 (1995) A.J. Rosenspire, A.L. Kindzelskii, H.R. Petty, J. Immunol. 169, 5396 (2002) A.L. Kindzelskii, J.-B. Huang, T. Chaiworapongsa et al., J. Clin. Invest. 110, 1801 (2002) A.L. Kindzelskii, T. Ueki, H. Michibata et al., J. Immunol. 172, 6373 (2004) A.L. Kindzelskii, R.G. Sitrin, H.R. Petty, J. Immunol. 172, 4681 (2004) A.L. Kindzelskii, H.R. Petty: J. Immunol. 170, 64 (2003) R.G. Worth, M.-K. Kim, A.L. Kindzelskii et al: Proc. Natl. Acad. Sci., USA 100, 4533 (2003) H.R. Petty: Opt. Photon. News, January, 34. (2004) H.R. Petty: ChemBioChem., in press (2004) J.-B. Huang, A.L. Kindzelskii, A.J. Clark et al., Cancer Res. 64, 2482 (2004)

Index

para-Hexaphenylene, 3 para-Hexaphenylene nanoﬁbers, 11 para-Phenylenes, 2, 6, 14 Accelerating waves Brusselator model, 101, 109 BZ-AOT system, 98, 100 Activationless hopping conductivity, 117, 148, 150, 151 Aerosol (OT) (AOT), 91 AOT microemulsion, 90, 91, 96 AOT–water–oil system, 92 Belousov–Zhabotinsky reactionaerosol OT system (BZ-AOT), 90, 93–95 BZ-AOT, see Aerosol (OT): Belousov–Zhabotinsky reactionaerosol OT system AFM, see Atomic force microscopy Aliphatic hydrocarbons, 33 Amorphous semiconductors current–voltage characteristics of, 117 resistivity behavior of, 116 Amorphous thin ﬁlms, 118 Anisotropic nanomaterials, 48, 52 Annealed carbon network, 133 Anthraquinone, nanostructures of, 3 Antibody molecules, 163 Anticapillary tweezer, 55, 62 Antimetastatic drug, 166 Antipercolation process, 50 Aromatic hydrocarbons, 33 Atomic force microscopy, 4, 69, 123 Autoimmune diseases, 160

Autonomous reaction–diﬀusion systems, 102 Avogadro’s number, 92 B´enard’s experiment, 59 Belousov–Zhabotinsky reaction, 90, 155 Belousov–Zhabotinsky reactionaerosol OT system (BZ-AOT), 90, 93–95 Benard–Marangoni instabilities, 68 Bicontinuous microemulsion, 48 Bifurcation, types of, 106 Bimodal distribution, 100 Binary solution, phase separation of, 74, 78, 85 Biochemical rate constants, 159 Boltzmann’s constant, 117, 134 Breath ﬁgures, formation of, 128, 130 Brownian motion, 49, 51 Brusselator model, 109 Bulk oscillations, 96 Buoyancy force, 82 BZ reaction, see Belousov-Zhabotinsky reaction Calcium signaling apparatus, structural heterogeneity of, 160 Calcium signaling matrix, 160 Calcium signaling waves, 160 Capillary attraction energy, 81 Carbon networks, 116, 122, 132, 133, 136, 141, 142 Carbon–carbon bonds, formation of, 42

170

Index

Carbon–carbon reductive elimination process, 38 Carboxyamido triazole (CAI), 166 Cell cytoplasm, 160 Cell metabolism, 156 Cell migration, 158, 163 Cell stimulation, aspect of, 158 Cell surface receptors, 163 Cellulose net structures, 133 Charge carrier transport, 151 Charge carriers, Coulomb interaction of, 138 Chemical dynamics, interaction of, 165 Chemotactic factor, 163 Cholorite-iodide-malonic acid (CIMA) oscillating reaction, 95 Closed batch reactor, 110 Cobalt nanocrystals, 60 Colloidal crystals, 116 Colloidal nanolithography, 61 Colloidal self-assemblies, structure of, 49, 53 Confocal resonance imaging techniques, 111 Copper nanocrystals production of, 53 various shapes of, 54 Copper nanorods, formation of, 52 CoPt3 nanoparticle rings, 72 Coulomb gap, 117, 134, 135, 138–140, 146–151 Coulomb interactions, 117 Current–voltage characteristics, 117, 132 Cut-oﬀ-ﬁlter, 11 Cytoskeleton, 159, 164 C–H bond activation, 19, 38, 40, 42 Dark ﬁeld microscopy, 10 Dash waves, 100 Degree of doping, 116 Dehydrogenative coupling, 38, 40, 42 Dewetting process, 81 Diblock copolymers, 116 Diethylhexyl sulfosuccinate, 49 Diﬀusion coeﬃcients, 92, 107, 159 Dipolar ﬂuids, 59 Dipolar interactions, 55 Dipole–dipole interactions, 48

Dipole-directed self-assembly, 68 Dipole-induced dipole forces, 7 Efros-Shklovskii (ES) law, 138 Eikonal equation, 97, 99 Electric conductivity, analysis of, 146 Electron diﬀraction spectrometer, 4 Electron-beam lithography, 67 Endoplasmic reticulum, 161, 163 Endothelial cells, 164 Epiﬂuorescence images, 9 microscopy, 8 Eukaryotic cells, 160 Evans method, 38 Extravascular tissues, 164 Face centered cubic (f.c.c.) crystal structure, 55 Fast-diﬀusing activator, patterns associated with, 97, 108, 109 Fast-diﬀusing inhibitor, 97, 108 Femtosecond laser, excitation by, 12 Fermi energy, 117, 138, 139 Fermi level, 148, 150 Field–Kur¨ os–Noyes (FKN) mechanism, 104, 105, 109 Fluid polymer layers of, 126–128 solutions of, 116 Fluorescent catalysts, use of, 111 Fractal-like geometry, 120 Ginzburg–Landau equation, 100 Granular thin ﬁlms, 116 Hamaker constant, ratio of, 79, 83, 84 HDA molecules, 83 HDA monolayer, 83 HDA pancake model for the formation of, 74–86 structures of, 69 Heterocyclic ligands, 31 Heterocyclic radical anions, 30 Hexaazatrinaphthylene (HATN) titanium complexes, 38 Hexadecylamine (HDA), 68 Hexane, evaporation rate of, 77 Hexaphenyl nanoﬁber, 7, 9

Index Hexose monophosphate shunt (HMS), 159 High-speed microscopy system, 155, 164 High-speed spectroscopy, 155 High-temperature conductivity, 141 Homogeneous lateral distribution, 82 Honeycomb carbon networks, 116 Honeycomb network structure, 118, 119 Hopping conductivity, 117, 146, 147 Hydrophilic and Hydrophobic interactions, 116 Hydrophobic hydrocarbon, 49 Infelta–Tachiya equation, kinetics of, 93 Inositol trisphosphate (IP3), 161 Intercellular communication, 164 Interconnected-cylinder microemulsion, 49 Interdigitated reverse micelles, 50 Interfacial tension, 78 Intermolecular separation, 84 Interoscillon distance, 111 Interparticle dipolar forces, 68 Intracellular membranes, 160 Intracellular waves, 158 Inverted epiﬂuorescence microscope, 12 Isotropic microemulsion, 50 Jacobian matrix, 107 Knudsen cell, 4 Laminar ﬂow, 83 Lateral friction force, 83 Leukocytes, 163, 164, 166 Linear stability analysis and types of bifurcations, 106, 107 results of, 107 Lithographic mask, 64 Living cells, propagating waves in, 155 Localized states density of, 117 radius of, 141 Low energy electron diﬀraction (LEED), 4 Maghemite nanocrystals, 59 Magnetic nanocrystals, 64 Magnetic resonance imaging techniques, 111

171

Magnetic susceptibility, 32 Malonic acid (MA), 90 Marangoni eﬀect, 59 Marangoni instabilities, 59 Mesoporous materials, 116 Mesoporous membranes, 116 Mesoporous solids, 115, 116 Mesoscopic structures, 49, 60, 64, 67 Mesostructured ﬂuids, 48 Metabolic activity, waves of, 156 Metabolic enzymes, 158 Metabolic matrix model, 159 Metabolic wave orientation, 158 Metal ion-catalyzed oxidation, 90 Metal nanoparticles, 52 Metal–insulator transition (MIT), 116, 150, 151 Microemulsion, 96 Molecular architectures, titanium-based C–C coupling reactions, 38, 41 formation of, 19 Molecular diﬀusion, 155 Monometallic compounds, 42 Monomolecular layer, 119 Mott’s hopping law, 134, 136–139, 147, 148, 150 Multinuclear transition metal complexes, 42 Muscovite mica, 4, 6, 14 Nano- and microoptics, applications in, 9 Nanoaggregates, mutual orientation and morphology control of, 6 Nanocrystals alignment of, 59 concentration of, 55, 59 morphology control of, 55 organization of, 48 production of, 51–55 rings of, 59 self-organization of, 55–61 Nanodroplets, patterns of, 90 Nanoﬁbers, orientation of, 8 Nanometer-sized metal particles, 68 Nanooptics, 1 Nanoparticles forces acting on, 82–84 ring formation of, 68–74, 81–85

172

Index

Nanoparticles (cont.) self-assembled rings of, 67 supra crystals of, 58 Nanoscaled optoelectronic elements, 2 NC-coated substrate, 81 Needle ﬁlms growth Au-modiﬁed mica, 7, 8 plain mica, 6, 7 water-treated mica, 8, 9 Neutrophil metabolic oscillations, 159 Neutrophils, 156, 158, 159, 163 Nitrocellulose network patterns in, 122, 131 structural forms of, 120 Nitrocellulose solution (NC), 68, 119, 128, 129 Nonlinear chemical dynamics, 90, 109, 165 Nonlinear optical eﬀects, 11 Nonmagnetic ﬂuid, 60 Nonpolar solvent, 55 Octanuclear compound, 34 Octanuclear molecular aggregate, 36 Ohmic conductivity, 117, 151 Oil to surfactant volume fraction ratio, 50 One-photon luminescence, 12 Optical band structure, 1 Optical far ﬁeld microscopy, 10 Optical microscopy, 9, 110 Optical transition dipole moment, 4, 12 Oregonator model, 105 Organic crystalline nanoﬁbers, generation and control of, 2 Organic ﬁeld eﬀect transistors (OFET), 3 Organic ﬁlm growth, 8, 9 Organic light-emitting diodes (OLED), 3 Organic molecular beam epitaxy, 14 Organic nanoaggregates, 14 Organic nanoﬁbers, future devices from, 14 Oriented needle growth, 6 Oscillons, 102, 106, 109, 111 Partial diﬀerential equations (PDE), 103

PDE models, 104 Pentacene, nanostructures of, 3 Periodic index modulation, 1 Photonic band gap (PBG) materials, 1 Plasma membrane, 155, 160, 163, 164 Polar molecules, diﬀusion of, 92 Polarizable organic molecules, 7 Poly(p-phenylenevinylene) and poly(3octylthiophene) networks, structural forms of, 123–125 Polymer honeycomb structures, experimental formation of, 118–123 Polymer network elementary cell of, 130 formation of 2D and 3D, 130–132 production of, 119, 120 Solidiﬁcation of, 131 Polymer thin ﬁlm, 118, 125, 150 Polymetallic supramolecular systems, 42 Polynuclear complexes, 37 Polynuclear titanium compounds, 18, 33 Quantum dots, 1 Quantum mechanics, 97 Quartz microbalance, 4 Quasicontinuous ﬁlms, 7 Rat basophilic leukemia (RBL), 163 Reaction–diﬀusion systems, 94, 100, 106, 107, 110 Reactive ion etching (RIE), 62 Redox-active acceptor ligands, 42 Reverse microemulsion, 89, 92 Reynolds coeﬃcient, 83 Scanning confocal microscopy, 163 Scanning electron microscope (SEM), 59, 63, 120, 121 Scanning near ﬁeld optical microscopy (SNOM), 12–14 Second harmonic generation (SHG), 11 Self-assembled materials, 115 Self-assembled nanocrystals and supraaggregates, 48 Semiautonomous cells, 166 Signal-to-noise-ratio, 12, 13

Index Silicon wafer, 62 single crystal X-ray analysis, 19 Single crystalline ﬁlm, orientation of, 4 Sodium bis(2-ethylhexyl) sulfosuccinate, 91 Solution–air interface, 74, 86 Solvent evaporation, 79, 129 Speciﬁc resistivity, analysis of, 136–138 Sphingosine-1-phosphate (S1P), 161 Stalagmometer, 119 Standing waves development of, 108 oscillation of, 97 Stochastic cellular automaton model, 104 Stokes–Einstein equation, 92 Substrate–cell interface, 156 Surface dipole orientations, 7 Surface hydrophobicity, 9 Surfactant molecules, 49, 92, 110 Surfactant–water interface, 50 Tetrahedral coordination geometry, 22, 37 Tetraline, structure of, 22 Tetrazine bridged complex, 23 Thin liquid ﬁlms, 68 Thin polymer ﬁlm, 121, 123, 126, 130 Threshold electrical ﬁeld, 144, 148, 149, 151 Titanium diamagnetic properties of, 18

173

polynuclear compounds of, 33 self-assembled polynuclear complexes, 18 supramolecular structures of, 17 trinuclear HATN titanium complexes, formation of, 39 Titanocene complexes, reactions of, 21 Transmission electron microscopy (TEM), 55, 69 Traveling metabolic wave, 159 Traveling waves, 97 Trojan peptides, 163 Turing bifurcation, 97, 107, 108 Turing patterns, 95, 102, 108, 110 Two-phase solvent, 77 Two-photon luminescence (TPL), 11 Two-photon signal intensity, 11 Ultrathin ﬁlms, growth of, 2–6, 85 van-der-Waals attraction, 59 van-der-Waals forces, 59 Variable range hopping (VRH), 116–118, 133–137, 147 Water surface transforms, 74 Water vapor condensation, 130 Water-in-oil droplets, 48–50 Water-in-oil microemulsion, 110 Weak dipolar interaction, 59 X-ray crystallography, 34 X-ray diﬀraction, 26, 39, 41

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